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<h4>
<code>Random</code>
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<div class="abstract">
<p>A pseudo-random number generator (PRNG) that implements
<a href="http://vigna.di.unimi.it/ftp/papers/xorshiftplus.pdf"><code>xorshift128+</code></a>, an efficient linear-feedback shift register.</p>
<div class="aside aside-warning">
<p class="aside-title">Warning</p>
Once seeded from cryptographically secure random bytes, an
instance of <code>Random</code> generates high-quality random numbers but is <em>not</em> a
cryptographically secure PRNG.
</div>
<p>To generate random numbers, create your own instance of <code>Random</code> with an
internal state seeded from cryptographically secure random bytes:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">random</span> <span class="o">=</span> <span class="kt">Random</span><span class="p">()</span><span class="o">!</span>
<span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">()</span> <span class="k">as</span> <span class="kt">Int</span>
<span class="c1">// You can also pass the desired result type as an argument.</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="kt">Int</span><span class="o">.</span><span class="k">self</span><span class="p">)</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between 0 and 42 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">42</span><span class="p">))</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between -42 and 0 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="o">-</span><span class="mi">42</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">0</span><span class="p">))</span>
<span class="p">}</span>
</code></pre>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="Classes/Random/Xoroshiro.html">Random.Xoroshiro</a></code>, <code><a href="Protocols/PRNG.html">PRNG</a></code>
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<a href="Classes/Random.html">See more…</a>
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<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">final</span> <span class="kd">class</span> <span class="kt">Random</span> <span class="p">:</span> <span class="kt"><a href="Protocols/PRNG.html">PRNG</a></span></code></pre>
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<article>
<section>
<div class="wrapper">
<div class="overview">
<h1>Random</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">final</span> <span class="kd">class</span> <span class="kt">Random</span> <span class="p">:</span> <span class="kt"><a href="../Protocols/PRNG.html">PRNG</a></span></code></pre>
</div>
</div>
<p>A pseudo-random number generator (PRNG) that implements
<a href="http://vigna.di.unimi.it/ftp/papers/xorshiftplus.pdf"><code>xorshift128+</code></a>, an efficient linear-feedback shift register.</p>
<div class="aside aside-warning">
<p class="aside-title">Warning</p>
Once seeded from cryptographically secure random bytes, an
instance of <code>Random</code> generates high-quality random numbers but is <em>not</em> a
cryptographically secure PRNG.
</div>
<p>To generate random numbers, create your own instance of <code>Random</code> with an
internal state seeded from cryptographically secure random bytes:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">random</span> <span class="o">=</span> <span class="kt">Random</span><span class="p">()</span><span class="o">!</span>
<span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">()</span> <span class="k">as</span> <span class="kt">Int</span>
<span class="c1">// You can also pass the desired result type as an argument.</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="kt">Int</span><span class="o">.</span><span class="k">self</span><span class="p">)</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between 0 and 42 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">42</span><span class="p">))</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between -42 and 0 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="o">-</span><span class="mi">42</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">0</span><span class="p">))</span>
<span class="p">}</span>
</code></pre>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Classes/Random/Xoroshiro.html">Random.Xoroshiro</a></code>, <code><a href="../Protocols/PRNG.html">PRNG</a></code>
</div>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex6RandomC5states6UInt64V_AFtv">state</a></li>
<li><a href="#/s:12NumericAnnex6RandomCACs6UInt64V_AEt5state_tcfc">init(state:)</a></li>
<li><a href="#/s:12NumericAnnex6RandomCACSgycfc">init()</a></li>
<li><a href="#/s:s16IteratorProtocolP4next7ElementQzSgyF">next()</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex6RandomC9XoroshiroC">Xoroshiro</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex6RandomC5states6UInt64V_AFtv"></a>
<a name="//apple_ref/swift/Property/state" class="dashAnchor"></a>
<h4>
<code>state</code>
</h4>
</div>
<div class="abstract">
<p>The internal state of the pseudo-random number generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="k">var</span> <span class="nv">state</span><span class="p">:</span> <span class="p">(</span><span class="kt">UInt64</span><span class="p">,</span> <span class="kt">UInt64</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex6RandomCACs6UInt64V_AEt5state_tcfc"></a>
<a name="//apple_ref/swift/Method/init(state:)" class="dashAnchor"></a>
<h4>
<code>init(state:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with the given internal state.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="nv">state</span><span class="p">:</span> <span class="p">(</span><span class="kt">UInt64</span><span class="p">,</span> <span class="kt">UInt64</span><span class="p">))</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>state</code></dt>
<dd>
<p>The value to be used as the generator&rsquo;s internal state.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex6RandomCACSgycfc"></a>
<a name="//apple_ref/swift/Method/init()" class="dashAnchor"></a>
<h4>
<code>init()</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with an internal state seeded
using cryptographically secure random bytes.</p>
<p>If cryptographically secure random bytes are unavailable, the result is
<code>nil</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="n">convenience</span> <span class="nf">init</span><span class="p">?()</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s16IteratorProtocolP4next7ElementQzSgyF"></a>
<a name="//apple_ref/swift/Method/next()" class="dashAnchor"></a>
<h4>
<code>next()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">next</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">UInt64</span><span class="p">?</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex6RandomC9XoroshiroC"></a>
<a name="//apple_ref/swift/Class/Xoroshiro" class="dashAnchor"></a>
<h4>
<code>Xoroshiro</code>
</h4>
</div>
<div class="abstract">
<p>A pseudo-random number generator (PRNG) that implements <code>xoroshiro128+</code>, a
successor to <code>xorshift128+</code> devised by S. Vigna and D. Blackman.</p>
<div class="aside aside-warning">
<p class="aside-title">Warning</p>
Once seeded from cryptographically secure random bytes, an
instance of <code>Random.Xoroshiro</code> generates high-quality random numbers but
is <em>not</em> a cryptographically secure PRNG.
</div>
<p>To generate random numbers, create your own instance of <code>Random.Xoroshiro</code>
with an internal state seeded from cryptographically secure random bytes:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">random</span> <span class="o">=</span> <span class="kt">Random</span><span class="o">.</span><span class="kt">Xoroshiro</span><span class="p">()</span><span class="o">!</span>
<span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">()</span> <span class="k">as</span> <span class="kt">Int</span>
<span class="c1">// You can also pass the desired result type as an argument.</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="kt">Int</span><span class="o">.</span><span class="k">self</span><span class="p">)</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between 0 and 42 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">42</span><span class="p">))</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between -42 and 0 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="o">-</span><span class="mi">42</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">0</span><span class="p">))</span>
<span class="p">}</span>
</code></pre>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Classes/Random.html">Random</a></code>, <code><a href="../Protocols/PRNG.html">PRNG</a></code>
</div>
<a href="../Classes/Random/Xoroshiro.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">final</span> <span class="kd">class</span> <span class="kt">Xoroshiro</span> <span class="p">:</span> <span class="kt"><a href="../Protocols/PRNG.html">PRNG</a></span></code></pre>
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<div class="overview">
<h1>Xoroshiro</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">final</span> <span class="kd">class</span> <span class="kt">Xoroshiro</span> <span class="p">:</span> <span class="kt"><a href="../../Protocols/PRNG.html">PRNG</a></span></code></pre>
</div>
</div>
<p>A pseudo-random number generator (PRNG) that implements <code>xoroshiro128+</code>, a
successor to <code>xorshift128+</code> devised by S. Vigna and D. Blackman.</p>
<div class="aside aside-warning">
<p class="aside-title">Warning</p>
Once seeded from cryptographically secure random bytes, an
instance of <code>Random.Xoroshiro</code> generates high-quality random numbers but
is <em>not</em> a cryptographically secure PRNG.
</div>
<p>To generate random numbers, create your own instance of <code>Random.Xoroshiro</code>
with an internal state seeded from cryptographically secure random bytes:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">random</span> <span class="o">=</span> <span class="kt">Random</span><span class="o">.</span><span class="kt">Xoroshiro</span><span class="p">()</span><span class="o">!</span>
<span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">()</span> <span class="k">as</span> <span class="kt">Int</span>
<span class="c1">// You can also pass the desired result type as an argument.</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="kt">Int</span><span class="o">.</span><span class="k">self</span><span class="p">)</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between 0 and 42 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">42</span><span class="p">))</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between -42 and 0 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="o">-</span><span class="mi">42</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">0</span><span class="p">))</span>
<span class="p">}</span>
</code></pre>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../../Classes/Random.html">Random</a></code>, <code><a href="../../Protocols/PRNG.html">PRNG</a></code>
</div>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex6RandomC9XoroshiroC5states6UInt64V_AHtv">state</a></li>
<li><a href="#/s:12NumericAnnex6RandomC9XoroshiroCAEs6UInt64V_AGt5state_tcfc">init(state:)</a></li>
<li><a href="#/s:12NumericAnnex6RandomC9XoroshiroCAESgycfc">init()</a></li>
<li><a href="#/s:s16IteratorProtocolP4next7ElementQzSgyF">next()</a></li>
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<li>
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<a name="/s:12NumericAnnex6RandomC9XoroshiroC5states6UInt64V_AHtv"></a>
<a name="//apple_ref/swift/Property/state" class="dashAnchor"></a>
<h4>
<code>state</code>
</h4>
</div>
<div class="abstract">
<p>The internal state of the pseudo-random number generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="k">var</span> <span class="nv">state</span><span class="p">:</span> <span class="p">(</span><span class="kt">UInt64</span><span class="p">,</span> <span class="kt">UInt64</span><span class="p">)</span></code></pre>
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<a name="/s:12NumericAnnex6RandomC9XoroshiroCAEs6UInt64V_AGt5state_tcfc"></a>
<a name="//apple_ref/swift/Method/init(state:)" class="dashAnchor"></a>
<h4>
<code>init(state:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with the given internal state.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="nv">state</span><span class="p">:</span> <span class="p">(</span><span class="kt">UInt64</span><span class="p">,</span> <span class="kt">UInt64</span><span class="p">))</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>state</code></dt>
<dd>
<p>The value to be used as the generator&rsquo;s internal state.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex6RandomC9XoroshiroCAESgycfc"></a>
<a name="//apple_ref/swift/Method/init()" class="dashAnchor"></a>
<h4>
<code>init()</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with an internal state seeded
using cryptographically secure random bytes.</p>
<p>If cryptographically secure random bytes are unavailable, the result is
<code>nil</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="n">convenience</span> <span class="nf">init</span><span class="p">?()</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s16IteratorProtocolP4next7ElementQzSgyF"></a>
<a name="//apple_ref/swift/Method/next()" class="dashAnchor"></a>
<h4>
<code>next()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">next</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">UInt64</span><span class="p">?</span></code></pre>
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<h1>Extensions</h1>
<p>The following extensions are available globally.</p>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:s13BinaryIntegerP">BinaryInteger</a></li>
<li><a href="#/s:s15UnsignedIntegerP">UnsignedInteger</a></li>
<li><a href="#/s:Si">Int</a></li>
<li><a href="#/s:s4Int8V">Int8</a></li>
<li><a href="#/s:s5Int16V">Int16</a></li>
<li><a href="#/s:s5Int32V">Int32</a></li>
<li><a href="#/s:s5Int64V">Int64</a></li>
<li><a href="#/s:Su">UInt</a></li>
<li><a href="#/s:s5UInt8V">UInt8</a></li>
<li><a href="#/s:s6UInt16V">UInt16</a></li>
<li><a href="#/s:s6UInt32V">UInt32</a></li>
<li><a href="#/s:s6UInt64V">UInt64</a></li>
<li><a href="#/">DoubleWidth</a></li>
</ul>
<ul>
<li><a href="#/s:Sf">Float</a></li>
<li><a href="#/s:Sd">Double</a></li>
</ul>
<ul>
<li><a href="#/s:s13FloatingPointP">FloatingPoint</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
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<li>
<div>
<a name="/s:s13BinaryIntegerP"></a>
<a name="//apple_ref/swift/Extension/BinaryInteger" class="dashAnchor"></a>
<h4>
<code>BinaryInteger</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/BinaryInteger.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">BinaryInteger</span> <span class="p">:</span> <span class="kt">Hashable</span><span class="p">,</span> <span class="kt">Numeric</span><span class="p">,</span> <span class="kt">CustomStringConvertible</span><span class="p">,</span> <span class="kt">Strideable</span></code></pre>
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<div>
<a name="/s:s15UnsignedIntegerP"></a>
<a name="//apple_ref/swift/Extension/UnsignedInteger" class="dashAnchor"></a>
<h4>
<code>UnsignedInteger</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/UnsignedInteger.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">UnsignedInteger</span> <span class="p">:</span> <span class="kt">BinaryInteger</span> <span class="k">where</span> <span class="k">Self</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">BinaryInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:Si"></a>
<a name="//apple_ref/swift/Extension/Int" class="dashAnchor"></a>
<h4>
<code>Int</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Int.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s4Int8V"></a>
<a name="//apple_ref/swift/Extension/Int8" class="dashAnchor"></a>
<h4>
<code>Int8</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Int8.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int8</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s5Int16V"></a>
<a name="//apple_ref/swift/Extension/Int16" class="dashAnchor"></a>
<h4>
<code>Int16</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Int16.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int16</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s5Int32V"></a>
<a name="//apple_ref/swift/Extension/Int32" class="dashAnchor"></a>
<h4>
<code>Int32</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Int32.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int32</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s5Int64V"></a>
<a name="//apple_ref/swift/Extension/Int64" class="dashAnchor"></a>
<h4>
<code>Int64</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Int64.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int64</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
</div>
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<li>
<div>
<a name="/s:Su"></a>
<a name="//apple_ref/swift/Extension/UInt" class="dashAnchor"></a>
<h4>
<code>UInt</code>
</h4>
</div>
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<a href="Extensions/UInt.html">See more…</a>
</div>
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<h5>Declaration</h5>
<div class="language">
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
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<h4>
<code>UInt8</code>
</h4>
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<a href="Extensions/UInt8.html">See more…</a>
</div>
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<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt8</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
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<div>
<a name="/s:s6UInt16V"></a>
<a name="//apple_ref/swift/Extension/UInt16" class="dashAnchor"></a>
<h4>
<code>UInt16</code>
</h4>
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<div class="abstract">
<a href="Extensions/UInt16.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt16</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
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</li>
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<div>
<a name="/s:s6UInt32V"></a>
<a name="//apple_ref/swift/Extension/UInt32" class="dashAnchor"></a>
<h4>
<code>UInt32</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/UInt32.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt32</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
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</li>
<li>
<div>
<a name="/s:s6UInt64V"></a>
<a name="//apple_ref/swift/Extension/UInt64" class="dashAnchor"></a>
<h4>
<code>UInt64</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/UInt64.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt64</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/"></a>
<a name="//apple_ref/swift/Extension/DoubleWidth" class="dashAnchor"></a>
<h4>
<code>DoubleWidth</code>
</h4>
</div>
<div class="abstract">
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</div>
</li>
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<div>
<a name="/s:Sf"></a>
<a name="//apple_ref/swift/Extension/Float" class="dashAnchor"></a>
<h4>
<code>Float</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Float.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:Sd"></a>
<a name="//apple_ref/swift/Extension/Double" class="dashAnchor"></a>
<h4>
<code>Double</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Double.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:s13FloatingPointP"></a>
<a name="//apple_ref/swift/Extension/FloatingPoint" class="dashAnchor"></a>
<h4>
<code>FloatingPoint</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/FloatingPoint.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">FloatingPoint</span> <span class="p">:</span> <span class="kt">SignedNumeric</span><span class="p">,</span> <span class="kt">Strideable</span><span class="p">,</span> <span class="kt">Hashable</span></code></pre>
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<a href="../Extensions/UInt32.html">UInt32</a>
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<a href="../Extensions/UInt64.html">UInt64</a>
</li>
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<a href="../Extensions/UInt8.html">UInt8</a>
</li>
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<a href="../Extensions/UnsignedInteger.html">UnsignedInteger</a>
</li>
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<p><a href="../Functions.html">Functions</a></p>
<ul>
<li>
<a href="../Functions.html#/s:12NumericAnnex3absAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">abs(_:)</a>
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<a href="../Functions.html#/s:12NumericAnnex3absAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">abs(_:)</a>
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<a href="../Functions.html#/s:12NumericAnnex4ceilAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">ceil(_:)</a>
</li>
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<a href="../Functions.html#/s:12NumericAnnex5floorAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">floor(_:)</a>
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<a href="../Functions.html#/s:12NumericAnnex5roundAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">round(_:)</a>
</li>
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<a href="../Functions.html#/s:12NumericAnnex4sqrtAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">sqrt(_:)</a>
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<a href="../Functions.html#/s:12NumericAnnex5truncAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">trunc(_:)</a>
</li>
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<p><a href="../Protocols.html">Protocols</a></p>
<ul>
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<a href="../Protocols/Math.html">Math</a>
</li>
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<a href="../Protocols/PRNG.html">PRNG</a>
</li>
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<a href="../Protocols/Real.html">Real</a>
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<p><a href="../Structs.html">Structures</a></p>
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<a href="../Structs/Complex.html">Complex</a>
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<a href="../Structs/Rational.html">Rational</a>
</li>
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<p><a href="../Typealiases.html">Type Aliases</a></p>
<ul>
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<a href="../Typealiases.html#/s:12NumericAnnex12RoundingRulea">RoundingRule</a>
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<a href="../Typealiases.html#/s:12NumericAnnex4Signa">Sign</a>
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<article>
<section>
<div class="wrapper">
<div class="overview">
<h1>BinaryInteger</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">BinaryInteger</span> <span class="p">:</span> <span class="kt">Hashable</span><span class="p">,</span> <span class="kt">Numeric</span><span class="p">,</span> <span class="kt">CustomStringConvertible</span><span class="p">,</span> <span class="kt">Strideable</span></code></pre>
</div>
</div>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<p><a href="#/Exponentiation">Exponentiation</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexE2mmoixx_xtFZ">**(_:_:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexE3mmeoiyxz_xtFZ">**=(_:_:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexE3powxx_xtFZ">pow(_:_:)</a></li>
</ul>
<p><a href="#/Square%20Root">Square Root</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexE4sqrtxxFZ">sqrt(_:)</a></li>
</ul>
<p><a href="#/Cube%20Root">Cube Root</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE4cbrtxxFZ">cbrt(_:)</a></li>
</ul>
<p><a href="#/Factoring">Factoring</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE3gcdxx_xtFZ">gcd(_:_:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE3lcmxx_xtFZ">lcm(_:_:)</a></li>
</ul>
<p><a href="#/Factoring%20(Fixed-Width)">Factoring (Fixed-Width)</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE20gcdReportingOverflowx12partialValue_Sb8overflowtx_xtFZ">gcdReportingOverflow(_:_:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE20lcmReportingOverflowx12partialValue_Sb8overflowtx_xtFZ">lcmReportingOverflow(_:_:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE07lcmFullF0x4high_AF3lowtx_xtFZ">lcmFullWidth(_:_:)</a></li>
</ul>
<p><a href="#/Initializers">Initializers</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexExSgAC8RationalVyqd__G7exactly_tcs9DecodableRd__s9EncodableRd__s06SignedB0Rd__s021_ExpressibleByBuiltinB7LiteralRd__s08UnsignedB09Magnitudes0C0PRpd__AnP_ANQYd__AQRSlufc">init(exactly:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexExAC8RationalVyqd__Gcs9DecodableRd__s9EncodableRd__s06SignedB0Rd__s021_ExpressibleByBuiltinB7LiteralRd__s08UnsignedB09Magnitudes0C0PRpd__AlN_ALQYd__AORSlufc">init(_:)</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<div>
<a name="/Exponentiation"></a>
<a name="//apple_ref/swift/Section/Exponentiation" class="dashAnchor"></a>
<h3>Exponentiation</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexE2mmoixx_xtFZ"></a>
<a name="//apple_ref/swift/Method/**(_:_:)" class="dashAnchor"></a>
<h4>
<code>**(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the result of raising <code>lhs</code> to the power of <code>rhs</code>, rounded to a
representable value.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexE3mmeoiyxz_xtFZ"></a>
<a name="//apple_ref/swift/Method/**=(_:_:)" class="dashAnchor"></a>
<h4>
<code>**=(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Raises <code>lhs</code> to the power of <code>rhs</code> and stores the result in <code>lhs</code>, rounded
to a representable value.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexE3powxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/pow(_:_:)" class="dashAnchor"></a>
<h4>
<code>pow(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the result of raising <code>base</code> to the power of <code>exponent</code>, rounded
to a representable value (deprecated).</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">pow</span><span class="p">(</span><span class="n">_</span> <span class="nv">base</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">exponent</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<div>
<a name="/Square%20Root"></a>
<a name="//apple_ref/swift/Section/Square Root" class="dashAnchor"></a>
<h3>Square Root</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexE4sqrtxxFZ"></a>
<a name="//apple_ref/swift/Method/sqrt(_:)" class="dashAnchor"></a>
<h4>
<code>sqrt(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the square root of <code>x</code>, rounding toward zero. If <code>x</code> is negative,
a runtime error may occur.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">sqrt</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<div>
<a name="/Cube%20Root"></a>
<a name="//apple_ref/swift/Section/Cube Root" class="dashAnchor"></a>
<h3>Cube Root</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE4cbrtxxFZ"></a>
<a name="//apple_ref/swift/Method/cbrt(_:)" class="dashAnchor"></a>
<h4>
<code>cbrt(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the cube root of <code>x</code>, rounding toward zero.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">cbrt</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<div>
<a name="/Factoring"></a>
<a name="//apple_ref/swift/Section/Factoring" class="dashAnchor"></a>
<h3>Factoring</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE3gcdxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/gcd(_:_:)" class="dashAnchor"></a>
<h4>
<code>gcd(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the greatest common divisor of <code>a</code> and <code>b</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">gcd</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE3lcmxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/lcm(_:_:)" class="dashAnchor"></a>
<h4>
<code>lcm(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the least common multiple of <code>a</code> and <code>b</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcm</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<div>
<a name="/Factoring%20(Fixed-Width)"></a>
<a name="//apple_ref/swift/Section/Factoring (Fixed-Width)" class="dashAnchor"></a>
<h3>Factoring (Fixed-Width)</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE20gcdReportingOverflowx12partialValue_Sb8overflowtx_xtFZ"></a>
<a name="//apple_ref/swift/Method/gcdReportingOverflow(_:_:)" class="dashAnchor"></a>
<h4>
<code>gcdReportingOverflow(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the greatest common divisor of <code>a</code> and <code>b</code> and a flag to indicate
whether overflow occurred during the operation.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">gcdReportingOverflow</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span>
<span class="o">-&gt;</span> <span class="p">(</span><span class="nv">partialValue</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">overflow</span><span class="p">:</span> <span class="kt">Bool</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE20lcmReportingOverflowx12partialValue_Sb8overflowtx_xtFZ"></a>
<a name="//apple_ref/swift/Method/lcmReportingOverflow(_:_:)" class="dashAnchor"></a>
<h4>
<code>lcmReportingOverflow(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the least common multiple of <code>a</code> and <code>b</code> and a flag to indicate
whether overflow occurred during the operation.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcmReportingOverflow</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span>
<span class="o">-&gt;</span> <span class="p">(</span><span class="nv">partialValue</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">overflow</span><span class="p">:</span> <span class="kt">Bool</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE07lcmFullF0x4high_AF3lowtx_xtFZ"></a>
<a name="//apple_ref/swift/Method/lcmFullWidth(_:_:)" class="dashAnchor"></a>
<h4>
<code>lcmFullWidth(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the high and low parts of the least common multiple of <code>a</code> and <code>b</code>
computed using full-width arithmetic.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcmFullWidth</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span>
<span class="o">-&gt;</span> <span class="p">(</span><span class="nv">high</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">low</span><span class="p">:</span> <span class="k">Self</span><span class="o">.</span><span class="kt">Magnitude</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<div>
<a name="/Initializers"></a>
<a name="//apple_ref/swift/Section/Initializers" class="dashAnchor"></a>
<h3>Initializers</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexExSgAC8RationalVyqd__G7exactly_tcs9DecodableRd__s9EncodableRd__s06SignedB0Rd__s021_ExpressibleByBuiltinB7LiteralRd__s08UnsignedB09Magnitudes0C0PRpd__AnP_ANQYd__AQRSlufc"></a>
<a name="//apple_ref/swift/Method/init(exactly:)" class="dashAnchor"></a>
<h4>
<code>init(exactly:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new binary integer from the given rational value, if it can be
represented exactly.</p>
<p>If <code>source</code> is not representable exactly, the result is <code>nil</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">init</span><span class="p">?</span><span class="o">&lt;</span><span class="kt">U</span><span class="o">&gt;</span><span class="p">(</span><span class="n">exactly</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">U</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>A rational value to convert to a binary integer.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexExAC8RationalVyqd__Gcs9DecodableRd__s9EncodableRd__s06SignedB0Rd__s021_ExpressibleByBuiltinB7LiteralRd__s08UnsignedB09Magnitudes0C0PRpd__AlN_ALQYd__AORSlufc"></a>
<a name="//apple_ref/swift/Method/init(_:)" class="dashAnchor"></a>
<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new binary integer from the given rational value, rounding
toward zero.</p>
<p>If <code>source</code> is outside the bounds of this type after rounding toward zero,
a runtime error may occur.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">init</span><span class="o">&lt;</span><span class="kt">U</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">U</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
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<h1>Double</h1>
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<div class="language">
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Double</span></code></pre>
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</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<p><a href="#/Real">Real</a></p>
<ul>
<li><a href="#/s:12NumericAnnex4MathP1exvZ">e</a></li>
<li><a href="#/s:12NumericAnnex4MathP3phixvZ">phi</a></li>
<li><a href="#/s:12NumericAnnex4MathP2mmoixx_xtFZ">**(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4MathP3mmeoiyxz_xtFZ">**=(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4MathP18naturalExponentialxyF">naturalExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17binaryExponentialxyF">binaryExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17commonExponentialxyF">commonExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP26naturalExponentialMinusOnexyF">naturalExponentialMinusOne()</a></li>
<li><a href="#/s:12NumericAnnex4MathP16naturalLogarithmxyF">naturalLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP15binaryLogarithmxyF">binaryLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP15commonLogarithmxyF">commonLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP23naturalLogarithmOnePlusxyF">naturalLogarithmOnePlus()</a></li>
<li><a href="#/s:12NumericAnnex4MathP8cubeRootxyF">cubeRoot()</a></li>
<li><a href="#/s:12NumericAnnex4MathP4sinexyF">sine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP6cosinexyF">cosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP7tangentxyF">tangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP11inverseSinexyF">inverseSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP13inverseCosinexyF">inverseCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP14inverseTangentxyF">inverseTangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP14hyperbolicSinexyF">hyperbolicSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP16hyperbolicCosinexyF">hyperbolicCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17hyperbolicTangentxyF">hyperbolicTangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP21inverseHyperbolicSinexyF">inverseHyperbolicSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP23inverseHyperbolicCosinexyF">inverseHyperbolicCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP24inverseHyperbolicTangentxyF">inverseHyperbolicTangent()</a></li>
<li><a href="#/s:12NumericAnnex4RealP5hypotxx_xtFZ">hypot(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF">inverseTangent(dividingBy:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP5errorxyF">error()</a></li>
<li><a href="#/s:12NumericAnnex4RealP18complementaryErrorxyF">complementaryError()</a></li>
<li><a href="#/s:12NumericAnnex4RealP5gammaxyF">gamma()</a></li>
<li><a href="#/s:12NumericAnnex4RealP16logarithmicGammaxyF">logarithmicGamma()</a></li>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="k">var</span> <span class="nv">e</span><span class="p">:</span> <span class="kt">Double</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="k">var</span> <span class="nv">phi</span><span class="p">:</span> <span class="kt">Double</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Double</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Double</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Double</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Double</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<div class="language">
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">binaryExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<h4>
<code>commonExponential()</code>
</h4>
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<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">commonExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<h4>
<code>naturalExponentialMinusOne()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalExponentialMinusOne</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<h4>
<code>naturalLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<h4>
<code>binaryLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">binaryLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<a name="//apple_ref/swift/Method/commonLogarithm()" class="dashAnchor"></a>
<h4>
<code>commonLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">commonLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<a name="//apple_ref/swift/Method/naturalLogarithmOnePlus()" class="dashAnchor"></a>
<h4>
<code>naturalLogarithmOnePlus()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalLogarithmOnePlus</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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</div>
</li>
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<div>
<a name="/s:12NumericAnnex4MathP8cubeRootxyF"></a>
<a name="//apple_ref/swift/Method/cubeRoot()" class="dashAnchor"></a>
<h4>
<code>cubeRoot()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">cubeRoot</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
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</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP4sinexyF"></a>
<a name="//apple_ref/swift/Method/sine()" class="dashAnchor"></a>
<h4>
<code>sine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">sine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP6cosinexyF"></a>
<a name="//apple_ref/swift/Method/cosine()" class="dashAnchor"></a>
<h4>
<code>cosine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">cosine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</li>
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<div>
<a name="/s:12NumericAnnex4MathP7tangentxyF"></a>
<a name="//apple_ref/swift/Method/tangent()" class="dashAnchor"></a>
<h4>
<code>tangent()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">tangent</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
</div>
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<div>
<a name="/s:12NumericAnnex4MathP11inverseSinexyF"></a>
<a name="//apple_ref/swift/Method/inverseSine()" class="dashAnchor"></a>
<h4>
<code>inverseSine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseSine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<div>
<a name="/s:12NumericAnnex4MathP13inverseCosinexyF"></a>
<a name="//apple_ref/swift/Method/inverseCosine()" class="dashAnchor"></a>
<h4>
<code>inverseCosine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
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<p><a href="../Typealiases.html">Type Aliases</a></p>
<ul>
<li>
<a href="../Typealiases.html#/s:12NumericAnnex12RoundingRulea">RoundingRule</a>
</li>
<li>
<a href="../Typealiases.html#/s:12NumericAnnex4Signa">Sign</a>
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</li>
</ul>
</nav>
<article>
<section>
<div class="wrapper">
<div class="overview">
<h1>Float</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<p><a href="#/Real">Real</a></p>
<ul>
<li><a href="#/s:12NumericAnnex4MathP1exvZ">e</a></li>
<li><a href="#/s:12NumericAnnex4MathP3phixvZ">phi</a></li>
<li><a href="#/s:12NumericAnnex4MathP2mmoixx_xtFZ">**(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4MathP3mmeoiyxz_xtFZ">**=(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4MathP18naturalExponentialxyF">naturalExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17binaryExponentialxyF">binaryExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17commonExponentialxyF">commonExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP26naturalExponentialMinusOnexyF">naturalExponentialMinusOne()</a></li>
<li><a href="#/s:12NumericAnnex4MathP16naturalLogarithmxyF">naturalLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP15binaryLogarithmxyF">binaryLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP15commonLogarithmxyF">commonLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP23naturalLogarithmOnePlusxyF">naturalLogarithmOnePlus()</a></li>
<li><a href="#/s:12NumericAnnex4MathP8cubeRootxyF">cubeRoot()</a></li>
<li><a href="#/s:12NumericAnnex4MathP4sinexyF">sine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP6cosinexyF">cosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP7tangentxyF">tangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP11inverseSinexyF">inverseSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP13inverseCosinexyF">inverseCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP14inverseTangentxyF">inverseTangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP14hyperbolicSinexyF">hyperbolicSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP16hyperbolicCosinexyF">hyperbolicCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17hyperbolicTangentxyF">hyperbolicTangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP21inverseHyperbolicSinexyF">inverseHyperbolicSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP23inverseHyperbolicCosinexyF">inverseHyperbolicCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP24inverseHyperbolicTangentxyF">inverseHyperbolicTangent()</a></li>
<li><a href="#/s:12NumericAnnex4RealP5hypotxx_xtFZ">hypot(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF">inverseTangent(dividingBy:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP5errorxyF">error()</a></li>
<li><a href="#/s:12NumericAnnex4RealP18complementaryErrorxyF">complementaryError()</a></li>
<li><a href="#/s:12NumericAnnex4RealP5gammaxyF">gamma()</a></li>
<li><a href="#/s:12NumericAnnex4RealP16logarithmicGammaxyF">logarithmicGamma()</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<div>
<a name="/Real"></a>
<a name="//apple_ref/swift/Section/Real" class="dashAnchor"></a>
<h3>Real</h3>
</div>
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4MathP1exvZ"></a>
<a name="//apple_ref/swift/Variable/e" class="dashAnchor"></a>
<h4>
<code>e</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="k">var</span> <span class="nv">e</span><span class="p">:</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP3phixvZ"></a>
<a name="//apple_ref/swift/Variable/phi" class="dashAnchor"></a>
<h4>
<code>phi</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="k">var</span> <span class="nv">phi</span><span class="p">:</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP2mmoixx_xtFZ"></a>
<a name="//apple_ref/swift/Method/**(_:_:)" class="dashAnchor"></a>
<h4>
<code>**(_:_:)</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Float</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Float</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP3mmeoiyxz_xtFZ"></a>
<a name="//apple_ref/swift/Method/**=(_:_:)" class="dashAnchor"></a>
<h4>
<code>**=(_:_:)</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Float</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Float</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP18naturalExponentialxyF"></a>
<a name="//apple_ref/swift/Method/naturalExponential()" class="dashAnchor"></a>
<h4>
<code>naturalExponential()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP17binaryExponentialxyF"></a>
<a name="//apple_ref/swift/Method/binaryExponential()" class="dashAnchor"></a>
<h4>
<code>binaryExponential()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">binaryExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP17commonExponentialxyF"></a>
<a name="//apple_ref/swift/Method/commonExponential()" class="dashAnchor"></a>
<h4>
<code>commonExponential()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">commonExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP26naturalExponentialMinusOnexyF"></a>
<a name="//apple_ref/swift/Method/naturalExponentialMinusOne()" class="dashAnchor"></a>
<h4>
<code>naturalExponentialMinusOne()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalExponentialMinusOne</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP16naturalLogarithmxyF"></a>
<a name="//apple_ref/swift/Method/naturalLogarithm()" class="dashAnchor"></a>
<h4>
<code>naturalLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP15binaryLogarithmxyF"></a>
<a name="//apple_ref/swift/Method/binaryLogarithm()" class="dashAnchor"></a>
<h4>
<code>binaryLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">binaryLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP15commonLogarithmxyF"></a>
<a name="//apple_ref/swift/Method/commonLogarithm()" class="dashAnchor"></a>
<h4>
<code>commonLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">commonLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP23naturalLogarithmOnePlusxyF"></a>
<a name="//apple_ref/swift/Method/naturalLogarithmOnePlus()" class="dashAnchor"></a>
<h4>
<code>naturalLogarithmOnePlus()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalLogarithmOnePlus</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP8cubeRootxyF"></a>
<a name="//apple_ref/swift/Method/cubeRoot()" class="dashAnchor"></a>
<h4>
<code>cubeRoot()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">cubeRoot</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP4sinexyF"></a>
<a name="//apple_ref/swift/Method/sine()" class="dashAnchor"></a>
<h4>
<code>sine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">sine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP6cosinexyF"></a>
<a name="//apple_ref/swift/Method/cosine()" class="dashAnchor"></a>
<h4>
<code>cosine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">cosine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP7tangentxyF"></a>
<a name="//apple_ref/swift/Method/tangent()" class="dashAnchor"></a>
<h4>
<code>tangent()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">tangent</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP11inverseSinexyF"></a>
<a name="//apple_ref/swift/Method/inverseSine()" class="dashAnchor"></a>
<h4>
<code>inverseSine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseSine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP13inverseCosinexyF"></a>
<a name="//apple_ref/swift/Method/inverseCosine()" class="dashAnchor"></a>
<h4>
<code>inverseCosine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseCosine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP14inverseTangentxyF"></a>
<a name="//apple_ref/swift/Method/inverseTangent()" class="dashAnchor"></a>
<h4>
<code>inverseTangent()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseTangent</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP14hyperbolicSinexyF"></a>
<a name="//apple_ref/swift/Method/hyperbolicSine()" class="dashAnchor"></a>
<h4>
<code>hyperbolicSine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">hyperbolicSine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP16hyperbolicCosinexyF"></a>
<a name="//apple_ref/swift/Method/hyperbolicCosine()" class="dashAnchor"></a>
<h4>
<code>hyperbolicCosine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">hyperbolicCosine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP17hyperbolicTangentxyF"></a>
<a name="//apple_ref/swift/Method/hyperbolicTangent()" class="dashAnchor"></a>
<h4>
<code>hyperbolicTangent()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">hyperbolicTangent</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP21inverseHyperbolicSinexyF"></a>
<a name="//apple_ref/swift/Method/inverseHyperbolicSine()" class="dashAnchor"></a>
<h4>
<code>inverseHyperbolicSine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseHyperbolicSine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP23inverseHyperbolicCosinexyF"></a>
<a name="//apple_ref/swift/Method/inverseHyperbolicCosine()" class="dashAnchor"></a>
<h4>
<code>inverseHyperbolicCosine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseHyperbolicCosine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP24inverseHyperbolicTangentxyF"></a>
<a name="//apple_ref/swift/Method/inverseHyperbolicTangent()" class="dashAnchor"></a>
<h4>
<code>inverseHyperbolicTangent()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseHyperbolicTangent</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP5hypotxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/hypot(_:_:)" class="dashAnchor"></a>
<h4>
<code>hypot(_:_:)</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">hypot</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt">Float</span><span class="p">,</span> <span class="n">_</span> <span class="nv">y</span><span class="p">:</span> <span class="kt">Float</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF"></a>
<a name="//apple_ref/swift/Method/inverseTangent(dividingBy:)" class="dashAnchor"></a>
<h4>
<code>inverseTangent(dividingBy:)</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseTangent</span><span class="p">(</span><span class="n">dividingBy</span> <span class="nv">other</span><span class="p">:</span> <span class="kt">Float</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">gamma</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">logarithmicGamma</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
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<h1>FloatingPoint</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">FloatingPoint</span> <span class="p">:</span> <span class="kt">SignedNumeric</span><span class="p">,</span> <span class="kt">Strideable</span><span class="p">,</span> <span class="kt">Hashable</span></code></pre>
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</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<p><a href="#/Initializers">Initializers</a></p>
<ul>
<li><a href="#/s:s13FloatingPointP12NumericAnnexExAC8RationalVySiGcfc">init(_:)</a></li>
<li><a href="#/s:s13FloatingPointP12NumericAnnexExAC8RationalVys4Int8VGcfc">init(_:)</a></li>
<li><a href="#/s:s13FloatingPointP12NumericAnnexExAC8RationalVys5Int16VGcfc">init(_:)</a></li>
<li><a href="#/s:s13FloatingPointP12NumericAnnexExAC8RationalVys5Int32VGcfc">init(_:)</a></li>
<li><a href="#/s:s13FloatingPointP12NumericAnnexExAC8RationalVys5Int64VGcfc">init(_:)</a></li>
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<h3>Initializers</h3>
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<a name="/s:s13FloatingPointP12NumericAnnexExAC8RationalVySiGcfc"></a>
<a name="//apple_ref/swift/Method/init(_:)" class="dashAnchor"></a>
<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new floating-point value from the given rational value, after
rounding the whole part, the numerator of the fractional part, and the
denominator of the fractional part each to the closest possible
representation.</p>
<p>If two representable values are equally close, the result of rounding is
the value with more trailing zeros in its significand bit pattern.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>The rational value to convert to a floating-point value.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:s13FloatingPointP12NumericAnnexExAC8RationalVys4Int8VGcfc"></a>
<a name="//apple_ref/swift/Method/init(_:)" class="dashAnchor"></a>
<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new floating-point value from the given rational value, after
rounding the whole part, the numerator of the fractional part, and the
denominator of the fractional part each to the closest possible
representation.</p>
<p>If two representable values are equally close, the result of rounding is
the value with more trailing zeros in its significand bit pattern.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">Int8</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>The rational value to convert to a floating-point value.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:s13FloatingPointP12NumericAnnexExAC8RationalVys5Int16VGcfc"></a>
<a name="//apple_ref/swift/Method/init(_:)" class="dashAnchor"></a>
<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new floating-point value from the given rational value, after
rounding the whole part, the numerator of the fractional part, and the
denominator of the fractional part each to the closest possible
representation.</p>
<p>If two representable values are equally close, the result of rounding is
the value with more trailing zeros in its significand bit pattern.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">Int16</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>The rational value to convert to a floating-point value.</p>
</dd>
</dl>
</div>
</li>
<li>
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<a name="/s:s13FloatingPointP12NumericAnnexExAC8RationalVys5Int32VGcfc"></a>
<a name="//apple_ref/swift/Method/init(_:)" class="dashAnchor"></a>
<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new floating-point value from the given rational value, after
rounding the whole part, the numerator of the fractional part, and the
denominator of the fractional part each to the closest possible
representation.</p>
<p>If two representable values are equally close, the result of rounding is
the value with more trailing zeros in its significand bit pattern.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">Int32</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
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<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>The rational value to convert to a floating-point value.</p>
</dd>
</dl>
</div>
</li>
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<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new floating-point value from the given rational value, after
rounding the whole part, the numerator of the fractional part, and the
denominator of the fractional part each to the closest possible
representation.</p>
<p>If two representable values are equally close, the result of rounding is
the value with more trailing zeros in its significand bit pattern.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">Int64</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>The rational value to convert to a floating-point value.</p>
</dd>
</dl>
</div>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Int16</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int16</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Int16</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Int16</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int16</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int32</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Int32</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int32</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Int32</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Int32</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int32</span><span class="p">)</span></code></pre>
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<h1>Int64</h1>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int64</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
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<li><a href="#/s:s5Int64V12NumericAnnexE2mmoiA2B_ABtFZ">**(_:_:)</a></li>
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<code>**(_:_:)</code>
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<p>Returns the result of raising <code>lhs</code> to the power of <code>rhs</code>, rounded to a
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Int64</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int64</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Int64</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Int64</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int64</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int8</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Int8</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int8</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Int8</span></code></pre>
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<p>Raises <code>lhs</code> to the power of <code>rhs</code> and stores the result in <code>lhs</code>, rounded
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Int8</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int8</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">UInt</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt16</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
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<code>**(_:_:)</code>
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<p>Returns the result of raising <code>lhs</code> to the power of <code>rhs</code>, rounded to a
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<h5>Declaration</h5>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">UInt16</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt16</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UInt16</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">UInt16</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt16</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt32</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">UInt32</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt32</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UInt32</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">UInt32</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt32</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt64</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">UInt64</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt64</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UInt64</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">UInt64</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt64</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt8</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">UInt8</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt8</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UInt8</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">UInt8</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt8</span><span class="p">)</span></code></pre>
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<a href="../Functions.html#/s:12NumericAnnex5roundAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">round(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex4sqrtAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">sqrt(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex5truncAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">trunc(_:)</a>
</li>
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<p><a href="../Protocols.html">Protocols</a></p>
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<a href="../Protocols/Math.html">Math</a>
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<a href="../Protocols/PRNG.html">PRNG</a>
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<a href="../Protocols/Real.html">Real</a>
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<p><a href="../Structs.html">Structures</a></p>
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<a href="../Structs/Complex.html">Complex</a>
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<a href="../Structs/Rational.html">Rational</a>
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<p><a href="../Typealiases.html">Type Aliases</a></p>
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<a href="../Typealiases.html#/s:12NumericAnnex12RoundingRulea">RoundingRule</a>
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<a href="../Typealiases.html#/s:12NumericAnnex4Signa">Sign</a>
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<div class="overview">
<h1>UnsignedInteger</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">UnsignedInteger</span> <span class="p">:</span> <span class="kt">BinaryInteger</span> <span class="k">where</span> <span class="k">Self</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">BinaryInteger</span></code></pre>
</div>
</div>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<p><a href="#/Cube%20Root">Cube Root</a></p>
<ul>
<li><a href="#/s:s15UnsignedIntegerP12NumericAnnexE4cbrtxxFZ">cbrt(_:)</a></li>
</ul>
<p><a href="#/Factoring">Factoring</a></p>
<ul>
<li><a href="#/s:s15UnsignedIntegerP12NumericAnnexE3gcdxx_xtFZ">gcd(_:_:)</a></li>
<li><a href="#/s:s15UnsignedIntegerP12NumericAnnexE3lcmxx_xtFZ">lcm(_:_:)</a></li>
</ul>
<p><a href="#/Factoring%20(Fixed-Width)">Factoring (Fixed-Width)</a></p>
<ul>
<li><a href="#/s:s15UnsignedIntegerP12NumericAnnexs010FixedWidthB0RzsAARzlE20lcmReportingOverflowx12partialValue_Sb8overflowtx_xtFZ">lcmReportingOverflow(_:_:)</a></li>
<li><a href="#/s:s15UnsignedIntegerP12NumericAnnexs010FixedWidthB0RzsAARzlE07lcmFullF0x4high_9Magnitudes0C0PQz3lowtx_xtFZ">lcmFullWidth(_:_:)</a></li>
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<a name="/Cube%20Root"></a>
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<h3>Cube Root</h3>
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<div>
<a name="/s:s15UnsignedIntegerP12NumericAnnexE4cbrtxxFZ"></a>
<a name="//apple_ref/swift/Method/cbrt(_:)" class="dashAnchor"></a>
<h4>
<code>cbrt(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the cube root of <code>x</code>, rounding toward zero.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">cbrt</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
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<a name="/Factoring"></a>
<a name="//apple_ref/swift/Section/Factoring" class="dashAnchor"></a>
<h3>Factoring</h3>
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<li>
<div>
<a name="/s:s15UnsignedIntegerP12NumericAnnexE3gcdxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/gcd(_:_:)" class="dashAnchor"></a>
<h4>
<code>gcd(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the greatest common divisor of <code>a</code> and <code>b</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">gcd</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
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</li>
<li>
<div>
<a name="/s:s15UnsignedIntegerP12NumericAnnexE3lcmxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/lcm(_:_:)" class="dashAnchor"></a>
<h4>
<code>lcm(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the least common multiple of <code>a</code> and <code>b</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcm</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
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<div>
<a name="/Factoring%20(Fixed-Width)"></a>
<a name="//apple_ref/swift/Section/Factoring (Fixed-Width)" class="dashAnchor"></a>
<h3>Factoring (Fixed-Width)</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s15UnsignedIntegerP12NumericAnnexs010FixedWidthB0RzsAARzlE20lcmReportingOverflowx12partialValue_Sb8overflowtx_xtFZ"></a>
<a name="//apple_ref/swift/Method/lcmReportingOverflow(_:_:)" class="dashAnchor"></a>
<h4>
<code>lcmReportingOverflow(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the least common multiple of <code>a</code> and <code>b</code> and a flag to indicate
whether overflow occurred during the operation.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcmReportingOverflow</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span>
<span class="o">-&gt;</span> <span class="p">(</span><span class="nv">partialValue</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">overflow</span><span class="p">:</span> <span class="kt">Bool</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s15UnsignedIntegerP12NumericAnnexs010FixedWidthB0RzsAARzlE07lcmFullF0x4high_9Magnitudes0C0PQz3lowtx_xtFZ"></a>
<a name="//apple_ref/swift/Method/lcmFullWidth(_:_:)" class="dashAnchor"></a>
<h4>
<code>lcmFullWidth(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the high and low parts of the least common multiple of <code>a</code> and <code>b</code>
computed using full-width arithmetic.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcmFullWidth</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span>
<span class="o">-&gt;</span> <span class="p">(</span><span class="nv">high</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">low</span><span class="p">:</span> <span class="k">Self</span><span class="o">.</span><span class="kt">Magnitude</span><span class="p">)</span></code></pre>
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<h1>Functions</h1>
<p>The following functions are available globally.</p>
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<details open>
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<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex3absAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">abs(_:)</a></li>
<li><a href="#/s:12NumericAnnex4sqrtAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">sqrt(_:)</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex3absAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">abs(_:)</a></li>
<li><a href="#/s:12NumericAnnex4ceilAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">ceil(_:)</a></li>
<li><a href="#/s:12NumericAnnex5floorAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">floor(_:)</a></li>
<li><a href="#/s:12NumericAnnex5roundAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">round(_:)</a></li>
<li><a href="#/s:12NumericAnnex5truncAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">trunc(_:)</a></li>
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<a name="//apple_ref/swift/Function/abs(_:)" class="dashAnchor"></a>
<h4>
<code>abs(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the absolute value (magnitude, modulus) of <code>z</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">abs</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">z</span><span class="p">:</span> <span class="kt"><a href="Structs/Complex.html">Complex</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Complex.html">Complex</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
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<a name="/s:12NumericAnnex4sqrtAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF"></a>
<a name="//apple_ref/swift/Function/sqrt(_:)" class="dashAnchor"></a>
<h4>
<code>sqrt(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the square root of <code>z</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">sqrt</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">z</span><span class="p">:</span> <span class="kt"><a href="Structs/Complex.html">Complex</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Complex.html">Complex</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
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<a name="/s:12NumericAnnex3absAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF"></a>
<a name="//apple_ref/swift/Function/abs(_:)" class="dashAnchor"></a>
<h4>
<code>abs(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the absolute value (magnitude) of <code>x</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">abs</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
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<a name="/s:12NumericAnnex4ceilAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF"></a>
<a name="//apple_ref/swift/Function/ceil(_:)" class="dashAnchor"></a>
<h4>
<code>ceil(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the closest integral value greater than or equal to <code>x</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">ceil</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
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<a name="/s:12NumericAnnex5floorAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF"></a>
<a name="//apple_ref/swift/Function/floor(_:)" class="dashAnchor"></a>
<h4>
<code>floor(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the closest integral value less than or equal to <code>x</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">floor</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
</div>
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<div>
<a name="/s:12NumericAnnex5roundAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF"></a>
<a name="//apple_ref/swift/Function/round(_:)" class="dashAnchor"></a>
<h4>
<code>round(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the closest integral value; if two values are equally close, returns
the one with greater magnitude.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">round</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
</div>
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<div>
<a name="/s:12NumericAnnex5truncAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF"></a>
<a name="//apple_ref/swift/Function/trunc(_:)" class="dashAnchor"></a>
<h4>
<code>trunc(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the closest integral value with magnitude less than or equal to that
of <code>x</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">trunc</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
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Copyright © 2017 Xiaodi Wu (https://github.com/xwu).
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
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7
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// swift-tools-version:3.1
import PackageDescription
let package = Package(
name: "NumericAnnex"
)

319
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<a href="Protocols/PRNG.html">PRNG</a>
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<section>
<div class="wrapper">
<div class="overview">
<h1>Protocols</h1>
<p>The following protocols are available globally.</p>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex4PRNGP">PRNG</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4RealP">Real</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4MathP">Math</a></li>
</ul>
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<section>
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<ul>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGP"></a>
<a name="//apple_ref/swift/Protocol/PRNG" class="dashAnchor"></a>
<h4>
<code>PRNG</code>
</h4>
</div>
<div class="abstract">
<p>A pseudo-random number generator (PRNG).</p>
<p>Reference types that conform to <code>PRNG</code> are infinite sequences of
pseudo-random elements. Protocol extension methods iterate over such a
sequence as necessary to generate pseudo-random values from the desired
distribution.</p>
<h2 id='considerations-for-conforming-types' class='heading'>Considerations for Conforming Types</h2>
<p>For clarity to end users, custom PRNGs may be implemented in an extension to
<code><a href="Classes/Random.html">Random</a></code>. For instance, the <code>xoroshiro128+</code> algorithm is implemented in a
final class named <code><a href="Classes/Random/Xoroshiro.html">Random.Xoroshiro</a></code>.</p>
<p>The static methods <code>_entropy(_:)</code> and <code>_entropy(_:count:)</code> return
cryptographically secure random bytes that may be useful for seeding your
custom PRNG. However, these methods may return <code>nil</code> if the requested number
of random bytes is not available, and they are not recommended as a routine
source of random data.</p>
<h2 id='adding-other-probability-distributions' class='heading'>Adding Other Probability Distributions</h2>
<p>Many built-in protocol extension methods make use of the primitive,
overloaded method <code>_random(_:bitCount:)</code>. You may wish to use the same
method in new protocol extension methods that return pseudo-random
values from other probability distributions.</p>
<p>The method <code>_random(_:bitCount:)</code> generates uniformly distributed binary
floating-point values in the half-open range [0, 1) with a precision of
either <code>bitCount</code> or the significand bit count of the floating-point type,
whichever is less. Additionally, this method generates uniformly distributed
unsigned integers in the half-open range [0, 2 ** <em>x</em>), where ** is the
exponentiation operator and <em>x</em> is the lesser of <code>bitCount</code> and the bit
width of the integer type.</p>
<p>For end users, however, the recommended spelling for a uniformly distributed
numeric value is <code>uniform()</code>; that method is overloaded to permit custom
minimum and maximum values for the uniform distribution.</p>
<a href="Protocols/PRNG.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">protocol</span> <span class="kt">PRNG</span> <span class="p">:</span> <span class="kd">class</span><span class="p">,</span> <span class="kt">IteratorProtocol</span><span class="p">,</span> <span class="kt">Sequence</span>
<span class="k">where</span> <span class="kt">Element</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="p">,</span>
<span class="kt">SubSequence</span> <span class="p">:</span> <span class="kt">Sequence</span><span class="p">,</span>
<span class="kt">Element</span> <span class="o">==</span> <span class="kt">SubSequence</span><span class="o">.</span><span class="kt">Element</span></code></pre>
</div>
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<ul>
<li>
<div>
<a name="/s:12NumericAnnex4RealP"></a>
<a name="//apple_ref/swift/Protocol/Real" class="dashAnchor"></a>
<h4>
<code>Real</code>
</h4>
</div>
<div class="abstract">
<p>A floating-point type that supports elementary functions and a selection of
special functions.</p>
<p>The <code>Real</code> protocol provides a suitable basis for writing functions that
work on any floating-point type that supports the required functions.</p>
<a href="Protocols/Real.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">protocol</span> <span class="kt">Real</span> <span class="p">:</span> <span class="kt"><a href="Protocols/Math.html">Math</a></span><span class="p">,</span> <span class="kt">FloatingPoint</span></code></pre>
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<li>
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<a name="/s:12NumericAnnex4MathP"></a>
<a name="//apple_ref/swift/Protocol/Math" class="dashAnchor"></a>
<h4>
<code>Math</code>
</h4>
</div>
<div class="abstract">
<p>A signed numeric type that supports elementary functions.</p>
<p>The <code>Math</code> protocol provides a suitable basis for writing functions that
work on any real or complex floating-point type that supports the required
functions.</p>
<a href="Protocols/Math.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">protocol</span> <span class="kt">Math</span> <span class="p">:</span> <span class="kt">SignedNumeric</span></code></pre>
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<a href="../Functions.html#/s:12NumericAnnex3absAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">abs(_:)</a>
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<a href="../Protocols/Real.html">Real</a>
</li>
</ul>
</li>
<li>
<p><a href="../Structs.html">Structures</a></p>
<ul>
<li>
<a href="../Structs/Complex.html">Complex</a>
</li>
<li>
<a href="../Structs/Rational.html">Rational</a>
</li>
</ul>
</li>
<li>
<p><a href="../Typealiases.html">Type Aliases</a></p>
<ul>
<li>
<a href="../Typealiases.html#/s:12NumericAnnex12RoundingRulea">RoundingRule</a>
</li>
<li>
<a href="../Typealiases.html#/s:12NumericAnnex4Signa">Sign</a>
</li>
</ul>
</li>
</ul>
</nav>
<article>
<section>
<div class="wrapper">
<div class="overview">
<h1>PRNG</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">protocol</span> <span class="kt">PRNG</span> <span class="p">:</span> <span class="kd">class</span><span class="p">,</span> <span class="kt">IteratorProtocol</span><span class="p">,</span> <span class="kt">Sequence</span>
<span class="k">where</span> <span class="kt">Element</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="p">,</span>
<span class="kt">SubSequence</span> <span class="p">:</span> <span class="kt">Sequence</span><span class="p">,</span>
<span class="kt">Element</span> <span class="o">==</span> <span class="kt">SubSequence</span><span class="o">.</span><span class="kt">Element</span></code></pre>
</div>
</div>
<p>A pseudo-random number generator (PRNG).</p>
<p>Reference types that conform to <code>PRNG</code> are infinite sequences of
pseudo-random elements. Protocol extension methods iterate over such a
sequence as necessary to generate pseudo-random values from the desired
distribution.</p>
<h2 id='considerations-for-conforming-types' class='heading'>Considerations for Conforming Types</h2>
<p>For clarity to end users, custom PRNGs may be implemented in an extension to
<code><a href="../Classes/Random.html">Random</a></code>. For instance, the <code>xoroshiro128+</code> algorithm is implemented in a
final class named <code><a href="../Classes/Random/Xoroshiro.html">Random.Xoroshiro</a></code>.</p>
<p>The static methods <code>_entropy(_:)</code> and <code>_entropy(_:count:)</code> return
cryptographically secure random bytes that may be useful for seeding your
custom PRNG. However, these methods may return <code>nil</code> if the requested number
of random bytes is not available, and they are not recommended as a routine
source of random data.</p>
<h2 id='adding-other-probability-distributions' class='heading'>Adding Other Probability Distributions</h2>
<p>Many built-in protocol extension methods make use of the primitive,
overloaded method <code>_random(_:bitCount:)</code>. You may wish to use the same
method in new protocol extension methods that return pseudo-random
values from other probability distributions.</p>
<p>The method <code>_random(_:bitCount:)</code> generates uniformly distributed binary
floating-point values in the half-open range [0, 1) with a precision of
either <code>bitCount</code> or the significand bit count of the floating-point type,
whichever is less. Additionally, this method generates uniformly distributed
unsigned integers in the half-open range [0, 2 ** <em>x</em>), where ** is the
exponentiation operator and <em>x</em> is the lesser of <code>bitCount</code> and the bit
width of the integer type.</p>
<p>For end users, however, the recommended spelling for a uniformly distributed
numeric value is <code>uniform()</code>; that method is overloaded to permit custom
minimum and maximum values for the uniform distribution.</p>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex4PRNGP5State">State</a></li>
<li><a href="#/s:12NumericAnnex4PRNGP5state5StateQzv">state</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPx5StateQz5state_tcfc">init(state:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPxSgycfc">init()</a></li>
<li><a href="#/s:12NumericAnnex4PRNGP3max7ElementQzvZ">max</a></li>
<li><a href="#/s:12NumericAnnex4PRNGP3min7ElementQzvZ">min</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4PRNGPAAE15_randomBitWidthSivZ">_randomBitWidth</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE8_entropyqd__Sgqd__ms17FixedWidthIntegerRd__s08UnsignedG0Rd__lFZ">_entropy(_:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE8_entropySayqd__GSgqd__m_Si5countts17FixedWidthIntegerRd__s08UnsignedH0Rd__lFZ">_entropy(_:count:)</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7_randomqd__qd__m_Si8bitCountts17FixedWidthIntegerRd__s08UnsignedI0Rd__lF">_random(_:bitCount:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__m_qd__1aqd__1bts17FixedWidthIntegerRd__s08UnsignedG0Rd__lF">uniform(_:a:b:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__ms17FixedWidthIntegerRd__s08UnsignedG0Rd__lF">uniform(_:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts17FixedWidthIntegerRd__s08UnsignedJ0Rd__lF">uniform(_:a:b:count:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts17FixedWidthIntegerRd__s08UnsignedJ0Rd__lF">uniform(_:count:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__m_qd__1aqd__1bts17FixedWidthIntegerRd__s06SignedG0Rd__sAG9Magnitudes0A0PRpd__s08UnsignedG0ALRQlF">uniform(_:a:b:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__ms17FixedWidthIntegerRd__s06SignedG0Rd__sAE9Magnitudes0A0PRpd__s08UnsignedG0AJRQlF">uniform(_:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts17FixedWidthIntegerRd__s06SignedJ0Rd__sAK9Magnitudes0A0PRpd__s08UnsignedJ0APRQlF">uniform(_:a:b:count:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts17FixedWidthIntegerRd__s06SignedJ0Rd__sAI9Magnitudes0A0PRpd__s08UnsignedJ0ANRQlF">uniform(_:count:)</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7_randomqd__qd__m_Si8bitCountts19BinaryFloatingPointRd__lF">_random(_:bitCount:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniformqd__qd__m_qd__1aqd__1bts19BinaryFloatingPointRd__lF">uniform(_:a:b:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniformqd__qd__ms19BinaryFloatingPointRd__lF">uniform(_:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts19BinaryFloatingPointRd__lF">uniform(_:a:b:count:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts19BinaryFloatingPointRd__lF">uniform(_:count:)</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGP5State"></a>
<a name="//apple_ref/swift/Alias/State" class="dashAnchor"></a>
<h4>
<code>State</code>
</h4>
</div>
<div class="abstract">
<p>A type that can represent the internal state of the pseudo-random number
generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="k">associatedtype</span> <span class="kt">State</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGP5state5StateQzv"></a>
<a name="//apple_ref/swift/Property/state" class="dashAnchor"></a>
<h4>
<code>state</code>
</h4>
</div>
<div class="abstract">
<p>The internal state of the pseudo-random number generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="k">var</span> <span class="nv">state</span><span class="p">:</span> <span class="kt"><a href="../Protocols/PRNG.html#/s:12NumericAnnex4PRNGP5State">State</a></span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPx5StateQz5state_tcfc"></a>
<a name="//apple_ref/swift/Method/init(state:)" class="dashAnchor"></a>
<h4>
<code>init(state:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with the given internal state.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="nf">init</span><span class="p">(</span><span class="nv">state</span><span class="p">:</span> <span class="kt"><a href="../Protocols/PRNG.html#/s:12NumericAnnex4PRNGP5State">State</a></span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>state</code></dt>
<dd>
<p>The value to be used as the generator&rsquo;s internal state.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPxSgycfc"></a>
<a name="//apple_ref/swift/Method/init()" class="dashAnchor"></a>
<h4>
<code>init()</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with an internal state seeded
using cryptographically secure random bytes.</p>
<p>If cryptographically secure random bytes are unavailable, the result is
<code>nil</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="nf">init</span><span class="p">?()</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGP3max7ElementQzvZ"></a>
<a name="//apple_ref/swift/Variable/max" class="dashAnchor"></a>
<h4>
<code>max</code>
<span class="declaration-note">Default implementation</span>
</h4>
</div>
<div class="abstract">
<p>The maximum value that may be generated by the pseudo-random number
generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">static</span> <span class="k">var</span> <span class="nv">max</span><span class="p">:</span> <span class="kt">Element</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGP3min7ElementQzvZ"></a>
<a name="//apple_ref/swift/Variable/min" class="dashAnchor"></a>
<h4>
<code>min</code>
<span class="declaration-note">Default implementation</span>
</h4>
</div>
<div class="abstract">
<p>The minimum value that may be generated by the pseudo-random number
generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">static</span> <span class="k">var</span> <span class="nv">min</span><span class="p">:</span> <span class="kt">Element</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE15_randomBitWidthSivZ"></a>
<a name="//apple_ref/swift/Variable/_randomBitWidth" class="dashAnchor"></a>
<h4>
<code>_randomBitWidth</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>The number of pseudo-random bits available from a value generated by the
pseudo-random number generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="k">var</span> <span class="nv">_randomBitWidth</span><span class="p">:</span> <span class="kt">Int</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE8_entropyqd__Sgqd__ms17FixedWidthIntegerRd__s08UnsignedG0Rd__lFZ"></a>
<a name="//apple_ref/swift/Method/_entropy(_:)" class="dashAnchor"></a>
<h4>
<code>_entropy(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns a value filled with data from a source of cryptographically secure
random bytes, or <code>nil</code> if a sufficient number of cryptographically secure
random bytes is unavailable.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="n">_entropy</span><span class="o">&lt;</span>
<span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span>
<span class="o">&gt;</span><span class="p">(</span><span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span><span class="p">?</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE8_entropySayqd__GSgqd__m_Si5countts17FixedWidthIntegerRd__s08UnsignedH0Rd__lFZ"></a>
<a name="//apple_ref/swift/Method/_entropy(_:count:)" class="dashAnchor"></a>
<h4>
<code>_entropy(_:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns an array of <code>count</code> values filled with data from a source of
cryptographically secure random bytes, or <code>nil</code> if a sufficient number of
cryptographically secure random bytes is unavailable.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="n">_entropy</span><span class="o">&lt;</span>
<span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span>
<span class="o">&gt;</span><span class="p">(</span><span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="p">[</span><span class="kt">T</span><span class="p">]?</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7_randomqd__qd__m_Si8bitCountts17FixedWidthIntegerRd__s08UnsignedI0Rd__lF"></a>
<a name="//apple_ref/swift/Method/_random(_:bitCount:)" class="dashAnchor"></a>
<h4>
<code>_random(_:bitCount:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random unsigned integer of type <code>T</code> in the range from 0
to <code>2 ** min(bitCount, T.bitWidth)</code> (exclusive), where <code>**</code> is the
exponentiation operator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">_random</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">bitCount</span><span class="p">:</span> <span class="kt">Int</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="n">bitWidth</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__m_qd__1aqd__1bts17FixedWidthIntegerRd__s08UnsignedG0Rd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random unsigned integer of type <code>T</code> in the range from
<code>a</code> through <code>b</code> (inclusive) from the discrete uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__ms17FixedWidthIntegerRd__s08UnsignedG0Rd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random unsigned integer of type <code>T</code> in the range from
<code>T.min</code> through <code>T.max</code> (inclusive) from the discrete uniform
distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts17FixedWidthIntegerRd__s08UnsignedJ0Rd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random unsigned integers of type
<code>T</code> in the range from <code>a</code> through <code>b</code> (inclusive) from the discrete
uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts17FixedWidthIntegerRd__s08UnsignedJ0Rd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random unsigned integers of type
<code>T</code> in the range from <code>T.min</code> through <code>T.max</code> (inclusive) from the
discrete uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__m_qd__1aqd__1bts17FixedWidthIntegerRd__s06SignedG0Rd__sAG9Magnitudes0A0PRpd__s08UnsignedG0ALRQlF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random signed integer of type <code>T</code> in the range from <code>a</code>
through <code>b</code> (inclusive) from the discrete uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">SignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span> <span class="k">where</span> <span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__ms17FixedWidthIntegerRd__s06SignedG0Rd__sAE9Magnitudes0A0PRpd__s08UnsignedG0AJRQlF"></a>
<a name="//apple_ref/swift/Method/uniform(_:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random signed integer of type <code>T</code> in the range from
<code>T.min</code> through <code>T.max</code> (inclusive) from the discrete uniform
distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">SignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span> <span class="k">where</span> <span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts17FixedWidthIntegerRd__s06SignedJ0Rd__sAK9Magnitudes0A0PRpd__s08UnsignedJ0APRQlF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random signed integers of type <code>T</code>
in the range from <code>a</code> through <code>b</code> (inclusive) from the discrete uniform
distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">SignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span>
<span class="k">where</span> <span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts17FixedWidthIntegerRd__s06SignedJ0Rd__sAI9Magnitudes0A0PRpd__s08UnsignedJ0ANRQlF"></a>
<a name="//apple_ref/swift/Method/uniform(_:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random signed integers of type <code>T</code>
in the range from <code>T.min</code> through <code>T.max</code> (inclusive) from the discrete
uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">SignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span>
<span class="k">where</span> <span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7_randomqd__qd__m_Si8bitCountts19BinaryFloatingPointRd__lF"></a>
<a name="//apple_ref/swift/Method/_random(_:bitCount:)" class="dashAnchor"></a>
<h4>
<code>_random(_:bitCount:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random binary floating-point value of type <code>T</code> in the
range from 0 to 1 (exclusive) with <code>min(bitCount, T.significandBitCount)</code>
bits of precision.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">_random</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">BinaryFloatingPoint</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">bitCount</span><span class="p">:</span> <span class="kt">Int</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="n">significandBitCount</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniformqd__qd__m_qd__1aqd__1bts19BinaryFloatingPointRd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random binary floating-point value of type <code>T</code> in the
range from <code>a</code> to <code>b</code> (exclusive) from the uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">BinaryFloatingPoint</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniformqd__qd__ms19BinaryFloatingPointRd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random binary floating-point value of type <code>T</code> in the
range from 0 to 1 (exclusive) from the uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">BinaryFloatingPoint</span><span class="o">&gt;</span><span class="p">(</span><span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts19BinaryFloatingPointRd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random binary floating-point values
of type <code>T</code> in the range from <code>a</code> to <code>b</code> (exclusive) from the uniform
distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">BinaryFloatingPoint</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts19BinaryFloatingPointRd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random binary floating-point values
of type <code>T</code> in the range from 0 to 1 (exclusive) from the uniform
distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">BinaryFloatingPoint</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span></code></pre>
</div>
</div>
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</ul>
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</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex4ceilAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">ceil(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex5floorAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">floor(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex5roundAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">round(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex4sqrtAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">sqrt(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex5truncAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">trunc(_:)</a>
</li>
</ul>
</li>
<li>
<p><a href="../Protocols.html">Protocols</a></p>
<ul>
<li>
<a href="../Protocols/Math.html">Math</a>
</li>
<li>
<a href="../Protocols/PRNG.html">PRNG</a>
</li>
<li>
<a href="../Protocols/Real.html">Real</a>
</li>
</ul>
</li>
<li>
<p><a href="../Structs.html">Structures</a></p>
<ul>
<li>
<a href="../Structs/Complex.html">Complex</a>
</li>
<li>
<a href="../Structs/Rational.html">Rational</a>
</li>
</ul>
</li>
<li>
<p><a href="../Typealiases.html">Type Aliases</a></p>
<ul>
<li>
<a href="../Typealiases.html#/s:12NumericAnnex12RoundingRulea">RoundingRule</a>
</li>
<li>
<a href="../Typealiases.html#/s:12NumericAnnex4Signa">Sign</a>
</li>
</ul>
</li>
</ul>
</nav>
<article>
<section>
<div class="wrapper">
<div class="overview">
<h1>Real</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">protocol</span> <span class="kt">Real</span> <span class="p">:</span> <span class="kt"><a href="../Protocols/Math.html">Math</a></span><span class="p">,</span> <span class="kt">FloatingPoint</span></code></pre>
</div>
</div>
<p>A floating-point type that supports elementary functions and a selection of
special functions.</p>
<p>The <code>Real</code> protocol provides a suitable basis for writing functions that
work on any floating-point type that supports the required functions.</p>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex4RealP5hypotxx_xtFZ">hypot(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF">inverseTangent(dividingBy:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP5errorxyF">error()</a></li>
<li><a href="#/s:12NumericAnnex4RealP18complementaryErrorxyF">complementaryError()</a></li>
<li><a href="#/s:12NumericAnnex4RealP5gammaxyF">gamma()</a></li>
<li><a href="#/s:12NumericAnnex4RealP16logarithmicGammaxyF">logarithmicGamma()</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4RealPAAE8cubeRootxyF">cubeRoot()</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4RealPAAE5atan2xx_xtFZ">atan2(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealPAAE3erfxxFZ">erf(_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealPAAE4erfcxxFZ">erfc(_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealPAAE6tgammaxxFZ">tgamma(_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealPAAE6lgammaxxFZ">lgamma(_:)</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4RealP5hypotxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/hypot(_:_:)" class="dashAnchor"></a>
<h4>
<code>hypot(_:_:)</code>
<span class="declaration-note">Default implementation</span>
</h4>
</div>
<div class="abstract">
<p>Returns the hypotenuse of a right-angle triangle with legs (catheti) of
length <code>x</code> and <code>y</code>, preventing avoidable arithmetic overflow and
underflow. The return value is the square root of the sum of squares of
the arguments.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">static</span> <span class="kd">func</span> <span class="nf">hypot</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">y</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>x</code></dt>
<dd>
<p>The length of one leg (cathetus) of a right-angle triangle.</p>
</dd>
<dt><code>y</code></dt>
<dd>
<p>The length of the other leg (cathetus) of a right-angle triangle.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF"></a>
<a name="//apple_ref/swift/Method/inverseTangent(dividingBy:)" class="dashAnchor"></a>
<h4>
<code>inverseTangent(dividingBy:)</code>
<span class="declaration-note">Default implementation</span>
</h4>
</div>
<div class="abstract">
<p>Returns the inverse tangent of <code>self / other</code>, using the signs of <code>self</code>
and <code>other</code> to determine the quadrant of the computed angle.</p>
<p>If <code>self == 0 &amp;&amp; other == 0</code>, the return value is still finite.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<p><code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealPAAE5atan2xx_xtFZ">atan2(_:_:)</a></code></p>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">func</span> <span class="nf">inverseTangent</span><span class="p">(</span><span class="n">dividingBy</span> <span class="nv">other</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>other</code></dt>
<dd>
<p>The divisor by which to divide <code>self</code>.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP5errorxyF"></a>
<a name="//apple_ref/swift/Method/error()" class="dashAnchor"></a>
<h4>
<code>error()</code>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/Erf.html">error function</a> of <code>self</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealPAAE3erfxxFZ">erf(_:)</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">func</span> <span class="nf">error</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP18complementaryErrorxyF"></a>
<a name="//apple_ref/swift/Method/complementaryError()" class="dashAnchor"></a>
<h4>
<code>complementaryError()</code>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/Erfc.html">complementary error function</a> of <code>self</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealPAAE4erfcxxFZ">erfc(_:)</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">func</span> <span class="nf">complementaryError</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP5gammaxyF"></a>
<a name="//apple_ref/swift/Method/gamma()" class="dashAnchor"></a>
<h4>
<code>gamma()</code>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/GammaFunction.html">gamma function</a> of <code>self</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealPAAE6tgammaxxFZ">tgamma(_:)</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">func</span> <span class="nf">gamma</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP16logarithmicGammaxyF"></a>
<a name="//apple_ref/swift/Method/logarithmicGamma()" class="dashAnchor"></a>
<h4>
<code>logarithmicGamma()</code>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/LogGammaFunction.html">logarithmic gamma function</a> of <code>self</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealPAAE6lgammaxxFZ">lgamma(_:)</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">func</span> <span class="nf">logarithmicGamma</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE8cubeRootxyF"></a>
<a name="//apple_ref/swift/Method/cubeRoot()" class="dashAnchor"></a>
<h4>
<code>cubeRoot()</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the cube root of the value, rounded to a representable value
(default implementation).</p>
<div class="aside aside-note">
<p class="aside-title">Note</p>
<p>This method is required by the protocol <code><a href="../Protocols/Math.html">Math</a></code>; it is not an
extension method.</p>
</div>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<p><code>cbrt(_:)</code></p>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">cubeRoot</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE5atan2xx_xtFZ"></a>
<a name="//apple_ref/swift/Method/atan2(_:_:)" class="dashAnchor"></a>
<h4>
<code>atan2(_:_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the inverse tangent of <code>y / x</code>, using the signs of <code>y</code> and <code>x</code> to
determine the quadrant of the computed angle.</p>
<p>If <code>x == 0 &amp;&amp; y == 0</code>, the return value is still finite.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<p><code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF">inverseTangent(dividingBy:)</a></code></p>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">atan2</span><span class="p">(</span><span class="n">_</span> <span class="nv">y</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>y</code></dt>
<dd>
<p>The value to divide.</p>
</dd>
<dt><code>x</code></dt>
<dd>
<p>The divisor by which to divide <code>y</code>.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE3erfxxFZ"></a>
<a name="//apple_ref/swift/Method/erf(_:)" class="dashAnchor"></a>
<h4>
<code>erf(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/Erf.html">error function</a> of <code>x</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealP5errorxyF">error()</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">erf</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE4erfcxxFZ"></a>
<a name="//apple_ref/swift/Method/erfc(_:)" class="dashAnchor"></a>
<h4>
<code>erfc(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/Erfc.html">complementary error function</a> of <code>x</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealP18complementaryErrorxyF">complementaryError()</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">erfc</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE6tgammaxxFZ"></a>
<a name="//apple_ref/swift/Method/tgamma(_:)" class="dashAnchor"></a>
<h4>
<code>tgamma(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/GammaFunction.html">gamma function</a> of <code>x</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealP5gammaxyF">gamma()</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">tgamma</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE6lgammaxxFZ"></a>
<a name="//apple_ref/swift/Method/lgamma(_:)" class="dashAnchor"></a>
<h4>
<code>lgamma(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/LogGammaFunction.html">logarithmic gamma function</a> of <code>x</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealP16logarithmicGammaxyF">logarithmicGamma()</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lgamma</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
</div>
</section>
</article>
</div>
<footer>
<p>&copy; 2017 <a class="link" href="https://diploid.ca" target="_blank" rel="external">Xiaodi Wu</a>. All rights reserved. (Last updated: 2017-09-30)</p>
<p>Generated by <a href="https://github.com/realm/jazzy" target="_blank" rel="external">Jazzy</a> v0.8.4, a <a href="http://realm.io" target="_blank" rel="external">Realm</a> project.</p>
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108
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@ -1,108 +0,0 @@
# NumericAnnex
NumericAnnex supplements the numeric facilities provided in the Swift standard
library.
## Features
- [x] `BinaryInteger` exponentiation, greatest common divisor, and least common
multiple functions.
- [x] `Math`, a protocol for types providing square root, cube root, and
elementary transcendental functions.
- [x] `FloatingPointMath`, a protocol for floating-point types providing certain
special functions.
- [x] `PRNG`, a protocol for pseudo-random number generators (PRNGs).
- [x] `Complex`, a value type to represent complex values in Cartesian form.
- [x] `Rational`, a value type to represent rational values, which supports
division by zero.
- [x] `Random` and `Random.Xoroshiro`, two reference types implementing
efficient non-cryptographically-secure PRNGs.
> Note: This project is in the early stages of development and is not
> production-ready at this time.
## Requirements
NumericAnnex now requires a recent development snapshot of Swift 4.0 that
includes the revised numeric protocols. It requires either `Darwin.C` or `Glibc`
for transcendental functions provided by the C standard library and either
`Security` (Apple platforms) or `Glibc` (Linux) for cryptographically secure
random bytes.
## Installation
After NumericAnnex has been cloned or downloaded locally, build the library by
invoking `swift build`, or run tests with `swift test`. An Xcode project can be
generated by invoking `swift package generate-xcodeproj`.
[Swift Package Manager](https://swift.org/package-manager/) can also be used to
add the package as a dependency for your own project. See Swift documentation
for details.
## Basic Usage
```swift
import NumericAnnex
var x: Ratio = 1 / 4
// Ratio is a type alias for Rational<Int>.
print(x.reciprocal())
// Prints "4".
x *= 8
print(x + x)
// Prints "4".
x = Ratio(Float.phi) // Golden ratio.
print(x)
// Prints "13573053/8388608".
var z: Complex64 = 42 * .i
// Complex64 is a type alias for Complex<Float>.
print(Complex.sqrt(z))
// Prints "4.58258 + 4.58258i".
z = .pi + .i * .log(2 - .sqrt(3))
print(Complex.cos(z).real)
// Prints "-2.0".
```
## Documentation
All public protocols, types, and functions have been carefully documented in
comments.
The project adheres to many design patterns found in the Swift standard library.
For example, `Math` types provide methods such as `cubeRoot()` and `tangent()`
just as `FloatingPoint` types provide methods such as `squareRoot()`.
No free functions are declared in this library unless they overload existing
ones in the Swift standard library. Instead, functions such as `cbrt(_:)` and
`tan(_:)` are provided as static members. This avoids collisions with C standard
library functions that you may wish to use. It also promotes clarity at the call
site when the result of a complex operation differs from that of its real
counterpart (e.g., `Complex128.cbrt(-8) != -2`).
## Future Directions
- [ ] Add more tests, including performance tests
- [ ] Design and implement additional methods on `PRNG`
- [ ] Design and implement `BigInt`
## License
All original work is released under the MIT license. See
[LICENSE](https://github.com/xwu/NumericAnnex/blob/master/LICENSE) for details.
Portions of the complex square root and elementary transcendental functions use
checks for special values adapted from libc++. Code in libc++ is dual-licensed
under the MIT and UIUC/NCSA licenses.

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Sources/Complex+Math.swift
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//
// Complex+Math.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 4/2/17.
//
// Note
// ====
//
// For maximum consistency with corresponding functions in C/C++, checks for
// special values in `naturalExponential()`, `squareRoot()`, trigonometric
// functions, and hyperbolic functions are adapted from libc++.
//
// Code in libc++ is dual-licensed under the MIT and UIUC/NCSA licenses.
// Copyright © 2009-2017 contributors to the LLVM/libc++ project.
extension Complex : Numeric {
@_transparent // @_inlineable
public init?<U>(exactly source: U) where U : BinaryInteger {
guard let t = T(exactly: source) else { return nil }
self.real = t
self.imaginary = 0
}
@_transparent // @_inlineable
public static func + (lhs: Complex, rhs: Complex) -> Complex {
return Complex(
real: lhs.real + rhs.real, imaginary: lhs.imaginary + rhs.imaginary
)
}
@_transparent // @_inlineable
public static func += (lhs: inout Complex, rhs: Complex) {
lhs.real += rhs.real
lhs.imaginary += rhs.imaginary
}
@_transparent // @_inlineable
public static func - (lhs: Complex, rhs: Complex) -> Complex {
return Complex(
real: lhs.real - rhs.real, imaginary: lhs.imaginary - rhs.imaginary
)
}
@_transparent // @_inlineable
public static func -= (lhs: inout Complex, rhs: Complex) {
lhs.real -= rhs.real
lhs.imaginary -= rhs.imaginary
}
@_transparent // @_inlineable
public static func * (lhs: Complex, rhs: Complex) -> Complex {
return Complex(
real: lhs.real * rhs.real - lhs.imaginary * rhs.imaginary,
imaginary: lhs.real * rhs.imaginary + lhs.imaginary * rhs.real
)
}
@_transparent // @_inlineable
public static func *= (lhs: inout Complex, rhs: Complex) {
let t = lhs.real
lhs.real = lhs.real * rhs.real - lhs.imaginary * rhs.imaginary
lhs.imaginary = t * rhs.imaginary + lhs.imaginary * rhs.real
}
}
extension Complex : SignedNumeric {
@_transparent // @_inlineable
public static prefix func - (operand: Complex) -> Complex {
return Complex(real: -operand.real, imaginary: -operand.imaginary)
}
@_transparent // @_inlineable
public mutating func negate() {
real.negate()
imaginary.negate()
}
}
extension Complex : Math {
@_transparent // @_inlineable
public static var pi: Complex {
return Complex(real: .pi)
}
@_transparent // @_inlineable
public static var e: Complex {
return Complex(real: .e)
}
@_transparent // @_inlineable
public static var phi: Complex {
return Complex(real: .phi)
}
@_transparent // @_inlineable
public static func / (lhs: Complex, rhs: Complex) -> Complex {
// Prevent avoidable overflow; see Numerical Recipes.
if rhs.real.magnitude >= rhs.imaginary.magnitude {
let ratio = rhs.imaginary / rhs.real
let denominator = rhs.real + rhs.imaginary * ratio
return Complex(
real: (lhs.real + lhs.imaginary * ratio) / denominator,
imaginary: (lhs.imaginary - lhs.real * ratio) / denominator
)
}
let ratio = rhs.real / rhs.imaginary
let denominator = rhs.real * ratio + rhs.imaginary
return Complex(
real: (lhs.real * ratio + lhs.imaginary) / denominator,
imaginary: (lhs.imaginary * ratio - lhs.real) / denominator
)
/*
let denominator = rhs.squaredMagnitude
return Complex(
real:
(lhs.real * rhs.real + lhs.imaginary * rhs.imaginary) / denominator,
imaginary:
(lhs.imaginary * rhs.real - lhs.real * rhs.imaginary) / denominator
)
*/
}
@_transparent // @_inlineable
public static func /= (lhs: inout Complex, rhs: Complex) {
// Prevent avoidable overflow; see Numerical Recipes.
let t = lhs.real
if rhs.real.magnitude >= rhs.imaginary.magnitude {
let ratio = rhs.imaginary / rhs.real
let denominator = rhs.real + rhs.imaginary * ratio
lhs.real = (lhs.real + lhs.imaginary * ratio) / denominator
lhs.imaginary = (lhs.imaginary - t * ratio) / denominator
} else {
let ratio = rhs.real / rhs.imaginary
let denominator = rhs.real * ratio + rhs.imaginary
lhs.real = (lhs.real * ratio + lhs.imaginary) / denominator
lhs.imaginary = (lhs.imaginary * ratio - t) / denominator
}
}
// @_transparent // @_inlineable
public func naturalExponential() -> Complex {
if real.isNaN && imaginary == 0 { return self }
var im = imaginary
if real.isInfinite {
if real < 0 && !im.isFinite {
im = 1
} else if im == 0 || !im.isFinite {
if im.isInfinite { im = .nan }
return Complex(real: real, imaginary: im)
}
}
return Complex(r: T.exp(real), theta: im)
}
@_transparent // @_inlineable
public func naturalLogarithm() -> Complex {
return Complex(real: T.log(magnitude), imaginary: argument)
}
@_transparent // @_inlineable
public func commonLogarithm() -> Complex {
return Complex.log(self) / Complex.log(10 as Complex)
}
// @_transparent // @_inlineable
public func squareRoot() -> Complex {
if imaginary.isInfinite {
return Complex(real: .infinity, imaginary: imaginary)
}
if real.isInfinite {
if real > 0 {
return Complex(
real: real,
imaginary: imaginary.isNaN ? imaginary :
T(signOf: imaginary, magnitudeOf: 0)
)
}
return Complex(
real: imaginary.isNaN ? imaginary : 0,
imaginary: T(signOf: imaginary, magnitudeOf: real)
)
}
// Guard intermediate results and enforce a branch cut; see Numerical
// Recipes.
if real == 0 && imaginary == 0 { return 0 }
let w: T
let x = abs(real), y = abs(imaginary)
if x >= y {
let r = y / x
w = T.sqrt(x) * T.sqrt((1 + T.sqrt(1 + r * r)) / 2)
} else {
let r = x / y
w = T.sqrt(y) * T.sqrt((r + T.sqrt(1 + r * r)) / 2)
}
if real >= 0 {
return Complex(real: w, imaginary: imaginary / (2 * w))
}
/* real < 0 */
return Complex(real: y / (2 * w), imaginary: imaginary >= 0 ? w : -w)
/*
return Complex(r: T.sqrt(magnitude), theta: argument / 2)
*/
}
@_transparent // @_inlineable
public func cubeRoot() -> Complex {
return Complex.exp(Complex.log(self) / 3)
/*
return Complex(r: T.cbrt(magnitude), theta: argument / 3)
*/
}
@_transparent // @_inlineable
public func power(of base: Complex) -> Complex {
return Complex.exp(self * Complex.log(base))
}
@_transparent // @_inlineable
public func sine() -> Complex {
let sinh = Complex.sinh(Complex(real: -imaginary, imaginary: real))
return Complex(real: sinh.imaginary, imaginary: -sinh.real)
}
@_transparent // @_inlineable
public func cosine() -> Complex {
return Complex.cosh(Complex(real: -imaginary, imaginary: real))
}
@_transparent // @_inlineable
public func tangent() -> Complex {
let tanh = Complex.tanh(Complex(real: -imaginary, imaginary: real))
return Complex(real: tanh.imaginary, imaginary: -tanh.real)
}
@_transparent // @_inlineable
public func inverseSine() -> Complex {
let asinh = Complex.asinh(Complex(real: -imaginary, imaginary: real))
return Complex(real: asinh.imaginary, imaginary: -asinh.real)
}
// @_transparent // @_inlineable
public func inverseCosine() -> Complex {
if real.isNaN {
if imaginary.isInfinite {
return Complex(real: real, imaginary: -imaginary)
}
return Complex(real: real, imaginary: real)
}
if real.isInfinite {
if imaginary.isNaN {
return Complex(real: imaginary, imaginary: real)
}
if imaginary.isInfinite {
return Complex(
real: real < 0 ? .pi * 3 / 4 : .pi / 4,
imaginary: -imaginary
)
}
return real < 0 ?
Complex(real: .pi, imaginary: imaginary.sign == .minus ? -real : real) :
Complex(real: 0, imaginary: imaginary.sign == .minus ? real : -real)
}
if real == 0 && (imaginary.isNaN || imaginary == 0) {
return Complex(real: .pi / 2, imaginary: -imaginary)
}
if imaginary.isInfinite {
return Complex(real: .pi / 2, imaginary: -imaginary)
}
let a = Complex.pow(self, 2) - 1
let b = Complex.log(self + Complex.sqrt(a))
return Complex(
real: abs(b.imaginary),
imaginary: imaginary.sign == .minus ? abs(b.real) : -abs(b.real)
)
}
@_transparent // @_inlineable
public func inverseTangent() -> Complex {
let atanh = Complex.atanh(Complex(real: -imaginary, imaginary: real))
return Complex(real: atanh.imaginary, imaginary: -atanh.real)
}
// @_transparent // @_inlineable
public func hyperbolicSine() -> Complex {
if (real.isInfinite || real == 0) && !imaginary.isFinite {
return Complex(real: real, imaginary: .nan)
}
if imaginary == 0 && !real.isFinite {
return self
}
return Complex(
real: T.sinh(real) * T.cos(imaginary),
imaginary: T.cosh(real) * T.sin(imaginary)
)
}
// @_transparent // @_inlineable
public func hyperbolicCosine() -> Complex {
if real.isInfinite && !imaginary.isFinite {
return Complex(real: abs(real), imaginary: .nan)
}
if real == 0 && imaginary == 0 {
return Complex(real: 1, imaginary: imaginary)
}
if real == 0 && !imaginary.isFinite {
return Complex(real: .nan, imaginary: real)
}
if imaginary == 0 && !real.isFinite {
return Complex(real: abs(real), imaginary: imaginary)
}
return Complex(
real: T.cosh(real) * T.cos(imaginary),
imaginary: T.sinh(real) * T.sin(imaginary)
)
}
// @_transparent // @_inlineable
public func hyperbolicTangent() -> Complex {
if real.isNaN && imaginary == 0 { return self }
if real.isInfinite {
if !imaginary.isFinite { return 1 }
return Complex(
real: 1, imaginary: T(signOf: T.sin(2 * imaginary), magnitudeOf: 0)
)
}
// See AMS55 4.5.51
let twiceReal = 2 * real, twiceImaginary = 2 * imaginary
let denominator = T.cosh(twiceReal) + T.cos(twiceImaginary)
let sinh = T.sinh(twiceReal)
if sinh.isInfinite && denominator.isInfinite {
return Complex(
real: sinh > 0 ? (1 as T) : -(1 as T),
imaginary: twiceImaginary > 0 ? (0 as T) : -(0 as T)
)
}
return Complex(
real: sinh / denominator,
imaginary: T.sin(twiceImaginary) / denominator
)
}
// @_transparent // @_inlineable
public func inverseHyperbolicSine() -> Complex {
if real.isNaN {
if imaginary.isInfinite {
return Complex(real: imaginary, imaginary: real)
}
if imaginary == 0 { return self }
return Complex(real: real, imaginary: real)
}
if real.isInfinite {
if imaginary.isNaN { return self }
if imaginary.isInfinite {
return Complex(
real: real, imaginary: T(signOf: imaginary, magnitudeOf: .pi / 4)
)
}
return Complex(
real: real, imaginary: T(signOf: imaginary, magnitudeOf: 0)
)
}
if imaginary.isInfinite {
return Complex(
real: T(signOf: real, magnitudeOf: imaginary),
imaginary: T(signOf: imaginary, magnitudeOf: .pi / 2)
)
}
let a = Complex.pow(self, 2) + 1
let b = Complex.log(self + Complex.sqrt(a))
return Complex(
real: T(signOf: real, magnitudeOf: b.real),
imaginary: T(signOf: imaginary, magnitudeOf: b.imaginary)
)
}
// @_transparent // @_inlineable
public func inverseHyperbolicCosine() -> Complex {
if real.isNaN {
if imaginary.isInfinite {
return Complex(real: .infinity, imaginary: real)
}
return Complex(real: real, imaginary: real)
}
if real.isInfinite {
if imaginary.isNaN {
return Complex(real: .infinity, imaginary: imaginary)
}
if imaginary.isInfinite {
switch (real.sign, imaginary.sign) {
case (.plus, .plus):
return Complex(real: .infinity, imaginary: .pi / 4)
case (.plus, .minus):
return Complex(real: .infinity, imaginary: -.pi / 4)
case (.minus, .plus):
return Complex(real: .infinity, imaginary: .pi * 3 / 4)
case (.minus, .minus):
return Complex(real: .infinity, imaginary: -.pi * 3 / 4)
}
}
switch (real.sign, imaginary.sign) {
case (.plus, .plus):
return Complex(real: .infinity, imaginary: 0)
case (.plus, .minus):
return Complex(real: .infinity, imaginary: -(0 as T))
case (.minus, .plus):
return Complex(real: .infinity, imaginary: .pi)
case (.minus, .minus):
return Complex(real: .infinity, imaginary: -.pi)
}
}
if imaginary.isNaN {
// See C11 DR 471.
return Complex(
real: imaginary,
imaginary: real == 0 ? .pi / 2 : imaginary
)
}
if imaginary.isInfinite {
return Complex(
real: .infinity, imaginary: T(signOf: imaginary, magnitudeOf: .pi / 2)
)
}
let a = Complex.pow(self, 2) - 1
let b = Complex.log(self + Complex.sqrt(a))
return Complex(
real: T(signOf: 0, magnitudeOf: b.real),
imaginary: T(signOf: imaginary, magnitudeOf: b.imaginary)
)
}
// @_transparent // @_inlineable
public func inverseHyperbolicTangent() -> Complex {
if imaginary.isNaN {
if real.isInfinite || real == 0 {
return Complex(
real: T(signOf: real, magnitudeOf: 0), imaginary: imaginary
)
}
return Complex(real: imaginary, imaginary: imaginary)
}
if imaginary.isInfinite {
return Complex(
real: T(signOf: real, magnitudeOf: 0),
imaginary: T(signOf: imaginary, magnitudeOf: .pi / 2)
)
}
if imaginary == 0 && abs(real) == 1 {
return Complex(
real: T(signOf: real, magnitudeOf: .infinity),
imaginary: T(signOf: imaginary, magnitudeOf: 0)
)
}
if real.isNaN {
return Complex(real: real, imaginary: real)
}
if real.isInfinite {
return Complex(
real: T(signOf: real, magnitudeOf: 0),
imaginary: T(signOf: imaginary, magnitudeOf: .pi / 2)
)
}
let a = Complex.log((1 + self) / (1 - self)) / 2
return Complex(
real: T(signOf: real, magnitudeOf: a.real),
imaginary: T(signOf: imaginary, magnitudeOf: a.imaginary)
)
}
}
/// Returns the absolute value (magnitude, modulus) of `z`.
@_transparent
public func abs<T>(_ z: Complex<T>) -> Complex<T> {
return Complex(real: z.magnitude)
}
/// Returns the square root of `z`.
@_transparent
public func sqrt<T>(_ z: Complex<T>) -> Complex<T> {
return z.squareRoot()
}

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Sources/Complex.swift
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//
// Complex.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 3/25/17.
//
/// A type to represent a complex value in Cartesian form.
///
/// Create new instances of `Complex<T>` using integer or floating-point
/// literals and the imaginary unit `.i`. For example:
///
/// ```swift
/// let x = 2 + 4 * .i // `x` is of type `Complex<Double>`
/// let y = 3.5 + 7 * .i // `y` is of type `Complex<Double>`
///
/// let z: Complex64 = .e + .pi * .i // `z` is of type `Complex<Float>`
/// ```
///
/// Additional Considerations
/// =========================
///
/// Floating-point types have special values that represent infinity or NaN
/// ("not a number"). Complex functions in different languages may return
/// different results when working with special values.
///
/// Many complex functions have [branch cuts][dfn], which are curves in the
/// complex plane across which a function is discontinuous. Different languages
/// may adopt different branch cut structures for the same complex function.
///
/// Implementations in `Complex<T>` adhere to the [C standard][std] (Annex G) as
/// closely as possible with respect to special values and branch cuts.
///
/// To users unfamiliar with complex functions, the principal value returned by
/// some complex functions may be unexpected. For example,
/// `Double.cbrt(-8) == -2`, which is the __real root__, while
/// `Complex.cbrt(-8) == 2 * Complex.exp(.i * .pi / 3)`, which is the
/// __principal root__.
///
/// [dfn]: http://mathworld.wolfram.com/BranchCut.html
/// [std]: http://www.open-std.org/JTC1/SC22/WG14/www/standards.html#9899
@_fixed_layout
public struct Complex<T : FloatingPointMath>
where T : _ExpressibleByBuiltinFloatLiteral {
/// The real component of the complex value.
public var real: T
/// The imaginary component of the complex value.
public var imaginary: T
/// Creates a new value from the given real and imaginary components.
///
/// - Parameters:
/// - real: The new value's real component.
/// - imaginary: The new value's imaginary component.
@_transparent // @_inlineable
public init(real: T = 0, imaginary: T = 0) {
self.real = real
self.imaginary = imaginary
}
/// The imaginary unit i.
@_transparent // @_inlineable
public static var i: Complex {
return Complex(real: 0, imaginary: 1)
}
}
extension Complex {
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - real: The integer to convert to a floating-point real component.
@_transparent // @_inlineable
init(_ real: Int) {
self.real = T(real)
self.imaginary = 0
}
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - real: The integer to convert to a floating-point real component.
@_transparent // @_inlineable
init(_ real: Int8) {
self.real = T(real)
self.imaginary = 0
}
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - real: The integer to convert to a floating-point real component.
@_transparent // @_inlineable
init(_ real: Int16) {
self.real = T(real)
self.imaginary = 0
}
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - real: The integer to convert to a floating-point real component.
@_transparent // @_inlineable
init(_ real: Int32) {
self.real = T(real)
self.imaginary = 0
}
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - real: The integer to convert to a floating-point real component.
@_transparent // @_inlineable
init(_ real: Int64) {
self.real = T(real)
self.imaginary = 0
}
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - real: The integer to convert to a floating-point real component.
@_transparent // @_inlineable
init(_ real: UInt) {
self.real = T(real)
self.imaginary = 0
}
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - real: The integer to convert to a floating-point real component.
@_transparent // @_inlineable
init(_ real: UInt8) {
self.real = T(real)
self.imaginary = 0
}
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - real: The integer to convert to a floating-point real component.
@_transparent // @_inlineable
init(_ real: UInt16) {
self.real = T(real)
self.imaginary = 0
}
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - real: The integer to convert to a floating-point real component.
@_transparent // @_inlineable
init(_ real: UInt32) {
self.real = T(real)
self.imaginary = 0
}
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - real: The integer to convert to a floating-point real component.
@_transparent // @_inlineable
init(_ real: UInt64) {
self.real = T(real)
self.imaginary = 0
}
// ---------------------------------------------------------------------------
// FIXME: If corresponding requirements are added to FloatingPoint
// add init<U : Integer>(_: U) as well as init?<U : Integer>(exactly: U).
// ---------------------------------------------------------------------------
}
extension Complex where T : BinaryFloatingPoint {
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// - Parameters:
/// - real: The value to convert to a real component of type `T`.
@_transparent // @_inlineable
public init(_ real: Float) {
self.real = T(real)
self.imaginary = 0
}
/// Creates a new value from the given real component, rounded to the closest
/// possible representation.
///
/// - Parameters:
/// - real: The value to convert to a real component of type `T`.
@_transparent // @_inlineable
public init(_ real: Double) {
self.real = T(real)
self.imaginary = 0
}
// ---------------------------------------------------------------------------
// FIXME: If corresponding requirements are added to BinaryFloatingPoint
// add init<U : BinaryFloatingPoint>(_: U) as well as
// init?<U : BinaryFloatingPoint>(exactly: U).
// ---------------------------------------------------------------------------
}
extension Complex {
/// Creates a new value from the given polar coordinates `(r, theta)`.
///
/// - Parameters:
/// - r: The new value's radial coordinate.
/// - theta: The new value's angular coordinate.
@_transparent // @_inlineable
public init(r: T, theta: T) {
self.real = r * T.cos(theta)
self.imaginary = r * T.sin(theta)
}
/// The principal argument (phase angle) of this value.
///
/// Special cases are handled as if calling `T.atan2(imaginary, real)`. The
/// result lies in the range [-π, π].
@_transparent // @_inlineable
public var argument: T {
return T.atan2(imaginary, real)
}
/// A Boolean value indicating whether the instance's real and imaginary
/// components are both in canonical form, as defined in the [IEEE 754
/// specification][spec]. Every `Float` or `Double` value is canonical.
///
/// [spec]: http://ieeexplore.ieee.org/servlet/opac?punumber=4610933
@_transparent // @_inlineable
public var isCanonical: Bool {
return real.isCanonical && imaginary.isCanonical
}
/// A Boolean value indicating whether the instance's real and imaginary
/// components are both finite.
///
/// All values other than NaN and infinity are considered finite.
@_transparent // @_inlineable
public var isFinite: Bool {
return real.isFinite && imaginary.isFinite
}
/// A Boolean value indicating whether the instance's real and/or imaginary
/// components are infinite.
///
/// Note that `isFinite` and `isInfinite` do not form a dichotomy because NaN
/// is neither finite nor infinite.
@_transparent // @_inlineable
public var isInfinite: Bool {
return real.isInfinite || imaginary.isInfinite
}
/// A Boolean value indicating whether the instance's real and/or imaginary
/// components are NaN ("not a number").
///
/// Because NaN is not equal to any value, including NaN, use this property
/// instead of the equal-to operator (`==`) or not-equal-to operator (`!=`) to
/// test whether a value is or is not NaN.
///
/// This property is `true` for both quiet and signaling NaNs.
@_transparent // @_inlineable
public var isNaN: Bool {
return real.isNaN || imaginary.isNaN
}
/// A Boolean value indicating whether the instance's real and/or imaginary
/// components are signaling NaNs.
///
/// Signaling NaNs typically raise the Invalid flag when used in general
/// computing operations.
@_transparent // @_inlineable
public var isSignalingNaN: Bool {
return real.isSignalingNaN || imaginary.isSignalingNaN
}
/// A Boolean value indicating whether the instance is equal to zero.
///
/// The `isZero` property of a value `z` is `true` when both real and
/// imaginary components represent either `-0.0` or `+0.0`.
@_transparent // @_inlineable
public var isZero: Bool {
return real.isZero && imaginary.isZero
}
/// The magnitude (modulus, absolute value) of this value.
///
/// Special cases are handled as if calling `T.hypot(real, imaginary)`.
@_transparent // @_inlineable
public var magnitude: T {
return T.hypot(real, imaginary)
}
/// The polar coordinates representing this value, equivalent to
/// `(r: magnitude, theta: argument)`.
@_transparent // @_inlineable
public var polar: (r: T, theta: T) {
return (r: magnitude, theta: argument)
}
/// The squared magnitude (field norm, absolute square) of this value.
///
/// This is less costly to compute than `magnitude` and, in some cases, can be
/// used in its place. For example, if `a.magnitude > b.magnitude`, then
/// `a.squaredMagnitude > b.squaredMagnitude`.
@_transparent // @_inlineable
public var squaredMagnitude: T {
if real.isInfinite { return abs(real) }
if imaginary.isInfinite { return abs(imaginary) }
return real * real + imaginary * imaginary
}
/// Returns the complex conjugate of this value, obtained by reversing the
/// sign of the imaginary component.
@_transparent // @_inlineable
public func conjugate() -> Complex {
return Complex(real: real, imaginary: -imaginary)
}
/// Returns the projection of this value onto the Riemann sphere.
///
/// For most values `z`, `z.projection() == z`. The projection of any complex
/// infinity is positive real infinity. The sign of the imaginary component is
/// preserved in the sign of zero; that is,
/// `z.projection().imaginary.sign == z.imaginary.sign`.
@_transparent // @_inlineable
public func projection() -> Complex {
if real.isInfinite || imaginary.isInfinite {
return Complex(
real: .infinity, imaginary: T(signOf: imaginary, magnitudeOf: 0)
)
}
return self
}
/// Returns the reciprocal (multiplicative inverse) of this value.
@_transparent // @_inlineable
public func reciprocal() -> Complex {
let denominator = squaredMagnitude
return Complex(
real: real / denominator, imaginary: -imaginary / denominator
)
}
}
extension Complex : ExpressibleByFloatLiteral {
@_transparent // @_inlineable
public init(floatLiteral value: T) {
self.real = value
self.imaginary = 0
}
}
extension Complex : ExpressibleByIntegerLiteral {
@_transparent // @_inlineable
public init(integerLiteral value: Int) {
self.real = T(value)
self.imaginary = 0
}
}
extension Complex : CustomStringConvertible {
@_transparent // @_inlineable
public var description: String {
return "\(real) + \(imaginary)i"
}
}
extension Complex : Equatable {
@_transparent // @_inlineable
public static func == (lhs: Complex, rhs: Complex) -> Bool {
return lhs.real == rhs.real && lhs.imaginary == rhs.imaginary
}
}
extension Complex : Hashable {
// @_transparent // @_inlineable
public var hashValue: Int {
return _fnv1a(real, imaginary)
}
}
public typealias Complex64 = Complex<Float>
public typealias Complex128 = Complex<Double>

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//
// Exponentiation.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 4/24/17.
//
extension BinaryInteger {
/// Returns the result of raising `base` to the power of `exponent`, rounded
/// to a representable value.
// @_transparent // @_inlineable
public static func pow(_ base: Self, _ exponent: Self) -> Self {
var x = base, n = exponent
if Self.isSigned && n < 0 {
x = 1 / x
n = 0 - n
} else if n == 0 {
return 1
}
// Exponentiate by iterative squaring.
var y = 1 as Self
while n > 1 {
if n % 2 == 1 {
y *= x
n -= 1
}
x *= x
n /= 2
}
return x * y
}
}

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//
// Factoring.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 4/15/17.
//
extension UnsignedInteger {
/// Returns the greatest common divisor of `a` and `b`.
// @_transparent // @_inlineable
public static func gcd(_ a: Self, _ b: Self) -> Self {
// An iterative version of Stein's algorithm.
if a == 0 { return b } // gcd(0, b) == b
if b == 0 { return a } // gcd(a, 0) == a
var a = a, b = b, shift = 0 as Self
while ((a | b) & 1) == 0 {
a &>>= 1
b &>>= 1
shift += 1
}
// Now, shift is equal to log2(k), where k is the greatest power of 2
// dividing a and b.
while (a & 1) == 0 { a &>>= 1 } // Now, a is odd.
repeat {
while (b & 1) == 0 { b &>>= 1 } // Now, b is odd.
if a > b { swap(&a, &b) } // Now, a < b.
b -= a
} while b != 0
// Restore common factors of 2.
return a &<< shift
}
/// Returns the least common multiple of `a` and `b`.
@_transparent // @_inlineable
public static func lcm(_ a: Self, _ b: Self) -> Self {
if a == 0 || b == 0 { return 0 }
return a / .gcd(a, b) * b
}
}
extension UnsignedInteger where Self : FixedWidthInteger {
/// Returns the least common multiple of `a` and `b` and a flag to indicate
/// whether overflow occurred during the operation.
// @_transparent // @_inlineable
public static func lcmReportingOverflow(_ a: Self, _ b: Self)
-> (partialValue: Self, overflow: ArithmeticOverflow) {
if a == 0 || b == 0 { return (partialValue: 0, overflow: .none) }
return (a / .gcd(a, b)).multipliedReportingOverflow(by: b)
}
/// Returns the high and low parts of the least common multiple of `a` and `b`
/// computed using full-width arithmetic.
// @_transparent // @_inlineable
public static func lcmFullWidth(_ a: Self, _ b: Self)
-> (high: Self, low: Self.Magnitude) {
if a == 0 || b == 0 { return (0, 0) }
return (a / .gcd(a, b)).multipliedFullWidth(by: b)
}
}
extension BinaryInteger where Magnitude : UnsignedInteger {
/// Returns the greatest common divisor of `a` and `b`.
@_transparent // @_inlineable
public static func gcd(_ a: Self, _ b: Self) -> Self {
return Self(Magnitude.gcd(a.magnitude, b.magnitude))
}
/// Returns the least common multiple of `a` and `b`.
@_transparent // @_inlineable
public static func lcm(_ a: Self, _ b: Self) -> Self {
return Self(Magnitude.lcm(a.magnitude, b.magnitude))
}
}
// `BinaryInteger where Self : FixedWidthInteger` may seem superfluous, but it
// is necessary to disambiguate calls to `Magnitude.lcmReportingOverflow(_:_:)`
// and `Magnitude.lcmFullWidth(_:_:)`.
extension BinaryInteger
where Self : FixedWidthInteger, Magnitude : FixedWidthInteger & UnsignedInteger,
Magnitude.Magnitude == Magnitude {
/// Returns the greatest common divisor of `a` and `b` and a flag to indicate
/// whether overflow occurred during the operation.
// @_transparent // @_inlineable
public static func gcdReportingOverflow(_ a: Self, _ b: Self)
-> (partialValue: Self, overflow: ArithmeticOverflow) {
let t = Self(extendingOrTruncating: Magnitude.gcd(a.magnitude, b.magnitude))
return (
partialValue: t,
overflow: ArithmeticOverflow(t < 0)
)
}
/// Returns the least common multiple of `a` and `b` and a flag to indicate
/// whether overflow occurred during the operation.
// @_transparent // @_inlineable
public static func lcmReportingOverflow(_ a: Self, _ b: Self)
-> (partialValue: Self, overflow: ArithmeticOverflow) {
let (t, overflow) = Magnitude.lcmReportingOverflow(a.magnitude, b.magnitude)
let u = Self(extendingOrTruncating: t)
return (
partialValue: u,
overflow: ArithmeticOverflow(overflow == .overflow || u < 0)
)
}
/// Returns the high and low parts of the least common multiple of `a` and `b`
/// computed using full-width arithmetic.
// @_transparent // @_inlineable
public static func lcmFullWidth(_ a: Self, _ b: Self)
-> (high: Self, low: Self.Magnitude) {
let t = Magnitude.lcmFullWidth(a.magnitude, b.magnitude)
return (high: Self(t.high), low: t.low)
}
}

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//
// FloatingPointMath.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 4/1/17.
//
#if os(Linux)
import Glibc
#else
import Darwin.C
#endif
/// A floating-point type that provides a selection of special functions.
///
/// The `FloatingPointMath` protocol provides a suitable basis for writing
/// functions that work on any floating-point type that provides the required
/// functions.
public protocol FloatingPointMath : Math, FloatingPoint /*, Hashable */ {
/// Creates a new value from the given rational value, after rounding the
/// whole part and the numerator and denominator of the fractional part each
/// to the closest possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - value: The rational value to convert to a floating-point value.
init(_ value: Rational<Int>)
/// Creates a new value from the given rational value, after rounding the
/// whole part and the numerator and denominator of the fractional part each
/// to the closest possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - value: The rational value to convert to a floating-point value.
init(_ value: Rational<Int8>)
/// Creates a new value from the given rational value, after rounding the
/// whole part and the numerator and denominator of the fractional part each
/// to the closest possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - value: The rational value to convert to a floating-point value.
init(_ value: Rational<Int16>)
/// Creates a new value from the given rational value, after rounding the
/// whole part and the numerator and denominator of the fractional part each
/// to the closest possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - value: The rational value to convert to a floating-point value.
init(_ value: Rational<Int32>)
/// Creates a new value from the given rational value, after rounding the
/// whole part and the numerator and denominator of the fractional part each
/// to the closest possible representation.
///
/// If two representable values are equally close, the result is the value
/// with more trailing zeros in its significand bit pattern.
///
/// - Parameters:
/// - value: The rational value to convert to a floating-point value.
init(_ value: Rational<Int64>)
// FIXME: If corresponding requirements are added to FloatingPoint
// add init<U : SignedInteger>(_: Rational<U>) as well as
// init?<U : SignedInteger>(exactly: Rational<U>).
/// Returns the hypotenuse of a right-angle triangle with legs (catheti) of
/// length `x` and `y`, preventing avoidable arithmetic overflow and
/// underflow. The return value is the square root of the sum of squares of
/// the arguments.
///
/// - Parameters:
/// - x: The length of one leg (cathetus) of a right-angle triangle.
/// - y: The length of the other leg (cathetus) of a right-angle triangle.
static func hypot(_ x: Self, _ y: Self) -> Self
/// Returns the inverse tangent of `self / other`, using the signs of `self`
/// and `other` to determine the quadrant of the computed angle.
///
/// If `self == 0 && other == 0`, the return value is still finite.
///
/// - Parameters:
/// - other: The divisor by which to divide `self`.
///
/// - SeeAlso: `atan2(_:_:)`
func inverseTangent(dividingBy other: Self) -> Self
/// Returns the value of the [error function][dfn] of `self`.
///
/// [dfn]: http://mathworld.wolfram.com/Erf.html
///
/// - SeeAlso: `erf(_:)`
func error() -> Self
/// Returns the value of the [complementary error function][dfn] of `self`.
///
/// [dfn]: http://mathworld.wolfram.com/Erfc.html
///
/// - SeeAlso: `erfc(_:)`
func complementaryError() -> Self
/// Returns the value of the [gamma function][dfn] of `self`.
///
/// [dfn]: http://mathworld.wolfram.com/GammaFunction.html
///
/// - SeeAlso: `tgamma(_:)`
func gamma() -> Self
/// Returns the value of the [logarithmic gamma function][dfn] of `self`.
///
/// [dfn]: http://mathworld.wolfram.com/LogGammaFunction.html
///
/// - SeeAlso: `lgamma(_:)`
func logarithmicGamma() -> Self
}
extension FloatingPointMath {
public init(_ value: Rational<Int>) {
let (whole, fraction) = value.mixed
self = Self(whole) + Self(fraction.numerator) / Self(fraction.denominator)
}
public init(_ value: Rational<Int8>) {
let (whole, fraction) = value.mixed
self = Self(whole) + Self(fraction.numerator) / Self(fraction.denominator)
}
public init(_ value: Rational<Int16>) {
let (whole, fraction) = value.mixed
self = Self(whole) + Self(fraction.numerator) / Self(fraction.denominator)
}
public init(_ value: Rational<Int32>) {
let (whole, fraction) = value.mixed
self = Self(whole) + Self(fraction.numerator) / Self(fraction.denominator)
}
public init(_ value: Rational<Int64>) {
let (whole, fraction) = value.mixed
self = Self(whole) + Self(fraction.numerator) / Self(fraction.denominator)
}
public static func hypot(_ x: Self, _ y: Self) -> Self {
var x = abs(x), y = abs(y)
if x < y {
swap(&x, &y)
}
let ratio = y / x
return x * .sqrt(1 + ratio * ratio)
}
public func inverseTangent(dividingBy other: Self) -> Self {
let y = self, x = other
if y.isInfinite {
if x.isInfinite {
switch (y.sign, x.sign) {
case (.minus, .minus): return -.pi * 3 / 4
case (.plus, .minus): return .pi * 3 / 4
case (.minus, .plus): return -.pi / 4
case (.plus, .plus): return .pi / 4
}
}
return y.sign == .minus ? -.pi / 2 : .pi / 2
}
if x > 0 {
return 2 * .atan(y / (.sqrt(x * x + y * y) + x))
}
if y != 0 {
return 2 * .atan((.sqrt(x * x + y * y) - x) / y)
}
if x.sign == .minus /* && y == 0 */ {
return y.sign == .minus ? -.pi : .pi
}
/* x.sign == .plus && y == 0 */
return y.sign == .minus ? -(0 as Self) : 0
}
}
extension FloatingPointMath {
/// Returns the inverse tangent of `y / x`, using the signs of `y` and `x` to
/// determine the quadrant of the computed angle.
///
/// If `x == 0 && y == 0`, the return value is still finite.
///
/// - Parameters:
/// - y: The value to divide.
/// - x: The divisor by which to divide `y`.
///
/// - SeeAlso: `inverseTangent(dividingBy:)`
@_transparent // @_inlineable
public static func atan2(_ y: Self, _ x: Self) -> Self {
// Note the order of the internal names `y` and `x`, which is deliberate: a
// minority of programming languages vend functions that reverse the order
// of the arguments, but all refer to the denominator as `x` and the
// numerator as `y`.
return y.inverseTangent(dividingBy: x)
}
/// Returns the value of the [error function][dfn] of `x`.
///
/// [dfn]: http://mathworld.wolfram.com/Erf.html
///
/// - SeeAlso: `error()`
@_transparent // @_inlineable
public static func erf(_ x: Self) -> Self {
return x.error()
}
/// Returns the value of the [complementary error function][dfn] of `x`.
///
/// [dfn]: http://mathworld.wolfram.com/Erfc.html
///
/// - SeeAlso: `complementaryError()`
@_transparent // @_inlineable
public static func erfc(_ x: Self) -> Self {
return x.complementaryError()
}
/// Returns the value of the [gamma function][dfn] of `x`.
///
/// [dfn]: http://mathworld.wolfram.com/GammaFunction.html
///
/// - SeeAlso: `gamma()`
@_transparent // @_inlineable
public static func tgamma(_ x: Self) -> Self {
return x.gamma()
}
/// Returns the value of the [logarithmic gamma function][dfn] of `x`.
///
/// [dfn]: http://mathworld.wolfram.com/LogGammaFunction.html
///
/// - SeeAlso: `logarithmicGamma()`
@_transparent // @_inlineable
public static func lgamma(_ x: Self) -> Self {
return x.logarithmicGamma()
}
}
extension Float : FloatingPointMath {
@_transparent
public static var e: Float {
return Float(0x1.5bf0a8p1)
}
@_transparent
public static var phi: Float {
return Float(0x1.9e377ap0)
}
@_transparent
public func naturalExponential() -> Float {
return expf(self)
}
@_transparent
public func binaryExponential() -> Float {
return exp2f(self)
}
@_transparent
public func commonExponential() -> Float {
#if os(Linux)
return exp10f(self)
#else
return __exp10f(self)
#endif
}
@_transparent
public func naturalExponentialMinusOne() -> Float {
return expm1f(self)
}
@_transparent
public func naturalLogarithm() -> Float {
return logf(self)
}
@_transparent
public func binaryLogarithm() -> Float {
return log2f(self)
}
@_transparent
public func commonLogarithm() -> Float {
return log10f(self)
}
@_transparent
public func naturalLogarithmOnePlus() -> Float {
return log1pf(self)
}
@_transparent
public func cubeRoot() -> Float {
return cbrtf(self)
}
@_transparent
public func power(of base: Float) -> Float {
return powf(base, self)
}
@_transparent
public func sine() -> Float {
return sinf(self)
}
@_transparent
public func cosine() -> Float {
return cosf(self)
}
@_transparent
public func tangent() -> Float {
return tanf(self)
}
@_transparent
public func inverseSine() -> Float {
return asinf(self)
}
@_transparent
public func inverseCosine() -> Float {
return acosf(self)
}
@_transparent
public func inverseTangent() -> Float {
return atanf(self)
}
@_transparent
public func hyperbolicSine() -> Float {
return sinhf(self)
}
@_transparent
public func hyperbolicCosine() -> Float {
return coshf(self)
}
@_transparent
public func hyperbolicTangent() -> Float {
return tanhf(self)
}
@_transparent
public func inverseHyperbolicSine() -> Float {
return asinhf(self)
}
@_transparent
public func inverseHyperbolicCosine() -> Float {
return acoshf(self)
}
@_transparent
public func inverseHyperbolicTangent() -> Float {
return atanhf(self)
}
@_transparent
public static func hypot(_ x: Float, _ y: Float) -> Float {
return hypotf(x, y)
}
@_transparent
public func inverseTangent(dividingBy other: Float) -> Float {
return atan2f(self, other)
}
@_transparent
public func error() -> Float {
return erff(self)
}
@_transparent
public func complementaryError() -> Float {
return erfcf(self)
}
@_transparent
public func gamma() -> Float {
return tgammaf(self)
}
@_transparent
public func logarithmicGamma() -> Float {
return lgammaf(self)
}
}
extension Double : FloatingPointMath {
@_transparent
public static var e: Double {
return Double(0x1.5bf0a8b145769p1)
}
@_transparent
public static var phi: Double {
return Double(0x1.9e3779b97f4a8p0)
}
@_transparent
public func naturalExponential() -> Double {
#if os(Linux)
return Glibc.exp(self)
#else
return Darwin.exp(self)
#endif
}
@_transparent
public func binaryExponential() -> Double {
#if os(Linux)
return Glibc.exp2(self)
#else
return Darwin.exp2(self)
#endif
}
@_transparent
public func commonExponential() -> Double {
#if os(Linux)
return Glibc.exp10(self)
#else
return __exp10(self)
#endif
}
@_transparent
public func naturalExponentialMinusOne() -> Double {
#if os(Linux)
return Glibc.expm1(self)
#else
return Darwin.expm1(self)
#endif
}
@_transparent
public func naturalLogarithm() -> Double {
#if os(Linux)
return Glibc.log(self)
#else
return Darwin.log(self)
#endif
}
@_transparent
public func binaryLogarithm() -> Double {
#if os(Linux)
return Glibc.log2(self)
#else
return Darwin.log2(self)
#endif
}
@_transparent
public func commonLogarithm() -> Double {
#if os(Linux)
return Glibc.log10(self)
#else
return Darwin.log10(self)
#endif
}
@_transparent
public func naturalLogarithmOnePlus() -> Double {
#if os(Linux)
return Glibc.log1p(self)
#else
return Darwin.log1p(self)
#endif
}
@_transparent
public func cubeRoot() -> Double {
#if os(Linux)
return Glibc.cbrt(self)
#else
return Darwin.cbrt(self)
#endif
}
@_transparent
public func power(of base: Double) -> Double {
#if os(Linux)
return Glibc.pow(base, self)
#else
return Darwin.pow(base, self)
#endif
}
@_transparent
public func sine() -> Double {
#if os(Linux)
return Glibc.sin(self)
#else
return Darwin.sin(self)
#endif
}
@_transparent
public func cosine() -> Double {
#if os(Linux)
return Glibc.cos(self)
#else
return Darwin.cos(self)
#endif
}
@_transparent
public func tangent() -> Double {
#if os(Linux)
return Glibc.tan(self)
#else
return Darwin.tan(self)
#endif
}
@_transparent
public func inverseSine() -> Double {
#if os(Linux)
return Glibc.asin(self)
#else
return Darwin.asin(self)
#endif
}
@_transparent
public func inverseCosine() -> Double {
#if os(Linux)
return Glibc.acos(self)
#else
return Darwin.acos(self)
#endif
}
@_transparent
public func inverseTangent() -> Double {
#if os(Linux)
return Glibc.atan(self)
#else
return Darwin.atan(self)
#endif
}
@_transparent
public func hyperbolicSine() -> Double {
#if os(Linux)
return Glibc.sinh(self)
#else
return Darwin.sinh(self)
#endif
}
@_transparent
public func hyperbolicCosine() -> Double {
#if os(Linux)
return Glibc.cosh(self)
#else
return Darwin.cosh(self)
#endif
}
@_transparent
public func hyperbolicTangent() -> Double {
#if os(Linux)
return Glibc.tanh(self)
#else
return Darwin.tanh(self)
#endif
}
@_transparent
public func inverseHyperbolicSine() -> Double {
#if os(Linux)
return Glibc.asinh(self)
#else
return Darwin.asinh(self)
#endif
}
@_transparent
public func inverseHyperbolicCosine() -> Double {
#if os(Linux)
return Glibc.acosh(self)
#else
return Darwin.acosh(self)
#endif
}
@_transparent
public func inverseHyperbolicTangent() -> Double {
#if os(Linux)
return Glibc.atanh(self)
#else
return Darwin.atanh(self)
#endif
}
@_transparent
public static func hypot(_ x: Double, _ y: Double) -> Double {
#if os(Linux)
return Glibc.hypot(x, y)
#else
return Darwin.hypot(x, y)
#endif
}
@_transparent
public func inverseTangent(dividingBy other: Double) -> Double {
#if os(Linux)
return Glibc.atan2(self, other)
#else
return Darwin.atan2(self, other)
#endif
}
@_transparent
public func error() -> Double {
#if os(Linux)
return Glibc.erf(self)
#else
return Darwin.erf(self)
#endif
}
@_transparent
public func complementaryError() -> Double {
#if os(Linux)
return Glibc.erfc(self)
#else
return Darwin.erfc(self)
#endif
}
@_transparent
public func gamma() -> Double {
#if os(Linux)
return Glibc.tgamma(self)
#else
return Darwin.tgamma(self)
#endif
}
@_transparent
public func logarithmicGamma() -> Double {
#if os(Linux)
return Glibc.lgamma(self)
#else
return Darwin.lgamma(self)
#endif
}
}

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//
// Hash.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 4/1/17.
//
// @_versioned
internal func _fnv1a(_ data: AnyHashable...) -> Int {
let basis, prime: UInt
switch MemoryLayout<UInt>.size {
case 4:
basis = 2166136261
prime = 16777619
case 8:
basis = 14695981039346656037
prime = 1099511628211
/*
case 16:
basis = 144066263297769815596495629667062367629
prime = 309485009821345068724781371
*/
default:
fatalError("Unsupported UInt bit width for FNV hash")
}
let result = data.reduce(basis) { partialResult, element in
var h = element.hashValue
return withUnsafeBytes(of: &h) { bytes in
bytes.reduce(partialResult) { ($0 ^ UInt($1)) &* prime }
}
}
return Int(bitPattern: result)
}

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//
// Math.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 3/31/17.
//
/// A signed numeric type that provides square root, cube root, and elementary
/// transcendental functions.
///
/// The `Math` protocol provides a suitable basis for writing functions that
/// work on any real or complex floating-point type which provides the required
/// functions.
public protocol Math : SignedNumeric {
/// The mathematical constant pi (_π_).
///
/// This value should be rounded toward zero to keep user computations with
/// angles from inadvertently ending up in the wrong quadrant. A type that
/// conforms to the `Math` protocol provides the value for `pi` at its best
/// possible precision.
static var pi: Self { get }
/// The mathematical constant _e_, or Euler's number.
static var e: Self { get }
/// The mathematical constant phi (_φ_), or golden ratio.
static var phi: Self { get }
/// Returns the quotient obtained by dividing the first value by the second,
/// rounded to a representable value.
///
/// - Parameters:
/// - lhs: The value to divide.
/// - rhs: The value by which to divide `lhs`.
static func / (lhs: Self, rhs: Self) -> Self
/// Divides the left-hand side by the right-hand side and stores the quotient
/// in the left-hand side, rounded to a representable value.
///
/// - Parameters:
/// - lhs: The value to divide.
/// - rhs: The value by which to divide `lhs`.
static func /= (lhs: inout Self, rhs: Self)
/// Returns the natural exponential of the value, rounded to a representable
/// value.
///
/// The natural exponential of a value `x` is _e_ (2.7182818...) raised to the
/// power of `x`.
///
/// - SeeAlso: `exp(_:)`
func naturalExponential() -> Self
/// Returns the binary exponential of the value, rounded to a representable
/// value.
///
/// The binary exponential of a value `x` is 2 raised to the power of `x`.
///
/// - SeeAlso: `exp2(_:)`
func binaryExponential() -> Self
/// Returns the common exponential of the value, rounded to a representable
/// value.
///
/// The common exponential of a value `x` is 10 raised to the power of `x`.
///
/// - SeeAlso: `exp10(_:)`
func commonExponential() -> Self
/// Returns the result of subtracting one from the natural exponential of the
/// value, rounded to a representable value.
///
/// The natural exponential of a value `x` is _e_ (2.7182818...) raised to the
/// power of `x`.
///
/// This function is more accurate than `x.naturalExponential() - 1` for
/// values of `x` close to zero.
///
/// - SeeAlso: `expm1(_:)`
func naturalExponentialMinusOne() -> Self
/// Returns the natural (base _e_) logarithm of the value, rounded to a
/// representable value.
///
/// - SeeAlso: `log(_:)`
func naturalLogarithm() -> Self
/// Returns the binary (base 2) logarithm of the value, rounded to a
/// representable value.
///
/// - SeeAlso: `log2(_:)`
func binaryLogarithm() -> Self
/// Returns the common (base 10) logarithm of the value, rounded to a
/// representable value.
///
/// - SeeAlso: `log10(_:)`
func commonLogarithm() -> Self
/// Returns the natural (base _e_) logarithm of the result of adding one to
/// the value, rounded to a representable value.
///
/// - SeeAlso: `log1p(_:)`
func naturalLogarithmOnePlus() -> Self
/// Returns the square root of the value, rounded to a representable value.
///
/// - SeeAlso: `sqrt(_:)`
func squareRoot() -> Self
/// Returns the cube root of the value, rounded to a representable value.
///
/// - SeeAlso: `cbrt(_:)`
func cubeRoot() -> Self
/// Returns the result of raising `base` to the power of the value, rounded to
/// a representable value.
///
/// - Note: The argument is the _base_ and the receiver is the _exponent_.
///
/// - Parameters:
/// - base: The base.
///
/// - SeeAlso: `pow(_:_:)`
func power(of base: Self) -> Self
/// Returns the sine of the value (given in radians), rounded to a
/// representable value.
///
/// - SeeAlso: `sin(_:)`
func sine() -> Self
/// Returns the cosine of the value (given in radians), rounded to a
/// representable value.
///
/// - SeeAlso: `cos(_:)`
func cosine() -> Self
/// Returns the tangent of the value (given in radians), rounded to a
/// representable value.
///
/// - SeeAlso: `tan(_:)`
func tangent() -> Self
/// Returns the principal value of the inverse sine of the value, rounded to a
/// representable value.
///
/// - SeeAlso: `asin(_:)`
func inverseSine() -> Self
/// Returns the principal value of the inverse cosine of the value, rounded to
/// a representable value.
///
/// - SeeAlso: `acos(_:)`
func inverseCosine() -> Self
/// Returns the principal value of the inverse tangent of the value, rounded
/// to a representable value.
///
/// - SeeAlso: `atan(_:)`
func inverseTangent() -> Self
/// Returns the hyperbolic sine of the value, rounded to a representable
/// value.
///
/// - SeeAlso: `sinh(_:)`
func hyperbolicSine() -> Self
/// Returns the hyperbolic cosine of the value, rounded to a representable
/// value.
///
/// - SeeAlso: `cosh(_:)`
func hyperbolicCosine() -> Self
/// Returns the hyperbolic tangent of the value, rounded to a representable
/// value.
///
/// - SeeAlso: `tanh(_:)`
func hyperbolicTangent() -> Self
/// Returns the principal value of the inverse hyperbolic sine of the value,
/// rounded to a representable value.
///
/// - SeeAlso: `asinh(_:)`
func inverseHyperbolicSine() -> Self
/// Returns the principal value of the inverse hyperbolic cosine of the value,
/// rounded to a representable value.
///
/// - SeeAlso: `acosh(_:)`
func inverseHyperbolicCosine() -> Self
/// Returns the principal value of the inverse hyperbolic tangent of the
/// value, rounded to a representable value.
///
/// - SeeAlso: `atanh(_:)`
func inverseHyperbolicTangent() -> Self
}
extension Math {
// @_transparent
public static var e: Self {
return Self.exp(1 as Self)
}
// @_transparent
public static var phi: Self {
return Self.sqrt(((1 as Self) + Self.sqrt(5 as Self)) / (2 as Self))
}
public func binaryExponential() -> Self {
return Self.exp(self * Self.log(2 as Self))
}
public func commonExponential() -> Self {
return Self.exp(self * Self.log(10 as Self))
}
public func naturalExponentialMinusOne() -> Self {
return Self.exp(self) - (1 as Self)
}
public func binaryLogarithm() -> Self {
return Self.log(self) / Self.log(2 as Self)
}
public func commonLogarithm() -> Self {
return Self.log(self) / Self.log(10 as Self)
}
public func naturalLogarithmOnePlus() -> Self {
return Self.log(self + (1 as Self))
}
public func cubeRoot() -> Self {
return Self.pow(self, 1 / 3 as Self)
}
public func tangent() -> Self {
return sine() / cosine()
}
public func hyperbolicTangent() -> Self {
return hyperbolicSine() / hyperbolicCosine()
}
}
extension Math {
/// Returns the natural exponential of `x`, rounded to a representable value.
///
/// The natural exponential of a value `x` is _e_ (2.7182818...) raised to the
/// power of `x`.
///
/// - SeeAlso: `naturalExponential()`
@_transparent // @_inlineable
public static func exp(_ x: Self) -> Self {
return x.naturalExponential()
}
/// Returns the binary exponential of `x`, rounded to a representable value.
///
/// The binary exponential of a value `x` is 2 raised to the power of `x`.
///
/// - SeeAlso: `binaryExponential()`
@_transparent // @_inlineable
public static func exp2(_ x: Self) -> Self {
return x.binaryExponential()
}
/// Returns the common exponential of `x`, rounded to a representable value.
///
/// The common exponential of a value `x` is 10 raised to the power of `x`.
///
/// - SeeAlso: `commonExponential()`
@_transparent // @_inlineable
public static func exp10(_ x: Self) -> Self {
return x.commonExponential()
}
/// Returns the result of subtracting one from the natural exponential of `x`,
/// rounded to a representable value.
///
/// The natural exponential of a value `x` is _e_ (2.7182818...) raised to the
/// power of `x`.
///
/// This function is more accurate than `.exp(x) - 1` for values of `x` close
/// to zero.
///
/// - SeeAlso: `naturalExponentialMinusOne()`
@_transparent // @_inlineable
public static func expm1(_ x: Self) -> Self {
return x.naturalExponentialMinusOne()
}
/// Returns the natural (base _e_) logarithm of `x`, rounded to a
/// representable value.
///
/// - SeeAlso: `naturalLogarithm()`
@_transparent // @_inlineable
public static func log(_ x: Self) -> Self {
return x.naturalLogarithm()
}
/// Returns the binary (base 2) logarithm of `x`, rounded to a representable
/// value.
///
/// - SeeAlso: `binaryLogarithm()`
@_transparent // @_inlineable
public static func log2(_ x: Self) -> Self {
return x.binaryLogarithm()
}
/// Returns the common (base 10) logarithm of `x`, rounded to a representable
/// value.
///
/// - SeeAlso: `commonLogarithm()`
@_transparent // @_inlineable
public static func log10(_ x: Self) -> Self {
return x.commonLogarithm()
}
/// Returns the natural (base _e_) logarithm of the result of adding one to
/// `x`, rounded to a representable value.
///
/// - SeeAlso: `naturalLogarithmOfOnePlus()`
@_transparent // @_inlineable
public static func log1p(_ x: Self) -> Self {
return x.naturalLogarithmOnePlus()
}
/// Returns the square root of `x`, rounded to a representable value.
///
/// - SeeAlso: `squareRoot()`
@_transparent // @_inlineable
public static func sqrt(_ x: Self) -> Self {
return x.squareRoot()
}
/// Returns the cube root of `x`, rounded to a representable value.
///
/// - SeeAlso: `cubeRoot()`
@_transparent // @_inlineable
public static func cbrt(_ x: Self) -> Self {
return x.cubeRoot()
}
/// Returns the result of raising `base` to the power of `exponent`, rounded
/// to a representable value.
///
/// - SeeAlso: `power(of:)`
@_transparent // @_inlineable
public static func pow(_ base: Self, _ exponent: Self) -> Self {
return exponent.power(of: base)
}
/// Returns the sine of `x` (given in radians), rounded to a representable
/// value.
///
/// - SeeAlso: `sine()`
@_transparent // @_inlineable
public static func sin(_ x: Self) -> Self {
return x.sine()
}
/// Returns the cosine of `x` (given in radians), rounded to a representable
/// value.
///
/// - SeeAlso: `cosine()`
@_transparent // @_inlineable
public static func cos(_ x: Self) -> Self {
return x.cosine()
}
/// Returns the tangent of `x` (given in radians), rounded to a representable
/// value.
///
/// - SeeAlso: `tangent()`
@_transparent // @_inlineable
public static func tan(_ x: Self) -> Self {
return x.tangent()
}
/// Returns the principal value of the inverse sine of `x`, rounded to a
/// representable value.
///
/// - SeeAlso: `inverseSine()`
@_transparent // @_inlineable
public static func asin(_ x: Self) -> Self {
return x.inverseSine()
}
/// Returns the principal value of the inverse cosine of `x`, rounded to a
/// representable value.
///
/// - SeeAlso: `inverseCosine()`
@_transparent // @_inlineable
public static func acos(_ x: Self) -> Self {
return x.inverseCosine()
}
/// Returns the principal value of the inverse tangent of `x`, rounded to a
/// representable value.
///
/// - SeeAlso: `inverseTangent()`
@_transparent // @_inlineable
public static func atan(_ x: Self) -> Self {
return x.inverseTangent()
}
/// Returns the hyperbolic sine of `x`, rounded to a representable value.
///
/// - SeeAlso: `hyperbolicSine()`
@_transparent // @_inlineable
public static func sinh(_ x: Self) -> Self {
return x.hyperbolicSine()
}
/// Returns the hyperbolic cosine of `x`, rounded to a representable value.
///
/// - SeeAlso: `hyperbolicCosine()`
@_transparent // @_inlineable
public static func cosh(_ x: Self) -> Self {
return x.hyperbolicCosine()
}
/// Returns the hyperbolic tangent of `x`, rounded to a representable value.
///
/// - SeeAlso: `hyperbolicTangent()`
@_transparent // @_inlineable
public static func tanh(_ x: Self) -> Self {
return x.hyperbolicTangent()
}
/// Returns the principal value of the inverse hyperbolic sine of `x`, rounded
/// to a representable value.
///
/// - SeeAlso: `inverseHyperbolicSine()`
@_transparent // @_inlineable
public static func asinh(_ x: Self) -> Self {
return x.inverseHyperbolicSine()
}
/// Returns the principal value of the inverse hyperbolic cosine of `x`,
/// rounded to a representable value.
///
/// - SeeAlso: `inverseHyperbolicCosine()`
@_transparent // @_inlineable
public static func acosh(_ x: Self) -> Self {
return x.inverseHyperbolicCosine()
}
/// Returns the principal value of the inverse hyperbolic tangent of `x`,
/// rounded to a representable value.
///
/// - SeeAlso: `inverseHyperbolicTangent()`
@_transparent // @_inlineable
public static func atanh(_ x: Self) -> Self {
return x.inverseHyperbolicTangent()
}
}

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Sources/PRNG.swift
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@ -1,460 +0,0 @@
//
// PRNG.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 5/15/17.
//
#if os(Linux)
import Glibc
#else
import Security
#endif
/// A pseudo-random number generator.
public protocol PRNG : class, IteratorProtocol, Sequence
where Element : FixedWidthInteger & UnsignedInteger, SubSequence : Sequence,
/* Element == Iterator.Element, Element == SubSequence.Iterator.Element, */
Element == SubSequence.Element {
/// A type that can represent the internal state of the pseudo-random number
/// generator.
associatedtype State
/// The maximum value that may be generated by the pseudo-random number
/// generator.
static var max: Element { get }
/// The minimum value that may be generated by the pseudo-random number
/// generator.
static var min: Element { get }
/// The internal state of the pseudo-random number generator.
var state: State { get set }
/// Creates a pseudo-random number generator with the given internal state.
///
/// - Parameters:
/// - state: The value to be used as the generator's internal state.
init(state: State)
/// Creates a pseudo-random number generator with an internal state seeded
/// using cryptographically secure random bytes.
///
/// If cryptographically secure random bytes are unavailable, the result is
/// `nil`.
init?()
}
extension PRNG {
public static var max: Element { return Element.max }
public static var min: Element { return Element.min }
/// The number of pseudo-random bits available from a value generated by the
/// pseudo-random number generator.
public static var _randomBitWidth: Int {
let difference = Self.max - Self.min
guard difference < Element.max else { return Element.bitWidth }
return Element.bitWidth - (difference + 1).leadingZeroBitCount - 1
}
/// Returns a value filled with data from a source of cryptographically secure
/// random bytes, or `nil` if a sufficient number of cryptographically secure
/// random bytes is unavailable.
public static func _entropy<
T : FixedWidthInteger & UnsignedInteger
>(_: T.Type = T.self) -> T? {
let size = MemoryLayout<T>.size
var value = T()
#if os(Linux)
// Read from `urandom`.
// https://sockpuppet.org/blog/2014/02/25/safely-generate-random-numbers/
guard let file = fopen("/dev/urandom", "rb") else { return nil }
defer { fclose(file) }
let read = fread(&value, size, 1, file)
guard read == 1 else { return nil }
#else
// Sandboxing can make `urandom` unavailable.
let result = withUnsafeMutableBytes(of: &value) { ptr -> Int32 in
let bytes = ptr.baseAddress!.bindMemory(to: UInt8.self, capacity: size)
defer { ptr.baseAddress!.bindMemory(to: T.self, capacity: 1) }
return SecRandomCopyBytes(nil, size, bytes)
}
guard result == errSecSuccess else { return nil }
#endif
return value
}
/// Returns an array of `count` values filled with data from a source of
/// cryptographically secure random bytes, or `nil` if a sufficient number of
/// cryptographically secure random bytes is unavailable.
public static func _entropy<
T : FixedWidthInteger & UnsignedInteger
>(_: T.Type = T.self, count: Int) -> [T]? {
let stride = MemoryLayout<T>.stride
var value = [T](repeating: 0, count: count)
#if os(Linux)
guard let file = fopen("/dev/urandom", "rb") else { return nil }
defer { fclose(file) }
let read = fread(&value, stride, count, file)
guard read == count else { return nil }
#else
let result = value.withUnsafeMutableBytes { ptr -> Int32 in
let n = stride * count
let bytes = ptr.baseAddress!.bindMemory(to: UInt8.self, capacity: n)
defer { ptr.baseAddress!.bindMemory(to: T.self, capacity: count) }
return SecRandomCopyBytes(nil, n, bytes)
}
guard result == errSecSuccess else { return nil }
#endif
return value
}
}
extension PRNG {
/// Generates a pseudo-random unsigned integer of type `T` in the range from 0
/// to `2 ** min(bitCount, T.bitWidth)` (exclusive), where `**` is the
/// exponentiation operator.
public func _random<T : FixedWidthInteger & UnsignedInteger>(
_: T.Type = T.self, bitCount: Int = T.bitWidth
) -> T {
let randomBitWidth = Self._randomBitWidth
let bitCount = Swift.min(bitCount, T.bitWidth)
if T.bitWidth == Element.bitWidth &&
randomBitWidth == Element.bitWidth &&
bitCount == T.bitWidth {
// It is an awkward way of spelling `next()`, but it is necessary.
guard let next = first(where: { _ in true }) else { fatalError() }
return T(extendingOrTruncating: next)
}
let (quotient, remainder) =
bitCount.quotientAndRemainder(dividingBy: randomBitWidth)
let max = (Element.max &>> (Element.bitWidth - randomBitWidth)) + Self.min
var temporary = 0 as T
// Call `next()` at least `quotient` times.
for i in 0..<quotient {
guard let next = first(where: { $0 <= max }) else { fatalError() }
temporary +=
T(extendingOrTruncating: next) &<< (randomBitWidth * i)
}
// If `remainder != 0`, call `next()` at least one more time.
if remainder != 0 {
guard let next = first(where: { $0 <= max }) else { fatalError() }
let mask = Element.max &>> (Element.bitWidth - remainder)
temporary +=
T(extendingOrTruncating: next & mask) &<< (randomBitWidth * quotient)
}
return temporary
}
/// Generates a pseudo-random unsigned integer of type `T` in the range from
/// `a` through `b` (inclusive) from the discrete uniform distribution.
public func uniform<T : FixedWidthInteger & UnsignedInteger>(
_: T.Type = T.self, a: T, b: T
) -> T {
precondition(
b >= a,
"Discrete uniform distribution parameter b should not be less than a"
)
guard a != b else { return a }
let difference = b - a
guard difference < T.max else {
return _random() + a
}
let bitCount = T.bitWidth - difference.leadingZeroBitCount
var temporary: T
repeat {
temporary = _random(bitCount: bitCount)
} while temporary > difference
return temporary + a
}
/// Generates a pseudo-random unsigned integer of type `T` in the range from
/// `T.min` through `T.max` (inclusive) from the discrete uniform
/// distribution.
@_transparent // @_inlineable
public func uniform<T : FixedWidthInteger & UnsignedInteger>(
_: T.Type = T.self
) -> T {
return uniform(a: T.min, b: T.max)
}
/// Generates a sequence of `count` pseudo-random unsigned integers of type
/// `T` in the range from `a` through `b` (inclusive) from the discrete
/// uniform distribution.
@_transparent // @_inlineable
public func uniform<T : FixedWidthInteger & UnsignedInteger>(
_: T.Type = T.self, a: T, b: T, count: Int
) -> UnfoldSequence<T, Int> {
precondition(count >= 0, "Element count should be non-negative")
return sequence(state: 0) { (state: inout Int) -> T? in
state += 1
return state > count ? nil : self.uniform(a: a, b: b)
}
}
/// Generates a sequence of `count` pseudo-random unsigned integers of type
/// `T` in the range from `T.min` through `T.max` (inclusive) from the
/// discrete uniform distribution.
@_transparent // @_inlineable
public func uniform<T : FixedWidthInteger & UnsignedInteger>(
_: T.Type = T.self, count: Int
) -> UnfoldSequence<T, Int> {
return uniform(a: T.min, b: T.max, count: count)
}
/// Generates a pseudo-random signed integer of type `T` in the range from `a`
/// through `b` (inclusive) from the discrete uniform distribution.
public func uniform<T : FixedWidthInteger & SignedInteger>(
_: T.Type = T.self, a: T, b: T
) -> T where T.Magnitude : FixedWidthInteger & UnsignedInteger {
precondition(
b >= a,
"Discrete uniform distribution parameter b should not be less than a"
)
guard a != b else { return a }
let test = a.signum() < 0
let difference = test
? (b.signum() < 0 ? a.magnitude - b.magnitude : b.magnitude + a.magnitude)
: b.magnitude - a.magnitude
guard difference < T.Magnitude.max else {
return test ? T(_random() - a.magnitude) : T(_random() + a.magnitude)
}
let bitCount = T.Magnitude.bitWidth - difference.leadingZeroBitCount
var temporary: T.Magnitude
repeat {
temporary = _random(bitCount: bitCount)
} while temporary > difference
return test ? T(temporary - a.magnitude) : T(temporary + a.magnitude)
}
/// Generates a pseudo-random signed integer of type `T` in the range from
/// `T.min` through `T.max` (inclusive) from the discrete uniform
/// distribution.
@_transparent // @_inlineable
public func uniform<T : FixedWidthInteger & SignedInteger>(
_: T.Type = T.self
) -> T where T.Magnitude : FixedWidthInteger & UnsignedInteger {
return uniform(a: T.min, b: T.max)
}
/// Generates a sequence of `count` pseudo-random signed integers of type `T`
/// in the range from `a` through `b` (inclusive) from the discrete uniform
/// distribution.
@_transparent // @_inlineable
public func uniform<T : FixedWidthInteger & SignedInteger>(
_: T.Type = T.self, a: T, b: T, count: Int
) -> UnfoldSequence<T, Int>
where T.Magnitude : FixedWidthInteger & UnsignedInteger {
precondition(count >= 0, "Element count should be non-negative")
return sequence(state: 0) { (state: inout Int) -> T? in
state += 1
return state > count ? nil : self.uniform(a: a, b: b)
}
}
/// Generates a sequence of `count` pseudo-random signed integers of type `T`
/// in the range from `T.min` through `T.max` (inclusive) from the discrete
/// uniform distribution.
@_transparent // @_inlineable
public func uniform<T : FixedWidthInteger & SignedInteger>(
_: T.Type = T.self, count: Int
) -> UnfoldSequence<T, Int>
where T.Magnitude : FixedWidthInteger & UnsignedInteger {
return uniform(a: T.min, b: T.max, count: count)
}
}
// FIXME: If `FloatingPoint.init(_: FixedWidthInteger)` is added
// then it becomes possible to remove the constraint `Element == UInt64`.
extension PRNG where Element == UInt64 {
/// Generates a pseudo-random binary floating-point value of type `T` in the
/// range from 0 to 1 (exclusive) with `min(bitCount, T.significandBitCount)`
/// bits of precision.
public func _random<T : BinaryFloatingPoint>(
_: T.Type = T.self, bitCount: Int = T.significandBitCount
) -> T {
let bitCount = Swift.min(bitCount, T.significandBitCount)
let (quotient, remainder) =
bitCount.quotientAndRemainder(dividingBy: Self._randomBitWidth)
let k = Swift.max(1, remainder == 0 ? quotient : quotient + 1)
let step = T(Self.max - Self.min)
let initial = (0 as T, 1 as T)
// Call `next()` exactly `k` times.
let (dividend, divisor) = prefix(k).reduce(initial) { partial, next in
let x = partial.0 + T(next - Self.min) * partial.1
let y = partial.1 + step * partial.1
return (x, y)
}
return dividend / divisor
}
/// Generates a pseudo-random binary floating-point value of type `T` in the
/// range from `a` to `b` (exclusive) from the uniform distribution.
public func uniform<T : BinaryFloatingPoint>(
_: T.Type = T.self, a: T, b: T
) -> T {
precondition(
b > a,
"Uniform distribution parameter b should be greater than a"
)
var temporary: T
repeat {
temporary = (b - a) * _random() + a
} while temporary == b
return temporary
}
/// Generates a pseudo-random binary floating-point value of type `T` in the
/// range from 0 to 1 (exclusive) from the uniform distribution.
@_transparent // @_inlineable
public func uniform<T : BinaryFloatingPoint>(_: T.Type = T.self) -> T {
return uniform(a: 0, b: 1)
}
/// Generates a sequence of `count` pseudo-random binary floating-point values
/// of type `T` in the range from `a` to `b` (exclusive) from the uniform
/// distribution.
@_transparent // @_inlineable
public func uniform<T : BinaryFloatingPoint>(
_: T.Type = T.self, a: T, b: T, count: Int
) -> UnfoldSequence<T, Int> {
precondition(count >= 0, "Element count should be non-negative")
return sequence(state: 0) { (state: inout Int) -> T? in
state += 1
return state > count ? nil : self.uniform(a: a, b: b)
}
}
/// Generates a sequence of `count` pseudo-random binary floating-point values
/// of type `T` in the range from 0 to 1 (exclusive) from the uniform
/// distribution.
@_transparent // @_inlineable
public func uniform<T : BinaryFloatingPoint>(
_: T.Type = T.self, count: Int
) -> UnfoldSequence<T, Int> {
return uniform(a: 0, b: 1, count: count)
}
}
#if false
extension PRNG where Element == UInt64 {
public func bernoulli<T : BinaryFloatingPoint>(
_: Bool.Type = Bool.self, p: T
) -> Bool {
precondition(
p >= 0 && p <= 1,
"Bernoulli distribution parameter p should be between zero and one"
)
var temporary: T
repeat {
temporary = _random()
} while temporary == 1
return temporary < p
}
@_transparent // @_inlineable
public func bernoulli(_: Bool.Type = Bool.self) -> Bool {
return bernoulli(p: 0.5)
}
@_transparent // @_inlineable
public func bernoulli<T : BinaryFloatingPoint>(
_: Bool.Type = Bool.self, p: T, count: Int
) -> UnfoldSequence<Bool, Int> {
precondition(count >= 0, "Element count should be non-negative")
return sequence(state: 0) { (state: inout Int) -> Bool? in
state += 1
return state > count ? nil : self.bernoulli(p: p)
}
}
@_transparent // @_inlineable
public func bernoulli(
_: Bool.Type = Bool.self, count: Int
) -> UnfoldSequence<Bool, Int> {
return bernoulli(p: 0.5, count: count)
}
public func exponential<T : BinaryFloatingPoint & FloatingPointMath>(
_: T.Type = T.self, lambda: T
) -> T {
precondition(
lambda > 0 && lambda < .infinity,
"Exponential distribution parameter lambda should be positive and finite"
)
var temporary: T
repeat {
temporary = -.log(1 - _random()) / lambda
} while temporary == .infinity
return temporary
}
@_transparent // @_inlineable
public func exponential<T : BinaryFloatingPoint & FloatingPointMath>(
_: T.Type = T.self
) -> T {
return exponential(lambda: 1)
}
@_transparent // @_inlineable
public func exponential<T : BinaryFloatingPoint & FloatingPointMath>(
_: T.Type = T.self, lambda: T, count: Int
) -> UnfoldSequence<T, Int> {
precondition(count >= 0, "Element count should be non-negative")
return sequence(state: 0) { (state: inout Int) -> T? in
state += 1
return state > count ? nil : self.exponential(lambda: lambda)
}
}
@_transparent // @_inlineable
public func exponential<T : BinaryFloatingPoint & FloatingPointMath>(
_: T.Type = T.self, count: Int
) -> UnfoldSequence<T, Int> {
return exponential(lambda: 1, count: count)
}
public func weibull<T : BinaryFloatingPoint & FloatingPointMath>(
_: T.Type = T.self, lambda: T, kappa: T
) -> T {
precondition(
lambda > 0 && lambda < .infinity && kappa > 0 && kappa < .infinity,
"Weibull distribution parameters should be positive and finite"
)
var temporary: T
repeat {
temporary = lambda * .pow(-.log(1 - _random()), 1 / kappa)
} while temporary == .infinity
return temporary
}
@_transparent // @_inlineable
public func weibull<T : BinaryFloatingPoint & FloatingPointMath>(
_: T.Type = T.self
) -> T {
return weibull(lambda: 1, kappa: 1)
}
@_transparent // @_inlineable
public func weibull<T : BinaryFloatingPoint & FloatingPointMath>(
_: T.Type = T.self, lambda: T, kappa: T, count: Int
) -> UnfoldSequence<T, Int> {
precondition(count >= 0, "Element count should be non-negative")
return sequence(state: 0) { (state: inout Int) -> T? in
state += 1
return state > count ? nil : self.weibull(lambda: lambda, kappa: kappa)
}
}
@_transparent // @_inlineable
public func weibull<T : BinaryFloatingPoint & FloatingPointMath>(
_: T.Type = T.self, count: Int
) -> UnfoldSequence<T, Int> {
return weibull(lambda: 1, kappa: 1, count: count)
}
}
#endif

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//
// Random.Xoroshiro.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 5/16/17.
//
extension Random {
/// A pseudo-random number generator that implements [`xoroshiro128+`], a
/// successor to `xorshift128+` devised by S. Vigna and D. Blackman.
public final class Xoroshiro : PRNG {
public var state: (UInt64, UInt64)
// TODO: Document this function.
// @_versioned
internal static func _rotl(_ value: UInt64, _ count: Int) -> UInt64 {
return (value &<< count) | (value &>> (64 &- count))
}
public func next() -> UInt64? {
let x = state.0
var y = state.1
let result = x &+ y
y ^= x
state.0 = Xoroshiro._rotl(x, 55) ^ y ^ (y &<< 14)
state.1 = Xoroshiro._rotl(y, 36)
return result
}
public init(state: (UInt64, UInt64)) {
self.state = state
}
public init?() {
repeat {
guard let entropy = Xoroshiro._entropy(UInt64.self, count: 2) else {
return nil
}
self.state = (entropy[0], entropy[1])
} while self.state == (0, 0)
}
}
}

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//
// Random.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 5/13/17.
//
/// A pseudo-random number generator that implements [`xorshift128+`][ref], an
/// efficient linear-feedback shift register.
///
/// [ref]: http://vigna.di.unimi.it/ftp/papers/xorshiftplus.pdf
public final class Random : PRNG {
public var state: (UInt64, UInt64)
public func next() -> UInt64? {
#if true
// An updated version of xorshift128+.
//
// This version adds the two halves of the current state, which allows for
// better parallelization.
var x = state.0
let y = state.1
let result = x &+ y
state.0 = y
x ^= x &<< 23
state.1 = x ^ y ^ (x &>> 18) ^ (y &>> 5)
return result
#else
// A previous version of xorshift128+.
var x = state.0
let y = state.1
state.0 = y
x ^= x &<< 23
state.1 = x ^ y ^ (x &>> 17) ^ (y &>> 26)
return state.1 &+ y
#endif
}
public init(state: (UInt64, UInt64)) {
self.state = state
}
public init?() {
repeat {
guard let entropy = Random._entropy(UInt64.self, count: 2) else {
return nil
}
self.state = (entropy[0], entropy[1])
} while self.state == (0, 0)
}
}

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//
// Rational+SignedNumeric.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 4/15/17.
//
extension Rational : Numeric {
// @_transparent // @_inlineable
public init?<U>(exactly source: U) where U : BinaryInteger {
guard let t = T(exactly: source) else { return nil }
// Ensure that `t.magnitude` is representable as a `T`.
guard let _ = T(exactly: t.magnitude) else { return nil }
self.numerator = t
self.denominator = 1
}
// @_transparent // @_inlineable
public static func + (lhs: Rational, rhs: Rational) -> Rational {
if lhs.denominator == 0 {
if rhs.denominator != 0 || lhs.numerator == 0 { return lhs }
if lhs.numerator > 0 { return rhs.numerator < 0 ? .nan : rhs }
return rhs.numerator > 0 ? .nan : rhs
}
if rhs.denominator == 0 { return rhs }
let ldm = lhs.denominator.magnitude
let rdm = rhs.denominator.magnitude
let gcd = T.Magnitude.gcd(ldm, rdm)
let a = T(rdm / gcd * lhs.numerator.magnitude)
let b = T(ldm / gcd * rhs.numerator.magnitude)
let n = lhs.sign == .plus
? (rhs.sign == .plus ? a + b : a - b)
: (rhs.sign == .plus ? b - a : -a - b)
let d = T(ldm / gcd * rdm)
return Rational(numerator: n, denominator: d).canonical
}
@_transparent // @_inlineable
public static func += (lhs: inout Rational, rhs: Rational) {
lhs = lhs + rhs
}
@_transparent // @_inlineable
public static func - (lhs: Rational, rhs: Rational) -> Rational {
return lhs + (-rhs)
}
@_transparent // @_inlineable
public static func -= (lhs: inout Rational, rhs: Rational) {
lhs = lhs + (-rhs)
}
// @_transparent // @_inlineable
public static func * (lhs: Rational, rhs: Rational) -> Rational {
if lhs.denominator == 0 {
if rhs.numerator == 0 { return .nan }
return rhs.sign == .plus ? lhs : -lhs
}
if rhs.denominator == 0 {
if lhs.numerator == 0 { return .nan }
return lhs.sign == .plus ? rhs : -rhs
}
let lnm = lhs.numerator.magnitude, ldm = lhs.denominator.magnitude
let rnm = rhs.numerator.magnitude, rdm = rhs.denominator.magnitude
// Note that if `T` is a signed fixed-width integer type, `gcd(lnm, rdm)` or
// `gcd(rnm, ldm)` could be equal to `-T.min`, which is not representable as
// a `T`. This is why the following arithmetic is performed with values of
// type `T.Magnitude`.
let a = T.Magnitude.gcd(lnm, rdm)
let b = T.Magnitude.gcd(rnm, ldm)
let n = lhs.sign == rhs.sign
? T(lnm / a * (rnm / b))
: -T(lnm / a * (rnm / b))
let d = T(ldm / b * (rdm / a))
return Rational(numerator: n, denominator: d)
}
@_transparent // @_inlineable
public static func *= (lhs: inout Rational, rhs: Rational) {
lhs = lhs * rhs
}
}
extension BinaryInteger {
/// Creates a new binary integer from the given rational value, if it can be
/// represented exactly.
///
/// If `source` is not representable exactly, the result is `nil`.
///
/// - Parameters:
/// - source: A rational value to convert to a binary integer.
@_transparent // @_inlineable
public init?<U>(exactly source: Rational<U>) {
let (whole, fraction) = source.mixed
guard fraction.isZero, let exact = Self(exactly: whole) else { return nil }
self = exact
}
/// Creates a new binary integer from the given rational value, truncating any
/// fractional part.
///
/// If `source` is outside the bounds of this type after truncation, a runtime
/// error may occur.
///
/// - Parameters:
/// - source: A rational value to convert to a binary integer.
@_transparent // @_inlineable
public init<U>(_ source: Rational<U>) {
self = Self(source.mixed.whole)
}
}
extension Rational : SignedNumeric {
@_transparent // @_inlineable
public static prefix func - (operand: Rational) -> Rational {
return Rational(
numerator: -operand.numerator, denominator: operand.denominator
)
}
@_transparent // @_inlineable
public mutating func negate() {
numerator.negate()
}
}
extension Rational {
/// Returns the quotient obtained by dividing the first value by the second,
/// trapping in case of arithmetic overflow.
///
/// - Parameters:
/// - lhs: The value to divide.
/// - rhs: The value by which to divide `lhs`.
@_transparent // @_inlineable
public static func / (lhs: Rational, rhs: Rational) -> Rational {
return lhs * rhs.reciprocal()
}
/// Divides the left-hand side by the right-hand side and stores the quotient
/// in the left-hand side, trapping in case of arithmetic overflow.
///
/// - Parameters:
/// - lhs: The value to divide.
/// - rhs: The value by which to divide `lhs`.
@_transparent // @_inlineable
public static func /= (lhs: inout Rational, rhs: Rational) {
lhs = lhs * rhs.reciprocal()
}
/// Returns this value rounded to an integral value using the specified
/// rounding rule.
///
/// ```swift
/// let x = 7 / 2 as Rational<Int>
/// print(x.rounded()) // Prints "4"
/// print(x.rounded(.towardZero)) // Prints "3"
/// print(x.rounded(.up)) // Prints "4"
/// print(x.rounded(.down)) // Prints "3"
/// ```
///
/// See the `FloatingPointRoundingRule` enumeration for more information about
/// the available rounding rules.
///
/// - Parameters:
/// - rule: The rounding rule to use.
///
/// - SeeAlso: `round(_:)`, `FloatingPointRoundingRule`
@_transparent // @_inlineable
public func rounded(
_ rule: RoundingRule = .toNearestOrAwayFromZero
) -> Rational {
var t = self
t.round(rule)
return t
}
/// Rounds the value to an integral value using the specified rounding rule.
///
/// ```swift
/// var x = 7 / 2 as Rational<Int>
/// x.round() // x == 4
///
/// var x = 7 / 2 as Rational<Int>
/// x.round(.towardZero) // x == 3
///
/// var x = 7 / 2 as Rational<Int>
/// x.round(.up) // x == 4
///
/// var x = 7 / 2 as Rational<Int>
/// x.round(.down) // x == 3
/// ```
///
/// See the `FloatingPointRoundingRule` enumeration for more information about
/// the available rounding rules.
///
/// - Parameters:
/// - rule: The rounding rule to use.
///
/// - SeeAlso: `round(_:)`, `FloatingPointRoundingRule`
@_transparent // @_inlineable
public mutating func round(_ rule: RoundingRule = .toNearestOrAwayFromZero) {
if denominator == 0 { return }
let f: T
(numerator, f) = numerator.quotientAndRemainder(dividingBy: denominator)
// Rounding rules only come into play if the fractional part is non-zero.
if f != 0 {
switch rule {
case .toNearestOrAwayFromZero:
fallthrough
case .toNearestOrEven:
switch denominator.magnitude.quotientAndRemainder(
dividingBy: f.magnitude
) {
case (2, 0): // Tie.
if rule == .toNearestOrEven && numerator % 2 == 0 { break }
fallthrough
case (1, _): // Nearest is away from zero.
if f > 0 { numerator += 1 } else { numerator -= 1 }
default: // Nearest is toward zero.
break
}
case .up:
if f > 0 { numerator += 1 }
case .down:
if f < 0 { numerator -= 1 }
case .towardZero:
break
case .awayFromZero:
if f > 0 { numerator += 1 } else { numerator -= 1 }
}
}
denominator = 1
}
}
/// Returns the absolute value (magnitude) of `x`.
@_transparent
public func abs<T>(_ x: Rational<T>) -> Rational<T> {
return x.magnitude
}
/// Returns the closest integral value greater than or equal to `x`.
@_transparent
public func ceil<T>(_ x: Rational<T>) -> Rational<T> {
return x.rounded(.up)
}
/// Returns the closest integral value less than or equal to `x`.
@_transparent
public func floor<T>(_ x: Rational<T>) -> Rational<T> {
return x.rounded(.down)
}
/// Returns the closest integral value; if two values are equally close, returns
/// the one with greater magnitude.
@_transparent
public func round<T>(_ x: Rational<T>) -> Rational<T> {
return x.rounded()
}
/// Returns the closest integral value with magnitude less than or equal to `x`.
@_transparent
public func trunc<T>(_ x: Rational<T>) -> Rational<T> {
return x.rounded(.towardZero)
}

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//
// Rational.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 4/15/17.
//
/// A type to represent a rational value.
///
/// - Note: `Ratio` is a type alias for `Rational<Int>`.
///
/// Create new instances of `Rational<T>` by using integer literals and the
/// division (`/`) operator. For example:
///
/// ```swift
/// let x = 1 / 3 as Ratio // `x` is of type `Rational<Int>`
/// let y = 2 as Ratio // `y` is of type `Rational<Int>`
/// let z: Ratio = 2 / 3 // `z` is also of type `Rational<Int>`
///
/// print(x + y + z) // Prints "3"
/// ```
///
/// You can create an unreduced fraction by using the initializer
/// `Rational<T>.init(numerator:denominator:)`. For example:
///
/// ```swift
/// let a = Ratio(numerator: 3, denominator: 3)
/// print(a) // Prints "3/3"
/// ```
///
/// All arithmetic operations with values in canonical form (i.e. reduced to
/// lowest terms) return results in canonical form. However, operations with
/// values not in canonical form may or may not return results that are
/// themselves in canonical form. The property `canonicalized` is the canonical
/// form of any value.
///
/// Additional Considerations
/// =========================
///
/// Special Values
/// --------------
///
/// `Rational<T>` does not prohibit zero as a denominator. Any instance with a
/// positive numerator and zero denominator represents (positive) infinity; any
/// instance with a negative numerator and zero denominator represents negative
/// infinity; and any instance with zero numerator and zero denominator
/// represents NaN ("not a number").
///
/// As with floating-point types, `Rational<T>.infinity` compares greater than
/// every finite value and negative infinity, and `-Rational<T>.infinity`
/// compares less than every finite value and positive infinity. Infinite values
/// of the same sign compare equal to each other.
///
/// As with floating-point types, `Rational<T>.nan` does not compare equal to
/// any other value, including another NaN. Use the property `isNaN` to test if
/// a value is NaN. `Rational<T>` arithmetic operations are intended to
/// propagate NaN in the same manner as analogous floating-point operations.
///
/// Fixed-Width Binary Parts
/// ------------------------
///
/// When a value of type `Rational<T>` is in canonical form, the sign of the
/// numerator is the sign of the value; that is, in canonical form, the sign of
/// the denominator is always positive. Therefore, `-1 / T.min` cannot be
/// represented as a value of type `Rational<T>` because `abs(T.min)` cannot be
/// represented as a value of type `T`.
///
/// To ensure that every representable value of type `Rational<T>` has a
/// representable magnitude and reciprocal of the same type, an overflow trap
/// occurs when the division (`/`) operator is used to create a value of type
/// `Rational<T>` with numerator `T.min`.
@_fixed_layout
public struct Rational<T : SignedInteger>
where T : _ExpressibleByBuiltinIntegerLiteral, T.Magnitude : UnsignedInteger,
T.Magnitude.Magnitude == T.Magnitude {
/// The numerator of the rational value.
public var numerator: T
/// The denominator of the rational value.
public var denominator: T
/// Creates a new value from the given numerator and denominator without
/// computing its canonical form (i.e., without reducing to lowest terms).
///
/// To create a value reduced to lowest terms, use the division (`/`)
/// operator. For example:
///
/// ```swift
/// let x = 3 / 3 as Rational<Int>
/// print(x) // Prints "1"
/// ```
///
/// - Parameters:
/// - numerator: The new value's numerator.
/// - denominator: The new value's denominator.
@_transparent // @_inlineable
public init(numerator: T, denominator: T) {
self.numerator = numerator
self.denominator = denominator
}
/// Positive infinity.
///
/// Infinity compares greater than all finite numbers and equal to other
/// (positive) infinite values.
@_transparent // @_inlineable
public static var infinity: Rational {
return Rational(numerator: 1, denominator: 0)
}
/// A quiet NaN ("not a number").
///
/// A NaN compares not equal, not greater than, and not less than every value,
/// including itself. Passing a NaN to an operation generally results in NaN.
@_transparent // @_inlineable
public static var nan: Rational {
return Rational(numerator: 0, denominator: 0)
}
}
extension Rational {
// TODO: Document this initializer.
// @_transparent // @_inlineable
public init<Source : BinaryInteger>(_ source: Source) {
let t = T(source)
// Ensure that `t.magnitude` is representable as a `T`.
_ = T(t.magnitude)
self.numerator = t
self.denominator = 1
}
// TODO: Document this initializer.
// @_transparent // @_inlineable
public init<Source : BinaryFloatingPoint>(_ source: Source) {
if source.isNaN { self = .nan; return }
if source == .infinity { self = .infinity; return }
if source == -.infinity { self = -.infinity; return }
if source.isZero { self = 0; return }
let exponent = source.exponent
let significandWidth = source.significandWidth
if significandWidth <= exponent {
self.numerator = T(source)
self.denominator = 1
return
}
let shift = significandWidth - Int(exponent)
let numerator = T(source * Source(1 &<< shift))
// Ensure that `numerator.magnitude` is representable as a `T`.
_ = T(numerator.magnitude)
let denominator = T(1 &<< shift)
// Ensure that `denominator.magnitude` is representable as a `T`.
_ = T(denominator.magnitude)
self.numerator = numerator
self.denominator = denominator
}
}
extension Rational {
/// Compares the (finite) magnitude of two finite values, returning -1 if
/// `lhs.magnitude` is less than `rhs.magnitude`, 0 if `lhs.magnitude` is
/// equal to `rhs.magnitude`, or 1 if `lhs.magnitude` is greater than
/// `rhs.magnitude`.
// @_versioned
internal static func _compareFiniteMagnitude(
_ lhs: Rational, _ rhs: Rational
) -> Int {
let ldm = lhs.denominator.magnitude
let rdm = rhs.denominator.magnitude
let gcd = T.Magnitude.gcd(ldm, rdm)
let a = rdm / gcd * lhs.numerator.magnitude
let b = ldm / gcd * rhs.numerator.magnitude
return a == b ? 0 : (a < b ? -1 : 1)
// FIXME: Use full-width multiplication to avoid trapping on overflow
// where `T : FixedWidthInteger, T.Magnitude : FixedWidthInteger`.
/*
let a = (rdm / gcd).multipliedFullWidth(by: lhs.numerator.magnitude)
let b = (ldm / gcd).multipliedFullWidth(by: rhs.numerator.magnitude)
return a.high == b.high
? (a.low == b.low ? 0 : (a.low < b.low ? -1 : 1))
: (a.high < b.high ? -1 : 1)
*/
}
/// The canonical representation of this value.
// @_transparent // @_inlineable
public var canonical: Rational {
let nm = numerator.magnitude, dm = denominator.magnitude
// Note that if `T` is a signed fixed-width integer type, `gcd(nm, dm)`
// could be equal to `-T.min`, which is not representable as a `T`. This is
// why the following arithmetic is performed with values of type
// `T.Magnitude`.
let gcd = T.Magnitude.gcd(nm, dm)
guard gcd != 0 else { return self }
let n = sign == .plus ? T(nm / gcd) : -T(nm / gcd)
let d = T(dm / gcd)
return Rational(numerator: n, denominator: d)
}
/// A Boolean value indicating whether the instance's representation is in
/// canonical form.
@_transparent // @_inlineable
public var isCanonical: Bool {
if denominator > 0 {
return T.Magnitude.gcd(numerator.magnitude, denominator.magnitude) == 1
}
return denominator == 0 && numerator.magnitude <= 1
}
/// A Boolean value indicating whether the instance is finite.
///
/// All values other than NaN and infinity are considered finite.
@_transparent // @_inlineable
public var isFinite: Bool {
return denominator != 0
}
/// A Boolean value indicating whether the instance is infinite.
///
/// Note that `isFinite` and `isInfinite` do not form a dichotomy because NaN
/// is neither finite nor infinite.
@_transparent // @_inlineable
public var isInfinite: Bool {
return denominator == 0 && numerator != 0
}
/// A Boolean value indicating whether the instance is NaN ("not a number").
///
/// Because NaN is not equal to any value, including NaN, use this property
/// instead of the equal-to operator (`==`) or not-equal-to operator (`!=`) to
/// test whether a value is or is not NaN.
@_transparent // @_inlineable
public var isNaN: Bool {
return denominator == 0 && numerator == 0
}
/// A Boolean value indicating whether the instance is a proper fraction.
///
/// A fraction `p / q` is proper iff `p > 0`, `q > 0`, and `p < q`.
@_transparent // @_inlineable
public var isProper: Bool {
return numerator > 0 && denominator > 0 && numerator < denominator
}
/// A Boolean value indicating whether the instance is equal to zero.
@_transparent // @_inlineable
public var isZero: Bool {
return denominator != 0 && numerator == 0
}
/// The magnitude (absolute value) of this value.
@_transparent // @_inlineable
public var magnitude: Rational {
return sign == .minus ? -self : self
}
/// The mixed form representing this value.
///
/// If the value is not finite, the mixed form has zero as its whole part and
/// the value as its fractional part.
@_transparent // @_inlineable
public var mixed: (whole: T, fractional: Rational) {
if denominator == 0 { return (whole: 0, fractional: self) }
let t = numerator.quotientAndRemainder(dividingBy: denominator)
return (
whole: t.quotient,
fractional: Rational(numerator: t.remainder, denominator: denominator)
)
}
/// The sign of this value.
@_transparent // @_inlineable
public var sign: Sign {
return numerator == 0 || (denominator < 0) == (numerator < 0)
? .plus
: .minus
}
/// Returns the reciprocal (multiplicative inverse) of this value.
@_transparent // @_inlineable
public func reciprocal() -> Rational {
return numerator < 0
? Rational(numerator: -denominator, denominator: -numerator)
: Rational(numerator: denominator, denominator: numerator)
}
}
extension Rational : ExpressibleByIntegerLiteral {
@_transparent // @_inlineable
public init(integerLiteral value: T) {
self.numerator = value
self.denominator = 1
}
}
extension Rational : CustomStringConvertible {
@_transparent // @_inlineable
public var description: String {
if numerator == 0 { return denominator == 0 ? "nan" : "0" }
if denominator == 0 { return numerator < 0 ? "-inf" : "inf" }
return denominator == 1 ? "\(numerator)" : "\(numerator)/\(denominator)"
}
}
extension Rational : Equatable {
// @_transparent // @_inlineable
public static func == (lhs: Rational, rhs: Rational) -> Bool {
if lhs.denominator == 0 {
if lhs.numerator == 0 { return false }
if lhs.numerator > 0 { return rhs.denominator == 0 && rhs.numerator > 0 }
return rhs.denominator == 0 && rhs.numerator < 0
}
if rhs.denominator == 0 { return false }
return lhs.sign == rhs.sign && _compareFiniteMagnitude(lhs, rhs) == 0
}
}
extension Rational : Hashable {
// @_transparent // @_inlineable
public var hashValue: Int {
let t = canonical
return _fnv1a(t.numerator, t.denominator)
}
}
extension Rational : Comparable {
// @_transparent // @_inlineable
public static func < (lhs: Rational, rhs: Rational) -> Bool {
if lhs.denominator == 0 {
if lhs.numerator >= 0 { return false }
return rhs.denominator != 0 || rhs.numerator > 0
}
if rhs.denominator == 0 { return rhs.numerator > 0 }
switch (lhs.sign, rhs.sign) {
case (.plus, .minus):
return false
case (.minus, .plus):
return true
case (.plus, .plus):
return _compareFiniteMagnitude(lhs, rhs) < 0
case (.minus, .minus):
return _compareFiniteMagnitude(lhs, rhs) > 0
}
}
@_transparent // @_inlineable
public static func > (lhs: Rational, rhs: Rational) -> Bool {
return rhs < lhs
}
// @_transparent // @_inlineable
public static func <= (lhs: Rational, rhs: Rational) -> Bool {
if lhs.denominator == 0 {
if lhs.numerator == 0 { return false }
if lhs.numerator > 0 { return rhs.denominator == 0 && rhs.numerator > 0 }
return rhs.denominator != 0 || rhs.numerator != 0
}
if rhs.denominator == 0 { return rhs.numerator > 0 }
switch (lhs.sign, rhs.sign) {
case (.plus, .minus):
return false
case (.minus, .plus):
return true
case (.plus, .plus):
return _compareFiniteMagnitude(lhs, rhs) <= 0
case (.minus, .minus):
return _compareFiniteMagnitude(lhs, rhs) >= 0
}
}
@_transparent // @_inlineable
public static func >= (lhs: Rational, rhs: Rational) -> Bool {
return rhs <= lhs
}
}
extension Rational : Strideable, _Strideable {
@_transparent // @_inlineable
public func distance(to other: Rational) -> Rational {
return other - self
}
@_transparent // @_inlineable
public func advanced(by amount: Rational) -> Rational {
return self + amount
}
}
extension Rational
where T : FixedWidthInteger, T.Magnitude : FixedWidthInteger {
// TODO: Document this initializer.
// @_transparent // @_inlineable
public init?<Source : BinaryFloatingPoint>(exactly source: Source) {
if source.isNaN { self = .nan; return }
if source == .infinity { self = .infinity; return }
if source == -.infinity { self = -.infinity; return }
if source.isZero { self = 0; return } // Consider -0.0 to be exactly 0.
let exponent = source.exponent
let significandWidth = source.significandWidth
let bitWidth = T.bitWidth
if significandWidth <= exponent {
guard exponent + 1 < bitWidth else { return nil }
self.numerator = T(source)
self.denominator = 1
return
}
let shift = significandWidth - Int(exponent)
guard significandWidth + 1 < bitWidth && shift < bitWidth else {
return nil
}
let numerator = T(source * Source(1 &<< shift))
// Ensure that `numerator.magnitude` is representable as a `T`.
guard let _ = T(exactly: numerator.magnitude) else { return nil }
let denominator = T(1 &<< shift)
// Ensure that `denominator.magnitude` is representable as a `T`.
guard let _ = T(exactly: denominator.magnitude) else { return nil }
self.numerator = numerator
self.denominator = denominator
}
}
public typealias Ratio = Rational<Int>

9
Sources/RoundingRule.swift
View File

@ -1,9 +0,0 @@
//
// RoundingRule.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 5/7/17.
//
/// A rule for rounding a number.
public typealias RoundingRule = FloatingPointRoundingRule

9
Sources/Sign.swift
View File

@ -1,9 +0,0 @@
//
// Sign.swift
// NumericAnnex
//
// Created by Xiaodi Wu on 4/30/17.
//
/// The mathematical sign of a number.
public typealias Sign = FloatingPointSign

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<h1>Structures</h1>
<p>The following structures are available globally.</p>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex7ComplexV">Complex</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex8RationalV">Rational</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex7ComplexV"></a>
<a name="//apple_ref/swift/Struct/Complex" class="dashAnchor"></a>
<h4>
<code>Complex</code>
</h4>
</div>
<div class="abstract">
<p>A type to represent a complex value in Cartesian form.</p>
<div class="aside aside-note">
<p class="aside-title">Note</p>
<code>Complex64</code> is a type alias for <code>Complex&lt;Float&gt;</code> and <code>Complex128</code> is
a type alias for <code>Complex&lt;Double&gt;</code>.
</div>
<p>Create new instances of <code>Complex&lt;T&gt;</code> using integer or floating-point
literals and the imaginary unit <code><a href="Structs/Complex.html#/s:12NumericAnnex7ComplexV1iACyxGvZ">Complex&lt;T&gt;.i</a></code>. For example:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">.</span><span class="n">i</span> <span class="c1">// `x` is of type `Complex&lt;Double&gt;`</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="mf">3.5</span> <span class="o">+</span> <span class="mi">7</span> <span class="o">*</span> <span class="o">.</span><span class="n">i</span> <span class="c1">// `y` is of type `Complex&lt;Double&gt;`</span>
<span class="k">let</span> <span class="nv">z</span><span class="p">:</span> <span class="kt">Complex64</span> <span class="o">=</span> <span class="o">.</span><span class="n">e</span> <span class="o">+</span> <span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="o">.</span><span class="n">i</span> <span class="c1">// `z` is of type `Complex&lt;Float&gt;`</span>
</code></pre>
<h2 id='additional-considerations' class='heading'>Additional Considerations</h2>
<p>Floating-point types have special values that represent infinity or NaN
(<q>not a number</q>). Complex functions in different languages may return
different results when working with special values.</p>
<p>Many complex functions have <a href="http://mathworld.wolfram.com/BranchCut.html">branch cuts</a>, which are curves in the
complex plane across which a function is discontinuous. Different languages
may adopt different branch cut structures for the same complex function.</p>
<p>Implementations in <code>Complex&lt;T&gt;</code> adhere to the <a href="http://www.open-std.org/JTC1/SC22/WG14/www/standards.html#9899">C standard</a> (Annex G) as
closely as possible with respect to special values and branch cuts.</p>
<p>To users unfamiliar with complex functions, the principal value returned by
some complex functions may be unexpected. For example,
<code>Double.cbrt(-8) == -2</code>, which is the <strong>real root</strong>, while
<code>Complex.cbrt(-8) == 2 * Complex.exp(.i * .pi / 3)</code>, which is the
<strong>principal root</strong>.</p>
<a href="Structs/Complex.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">struct</span> <span class="kt">Complex</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt"><a href="Protocols/Real.html">Real</a></span><span class="o">&gt;</span> <span class="p">:</span> <span class="kt">Codable</span>
<span class="k">where</span> <span class="kt">T</span> <span class="p">:</span> <span class="kt">Codable</span> <span class="o">&amp;</span> <span class="n">_ExpressibleByBuiltinFloatLiteral</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex8RationalV"></a>
<a name="//apple_ref/swift/Struct/Rational" class="dashAnchor"></a>
<h4>
<code>Rational</code>
</h4>
</div>
<div class="abstract">
<p>A type to represent a rational value.</p>
<div class="aside aside-note">
<p class="aside-title">Note</p>
<code>Ratio</code> is a type alias for <code>Rational&lt;Int&gt;</code>.
</div>
<p>Create new instances of <code>Rational&lt;T&gt;</code> by using integer literals and the
division (<code>/</code>) operator. For example:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="mi">3</span> <span class="k">as</span> <span class="kt">Ratio</span> <span class="c1">// `x` is of type `Rational&lt;Int&gt;`</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="mi">2</span> <span class="k">as</span> <span class="kt">Ratio</span> <span class="c1">// `y` is of type `Rational&lt;Int&gt;`</span>
<span class="k">let</span> <span class="nv">z</span><span class="p">:</span> <span class="kt">Ratio</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">/</span> <span class="mi">3</span> <span class="c1">// `z` is also of type `Rational&lt;Int&gt;`</span>
<span class="nf">print</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span> <span class="c1">// Prints "3"</span>
</code></pre>
<p>You can create an unreduced fraction by using the initializer
<code><a href="Structs/Rational.html#/s:12NumericAnnex8RationalVACyxGx9numerator_x11denominatortcfc">Rational&lt;T&gt;.init(numerator:denominator:)</a></code>. For example:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">a</span> <span class="o">=</span> <span class="kt">Ratio</span><span class="p">(</span><span class="nv">numerator</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span> <span class="nv">denominator</span><span class="p">:</span> <span class="mi">3</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="c1">// Prints "3/3"</span>
</code></pre>
<p>All arithmetic operations on values in canonical form (i.e. reduced to
lowest terms) return results in canonical form. However, operations on
values not in canonical form may or may not return results that are
themselves in canonical form. The property <code>canonicalized</code> is the canonical
form of any value.</p>
<h2 id='additional-considerations' class='heading'>Additional Considerations</h2>
<h3 id='special-values' class='heading'>Special Values</h3>
<p><code>Rational&lt;T&gt;</code> does not prohibit zero as a denominator. Any instance with a
positive numerator and zero denominator represents (positive) infinity; any
instance with a negative numerator and zero denominator represents negative
infinity; and any instance with zero numerator and zero denominator
represents NaN (<q>not a number</q>).</p>
<p>As with floating-point types, <code><a href="Structs/Rational.html#/s:12NumericAnnex8RationalV8infinityACyxGvZ">Rational&lt;T&gt;.infinity</a></code> compares greater than
every finite value and negative infinity, and <code>-Rational&lt;T&gt;.infinity</code>
compares less than every finite value and positive infinity. Infinite values
of the same sign compare equal to each other.</p>
<p>As with floating-point types, <code><a href="Structs/Rational.html#/s:12NumericAnnex8RationalV3nanACyxGvZ">Rational&lt;T&gt;.nan</a></code> does not compare equal to
any other value, including another NaN. Use the property <code>isNaN</code> to test if
a value is NaN. <code>Rational&lt;T&gt;</code> arithmetic operations are intended to
propagate NaN in the same manner as analogous floating-point operations.</p>
<h3 id='numerical-limits' class='heading'>Numerical Limits</h3>
<p>When a value of type <code>Rational&lt;T&gt;</code> is in canonical form, the sign of the
numerator is the sign of the value; that is, in canonical form, the sign of
the denominator is always positive. Therefore, <code>-1 / T.min</code> cannot be
represented as a value of type <code>Rational&lt;T&gt;</code> because <code>abs(T.min)</code> cannot be
represented as a value of type <code>T</code>.</p>
<p>To ensure that every representable value of type <code>Rational&lt;T&gt;</code> has a
representable magnitude and reciprocal of the same type, an overflow trap
occurs when the division (<code>/</code>) operator is used to create a value of type
<code>Rational&lt;T&gt;</code> with numerator <code>T.min</code>.</p>
<a href="Structs/Rational.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">struct</span> <span class="kt">Rational</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">SignedInteger</span><span class="o">&gt;</span> <span class="p">:</span> <span class="kt">Codable</span>
<span class="k">where</span> <span class="kt">T</span> <span class="p">:</span> <span class="kt">Codable</span> <span class="o">&amp;</span> <span class="n">_ExpressibleByBuiltinIntegerLiteral</span><span class="p">,</span>
<span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">UnsignedInteger</span><span class="p">,</span>
<span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="o">==</span> <span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span></code></pre>
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<p>&copy; 2017 <a class="link" href="https://diploid.ca" target="_blank" rel="external">Xiaodi Wu</a>. All rights reserved. (Last updated: 2017-09-30)</p>
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import XCTest
@testable import NumericAnnexTests
XCTMain([
testCase(FactoringTests.allTests),
testCase(RationalTests.allTests),
testCase(ComplexTests.allTests),
testCase(RandomXorshiftTests.allTests),
testCase(RandomXoroshiroTests.allTests),
])

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import XCTest
@testable import NumericAnnex
class ComplexTests: XCTestCase {
let pzpz = Complex(real: +0.0, imaginary: +0.0)
let pznz = Complex(real: +0.0, imaginary: -0.0)
let nzpz = Complex(real: -0.0, imaginary: +0.0)
let nznz = Complex(real: -0.0, imaginary: -0.0)
let pipi = Complex(real: Double.infinity, imaginary: .infinity)
let pini = Complex(real: Double.infinity, imaginary: -.infinity)
let nipi = Complex(real: -Double.infinity, imaginary: .infinity)
let nini = Complex(real: -Double.infinity, imaginary: -.infinity)
let pipz = Complex(real: .infinity, imaginary: +0.0)
let nipz = Complex(real: -.infinity, imaginary: +0.0)
let pinz = Complex(real: .infinity, imaginary: -0.0)
let ninz = Complex(real: -.infinity, imaginary: -0.0)
let pzpi = Complex(real: +0.0, imaginary: .infinity)
let pxpi = Complex(real: +2.0, imaginary: .infinity)
let pxni = Complex(real: +2.0, imaginary: -.infinity)
let pipy = Complex(real: .infinity, imaginary: +2.0)
let nipy = Complex(real: -.infinity, imaginary: +2.0)
let pzpn = Complex(real: +0.0, imaginary: .nan)
let nzpn = Complex(real: -0.0, imaginary: .nan)
let pnpz = Complex(real: .nan, imaginary: +0.0)
let pnnz = Complex(real: .nan, imaginary: -0.0)
let pxpn = Complex(real: +2.0, imaginary: .nan)
let nxpn = Complex(real: -2.0, imaginary: .nan)
let pnpy = Complex(real: .nan, imaginary: +2.0)
let pnny = Complex(real: .nan, imaginary: -2.0)
let pipn = Complex(real: Double.infinity, imaginary: .nan)
let nipn = Complex(real: -Double.infinity, imaginary: .nan)
let pnpi = Complex(real: Double.nan, imaginary: .infinity)
let pnni = Complex(real: Double.nan, imaginary: -.infinity)
let pnpn = Complex(real: Double.nan, imaginary: .nan)
func testComplexInitialization() {
let foo = Complex128(42 as Double)
let bar = 42 as Complex128
XCTAssertEqual(foo, bar)
}
func testComplexAddition() {
let foo: Complex128 = 1.0 + 2.0 * .i
let bar: Complex128 = 2 + 4 * .i
XCTAssertEqual(foo + bar, 3 + 6 * .i)
let baz: Complex64 = 1 + 2 * .i
let boo: Complex64 = 2.0 + 4.0 * .i
XCTAssertEqual((foo + bar).real, Double((baz + boo).real))
XCTAssertEqual((foo + bar).imaginary, Double((baz + boo).imaginary))
let moo: Complex64 = .pi + .e * .i
XCTAssertEqual(moo.real, .pi)
XCTAssertEqual(moo.imaginary, .e)
}
func testComplexDivision() {
let a = 3 + 2 * Complex128.i
let b = 4 - 3 * Complex128.i
let c = a / b
XCTAssertEqual(c.real, 6/25)
XCTAssertEqual(c.imaginary, 17/25)
}
func testComplexLogarithm() {
let a = Complex128(r: 1, theta: .pi / 4)
XCTAssertEqual(Complex.log(a).real, 0)
XCTAssertEqual(Complex.log(a).imaginary, a.argument)
let b = Complex128(r: 1, theta: .pi / 2)
XCTAssertEqual(Complex.log(b).real, 0)
XCTAssertEqual(Complex.log(b).imaginary, b.argument)
let c = Complex128(r: 1, theta: .pi)
XCTAssertEqual(Complex.log(c).real, 0)
XCTAssertEqual(Complex.log(c).imaginary, c.argument)
let d = Complex128(real: -1, imaginary: 0)
XCTAssertEqual(Complex.log(d).real, 0)
XCTAssertEqual(Complex.log(d).imaginary, d.argument)
let e = d.conjugate()
XCTAssertEqual(Complex.log(e), Complex.log(d).conjugate())
// Test special values.
var result: Complex128
result = Complex.log(nzpz)
XCTAssertEqual(result.real, -.infinity)
XCTAssertEqual(result.imaginary, .pi)
// Divide-by-zero flag should be raised.
result = Complex.log(pzpz)
XCTAssertEqual(result.real, -.infinity)
XCTAssertTrue(result.imaginary.isZero)
XCTAssertTrue(result.imaginary.sign == .plus)
// Divide-by-zero flag should be raised.
result = Complex.log(pxpi)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .pi / 2)
result = Complex.log(pxpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.log(nipy)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .pi)
result = Complex.log(pipy)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.isZero)
XCTAssertTrue(result.imaginary.sign == .plus)
result = Complex.log(nipi)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .pi * 3 / 4)
result = Complex.log(pipi)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .pi / 4)
result = Complex.log(pipn)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.log(nipn)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.log(pnpy)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.log(pnpi)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.log(pnpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
}
func testComplexSquareRoot() {
let a: Complex128 = -4
XCTAssertEqual(Complex.sqrt(a), 2 * .i)
let b: Complex128 = 4
XCTAssertEqual(Complex.sqrt(b), 2)
// Test special values.
var result: Complex128
result = Complex.sqrt(pzpz)
XCTAssertTrue(result.isZero)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.imaginary.sign == .plus)
result = Complex.sqrt(nzpz)
XCTAssertTrue(result.isZero)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.imaginary.sign == .plus)
result = Complex.sqrt(pxpi)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .infinity)
result = Complex.sqrt(pnpi)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .infinity)
result = Complex.sqrt(pxpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.sqrt(nipy)
XCTAssertEqual(result.real, 0)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertEqual(result.imaginary, .infinity)
result = Complex.sqrt(pipy)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, 0)
XCTAssertTrue(result.imaginary.sign == .plus)
result = Complex.sqrt(nipn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isInfinite)
// The sign of the imaginary part is unspecified.
result = Complex.sqrt(pipn)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.sqrt(pnpy)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.sqrt(pnpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
}
func testComplexCubeRoot() {
// Note that the principal cube root is not necessarily the real-valued
// cube root.
let a: Complex64 = -8
XCTAssertEqualWithAccuracy(Complex.cbrt(a).real, 1, accuracy: 0.000001)
XCTAssertEqualWithAccuracy(Complex.cbrt(a).imaginary,
2 * Float.sin(.pi / 3), accuracy: 0.000001)
let b: Complex64 = 8
XCTAssertEqual(Complex.cbrt(b), 2)
let c: Complex64 = -27 * .i
XCTAssertEqualWithAccuracy(Complex.cbrt(c).real,
3 * Float.cos(-.pi / 6), accuracy: 0.000001)
XCTAssertEqualWithAccuracy(Complex.cbrt(c).imaginary,
-1.5, accuracy: 0.000001)
let d: Complex64 = 27 * .i
XCTAssertEqual(Complex.cbrt(d), Complex.cbrt(c).conjugate())
let e: Complex128 = -8
XCTAssertEqual(Complex.cbrt(e), 2 * Complex.exp(.i * .pi / 3))
}
func testComplexExponentiation() {
let a = Complex(real: Double.log(42))
let b = Complex.exp(a)
XCTAssertEqualWithAccuracy(b.real, 42, accuracy: 0.00000000000001)
let i: Complex128 = .i
let actual = i.power(of: i)
XCTAssertEqual(actual.real, Double.exp(-Double.pi / 2))
XCTAssertEqual(actual.imaginary, 0)
// Test special values.
var result: Complex128
var expected: Complex128
result = Complex.exp(pzpz)
XCTAssertEqual(result.real, 1)
XCTAssertEqual(result.imaginary, 0)
XCTAssertTrue(result.imaginary.sign == .plus)
result = Complex.exp(nzpz)
XCTAssertEqual(result.real, 1)
XCTAssertEqual(result.imaginary, 0)
XCTAssertTrue(result.imaginary.sign == .plus)
result = Complex.exp(pxpi)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
// Invalid flag should be raised.
result = Complex.exp(pxpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.exp(pipz)
XCTAssertTrue(result == pipz)
XCTAssertTrue(result.imaginary.sign == .plus)
result = Complex.exp(nipy)
expected = Complex(r: 0, theta: nipy.imaginary)
XCTAssertEqual(result.real, expected.real)
XCTAssertEqual(result.imaginary, expected.imaginary)
result = Complex.exp(pipy)
expected = Complex(r: .infinity, theta: pipy.imaginary)
XCTAssertEqual(result.real, expected.real)
XCTAssertEqual(result.imaginary, expected.imaginary)
result = Complex.exp(nipi)
XCTAssertTrue(result.isZero)
// The signs of zero are unspecified.
result = Complex.exp(pipi)
XCTAssertTrue(result.real.isInfinite)
// The sign of the real part is unspecified.
XCTAssertTrue(result.imaginary.isNaN)
// Invalid flag should be raised.
result = Complex.exp(nipn)
XCTAssertTrue(result.isZero)
// The signs of zero are unspecified.
result = Complex.exp(pipn)
XCTAssertTrue(result.real.isInfinite)
// The sign of the real part is unspecified.
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.exp(pnpz)
XCTAssertTrue(result.real.isNaN)
XCTAssertEqual(result.imaginary, pnpz.imaginary)
result = Complex.exp(pnpy)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.exp(pnpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
}
func testComplexTrigonometry() {
let a: Complex128 = 1
XCTAssertEqual(Complex.sin(a).real, Double.sin(1))
XCTAssertTrue(Complex.sin(a).imaginary.isZero)
XCTAssertEqual(Complex.cos(a).real, Double.cos(1))
XCTAssertTrue(Complex.cos(a).imaginary.isZero)
XCTAssertEqual(Complex.tan(a).real, Double.tan(1))
XCTAssertTrue(Complex.tan(a).imaginary.isZero)
let b: Complex128 = a * .i
XCTAssertTrue(Complex.sin(b).real.isZero)
XCTAssertEqual(Complex.sin(b).imaginary, Double.sinh(1))
XCTAssertEqual(Complex.cos(b).real, Double.cosh(1))
XCTAssertTrue(Complex.cos(b).imaginary.isZero)
XCTAssertTrue(Complex.tan(b).real.isZero)
XCTAssertEqual(Complex.tan(b).imaginary, Double.tanh(1))
let c: Complex128 = -2
let d: Complex128 = .pi + .i * Complex.log(2 - Complex.sqrt(3))
let e: Complex128 = .pi / 2 - .i * Complex.log(2 + Complex.sqrt(3))
XCTAssertTrue(Complex.acos(c).real.sign == .plus)
XCTAssertEqualWithAccuracy(abs(Complex.acos(c).real), abs(d.real),
accuracy: 0.00000000000001)
XCTAssertTrue(Complex.acos(c).imaginary.sign == .minus)
XCTAssertEqualWithAccuracy(abs(Complex.acos(c).imaginary), abs(d.imaginary),
accuracy: 0.00000000000001)
XCTAssertTrue(Complex.asin(c).real.sign == .minus)
XCTAssertEqualWithAccuracy(abs(Complex.asin(c).real), abs(e.real),
accuracy: 0.00000000000001)
XCTAssertTrue(Complex.asin(c).imaginary.sign == .plus)
XCTAssertEqualWithAccuracy(abs(Complex.asin(c).imaginary), abs(e.imaginary),
accuracy: 0.00000000000001)
let f: Complex128 = c.conjugate()
XCTAssertEqual(Complex.acos(f), Complex.acos(c).conjugate())
XCTAssertEqual(Complex.asin(f), Complex.asin(c).conjugate())
let g: Complex128 = Complex(real: +0.0, imaginary: 2.0)
let h: Complex128 = Complex(real: -0.0, imaginary: 2.0)
let i: Complex128 = .pi / 2 + .i * Complex.log(3) / 2
XCTAssertTrue(Complex.atan(g).real.sign == .plus)
XCTAssertEqualWithAccuracy(abs(Complex.atan(g).real), abs(i.real),
accuracy: 0.00000000000001)
XCTAssertTrue(Complex.atan(g).imaginary.sign == .plus)
XCTAssertEqualWithAccuracy(abs(Complex.atan(g).imaginary), abs(i.imaginary),
accuracy: 0.00000000000001)
XCTAssertEqual(.i * Complex.atan(h), (.i * Complex.atan(g)).conjugate())
// Test special values.
var result: Complex128
result = Complex.acos(pzpz)
XCTAssertEqual(result.real, .pi / 2)
XCTAssertTrue(result.imaginary.isZero)
XCTAssertTrue(result.imaginary.sign == .minus)
result = Complex.acos(nzpz)
XCTAssertEqual(result.real, .pi / 2)
XCTAssertTrue(result.imaginary.isZero)
XCTAssertTrue(result.imaginary.sign == .minus)
result = Complex.acos(pzpn)
XCTAssertEqual(result.real, .pi / 2)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.acos(nzpn)
XCTAssertEqual(result.real, .pi / 2)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.acos(pxpi)
XCTAssertEqual(result.real, .pi / 2)
XCTAssertEqual(result.imaginary, -.infinity)
result = Complex.acos(pxpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.acos(nipy)
XCTAssertEqual(result.real, .pi)
XCTAssertEqual(result.imaginary, -.infinity)
result = Complex.acos(pipy)
XCTAssertTrue(result.real.isZero)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertEqual(result.imaginary, -.infinity)
result = Complex.acos(nipi)
XCTAssertEqual(result.real, .pi * 3 / 4)
XCTAssertEqual(result.imaginary, -.infinity)
result = Complex.acos(pipi)
XCTAssertEqual(result.real, .pi / 4)
XCTAssertEqual(result.imaginary, -.infinity)
result = Complex.acos(pipn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isInfinite)
// The sign of the imaginary part is unspecified.
result = Complex.acos(nipn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isInfinite)
// The sign of the imaginary part is unspecified.
result = Complex.acos(pnpy)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.acos(pnpi)
XCTAssertTrue(result.real.isNaN)
XCTAssertEqual(result.imaginary, -.infinity)
result = Complex.acos(pnpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
}
func testComplexHyperbolicFunctions() {
let a: Complex128 = Complex(real: 0.0, imaginary: -2.0)
XCTAssertEqualWithAccuracy(Complex.asinh(a).real,
Double.log(2 + Double.sqrt(3)),
accuracy: 0.00000000000001)
XCTAssertEqual(Complex.asinh(a).imaginary, -.pi / 2)
let b: Complex128 = Complex(real: -0.0, imaginary: -2.0)
XCTAssertEqualWithAccuracy(Complex.asinh(b).real,
-Double.log(2 + Double.sqrt(3)),
accuracy: 0.00000000000001)
XCTAssertEqual(Complex.asinh(b).imaginary, -.pi / 2)
let c: Complex128 = 1 + 2 * .i
let d: Complex128 = Complex.log(c + Complex.sqrt(-2 + 4 * .i))
XCTAssertEqualWithAccuracy(Complex.asinh(c).real, d.real,
accuracy: 0.00000000000001)
XCTAssertEqualWithAccuracy(Complex.asinh(c).imaginary, d.imaginary,
accuracy: 0.00000000000001)
XCTAssertEqualWithAccuracy(Complex.atanh(c).real,
Double.log(8) / 4 - Double.log(2) / 2,
accuracy: 0.00000000000001)
XCTAssertEqual(Complex.atanh(c).imaginary, .pi * 3 / 8)
let e: Complex128 = 0.5
XCTAssertEqual(Complex.acosh(e).real, 0)
XCTAssertEqualWithAccuracy(Complex.acosh(e).imaginary, 1.047197551196597746,
accuracy: 0.00000000000001)
let f: Complex128 = e.conjugate()
XCTAssertTrue(Complex.acosh(f).real.sign == Complex.acosh(e).real.sign)
XCTAssertEqual(Complex.acosh(f).real, 0)
XCTAssertEqual(Complex.acosh(f).imaginary, -Complex.acosh(e).imaginary)
let g: Complex128 = 1 + .i
let h: Complex128 =
Complex.log(g + Complex.pow(-1, 1 / 4) * Complex.sqrt(2 + .i))
XCTAssertEqual(Complex.acosh(g).real, h.real)
XCTAssertEqual(Complex.acosh(g).imaginary, h.imaginary)
let i: Complex128 = .i * Complex.acos(g)
XCTAssertEqual(Complex.acosh(g), i)
let j: Complex128 = 2
XCTAssertEqual(Complex.atanh(j).real, Double.log(3) / 2)
XCTAssertEqual(Complex.atanh(j).imaginary, .pi / 2)
let k: Complex128 = j.conjugate()
XCTAssertEqual(Complex.atanh(k).real, Double.log(3) / 2)
XCTAssertEqual(Complex.atanh(k).imaginary, -.pi / 2)
// -------------------------------------------------------------------------
// TODO: Test sinh, cosh, tanh.
//
// Per C11 DR 471:
// ctanh(0, +inf) = (0, NaN) and raises the invalid flag
// ctanh(0, NaN) = (0, NaN) and does not raise the invalid flag
// -------------------------------------------------------------------------
// Test special values.
var result: Complex128
var expected: Complex128
result = Complex.asinh(pzpz)
XCTAssertEqual(result, pzpz)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.imaginary.sign == .plus)
result = Complex.asinh(pxpi)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .pi / 2)
result = Complex.asinh(pxpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.asinh(pipy)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.sign == .plus)
XCTAssertTrue(result.imaginary.isZero)
result = Complex.asinh(pipi)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .pi / 4)
result = Complex.asinh(pipn)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.asinh(pnpz)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.sign == .plus)
XCTAssertTrue(result.imaginary.isZero)
result = Complex.asinh(pnpy)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.asinh(pnpi)
XCTAssertTrue(result.real.isInfinite)
// The sign of the real part is unspecified.
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.asinh(pnpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.acosh(pzpz)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.real.isZero)
XCTAssertEqual(result.imaginary, .pi / 2)
result = Complex.acosh(nzpz)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.real.isZero)
XCTAssertEqual(result.imaginary, .pi / 2)
result = Complex.acosh(pxpi)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .pi / 2)
result = Complex.acosh(pxpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.acosh(pzpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertEqual(result.imaginary, .pi / 2) // See C11 DR 471.
result = Complex.acosh(nipy)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .pi)
result = Complex.acosh(pipy)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.sign == .plus)
XCTAssertTrue(result.imaginary.isZero)
result = Complex.acosh(nipi)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .pi * 3 / 4)
result = Complex.acosh(pipi)
XCTAssertEqual(result.real, .infinity)
XCTAssertEqual(result.imaginary, .pi / 4)
result = Complex.acosh(pipn)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.acosh(nipn)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.acosh(pnpy)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.acosh(pnpi)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.acosh(pnpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.atanh(pzpz)
XCTAssertEqual(result, pzpz)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.imaginary.sign == .plus)
result = Complex.atanh(pzpn)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.real.isZero)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.atanh(1)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.sign == .plus)
XCTAssertTrue(result.imaginary.isZero)
// Divide-by-zero flag should be raised.
result = Complex.atanh(pxpi)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.real.isZero)
XCTAssertEqual(result.imaginary, .pi / 2)
result = Complex.atanh(pxpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.atanh(pipy)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.real.isZero)
XCTAssertEqual(result.imaginary, .pi / 2)
result = Complex.atanh(pipi)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.real.isZero)
XCTAssertEqual(result.imaginary, .pi / 2)
result = Complex.atanh(pipn)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.real.isZero)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.atanh(pnpy)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.atanh(pnpi)
XCTAssertTrue(result.real.isZero)
// The sign of the real part is unspecified.
XCTAssertEqual(result.imaginary, .pi / 2)
result = Complex.atanh(pnpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.sinh(pzpz)
XCTAssertTrue(result.real.sign == .plus)
XCTAssertTrue(result.real.isZero)
XCTAssertTrue(result.imaginary.sign == .plus)
XCTAssertTrue(result.imaginary.isZero)
result = Complex.sinh(pzpi)
XCTAssertTrue(result.real.isZero)
// The sign of the real part is unspecified.
XCTAssertTrue(result.imaginary.isNaN)
// Invalid flag should be raised.
result = Complex.sinh(pzpn)
XCTAssertTrue(result.real.isZero)
// The sign of the real part is unspecified (?).
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.sinh(pxpi)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
// Invalid flag should be raised.
result = Complex.sinh(pxpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.sinh(pipz)
XCTAssertEqual(result.real, .infinity)
XCTAssertTrue(result.imaginary.sign == .plus)
XCTAssertTrue(result.imaginary.isZero)
result = Complex.sinh(pipy)
expected = Complex(r: .infinity, theta: pipy.imaginary)
XCTAssertEqual(result.real, expected.real)
XCTAssertEqual(result.imaginary, expected.imaginary)
result = Complex.sinh(pipi)
XCTAssertTrue(result.real.isInfinite)
// The sign of the real part is unspecified.
XCTAssertTrue(result.imaginary.isNaN)
// Invalid flag should be raised.
result = Complex.sinh(pipn)
XCTAssertTrue(result.real.isInfinite)
// The sign of the real part is unspecified.
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.sinh(pnpz)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.sign == .plus)
XCTAssertTrue(result.imaginary.isZero)
result = Complex.sinh(pnpy)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
result = Complex.sinh(pnpn)
XCTAssertTrue(result.real.isNaN)
XCTAssertTrue(result.imaginary.isNaN)
}
static var allTests = [
("testComplexInitialization", testComplexInitialization),
("testComplexAddition", testComplexAddition),
("testComplexDivision", testComplexDivision),
("testComplexLogarithm", testComplexLogarithm),
("testComplexSquareRoot", testComplexSquareRoot),
("testComplexCubeRoot", testComplexCubeRoot),
("testComplexExponentiation", testComplexExponentiation),
("testComplexTrigonometry", testComplexTrigonometry),
("testComplexHyperbolicFunctions", testComplexHyperbolicFunctions),
]
}

32
Tests/NumericAnnexTests/FactoringTests.swift
View File

@ -1,32 +0,0 @@
import XCTest
@testable import NumericAnnex
class FactoringTests: XCTestCase {
func testGCD() {
XCTAssertEqual(UInt.gcd(18, 84), 6)
XCTAssertEqual(UInt.gcd(24, 60), 12)
XCTAssertEqual(UInt.gcd(42, 56), 14)
// Test special values.
XCTAssertEqual(UInt.gcd(0, 42), 42)
XCTAssertEqual(UInt.gcd(42, 0), 42)
XCTAssertEqual(UInt.gcd(0, 0), 0)
XCTAssertTrue(Int8.gcdReportingOverflow(-128, -128).overflow == .overflow)
}
func testLCM() {
XCTAssertEqual(UInt.lcm(4, 6), 12)
XCTAssertEqual(UInt.lcm(6, 21), 42)
// Test special values.
XCTAssertEqual(UInt.lcm(0, 42), 0)
XCTAssertEqual(UInt.lcm(42, 0), 0)
XCTAssertEqual(UInt.lcm(0, 0), 0)
}
static var allTests = [
("testGCD", testGCD),
("testLCM", testLCM),
]
}

433
Tests/NumericAnnexTests/RandomXoroshiroTests.swift
View File

@ -1,433 +0,0 @@
import XCTest
@testable import NumericAnnex
class RandomXoroshiroTests: XCTestCase {
func test_1_2() {
let rng = Random.Xoroshiro(state: (1, 2))
// Values were generated using the reference implementation found at:
// http://xoroshiro.di.unimi.it/xoroshiro128plus.c
let expected: [UInt64] = [
3,
36029003177443331,
78883775479546723,
11565523463456473958,
4242646275387589636,
256749404433942852,
11014892026844973196,
9059353499452950543,
8597521241247625872,
4693915028112570637,
7013278354776644264,
7324551527405749817,
5536070250743279847,
14551654802141766561,
16502220822753139053,
9169458631996292866,
8255063232684103966,
4985613803055290144,
8873762385537109026,
7955442200162942054,
10038983008978924840,
16556838790008055993,
1445216410056612069,
11648404329028311521,
11968528621191360117,
378994045525309041,
15533848123981932311,
17810787280139321748,
8698105432054197087,
3039374059427402804,
6000419305548865286,
5555673801313513616,
14832631808895921712,
15269786733006210781,
18192816199346518081,
9227630901375587864,
13636270006546584644,
10048331387000224533,
5881842723816098216,
2759740862732559190,
7198729811037598577,
2536130127161171247,
9244050483868404095,
4471143257962844606,
6826378813747079941,
8311102247454864474,
8030254548075018734,
4856383866562462327,
2146425163792814156,
678709922273904935,
14153046072826725166,
13096693794621671407,
3167365684852107430,
7035944759741560501,
4327331733683539348,
4331384222358257785,
11778493022454179159,
6436799581831663489,
5199453391521163130,
13866358363722915963,
10490459205808868365,
8942744637306013108,
13330606527360361965,
15114450932505609455,
10484962763089362709,
8486043591433584489,
1244312731768254023,
1423652513934467700,
7202410701832789339,
4247336634893478223,
2761260300685838243,
3026641667955338198,
11726045602658466327,
15986288544570028551,
1140902651811776482,
3031375389759955929,
10251211885453078785,
3805832680344220010,
6901351005802427116,
327749723378156456,
3850130685063866272,
395616431473633113,
1006002813608848610,
13567214008324118923,
8234626684467576068,
12325748354848861738,
13876857058316077328,
5575055431371035091,
1001747845116129793,
1911249420570102675,
11307478254871521778,
11741813699519564532,
18249690205084409006,
4141904051068235945,
8007541499524410059,
11522483542545463294,
12843807877472758944,
4036354428928582882,
3568160883062227481,
7069228111761650618,
12800820881038955908,
12096397496417421560,
18331638049296705373,
14080565232358176996,
11717911967111159659,
5236417326714399775,
896860817366679607,
1103175799875588730,
5624452808649533921,
14074612223815901137,
12560226745213461489,
8693158002233621656,
13522197534225561183,
17543423602239517471,
9007394771080283032,
13751257602651261054,
10523437980156934525,
3939326514580989433,
4009481458773474577,
5605944737217646703,
11357362020001843015,
6496864033205151805,
8784505647222405488,
6539787485601627798,
14223996719628965531,
11820130316034487593,
9216825067157709454,
13450820975868246163,
13582660326201054082,
7274082025836619686,
5139824568063119565,
15809486898860438280,
10511273953872056519,
8550850887347653734,
4982865717503297021,
16768400430211288709,
5532507867727508336,
3461217932251647116,
7937630382825106010,
8428419650246743539,
13756581925356151789,
12850941715331890900,
17583375651346257957,
14212940247871282486,
2939312478589591003,
9113217020860214630,
15037155367836561944,
5106936154711971714,
7731881123454188381,
8800799748666791518,
5334921694098114980,
16976572882617606055,
10733445809727535171,
18071763238883705179,
12390417336872755007,
4675767129793818894,
4624990676514788078,
13985922235308181354,
18199410361023570910,
11100295634832328716,
14349833775752484419,
2635828052976710695,
13494215123457180093,
824431475002587115,
16362343995287574060,
12565374284554722892,
3400506694682569059,
9678328826858469760,
2356773772080043977,
6660327511171829851,
7003621798856571358,
13678900301532284938,
10328133941778183218,
18415851989001661946,
164983221740675978,
12091409703168792651,
16023813642563214154,
13146064085390259297,
16830016805105752873,
12909765344649172761,
18132040913220627960,
13428079187771868408,
13253065382531928412,
5597099400358632931,
15048470487608189234,
13965835027251694013,
16488001888584899979,
6774957755200087230,
9420290878387669954,
18103357232511502336,
7958521771342382343,
7946005055385189225,
7459466188094139112,
12160219491927961419,
10303458678974273963,
15164880279874615320,
8229476507371032858,
8299451873416872397,
14240369133888159195,
14259674009027384056,
]
for (index, value) in zip(0..., rng.prefix(expected.count)) {
XCTAssertEqual(expected[index], value)
}
}
func test_42_42() {
let rng = Random.Xoroshiro(state: (42, 42))
// Values were generated using the reference implementation found at:
// http://xoroshiro.di.unimi.it/xoroshiro128plus.c
let expected: [UInt64] = [
84,
1513209474796486656,
1516164967689093120,
13042513867470012437,
12360923703050654741,
1540629971893098580,
16526710497398440032,
6385704233003790657,
11168422736538991365,
4445263159897632708,
3623790067483698302,
16679083957109345764,
2328362623729376304,
11959586146887887947,
1397979073127807009,
2057082927736384432,
9405332975393492225,
8930172824755803710,
15806946208844633572,
13763125990787205877,
3588104897035227753,
16535253711858701559,
1756777986788515775,
3260086412739166738,
2968118814995357638,
10401021566468687067,
10396793383754549895,
7751583068336470633,
18199822703658045417,
18106278619996331558,
11555865454533205083,
11019066312200696059,
14691523452735343350,
8374833488214025190,
5144857991660071778,
4303754754246322002,
7833483779443332279,
9537057695470270940,
15360815301067540763,
8680872279790622784,
5061961432584221634,
13845413006391880255,
9347406936646547437,
9936035123032424105,
7762742561483885717,
6225100736029176021,
12384612707518261907,
6932426897081241028,
12622735902618443507,
16497181272706281681,
4175144421792863622,
7187519007501887217,
1296625689586954249,
9756008725656319610,
13415925134280955255,
7869611265653995342,
15802320767156340368,
9917057770149375021,
6602199518275499834,
15985761826175804236,
6348789404701085358,
13882066392163471251,
6823376289545817014,
2440555314667442152,
8120581271152580867,
5962305562057075398,
8624786735003738880,
17278456019734296413,
8810594517455747061,
3308252836812617218,
11509046657648993076,
4526424802255960286,
11368279899573388309,
17164649625755550542,
17756655439803178301,
1790544258914355261,
16936001771718971059,
6604464130005614382,
4599098751468689332,
5595709687219398019,
7782673445638093156,
11625849522136425098,
17545728945788535376,
1086103391208514655,
6757121107432662022,
7110671459349670034,
14243248346778391217,
6293180738376339946,
82698844051683738,
9016754545624797992,
12686910675574395757,
11377577519711600987,
2421687222153439150,
5062679022650701181,
5366991913210349668,
17763749207842542169,
15150310935153524353,
8753573200416713058,
14717915573524068736,
4976419370494142329,
15950649534319229916,
14729183728516038958,
14130494893011053825,
484607939216453498,
11916622920760767819,
6235630645739181768,
3311677611128604917,
15387119495745197491,
18407483842959607091,
13443654906141944091,
7336989792155624100,
4969837398812759230,
4690320617290695452,
7922943443598648642,
7261933739777886868,
1428233557480945068,
13405045269283977471,
8264480103292844317,
3636005412915626226,
14531131283728793534,
1554317509710298241,
5671084098303337046,
2329626881373043482,
9843173781615930848,
11631854473490224468,
1971235111135032481,
16500865010222116014,
15716915215378368203,
9800201135256065410,
15826078301157734671,
8086394309479137252,
4024335468016869419,
9607553477750614063,
8476344891763358613,
9598206907093120703,
2222640063630506567,
4748064424880786781,
16133608208331438329,
4841284499098320283,
14366219220792192136,
12340209151804320950,
9394640533303495521,
9967177845151229775,
1422411866393454331,
1700294099603155803,
3169743091038458854,
1730162735865459074,
7076023242537824502,
9751492838437846300,
5563389164091569554,
14685081858851293124,
17127988883180719048,
13139712666856391346,
16133760556311029769,
16124042929298679518,
15244317750072181250,
16865140868997569773,
3382895046112672233,
3311493913432858434,
6972003746672027210,
10860730320333834349,
13157555520059317901,
18049794823806643433,
4096188303889068959,
5848535283123888736,
14097084481779687678,
5923531794703750167,
8671735658329475473,
12796338774024443025,
17688950762314547996,
17545970295496335826,
18087282577194630790,
6111642868515581430,
9663837423353030300,
5519877375598539571,
198283696489934212,
639923763610710485,
9642279419743806424,
12740328497763347702,
5689966802246858055,
2542424879537644225,
4148468791004988526,
15776468824412039713,
9064289572460882470,
8024516222176589607,
13357280224400389184,
6416386861523146761,
2944019619063193966,
15840065439036381038,
8319915330237311105,
12425399842755045103,
11362128272172250003,
7666439957252188840,
17584644503297617903,
18199845540565038844,
16892202086104407413,
11581668723621517396,
7063743100798932176,
16723393516470594016,
15389111121968225194,
]
for (index, value) in zip(0..., rng.prefix(expected.count)) {
XCTAssertEqual(expected[index], value)
}
}
static var allTests = [
("test_1_2", test_1_2),
("test_42_42", test_42_42),
]
}

433
Tests/NumericAnnexTests/RandomXorshiftTests.swift
View File

@ -1,433 +0,0 @@
import XCTest
@testable import NumericAnnex
class RandomXorshiftTests: XCTestCase {
func test_1_2() {
let rng = Random(state: (1, 2))
// Values were generated using the reference implementation found at:
// http://xoroshiro.di.unimi.it/xorshift128plus.c
let expected: [UInt64] = [
3,
8388645,
33816707,
70368778527840,
211106267172129,
281552312399723,
352084508939685,
648800200157934532,
2540241598295339419,
2648407162339308712,
2668752799858137237,
2693699070923777171,
4963962142036894822,
11906738374415673757,
9632391838565739090,
2066554175270807792,
2494022894752328945,
1356498747469623086,
11211414770921777427,
10429009827004399351,
2323958476090753392,
3568022355029630745,
66653323867787511,
2202670577644624798,
12857773250674142713,
4201717447333134628,
15638699059651815979,
4364251867564744889,
7660026492187243788,
3800851612386782149,
9605385339319476694,
2406190703679268226,
846469172938628641,
5337113899023465296,
15781852488115071549,
15354462637857246627,
11103786004116532909,
10104343325409002424,
12295544783831320534,
9091324406814034890,
6628842478985115447,
12899089858988772368,
16555193421103270775,
5182223508147526313,
1789523054416386623,
8305247958687497212,
4332901280696262112,
4830599505179982300,
16562699778378001540,
4301002138587088699,
16887648884050224867,
8961088417938801842,
18120357346625319699,
5062957422512203146,
14533691471387576319,
17893283120539033726,
8933729491225170482,
7703252517382402300,
4527677543428710360,
11852647525519638421,
14323485818621793660,
1603981125980772416,
10667357879029765177,
12618821681808242380,
15688459637854892994,
16858336139413885781,
8192223518607711332,
15086078044025798325,
7820396459134496922,
10260666624710253760,
12864353747813168684,
3349112384258896183,
17079950137619159659,
13693940288920244054,
4120914027243693443,
16513947166147576873,
7753202820822651981,
6626804775540638270,
6691531028144752467,
1848899687106318854,
13681393795236538282,
16157808923290844879,
14402137377069136975,
10721861038970349125,
11713871874537672148,
17636397933591539986,
4203420467318374,
14279516174659940213,
5030216290135979793,
12431123221355774434,
3353451983739559286,
1982730773867159337,
4287093158672282079,
10427894759339841969,
10992708284980487886,
9826778016246262346,
598098170064759435,
5132675295537973900,
2977953682469803199,
7095603002820811521,
14473363250705030589,
5815164147694210911,
893364492343289163,
7223131833749622824,
8283693734360236205,
18098811217686597526,
13472185143670172098,
4921989762323275340,
1122032220579367383,
1580787425287478212,
12971777770367737930,
14141602702208488443,
8934944843784058286,
6607626240826326523,
6849503334803588545,
8945087483095516625,
11138928521629212103,
17790276629890369045,
1578334294751373582,
2806185787680672391,
2659421138426664381,
5725303483961243766,
6184936954810297634,
939427805219677289,
7877825174927605138,
17042809939415329651,
7007032053414486723,
708948147059678574,
5807695797323831903,
10654984825451839699,
658076860194383956,
6054460744441943985,
18125771188890900160,
12455829269468258300,
7801377697876303864,
1155650389744014674,
14837203265123196363,
14611150480548774776,
11295561405428659877,
14652736317255425712,
10357309030900164003,
11362376726255131814,
17481625516190528824,
10063769529741033785,
10070196982447573973,
10858208953628516990,
8316996154212129605,
1966992145466461251,
9534492825751960063,
17331020003238783005,
4055301956728939736,
15747646385425363016,
5713890891727580694,
16363438003468365099,
6059680446507887551,
15805989924395708713,
7701919984793556335,
10881930014020511227,
16418946584288169403,
149521899493351325,
12534253806982171802,
11139483896115829864,
308345381091007206,
11578160645028040120,
16035213593244811876,
2483577470144144779,
2382343175518057727,
4467734399270823599,
6604294351011625227,
8344523632028305909,
13176362180825574816,
5057251926377527702,
4615034777510676815,
716527652147291657,
2255853362627944573,
1040521049739732381,
8767117186808380895,
4542310411533510448,
17298777313319151232,
6001563173036790655,
6632155663823324745,
260008012886308771,
8125529425889733497,
8459386663958119024,
478351272518501156,
16245194212119991188,
16067636681512186800,
13291772728582610504,
977325433216839797,
316583137798752960,
16331330055435573991,
13872944562523015302,
14073886698853620462,
10932979021145220513,
9767459346543799817,
3499076079183291818,
16794441505776618298,
2728580826238923060,
9982312178378568539,
10174783523235389589,
]
for (index, value) in zip(0..., rng.prefix(expected.count)) {
XCTAssertEqual(expected[index], value)
}
}
func test_42_42() {
let rng = Random(state: (42, 42))
// Values were generated using the reference implementation found at:
// http://xoroshiro.di.unimi.it/xorshift128plus.c
let expected: [UInt64] = [
84,
352322923,
363332930,
2955487621802017,
5910974877560714,
5984590265422244,
3102369235448484,
8794055518707935431,
11715271280642458577,
11121097408281568494,
11123324466519438486,
11813657815484460077,
15967982470865911351,
16027149834536272202,
14579330570455427858,
17901094324785512346,
7299173581280891606,
9598572314193889241,
6321253227128400616,
12069832686681671119,
8362407854952761049,
4240112657210925663,
8298289508193808350,
5492533585219385488,
3424077862959065976,
16763914594806242130,
1218199477047421836,
7673362999398496465,
5794997430478554395,
7270864472693218401,
6217237324811287380,
5931838762482735489,
2650854048920189469,
15912113305156886344,
12742758016124380936,
4295159997148736665,
377715972074097257,
2850808977988091172,
8491599958789257052,
9324795365995984871,
8211242245495735455,
17422811286952683550,
13418142500587507919,
5930788801560645895,
6741187258797332316,
10118554851581664695,
115595561983548771,
8043906809085936116,
11823913423613696098,
14518129458877160117,
8343257749050278180,
13733556794904034707,
8684565082910747967,
3120612299320189264,
12303575766779633627,
12795815681716387847,
9809457554914526522,
8920375781289094670,
1993124226495641609,
1733459562229749526,
17111289711044275575,
17744781264274534442,
8365870118406280068,
3394993270264399372,
17743599368353179843,
10866568771989044846,
5256627558940251261,
1523012312127918050,
4830133096723200725,
15678901288667906426,
12456461226754166128,
11853493959818320069,
12499595671348820672,
1087309203085082628,
8913015614641765480,
7628423184432057398,
3580904369680909773,
803445358366289622,
11594309456042139689,
4033442513255026894,
1433661196805234850,
12192207108666652295,
14586722686201389331,
6181944170276831130,
764669472652961289,
889048737081245825,
12138836305257926083,
888086957463834642,
2390515631187897836,
2446868417045655461,
11015187241185530986,
10036793306822931734,
12334486703513775025,
5108691882609665754,
9514819673388488194,
2273142206574835406,
14437507437202006078,
3730714957717498326,
2872726062891204564,
3798333342152566218,
16669861045337712380,
13102670696856779254,
10908263394212623530,
10676532138494980932,
15489857964433950539,
7937142546638092754,
13417642705898345692,
13600811864162143445,
9477162617086519863,
10624389419236725677,
7441007177458375221,
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for (index, value) in zip(0..., rng.prefix(expected.count)) {
XCTAssertEqual(expected[index], value)
}
}
static var allTests = [
("test_1_2", test_1_2),
("test_42_42", test_42_42),
]
}

266
Tests/NumericAnnexTests/RationalTests.swift
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@ -1,266 +0,0 @@
import XCTest
@testable import NumericAnnex
class RationalTests: XCTestCase {
func testRational() {
let a = 6 / 4 as Rational<Int>
XCTAssertEqual(a.description, "3/2")
XCTAssertEqual(a, 3 / 2 as Rational<Int>)
let b = 5 / 3 as Rational<Int>
XCTAssertEqual(b.description, "5/3")
XCTAssertLessThan(a, b)
XCTAssertEqual(a + b, 19 / 6 as Ratio)
XCTAssertEqual(b + a, 19 / 6 as Ratio)
XCTAssertEqual(a - b, -1 / 6 as Ratio)
XCTAssertEqual(b - a, 1 / 6 as Ratio)
XCTAssertEqual(a * b, 5 / 2 as Ratio)
XCTAssertEqual(b * a, 5 / 2 as Ratio)
XCTAssertEqual(a / b, 9 / 10 as Ratio)
XCTAssertEqual(b / a, 10 / 9 as Ratio)
XCTAssertEqual((10 / 9 as Ratio).mixed.whole, 1)
XCTAssertEqual((10 / 9 as Ratio).mixed.fractional, 1 / 9)
// Test special values.
let pn = Ratio.nan
let pi = Ratio.infinity
let ni = -Ratio.infinity
XCTAssertTrue((pn + pn).isNaN)
XCTAssertTrue((pn - pn).isNaN)
XCTAssertTrue((pn * pn).isNaN)
XCTAssertTrue((pn / pn).isNaN)
XCTAssertEqual(pi + pi, .infinity)
XCTAssertEqual(ni + ni, -.infinity)
XCTAssertEqual(pi + 0, .infinity)
XCTAssertEqual(ni + 0, -.infinity)
XCTAssertEqual(pi + 42, .infinity)
XCTAssertEqual(pi - 42, .infinity)
XCTAssertEqual(ni + 42, -.infinity)
XCTAssertEqual(ni - 42, -.infinity)
XCTAssertTrue((pi + ni).isNaN)
XCTAssertTrue((ni + pi).isNaN)
XCTAssertTrue((pi - pi).isNaN)
XCTAssertTrue((-pi + pi).isNaN)
XCTAssertTrue((0 / 0 as Ratio).isNaN)
XCTAssert((0 / 0 as Ratio) != .nan) // NaN compares unequal to everything.
XCTAssertTrue((42 / 0 as Ratio).isInfinite)
XCTAssertTrue((-42 / 0 as Ratio).isInfinite)
XCTAssert(42 / 0 as Ratio == .infinity)
XCTAssert(-42 / 0 as Ratio == -.infinity)
XCTAssertEqual((42 / 0 as Ratio).description, "inf")
XCTAssertEqual((-42 / 0 as Ratio).description, "-inf")
XCTAssertEqual(pi * pi, .infinity)
XCTAssertEqual(pi * ni, -.infinity)
XCTAssertEqual(ni * pi, -.infinity)
XCTAssertEqual(ni * ni, .infinity)
XCTAssertTrue((pi * 0).isNaN)
XCTAssertTrue((ni * 0).isNaN)
XCTAssertTrue((0 * pi).isNaN)
XCTAssertTrue((0 * ni).isNaN)
XCTAssertTrue((pn * pi).isNaN)
XCTAssertTrue((pi * pn).isNaN)
XCTAssertTrue((pn * ni).isNaN)
XCTAssertTrue((ni * pn).isNaN)
}
func testRationalConversion() {
var d: Double, r: Ratio
r = Ratio(42)
XCTAssertEqual(r, 42)
XCTAssertEqual(Int(r), 42)
r = Ratio(UInt8.max)
XCTAssertEqual(UInt8(r), .max)
r = Ratio(Int16.min)
XCTAssertEqual(Int16(r), .min)
XCTAssertNil(UInt16(exactly: r))
XCTAssertNil(Ratio(exactly: Int.min))
r = Ratio(Double.pi)
XCTAssertEqual(Double(r), .pi)
XCTAssertTrue(r.isCanonical)
r = Ratio(Double.e)
XCTAssertEqual(Double(r), .e)
XCTAssertTrue(r.isCanonical)
r = Ratio(Double.phi)
XCTAssertEqual(Double(r), .phi)
XCTAssertTrue(r.isCanonical)
r = Ratio(Float.pi)
XCTAssertEqual(Float(r), .pi)
XCTAssertTrue(r.isCanonical)
r = Ratio(Float.e)
XCTAssertEqual(Float(r), .e)
XCTAssertTrue(r.isCanonical)
r = Ratio(Float.phi)
XCTAssertEqual(Float(r), .phi)
XCTAssertTrue(r.isCanonical)
d = 0
XCTAssertEqual(Ratio(d), 0)
XCTAssertEqual(Double(Ratio(d)), 0)
d = 0.ulp
XCTAssertTrue(d.isSubnormal)
XCTAssertEqual(d.significand, 1)
XCTAssertEqual(d.exponent, -1074)
XCTAssertNil(Ratio(exactly: d))
d = .infinity
XCTAssertEqual(Ratio(d), .infinity)
XCTAssertEqual(Double(Ratio(d)), .infinity)
d = -.infinity
XCTAssertEqual(Ratio(d), -.infinity)
XCTAssertEqual(Double(Ratio(d)), -.infinity)
d = .nan
XCTAssertTrue(Ratio(d).isNaN)
XCTAssertTrue(Double(Ratio(d)).isNaN)
}
func testRationalComparison() {
let a = 1 / 2 as Ratio
let b = 1 / 4 as Ratio
XCTAssert(a != b)
XCTAssert(a == a)
XCTAssert(b == b)
XCTAssert(a > b)
XCTAssert(b < a)
XCTAssert(!(a < a))
let c = -1 / 2 as Ratio
let d = -1 / 4 as Ratio
XCTAssert(c != d)
XCTAssert(c == c)
XCTAssert(d == d)
XCTAssert(c < d)
XCTAssert(d > c)
XCTAssert(!(c < c))
let e = Ratio(numerator: 0, denominator: 1)
let f = Ratio(numerator: 0, denominator: 2)
let g = Ratio(numerator: 0, denominator: -1)
let h = Ratio(numerator: 0, denominator: -2)
let i = Ratio(numerator: 0, denominator: 0)
XCTAssert(e.sign == .plus)
XCTAssert(e == f)
XCTAssert(f.sign == .plus)
XCTAssert(e == g)
XCTAssert(g.sign == .plus)
XCTAssert(e == h)
XCTAssert(h.sign == .plus)
XCTAssert(e != i)
XCTAssert(i != i)
let j = Ratio.infinity
let k = Ratio(numerator: 2, denominator: 0)
let l = -Ratio.infinity
let m = Ratio(numerator: -2, denominator: 0)
let n = Ratio.nan
XCTAssert(j == k)
XCTAssert(j >= k)
XCTAssert(!(j < k))
XCTAssert(j != l)
XCTAssert(j > l)
XCTAssert(!(j < l))
XCTAssert(j != m)
XCTAssert(j > m)
XCTAssert(!(j < m))
XCTAssert(l != j)
XCTAssert(l < j)
XCTAssert(!(l > j))
XCTAssert(l != k)
XCTAssert(l < k)
XCTAssert(!(l > k))
XCTAssert(l == m)
XCTAssert(l >= m)
XCTAssert(!(l < m))
XCTAssert(j != n)
XCTAssert(!(j < n))
XCTAssert(!(j <= n))
XCTAssert(!(j > n))
XCTAssert(!(j >= n))
XCTAssert(l != n)
XCTAssert(!(l < n))
XCTAssert(!(l <= n))
XCTAssert(!(l > n))
XCTAssert(!(l >= n))
XCTAssert(n != n)
XCTAssert(!(n < n))
XCTAssert(!(n <= n))
XCTAssert(!(n > n))
XCTAssert(!(n >= n))
}
func testRationalRounding() {
let a = 10 / 3 as Ratio
XCTAssertEqual(a.rounded(), 3)
XCTAssertEqual(a.rounded(.up), 4)
XCTAssertEqual(a.rounded(.down), 3)
XCTAssertEqual(a.rounded(.towardZero), 3)
XCTAssertEqual(a.rounded(.awayFromZero), 4)
XCTAssertEqual(a.rounded(.toNearestOrEven), 3)
XCTAssertEqual(a.rounded(.toNearestOrAwayFromZero), 3)
let b = -a
XCTAssertEqual(b, -10 / 3)
XCTAssertEqual(b.rounded(), -3)
XCTAssertEqual(b.rounded(.up), -3)
XCTAssertEqual(b.rounded(.down), -4)
XCTAssertEqual(b.rounded(.towardZero), -3)
XCTAssertEqual(b.rounded(.awayFromZero), -4)
XCTAssertEqual(b.rounded(.toNearestOrEven), -3)
XCTAssertEqual(b.rounded(.toNearestOrAwayFromZero), -3)
let c = 5 / 2 as Ratio
XCTAssertEqual(c.rounded(), 3)
XCTAssertEqual(c.rounded(.up), 3)
XCTAssertEqual(c.rounded(.down), 2)
XCTAssertEqual(c.rounded(.towardZero), 2)
XCTAssertEqual(c.rounded(.awayFromZero), 3)
XCTAssertEqual(c.rounded(.toNearestOrEven), 2)
XCTAssertEqual(c.rounded(.toNearestOrAwayFromZero), 3)
let d = -c
XCTAssertEqual(d, -5 / 2)
XCTAssertEqual(d.rounded(), -3)
XCTAssertEqual(d.rounded(.up), -2)
XCTAssertEqual(d.rounded(.down), -3)
XCTAssertEqual(d.rounded(.towardZero), -2)
XCTAssertEqual(d.rounded(.awayFromZero), -3)
XCTAssertEqual(d.rounded(.toNearestOrEven), -2)
XCTAssertEqual(d.rounded(.toNearestOrAwayFromZero), -3)
let e = 1 / 9 as Ratio
XCTAssertEqual(e.rounded(), 0)
XCTAssertEqual(e.rounded(.up), 1)
XCTAssertEqual(e.rounded(.down), 0)
XCTAssertEqual(e.rounded(.towardZero), 0)
XCTAssertEqual(e.rounded(.awayFromZero), 1)
XCTAssertEqual(e.rounded(.toNearestOrEven), 0)
XCTAssertEqual(e.rounded(.toNearestOrAwayFromZero), 0)
}
static var allTests = [
("testRational", testRational),
("testRationalConversion", testRationalConversion),
("testRationalComparison", testRationalComparison),
("testRationalRounding", testRationalRounding),
]
}

243
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<!DOCTYPE html>
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<h1>Type Aliases</h1>
<p>The following type aliases are available globally.</p>
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<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
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<li><a href="#/s:12NumericAnnex4Signa">Sign</a></li>
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<a name="/s:12NumericAnnex4Signa"></a>
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<h4>
<code>Sign</code>
</h4>
</div>
<div class="abstract">
<p>The mathematical sign of a number.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">typealias</span> <span class="kt">Sign</span> <span class="o">=</span> <span class="kt">FloatingPointSign</span></code></pre>
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<h4>
<code>RoundingRule</code>
</h4>
</div>
<div class="abstract">
<p>A rule for rounding a number.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">typealias</span> <span class="kt">RoundingRule</span> <span class="o">=</span> <span class="kt">FloatingPointRoundingRule</span></code></pre>
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<h1>Classes</h1>
<p>The following classes are available globally.</p>
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<summary>Topics</summary>
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<li><a href="#/s:12NumericAnnex6RandomC">Random</a></li>
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<a name="/s:12NumericAnnex6RandomC"></a>
<a name="//apple_ref/swift/Class/Random" class="dashAnchor"></a>
<h4>
<code>Random</code>
</h4>
</div>
<div class="abstract">
<p>A pseudo-random number generator (PRNG) that implements
<a href="http://vigna.di.unimi.it/ftp/papers/xorshiftplus.pdf"><code>xorshift128+</code></a>, an efficient linear-feedback shift register.</p>
<div class="aside aside-warning">
<p class="aside-title">Warning</p>
Once seeded from cryptographically secure random bytes, an
instance of <code>Random</code> generates high-quality random numbers but is <em>not</em> a
cryptographically secure PRNG.
</div>
<p>To generate random numbers, create your own instance of <code>Random</code> with an
internal state seeded from cryptographically secure random bytes:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">random</span> <span class="o">=</span> <span class="kt">Random</span><span class="p">()</span><span class="o">!</span>
<span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">()</span> <span class="k">as</span> <span class="kt">Int</span>
<span class="c1">// You can also pass the desired result type as an argument.</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="kt">Int</span><span class="o">.</span><span class="k">self</span><span class="p">)</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between 0 and 42 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">42</span><span class="p">))</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between -42 and 0 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="o">-</span><span class="mi">42</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">0</span><span class="p">))</span>
<span class="p">}</span>
</code></pre>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="Classes/Random/Xoroshiro.html">Random.Xoroshiro</a></code>, <code><a href="Protocols/PRNG.html">PRNG</a></code>
</div>
<a href="Classes/Random.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">final</span> <span class="kd">class</span> <span class="kt">Random</span> <span class="p">:</span> <span class="kt"><a href="Protocols/PRNG.html">PRNG</a></span></code></pre>
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<article>
<section>
<div class="wrapper">
<div class="overview">
<h1>Random</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">final</span> <span class="kd">class</span> <span class="kt">Random</span> <span class="p">:</span> <span class="kt"><a href="../Protocols/PRNG.html">PRNG</a></span></code></pre>
</div>
</div>
<p>A pseudo-random number generator (PRNG) that implements
<a href="http://vigna.di.unimi.it/ftp/papers/xorshiftplus.pdf"><code>xorshift128+</code></a>, an efficient linear-feedback shift register.</p>
<div class="aside aside-warning">
<p class="aside-title">Warning</p>
Once seeded from cryptographically secure random bytes, an
instance of <code>Random</code> generates high-quality random numbers but is <em>not</em> a
cryptographically secure PRNG.
</div>
<p>To generate random numbers, create your own instance of <code>Random</code> with an
internal state seeded from cryptographically secure random bytes:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">random</span> <span class="o">=</span> <span class="kt">Random</span><span class="p">()</span><span class="o">!</span>
<span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">()</span> <span class="k">as</span> <span class="kt">Int</span>
<span class="c1">// You can also pass the desired result type as an argument.</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="kt">Int</span><span class="o">.</span><span class="k">self</span><span class="p">)</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between 0 and 42 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">42</span><span class="p">))</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between -42 and 0 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="o">-</span><span class="mi">42</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">0</span><span class="p">))</span>
<span class="p">}</span>
</code></pre>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Classes/Random/Xoroshiro.html">Random.Xoroshiro</a></code>, <code><a href="../Protocols/PRNG.html">PRNG</a></code>
</div>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex6RandomC5states6UInt64V_AFtv">state</a></li>
<li><a href="#/s:12NumericAnnex6RandomCACs6UInt64V_AEt5state_tcfc">init(state:)</a></li>
<li><a href="#/s:12NumericAnnex6RandomCACSgycfc">init()</a></li>
<li><a href="#/s:s16IteratorProtocolP4next7ElementQzSgyF">next()</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex6RandomC9XoroshiroC">Xoroshiro</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex6RandomC5states6UInt64V_AFtv"></a>
<a name="//apple_ref/swift/Property/state" class="dashAnchor"></a>
<h4>
<code>state</code>
</h4>
</div>
<div class="abstract">
<p>The internal state of the pseudo-random number generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="k">var</span> <span class="nv">state</span><span class="p">:</span> <span class="p">(</span><span class="kt">UInt64</span><span class="p">,</span> <span class="kt">UInt64</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex6RandomCACs6UInt64V_AEt5state_tcfc"></a>
<a name="//apple_ref/swift/Method/init(state:)" class="dashAnchor"></a>
<h4>
<code>init(state:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with the given internal state.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="nv">state</span><span class="p">:</span> <span class="p">(</span><span class="kt">UInt64</span><span class="p">,</span> <span class="kt">UInt64</span><span class="p">))</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>state</code></dt>
<dd>
<p>The value to be used as the generator&rsquo;s internal state.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex6RandomCACSgycfc"></a>
<a name="//apple_ref/swift/Method/init()" class="dashAnchor"></a>
<h4>
<code>init()</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with an internal state seeded
using cryptographically secure random bytes.</p>
<p>If cryptographically secure random bytes are unavailable, the result is
<code>nil</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="n">convenience</span> <span class="nf">init</span><span class="p">?()</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s16IteratorProtocolP4next7ElementQzSgyF"></a>
<a name="//apple_ref/swift/Method/next()" class="dashAnchor"></a>
<h4>
<code>next()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">next</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">UInt64</span><span class="p">?</span></code></pre>
</div>
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</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex6RandomC9XoroshiroC"></a>
<a name="//apple_ref/swift/Class/Xoroshiro" class="dashAnchor"></a>
<h4>
<code>Xoroshiro</code>
</h4>
</div>
<div class="abstract">
<p>A pseudo-random number generator (PRNG) that implements <code>xoroshiro128+</code>, a
successor to <code>xorshift128+</code> devised by S. Vigna and D. Blackman.</p>
<div class="aside aside-warning">
<p class="aside-title">Warning</p>
Once seeded from cryptographically secure random bytes, an
instance of <code>Random.Xoroshiro</code> generates high-quality random numbers but
is <em>not</em> a cryptographically secure PRNG.
</div>
<p>To generate random numbers, create your own instance of <code>Random.Xoroshiro</code>
with an internal state seeded from cryptographically secure random bytes:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">random</span> <span class="o">=</span> <span class="kt">Random</span><span class="o">.</span><span class="kt">Xoroshiro</span><span class="p">()</span><span class="o">!</span>
<span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">()</span> <span class="k">as</span> <span class="kt">Int</span>
<span class="c1">// You can also pass the desired result type as an argument.</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="kt">Int</span><span class="o">.</span><span class="k">self</span><span class="p">)</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between 0 and 42 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">42</span><span class="p">))</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between -42 and 0 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="o">-</span><span class="mi">42</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">0</span><span class="p">))</span>
<span class="p">}</span>
</code></pre>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Classes/Random.html">Random</a></code>, <code><a href="../Protocols/PRNG.html">PRNG</a></code>
</div>
<a href="../Classes/Random/Xoroshiro.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">final</span> <span class="kd">class</span> <span class="kt">Xoroshiro</span> <span class="p">:</span> <span class="kt"><a href="../Protocols/PRNG.html">PRNG</a></span></code></pre>
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<div class="overview">
<h1>Xoroshiro</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">final</span> <span class="kd">class</span> <span class="kt">Xoroshiro</span> <span class="p">:</span> <span class="kt"><a href="../../Protocols/PRNG.html">PRNG</a></span></code></pre>
</div>
</div>
<p>A pseudo-random number generator (PRNG) that implements <code>xoroshiro128+</code>, a
successor to <code>xorshift128+</code> devised by S. Vigna and D. Blackman.</p>
<div class="aside aside-warning">
<p class="aside-title">Warning</p>
Once seeded from cryptographically secure random bytes, an
instance of <code>Random.Xoroshiro</code> generates high-quality random numbers but
is <em>not</em> a cryptographically secure PRNG.
</div>
<p>To generate random numbers, create your own instance of <code>Random.Xoroshiro</code>
with an internal state seeded from cryptographically secure random bytes:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">random</span> <span class="o">=</span> <span class="kt">Random</span><span class="o">.</span><span class="kt">Xoroshiro</span><span class="p">()</span><span class="o">!</span>
<span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">()</span> <span class="k">as</span> <span class="kt">Int</span>
<span class="c1">// You can also pass the desired result type as an argument.</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="kt">Int</span><span class="o">.</span><span class="k">self</span><span class="p">)</span>
<span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between 0 and 42 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">42</span><span class="p">))</span>
<span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
<span class="nf">print</span><span class="p">(</span><span class="s">"Here's a random value between -42 and 0 (inclusive):"</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="nf">uniform</span><span class="p">(</span><span class="nv">a</span><span class="p">:</span> <span class="o">-</span><span class="mi">42</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="mi">0</span><span class="p">))</span>
<span class="p">}</span>
</code></pre>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../../Classes/Random.html">Random</a></code>, <code><a href="../../Protocols/PRNG.html">PRNG</a></code>
</div>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex6RandomC9XoroshiroC5states6UInt64V_AHtv">state</a></li>
<li><a href="#/s:12NumericAnnex6RandomC9XoroshiroCAEs6UInt64V_AGt5state_tcfc">init(state:)</a></li>
<li><a href="#/s:12NumericAnnex6RandomC9XoroshiroCAESgycfc">init()</a></li>
<li><a href="#/s:s16IteratorProtocolP4next7ElementQzSgyF">next()</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex6RandomC9XoroshiroC5states6UInt64V_AHtv"></a>
<a name="//apple_ref/swift/Property/state" class="dashAnchor"></a>
<h4>
<code>state</code>
</h4>
</div>
<div class="abstract">
<p>The internal state of the pseudo-random number generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="k">var</span> <span class="nv">state</span><span class="p">:</span> <span class="p">(</span><span class="kt">UInt64</span><span class="p">,</span> <span class="kt">UInt64</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex6RandomC9XoroshiroCAEs6UInt64V_AGt5state_tcfc"></a>
<a name="//apple_ref/swift/Method/init(state:)" class="dashAnchor"></a>
<h4>
<code>init(state:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with the given internal state.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="nv">state</span><span class="p">:</span> <span class="p">(</span><span class="kt">UInt64</span><span class="p">,</span> <span class="kt">UInt64</span><span class="p">))</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>state</code></dt>
<dd>
<p>The value to be used as the generator&rsquo;s internal state.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex6RandomC9XoroshiroCAESgycfc"></a>
<a name="//apple_ref/swift/Method/init()" class="dashAnchor"></a>
<h4>
<code>init()</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with an internal state seeded
using cryptographically secure random bytes.</p>
<p>If cryptographically secure random bytes are unavailable, the result is
<code>nil</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="n">convenience</span> <span class="nf">init</span><span class="p">?()</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s16IteratorProtocolP4next7ElementQzSgyF"></a>
<a name="//apple_ref/swift/Method/next()" class="dashAnchor"></a>
<h4>
<code>next()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">next</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">UInt64</span><span class="p">?</span></code></pre>
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<article>
<section>
<div class="wrapper">
<div class="overview">
<h1>Extensions</h1>
<p>The following extensions are available globally.</p>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:s13BinaryIntegerP">BinaryInteger</a></li>
<li><a href="#/s:s15UnsignedIntegerP">UnsignedInteger</a></li>
<li><a href="#/s:Si">Int</a></li>
<li><a href="#/s:s4Int8V">Int8</a></li>
<li><a href="#/s:s5Int16V">Int16</a></li>
<li><a href="#/s:s5Int32V">Int32</a></li>
<li><a href="#/s:s5Int64V">Int64</a></li>
<li><a href="#/s:Su">UInt</a></li>
<li><a href="#/s:s5UInt8V">UInt8</a></li>
<li><a href="#/s:s6UInt16V">UInt16</a></li>
<li><a href="#/s:s6UInt32V">UInt32</a></li>
<li><a href="#/s:s6UInt64V">UInt64</a></li>
<li><a href="#/">DoubleWidth</a></li>
</ul>
<ul>
<li><a href="#/s:Sf">Float</a></li>
<li><a href="#/s:Sd">Double</a></li>
</ul>
<ul>
<li><a href="#/s:s13FloatingPointP">FloatingPoint</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP"></a>
<a name="//apple_ref/swift/Extension/BinaryInteger" class="dashAnchor"></a>
<h4>
<code>BinaryInteger</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/BinaryInteger.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">BinaryInteger</span> <span class="p">:</span> <span class="kt">Hashable</span><span class="p">,</span> <span class="kt">Numeric</span><span class="p">,</span> <span class="kt">CustomStringConvertible</span><span class="p">,</span> <span class="kt">Strideable</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s15UnsignedIntegerP"></a>
<a name="//apple_ref/swift/Extension/UnsignedInteger" class="dashAnchor"></a>
<h4>
<code>UnsignedInteger</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/UnsignedInteger.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">UnsignedInteger</span> <span class="p">:</span> <span class="kt">BinaryInteger</span> <span class="k">where</span> <span class="k">Self</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">BinaryInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:Si"></a>
<a name="//apple_ref/swift/Extension/Int" class="dashAnchor"></a>
<h4>
<code>Int</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Int.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s4Int8V"></a>
<a name="//apple_ref/swift/Extension/Int8" class="dashAnchor"></a>
<h4>
<code>Int8</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Int8.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int8</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s5Int16V"></a>
<a name="//apple_ref/swift/Extension/Int16" class="dashAnchor"></a>
<h4>
<code>Int16</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Int16.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int16</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s5Int32V"></a>
<a name="//apple_ref/swift/Extension/Int32" class="dashAnchor"></a>
<h4>
<code>Int32</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Int32.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int32</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s5Int64V"></a>
<a name="//apple_ref/swift/Extension/Int64" class="dashAnchor"></a>
<h4>
<code>Int64</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Int64.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int64</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:Su"></a>
<a name="//apple_ref/swift/Extension/UInt" class="dashAnchor"></a>
<h4>
<code>UInt</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/UInt.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
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</li>
<li>
<div>
<a name="/s:s5UInt8V"></a>
<a name="//apple_ref/swift/Extension/UInt8" class="dashAnchor"></a>
<h4>
<code>UInt8</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/UInt8.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt8</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s6UInt16V"></a>
<a name="//apple_ref/swift/Extension/UInt16" class="dashAnchor"></a>
<h4>
<code>UInt16</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/UInt16.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt16</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s6UInt32V"></a>
<a name="//apple_ref/swift/Extension/UInt32" class="dashAnchor"></a>
<h4>
<code>UInt32</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/UInt32.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt32</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s6UInt64V"></a>
<a name="//apple_ref/swift/Extension/UInt64" class="dashAnchor"></a>
<h4>
<code>UInt64</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/UInt64.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt64</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/"></a>
<a name="//apple_ref/swift/Extension/DoubleWidth" class="dashAnchor"></a>
<h4>
<code>DoubleWidth</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/DoubleWidth.html">See more…</a>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:Sf"></a>
<a name="//apple_ref/swift/Extension/Float" class="dashAnchor"></a>
<h4>
<code>Float</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Float.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:Sd"></a>
<a name="//apple_ref/swift/Extension/Double" class="dashAnchor"></a>
<h4>
<code>Double</code>
</h4>
</div>
<div class="abstract">
<a href="Extensions/Double.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Double</span></code></pre>
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<code>FloatingPoint</code>
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<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">FloatingPoint</span> <span class="p">:</span> <span class="kt">SignedNumeric</span><span class="p">,</span> <span class="kt">Strideable</span><span class="p">,</span> <span class="kt">Hashable</span></code></pre>
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<h1>BinaryInteger</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">BinaryInteger</span> <span class="p">:</span> <span class="kt">Hashable</span><span class="p">,</span> <span class="kt">Numeric</span><span class="p">,</span> <span class="kt">CustomStringConvertible</span><span class="p">,</span> <span class="kt">Strideable</span></code></pre>
</div>
</div>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<p><a href="#/Exponentiation">Exponentiation</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexE2mmoixx_xtFZ">**(_:_:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexE3mmeoiyxz_xtFZ">**=(_:_:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexE3powxx_xtFZ">pow(_:_:)</a></li>
</ul>
<p><a href="#/Square%20Root">Square Root</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexE4sqrtxxFZ">sqrt(_:)</a></li>
</ul>
<p><a href="#/Cube%20Root">Cube Root</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE4cbrtxxFZ">cbrt(_:)</a></li>
</ul>
<p><a href="#/Factoring">Factoring</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE3gcdxx_xtFZ">gcd(_:_:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE3lcmxx_xtFZ">lcm(_:_:)</a></li>
</ul>
<p><a href="#/Factoring%20(Fixed-Width)">Factoring (Fixed-Width)</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE20gcdReportingOverflowx12partialValue_Sb8overflowtx_xtFZ">gcdReportingOverflow(_:_:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE20lcmReportingOverflowx12partialValue_Sb8overflowtx_xtFZ">lcmReportingOverflow(_:_:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE07lcmFullF0x4high_AF3lowtx_xtFZ">lcmFullWidth(_:_:)</a></li>
</ul>
<p><a href="#/Initializers">Initializers</a></p>
<ul>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexExSgAC8RationalVyqd__G7exactly_tcs9DecodableRd__s9EncodableRd__s06SignedB0Rd__s021_ExpressibleByBuiltinB7LiteralRd__s08UnsignedB09Magnitudes0C0PRpd__AnP_ANQYd__AQRSlufc">init(exactly:)</a></li>
<li><a href="#/s:s13BinaryIntegerP12NumericAnnexExAC8RationalVyqd__Gcs9DecodableRd__s9EncodableRd__s06SignedB0Rd__s021_ExpressibleByBuiltinB7LiteralRd__s08UnsignedB09Magnitudes0C0PRpd__AlN_ALQYd__AORSlufc">init(_:)</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<div>
<a name="/Exponentiation"></a>
<a name="//apple_ref/swift/Section/Exponentiation" class="dashAnchor"></a>
<h3>Exponentiation</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexE2mmoixx_xtFZ"></a>
<a name="//apple_ref/swift/Method/**(_:_:)" class="dashAnchor"></a>
<h4>
<code>**(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the result of raising <code>lhs</code> to the power of <code>rhs</code>, rounded to a
representable value.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexE3mmeoiyxz_xtFZ"></a>
<a name="//apple_ref/swift/Method/**=(_:_:)" class="dashAnchor"></a>
<h4>
<code>**=(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Raises <code>lhs</code> to the power of <code>rhs</code> and stores the result in <code>lhs</code>, rounded
to a representable value.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexE3powxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/pow(_:_:)" class="dashAnchor"></a>
<h4>
<code>pow(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the result of raising <code>base</code> to the power of <code>exponent</code>, rounded
to a representable value (deprecated).</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">pow</span><span class="p">(</span><span class="n">_</span> <span class="nv">base</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">exponent</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
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<div>
<a name="/Square%20Root"></a>
<a name="//apple_ref/swift/Section/Square Root" class="dashAnchor"></a>
<h3>Square Root</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexE4sqrtxxFZ"></a>
<a name="//apple_ref/swift/Method/sqrt(_:)" class="dashAnchor"></a>
<h4>
<code>sqrt(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the square root of <code>x</code>, rounding toward zero. If <code>x</code> is negative,
a runtime error may occur.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">sqrt</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<div>
<a name="/Cube%20Root"></a>
<a name="//apple_ref/swift/Section/Cube Root" class="dashAnchor"></a>
<h3>Cube Root</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE4cbrtxxFZ"></a>
<a name="//apple_ref/swift/Method/cbrt(_:)" class="dashAnchor"></a>
<h4>
<code>cbrt(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the cube root of <code>x</code>, rounding toward zero.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">cbrt</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<div>
<a name="/Factoring"></a>
<a name="//apple_ref/swift/Section/Factoring" class="dashAnchor"></a>
<h3>Factoring</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE3gcdxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/gcd(_:_:)" class="dashAnchor"></a>
<h4>
<code>gcd(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the greatest common divisor of <code>a</code> and <code>b</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">gcd</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexsAARzs08UnsignedB09MagnitudeRpzlE3lcmxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/lcm(_:_:)" class="dashAnchor"></a>
<h4>
<code>lcm(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the least common multiple of <code>a</code> and <code>b</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcm</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<div>
<a name="/Factoring%20(Fixed-Width)"></a>
<a name="//apple_ref/swift/Section/Factoring (Fixed-Width)" class="dashAnchor"></a>
<h3>Factoring (Fixed-Width)</h3>
</div>
<ul>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE20gcdReportingOverflowx12partialValue_Sb8overflowtx_xtFZ"></a>
<a name="//apple_ref/swift/Method/gcdReportingOverflow(_:_:)" class="dashAnchor"></a>
<h4>
<code>gcdReportingOverflow(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the greatest common divisor of <code>a</code> and <code>b</code> and a flag to indicate
whether overflow occurred during the operation.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">gcdReportingOverflow</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span>
<span class="o">-&gt;</span> <span class="p">(</span><span class="nv">partialValue</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">overflow</span><span class="p">:</span> <span class="kt">Bool</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE20lcmReportingOverflowx12partialValue_Sb8overflowtx_xtFZ"></a>
<a name="//apple_ref/swift/Method/lcmReportingOverflow(_:_:)" class="dashAnchor"></a>
<h4>
<code>lcmReportingOverflow(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the least common multiple of <code>a</code> and <code>b</code> and a flag to indicate
whether overflow occurred during the operation.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcmReportingOverflow</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span>
<span class="o">-&gt;</span> <span class="p">(</span><span class="nv">partialValue</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">overflow</span><span class="p">:</span> <span class="kt">Bool</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexs010FixedWidthB0RzsAD9MagnitudeRpzs08UnsignedB0AFRQAE_AEs0C0PQZAFRSlE07lcmFullF0x4high_AF3lowtx_xtFZ"></a>
<a name="//apple_ref/swift/Method/lcmFullWidth(_:_:)" class="dashAnchor"></a>
<h4>
<code>lcmFullWidth(_:_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the high and low parts of the least common multiple of <code>a</code> and <code>b</code>
computed using full-width arithmetic.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcmFullWidth</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span>
<span class="o">-&gt;</span> <span class="p">(</span><span class="nv">high</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">low</span><span class="p">:</span> <span class="k">Self</span><span class="o">.</span><span class="kt">Magnitude</span><span class="p">)</span></code></pre>
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<a name="/s:s13BinaryIntegerP12NumericAnnexExSgAC8RationalVyqd__G7exactly_tcs9DecodableRd__s9EncodableRd__s06SignedB0Rd__s021_ExpressibleByBuiltinB7LiteralRd__s08UnsignedB09Magnitudes0C0PRpd__AnP_ANQYd__AQRSlufc"></a>
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<code>init(exactly:)</code>
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<div class="abstract">
<p>Creates a new binary integer from the given rational value, if it can be
represented exactly.</p>
<p>If <code>source</code> is not representable exactly, the result is <code>nil</code>.</p>
</div>
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<h5>Declaration</h5>
<div class="language">
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">init</span><span class="p">?</span><span class="o">&lt;</span><span class="kt">U</span><span class="o">&gt;</span><span class="p">(</span><span class="n">exactly</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">U</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
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<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>A rational value to convert to a binary integer.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:s13BinaryIntegerP12NumericAnnexExAC8RationalVyqd__Gcs9DecodableRd__s9EncodableRd__s06SignedB0Rd__s021_ExpressibleByBuiltinB7LiteralRd__s08UnsignedB09Magnitudes0C0PRpd__AlN_ALQYd__AORSlufc"></a>
<a name="//apple_ref/swift/Method/init(_:)" class="dashAnchor"></a>
<h4>
<code>init(_:)</code>
</h4>
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<div class="abstract">
<p>Creates a new binary integer from the given rational value, rounding
toward zero.</p>
<p>If <code>source</code> is outside the bounds of this type after rounding toward zero,
a runtime error may occur.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">init</span><span class="o">&lt;</span><span class="kt">U</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">U</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>A rational value to convert to a binary integer.</p>
</dd>
</dl>
</div>
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<h1>Double</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<p><a href="#/Real">Real</a></p>
<ul>
<li><a href="#/s:12NumericAnnex4MathP1exvZ">e</a></li>
<li><a href="#/s:12NumericAnnex4MathP3phixvZ">phi</a></li>
<li><a href="#/s:12NumericAnnex4MathP2mmoixx_xtFZ">**(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4MathP3mmeoiyxz_xtFZ">**=(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4MathP18naturalExponentialxyF">naturalExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17binaryExponentialxyF">binaryExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17commonExponentialxyF">commonExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP26naturalExponentialMinusOnexyF">naturalExponentialMinusOne()</a></li>
<li><a href="#/s:12NumericAnnex4MathP16naturalLogarithmxyF">naturalLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP15binaryLogarithmxyF">binaryLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP15commonLogarithmxyF">commonLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP23naturalLogarithmOnePlusxyF">naturalLogarithmOnePlus()</a></li>
<li><a href="#/s:12NumericAnnex4MathP8cubeRootxyF">cubeRoot()</a></li>
<li><a href="#/s:12NumericAnnex4MathP4sinexyF">sine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP6cosinexyF">cosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP7tangentxyF">tangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP11inverseSinexyF">inverseSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP13inverseCosinexyF">inverseCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP14inverseTangentxyF">inverseTangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP14hyperbolicSinexyF">hyperbolicSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP16hyperbolicCosinexyF">hyperbolicCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17hyperbolicTangentxyF">hyperbolicTangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP21inverseHyperbolicSinexyF">inverseHyperbolicSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP23inverseHyperbolicCosinexyF">inverseHyperbolicCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP24inverseHyperbolicTangentxyF">inverseHyperbolicTangent()</a></li>
<li><a href="#/s:12NumericAnnex4RealP5hypotxx_xtFZ">hypot(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF">inverseTangent(dividingBy:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP5errorxyF">error()</a></li>
<li><a href="#/s:12NumericAnnex4RealP18complementaryErrorxyF">complementaryError()</a></li>
<li><a href="#/s:12NumericAnnex4RealP5gammaxyF">gamma()</a></li>
<li><a href="#/s:12NumericAnnex4RealP16logarithmicGammaxyF">logarithmicGamma()</a></li>
</ul>
</div>
</details>
</section>
<section>
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<div>
<a name="/Real"></a>
<a name="//apple_ref/swift/Section/Real" class="dashAnchor"></a>
<h3>Real</h3>
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<ul>
<li>
<div>
<a name="/s:12NumericAnnex4MathP1exvZ"></a>
<a name="//apple_ref/swift/Variable/e" class="dashAnchor"></a>
<h4>
<code>e</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="k">var</span> <span class="nv">e</span><span class="p">:</span> <span class="kt">Double</span></code></pre>
</div>
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<li>
<div>
<a name="/s:12NumericAnnex4MathP3phixvZ"></a>
<a name="//apple_ref/swift/Variable/phi" class="dashAnchor"></a>
<h4>
<code>phi</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="k">var</span> <span class="nv">phi</span><span class="p">:</span> <span class="kt">Double</span></code></pre>
</div>
</div>
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<a name="//apple_ref/swift/Method/**(_:_:)" class="dashAnchor"></a>
<h4>
<code>**(_:_:)</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Double</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Double</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<a name="/s:12NumericAnnex4MathP3mmeoiyxz_xtFZ"></a>
<a name="//apple_ref/swift/Method/**=(_:_:)" class="dashAnchor"></a>
<h4>
<code>**=(_:_:)</code>
</h4>
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</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Double</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Double</span><span class="p">)</span></code></pre>
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<h4>
<code>naturalExponential()</code>
</h4>
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</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<div>
<a name="/s:12NumericAnnex4MathP17binaryExponentialxyF"></a>
<a name="//apple_ref/swift/Method/binaryExponential()" class="dashAnchor"></a>
<h4>
<code>binaryExponential()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">binaryExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP17commonExponentialxyF"></a>
<a name="//apple_ref/swift/Method/commonExponential()" class="dashAnchor"></a>
<h4>
<code>commonExponential()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">commonExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP26naturalExponentialMinusOnexyF"></a>
<a name="//apple_ref/swift/Method/naturalExponentialMinusOne()" class="dashAnchor"></a>
<h4>
<code>naturalExponentialMinusOne()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalExponentialMinusOne</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP16naturalLogarithmxyF"></a>
<a name="//apple_ref/swift/Method/naturalLogarithm()" class="dashAnchor"></a>
<h4>
<code>naturalLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP15binaryLogarithmxyF"></a>
<a name="//apple_ref/swift/Method/binaryLogarithm()" class="dashAnchor"></a>
<h4>
<code>binaryLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">binaryLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP15commonLogarithmxyF"></a>
<a name="//apple_ref/swift/Method/commonLogarithm()" class="dashAnchor"></a>
<h4>
<code>commonLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">commonLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP23naturalLogarithmOnePlusxyF"></a>
<a name="//apple_ref/swift/Method/naturalLogarithmOnePlus()" class="dashAnchor"></a>
<h4>
<code>naturalLogarithmOnePlus()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalLogarithmOnePlus</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP8cubeRootxyF"></a>
<a name="//apple_ref/swift/Method/cubeRoot()" class="dashAnchor"></a>
<h4>
<code>cubeRoot()</code>
</h4>
</div>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">cubeRoot</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">sine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
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<h4>
<code>inverseCosine()</code>
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<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">hyperbolicSine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">hyperbolicCosine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<div class="language">
<p class="aside-title">Swift</p>
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<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">hypot</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt">Double</span><span class="p">,</span> <span class="n">_</span> <span class="nv">y</span><span class="p">:</span> <span class="kt">Double</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<h4>
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<div class="abstract">
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<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseTangent</span><span class="p">(</span><span class="n">dividingBy</span> <span class="nv">other</span><span class="p">:</span> <span class="kt">Double</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<h4>
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<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">error</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<code>complementaryError()</code>
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<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
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<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">gamma</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<div class="declaration">
<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">logarithmicGamma</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Double</span></code></pre>
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<article>
<section>
<div class="wrapper">
<div class="overview">
<h1>Float</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<p><a href="#/Real">Real</a></p>
<ul>
<li><a href="#/s:12NumericAnnex4MathP1exvZ">e</a></li>
<li><a href="#/s:12NumericAnnex4MathP3phixvZ">phi</a></li>
<li><a href="#/s:12NumericAnnex4MathP2mmoixx_xtFZ">**(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4MathP3mmeoiyxz_xtFZ">**=(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4MathP18naturalExponentialxyF">naturalExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17binaryExponentialxyF">binaryExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17commonExponentialxyF">commonExponential()</a></li>
<li><a href="#/s:12NumericAnnex4MathP26naturalExponentialMinusOnexyF">naturalExponentialMinusOne()</a></li>
<li><a href="#/s:12NumericAnnex4MathP16naturalLogarithmxyF">naturalLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP15binaryLogarithmxyF">binaryLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP15commonLogarithmxyF">commonLogarithm()</a></li>
<li><a href="#/s:12NumericAnnex4MathP23naturalLogarithmOnePlusxyF">naturalLogarithmOnePlus()</a></li>
<li><a href="#/s:12NumericAnnex4MathP8cubeRootxyF">cubeRoot()</a></li>
<li><a href="#/s:12NumericAnnex4MathP4sinexyF">sine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP6cosinexyF">cosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP7tangentxyF">tangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP11inverseSinexyF">inverseSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP13inverseCosinexyF">inverseCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP14inverseTangentxyF">inverseTangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP14hyperbolicSinexyF">hyperbolicSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP16hyperbolicCosinexyF">hyperbolicCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP17hyperbolicTangentxyF">hyperbolicTangent()</a></li>
<li><a href="#/s:12NumericAnnex4MathP21inverseHyperbolicSinexyF">inverseHyperbolicSine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP23inverseHyperbolicCosinexyF">inverseHyperbolicCosine()</a></li>
<li><a href="#/s:12NumericAnnex4MathP24inverseHyperbolicTangentxyF">inverseHyperbolicTangent()</a></li>
<li><a href="#/s:12NumericAnnex4RealP5hypotxx_xtFZ">hypot(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF">inverseTangent(dividingBy:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP5errorxyF">error()</a></li>
<li><a href="#/s:12NumericAnnex4RealP18complementaryErrorxyF">complementaryError()</a></li>
<li><a href="#/s:12NumericAnnex4RealP5gammaxyF">gamma()</a></li>
<li><a href="#/s:12NumericAnnex4RealP16logarithmicGammaxyF">logarithmicGamma()</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<div>
<a name="/Real"></a>
<a name="//apple_ref/swift/Section/Real" class="dashAnchor"></a>
<h3>Real</h3>
</div>
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4MathP1exvZ"></a>
<a name="//apple_ref/swift/Variable/e" class="dashAnchor"></a>
<h4>
<code>e</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="k">var</span> <span class="nv">e</span><span class="p">:</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP3phixvZ"></a>
<a name="//apple_ref/swift/Variable/phi" class="dashAnchor"></a>
<h4>
<code>phi</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="k">var</span> <span class="nv">phi</span><span class="p">:</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP2mmoixx_xtFZ"></a>
<a name="//apple_ref/swift/Method/**(_:_:)" class="dashAnchor"></a>
<h4>
<code>**(_:_:)</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Float</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Float</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
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<div>
<a name="/s:12NumericAnnex4MathP3mmeoiyxz_xtFZ"></a>
<a name="//apple_ref/swift/Method/**=(_:_:)" class="dashAnchor"></a>
<h4>
<code>**=(_:_:)</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Float</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Float</span><span class="p">)</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP18naturalExponentialxyF"></a>
<a name="//apple_ref/swift/Method/naturalExponential()" class="dashAnchor"></a>
<h4>
<code>naturalExponential()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP17binaryExponentialxyF"></a>
<a name="//apple_ref/swift/Method/binaryExponential()" class="dashAnchor"></a>
<h4>
<code>binaryExponential()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">binaryExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP17commonExponentialxyF"></a>
<a name="//apple_ref/swift/Method/commonExponential()" class="dashAnchor"></a>
<h4>
<code>commonExponential()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">commonExponential</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP26naturalExponentialMinusOnexyF"></a>
<a name="//apple_ref/swift/Method/naturalExponentialMinusOne()" class="dashAnchor"></a>
<h4>
<code>naturalExponentialMinusOne()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalExponentialMinusOne</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP16naturalLogarithmxyF"></a>
<a name="//apple_ref/swift/Method/naturalLogarithm()" class="dashAnchor"></a>
<h4>
<code>naturalLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP15binaryLogarithmxyF"></a>
<a name="//apple_ref/swift/Method/binaryLogarithm()" class="dashAnchor"></a>
<h4>
<code>binaryLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">binaryLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP15commonLogarithmxyF"></a>
<a name="//apple_ref/swift/Method/commonLogarithm()" class="dashAnchor"></a>
<h4>
<code>commonLogarithm()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">commonLogarithm</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP23naturalLogarithmOnePlusxyF"></a>
<a name="//apple_ref/swift/Method/naturalLogarithmOnePlus()" class="dashAnchor"></a>
<h4>
<code>naturalLogarithmOnePlus()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">naturalLogarithmOnePlus</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP8cubeRootxyF"></a>
<a name="//apple_ref/swift/Method/cubeRoot()" class="dashAnchor"></a>
<h4>
<code>cubeRoot()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">cubeRoot</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP4sinexyF"></a>
<a name="//apple_ref/swift/Method/sine()" class="dashAnchor"></a>
<h4>
<code>sine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">sine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP6cosinexyF"></a>
<a name="//apple_ref/swift/Method/cosine()" class="dashAnchor"></a>
<h4>
<code>cosine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">cosine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP7tangentxyF"></a>
<a name="//apple_ref/swift/Method/tangent()" class="dashAnchor"></a>
<h4>
<code>tangent()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">tangent</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP11inverseSinexyF"></a>
<a name="//apple_ref/swift/Method/inverseSine()" class="dashAnchor"></a>
<h4>
<code>inverseSine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseSine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP13inverseCosinexyF"></a>
<a name="//apple_ref/swift/Method/inverseCosine()" class="dashAnchor"></a>
<h4>
<code>inverseCosine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseCosine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP14inverseTangentxyF"></a>
<a name="//apple_ref/swift/Method/inverseTangent()" class="dashAnchor"></a>
<h4>
<code>inverseTangent()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">inverseTangent</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP14hyperbolicSinexyF"></a>
<a name="//apple_ref/swift/Method/hyperbolicSine()" class="dashAnchor"></a>
<h4>
<code>hyperbolicSine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">hyperbolicSine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP16hyperbolicCosinexyF"></a>
<a name="//apple_ref/swift/Method/hyperbolicCosine()" class="dashAnchor"></a>
<h4>
<code>hyperbolicCosine()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">hyperbolicCosine</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4MathP17hyperbolicTangentxyF"></a>
<a name="//apple_ref/swift/Method/hyperbolicTangent()" class="dashAnchor"></a>
<h4>
<code>hyperbolicTangent()</code>
</h4>
</div>
<div class="abstract">
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">hyperbolicTangent</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
</div>
</div>
</li>
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<div>
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<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
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<h4>
<code>hypot(_:_:)</code>
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</div>
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<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">hypot</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt">Float</span><span class="p">,</span> <span class="n">_</span> <span class="nv">y</span><span class="p">:</span> <span class="kt">Float</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
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<a name="//apple_ref/swift/Method/inverseTangent(dividingBy:)" class="dashAnchor"></a>
<h4>
<code>inverseTangent(dividingBy:)</code>
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<h5>Declaration</h5>
<div class="language">
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<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">error</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
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<a name="//apple_ref/swift/Method/complementaryError()" class="dashAnchor"></a>
<h4>
<code>complementaryError()</code>
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<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">complementaryError</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
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<a name="/s:12NumericAnnex4RealP5gammaxyF"></a>
<a name="//apple_ref/swift/Method/gamma()" class="dashAnchor"></a>
<h4>
<code>gamma()</code>
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<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">gamma</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
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<a name="//apple_ref/swift/Method/logarithmicGamma()" class="dashAnchor"></a>
<h4>
<code>logarithmicGamma()</code>
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<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">logarithmicGamma</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="kt">Float</span></code></pre>
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<h1>FloatingPoint</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">FloatingPoint</span> <span class="p">:</span> <span class="kt">SignedNumeric</span><span class="p">,</span> <span class="kt">Strideable</span><span class="p">,</span> <span class="kt">Hashable</span></code></pre>
</div>
</div>
</div>
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</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<p><a href="#/Initializers">Initializers</a></p>
<ul>
<li><a href="#/s:s13FloatingPointP12NumericAnnexExAC8RationalVySiGcfc">init(_:)</a></li>
<li><a href="#/s:s13FloatingPointP12NumericAnnexExAC8RationalVys4Int8VGcfc">init(_:)</a></li>
<li><a href="#/s:s13FloatingPointP12NumericAnnexExAC8RationalVys5Int16VGcfc">init(_:)</a></li>
<li><a href="#/s:s13FloatingPointP12NumericAnnexExAC8RationalVys5Int32VGcfc">init(_:)</a></li>
<li><a href="#/s:s13FloatingPointP12NumericAnnexExAC8RationalVys5Int64VGcfc">init(_:)</a></li>
</ul>
</div>
</details>
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<div class="group">
<div>
<a name="/Initializers"></a>
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<h3>Initializers</h3>
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<li>
<div>
<a name="/s:s13FloatingPointP12NumericAnnexExAC8RationalVySiGcfc"></a>
<a name="//apple_ref/swift/Method/init(_:)" class="dashAnchor"></a>
<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new floating-point value from the given rational value, after
rounding the whole part, the numerator of the fractional part, and the
denominator of the fractional part each to the closest possible
representation.</p>
<p>If two representable values are equally close, the result of rounding is
the value with more trailing zeros in its significand bit pattern.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>The rational value to convert to a floating-point value.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:s13FloatingPointP12NumericAnnexExAC8RationalVys4Int8VGcfc"></a>
<a name="//apple_ref/swift/Method/init(_:)" class="dashAnchor"></a>
<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new floating-point value from the given rational value, after
rounding the whole part, the numerator of the fractional part, and the
denominator of the fractional part each to the closest possible
representation.</p>
<p>If two representable values are equally close, the result of rounding is
the value with more trailing zeros in its significand bit pattern.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">Int8</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>The rational value to convert to a floating-point value.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:s13FloatingPointP12NumericAnnexExAC8RationalVys5Int16VGcfc"></a>
<a name="//apple_ref/swift/Method/init(_:)" class="dashAnchor"></a>
<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new floating-point value from the given rational value, after
rounding the whole part, the numerator of the fractional part, and the
denominator of the fractional part each to the closest possible
representation.</p>
<p>If two representable values are equally close, the result of rounding is
the value with more trailing zeros in its significand bit pattern.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">Int16</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>The rational value to convert to a floating-point value.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:s13FloatingPointP12NumericAnnexExAC8RationalVys5Int32VGcfc"></a>
<a name="//apple_ref/swift/Method/init(_:)" class="dashAnchor"></a>
<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new floating-point value from the given rational value, after
rounding the whole part, the numerator of the fractional part, and the
denominator of the fractional part each to the closest possible
representation.</p>
<p>If two representable values are equally close, the result of rounding is
the value with more trailing zeros in its significand bit pattern.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">Int32</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>The rational value to convert to a floating-point value.</p>
</dd>
</dl>
</div>
</li>
<li>
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<h4>
<code>init(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a new floating-point value from the given rational value, after
rounding the whole part, the numerator of the fractional part, and the
denominator of the fractional part each to the closest possible
representation.</p>
<p>If two representable values are equally close, the result of rounding is
the value with more trailing zeros in its significand bit pattern.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="nf">init</span><span class="p">(</span><span class="n">_</span> <span class="nv">source</span><span class="p">:</span> <span class="kt"><a href="../Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">Int64</span><span class="o">&gt;</span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>source</code></dt>
<dd>
<p>The rational value to convert to a floating-point value.</p>
</dd>
</dl>
</div>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int16</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
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<h5>Declaration</h5>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Int16</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int16</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Int16</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Int16</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int16</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Int32</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int32</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">Int64</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">SignedInteger</span></code></pre>
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<h5>Declaration</h5>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Int64</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int64</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Int64</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Int64</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int64</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">Int8</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int8</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">Int8</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">Int8</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">Int8</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt16</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
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<h5>Declaration</h5>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">UInt16</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt16</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UInt16</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">UInt16</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt16</span><span class="p">)</span></code></pre>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">UInt32</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt32</span><span class="p">)</span></code></pre>
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<h1>UInt64</h1>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt64</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
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<p>Returns the result of raising <code>lhs</code> to the power of <code>rhs</code>, rounded to a
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<h5>Declaration</h5>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">UInt64</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt64</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UInt64</span></code></pre>
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<p>Raises <code>lhs</code> to the power of <code>rhs</code> and stores the result in <code>lhs</code>, rounded
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">UInt64</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt64</span><span class="p">)</span></code></pre>
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<h1>UInt8</h1>
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<pre class="highlight"><code><span class="kd">struct</span> <span class="kt">UInt8</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span><span class="p">,</span> <span class="kt">UnsignedInteger</span></code></pre>
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<p>Returns the result of raising <code>lhs</code> to the power of <code>rhs</code>, rounded to a
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<h5>Declaration</h5>
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="kt">UInt8</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt8</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UInt8</span></code></pre>
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<p>Raises <code>lhs</code> to the power of <code>rhs</code> and stores the result in <code>lhs</code>, rounded
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<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="o">**=</span> <span class="p">(</span><span class="nv">lhs</span><span class="p">:</span> <span class="k">inout</span> <span class="kt">UInt8</span><span class="p">,</span> <span class="nv">rhs</span><span class="p">:</span> <span class="kt">UInt8</span><span class="p">)</span></code></pre>
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<h1>UnsignedInteger</h1>
<div class="declaration">
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<pre class="highlight"><code><span class="kd">protocol</span> <span class="kt">UnsignedInteger</span> <span class="p">:</span> <span class="kt">BinaryInteger</span> <span class="k">where</span> <span class="k">Self</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">BinaryInteger</span></code></pre>
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<details open>
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<p><a href="#/Cube%20Root">Cube Root</a></p>
<ul>
<li><a href="#/s:s15UnsignedIntegerP12NumericAnnexE4cbrtxxFZ">cbrt(_:)</a></li>
</ul>
<p><a href="#/Factoring">Factoring</a></p>
<ul>
<li><a href="#/s:s15UnsignedIntegerP12NumericAnnexE3gcdxx_xtFZ">gcd(_:_:)</a></li>
<li><a href="#/s:s15UnsignedIntegerP12NumericAnnexE3lcmxx_xtFZ">lcm(_:_:)</a></li>
</ul>
<p><a href="#/Factoring%20(Fixed-Width)">Factoring (Fixed-Width)</a></p>
<ul>
<li><a href="#/s:s15UnsignedIntegerP12NumericAnnexs010FixedWidthB0RzsAARzlE20lcmReportingOverflowx12partialValue_Sb8overflowtx_xtFZ">lcmReportingOverflow(_:_:)</a></li>
<li><a href="#/s:s15UnsignedIntegerP12NumericAnnexs010FixedWidthB0RzsAARzlE07lcmFullF0x4high_9Magnitudes0C0PQz3lowtx_xtFZ">lcmFullWidth(_:_:)</a></li>
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<h4>
<code>cbrt(_:)</code>
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<div class="abstract">
<p>Returns the cube root of <code>x</code>, rounding toward zero.</p>
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<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">cbrt</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
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<h4>
<code>gcd(_:_:)</code>
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<p>Returns the greatest common divisor of <code>a</code> and <code>b</code>.</p>
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<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">gcd</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
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<h4>
<code>lcm(_:_:)</code>
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<div class="abstract">
<p>Returns the least common multiple of <code>a</code> and <code>b</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcm</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
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<code>lcmReportingOverflow(_:_:)</code>
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<p>Returns the least common multiple of <code>a</code> and <code>b</code> and a flag to indicate
whether overflow occurred during the operation.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcmReportingOverflow</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span>
<span class="o">-&gt;</span> <span class="p">(</span><span class="nv">partialValue</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">overflow</span><span class="p">:</span> <span class="kt">Bool</span><span class="p">)</span></code></pre>
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<h4>
<code>lcmFullWidth(_:_:)</code>
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<p>Returns the high and low parts of the least common multiple of <code>a</code> and <code>b</code>
computed using full-width arithmetic.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lcmFullWidth</span><span class="p">(</span><span class="n">_</span> <span class="nv">a</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">b</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span>
<span class="o">-&gt;</span> <span class="p">(</span><span class="nv">high</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="nv">low</span><span class="p">:</span> <span class="k">Self</span><span class="o">.</span><span class="kt">Magnitude</span><span class="p">)</span></code></pre>
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<h1>Functions</h1>
<p>The following functions are available globally.</p>
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<li><a href="#/s:12NumericAnnex3absAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">abs(_:)</a></li>
<li><a href="#/s:12NumericAnnex4sqrtAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">sqrt(_:)</a></li>
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<li><a href="#/s:12NumericAnnex3absAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">abs(_:)</a></li>
<li><a href="#/s:12NumericAnnex4ceilAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">ceil(_:)</a></li>
<li><a href="#/s:12NumericAnnex5floorAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">floor(_:)</a></li>
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<code>abs(_:)</code>
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<p>Returns the absolute value (magnitude, modulus) of <code>z</code>.</p>
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<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">abs</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">z</span><span class="p">:</span> <span class="kt"><a href="Structs/Complex.html">Complex</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Complex.html">Complex</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
</div>
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<div>
<a name="/s:12NumericAnnex4sqrtAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF"></a>
<a name="//apple_ref/swift/Function/sqrt(_:)" class="dashAnchor"></a>
<h4>
<code>sqrt(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the square root of <code>z</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">sqrt</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">z</span><span class="p">:</span> <span class="kt"><a href="Structs/Complex.html">Complex</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Complex.html">Complex</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
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<a name="//apple_ref/swift/Function/abs(_:)" class="dashAnchor"></a>
<h4>
<code>abs(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the absolute value (magnitude) of <code>x</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">abs</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
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<a name="/s:12NumericAnnex4ceilAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF"></a>
<a name="//apple_ref/swift/Function/ceil(_:)" class="dashAnchor"></a>
<h4>
<code>ceil(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the closest integral value greater than or equal to <code>x</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">ceil</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
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<a name="/s:12NumericAnnex5floorAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF"></a>
<a name="//apple_ref/swift/Function/floor(_:)" class="dashAnchor"></a>
<h4>
<code>floor(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the closest integral value less than or equal to <code>x</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">floor</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
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<div>
<a name="/s:12NumericAnnex5roundAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF"></a>
<a name="//apple_ref/swift/Function/round(_:)" class="dashAnchor"></a>
<h4>
<code>round(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the closest integral value; if two values are equally close, returns
the one with greater magnitude.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">round</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
</div>
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<a name="/s:12NumericAnnex5truncAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF"></a>
<a name="//apple_ref/swift/Function/trunc(_:)" class="dashAnchor"></a>
<h4>
<code>trunc(_:)</code>
</h4>
</div>
<div class="abstract">
<p>Returns the closest integral value with magnitude less than or equal to that
of <code>x</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">trunc</span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt"><a href="Structs/Rational.html">Rational</a></span><span class="o">&lt;</span><span class="kt">T</span><span class="o">&gt;</span></code></pre>
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<div class="overview">
<h1>Protocols</h1>
<p>The following protocols are available globally.</p>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex4PRNGP">PRNG</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4RealP">Real</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4MathP">Math</a></li>
</ul>
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<li>
<div>
<a name="/s:12NumericAnnex4PRNGP"></a>
<a name="//apple_ref/swift/Protocol/PRNG" class="dashAnchor"></a>
<h4>
<code>PRNG</code>
</h4>
</div>
<div class="abstract">
<p>A pseudo-random number generator (PRNG).</p>
<p>Reference types that conform to <code>PRNG</code> are infinite sequences of
pseudo-random elements. Protocol extension methods iterate over such a
sequence as necessary to generate pseudo-random values from the desired
distribution.</p>
<h2 id='considerations-for-conforming-types' class='heading'>Considerations for Conforming Types</h2>
<p>For clarity to end users, custom PRNGs may be implemented in an extension to
<code><a href="Classes/Random.html">Random</a></code>. For instance, the <code>xoroshiro128+</code> algorithm is implemented in a
final class named <code><a href="Classes/Random/Xoroshiro.html">Random.Xoroshiro</a></code>.</p>
<p>The static methods <code>_entropy(_:)</code> and <code>_entropy(_:count:)</code> return
cryptographically secure random bytes that may be useful for seeding your
custom PRNG. However, these methods may return <code>nil</code> if the requested number
of random bytes is not available, and they are not recommended as a routine
source of random data.</p>
<h2 id='adding-other-probability-distributions' class='heading'>Adding Other Probability Distributions</h2>
<p>Many built-in protocol extension methods make use of the primitive,
overloaded method <code>_random(_:bitCount:)</code>. You may wish to use the same
method in new protocol extension methods that return pseudo-random
values from other probability distributions.</p>
<p>The method <code>_random(_:bitCount:)</code> generates uniformly distributed binary
floating-point values in the half-open range [0, 1) with a precision of
either <code>bitCount</code> or the significand bit count of the floating-point type,
whichever is less. Additionally, this method generates uniformly distributed
unsigned integers in the half-open range [0, 2 ** <em>x</em>), where ** is the
exponentiation operator and <em>x</em> is the lesser of <code>bitCount</code> and the bit
width of the integer type.</p>
<p>For end users, however, the recommended spelling for a uniformly distributed
numeric value is <code>uniform()</code>; that method is overloaded to permit custom
minimum and maximum values for the uniform distribution.</p>
<a href="Protocols/PRNG.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">protocol</span> <span class="kt">PRNG</span> <span class="p">:</span> <span class="kd">class</span><span class="p">,</span> <span class="kt">IteratorProtocol</span><span class="p">,</span> <span class="kt">Sequence</span>
<span class="k">where</span> <span class="kt">Element</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="p">,</span>
<span class="kt">SubSequence</span> <span class="p">:</span> <span class="kt">Sequence</span><span class="p">,</span>
<span class="kt">Element</span> <span class="o">==</span> <span class="kt">SubSequence</span><span class="o">.</span><span class="kt">Element</span></code></pre>
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<div>
<a name="/s:12NumericAnnex4RealP"></a>
<a name="//apple_ref/swift/Protocol/Real" class="dashAnchor"></a>
<h4>
<code>Real</code>
</h4>
</div>
<div class="abstract">
<p>A floating-point type that supports elementary functions and a selection of
special functions.</p>
<p>The <code>Real</code> protocol provides a suitable basis for writing functions that
work on any floating-point type that supports the required functions.</p>
<a href="Protocols/Real.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">protocol</span> <span class="kt">Real</span> <span class="p">:</span> <span class="kt"><a href="Protocols/Math.html">Math</a></span><span class="p">,</span> <span class="kt">FloatingPoint</span></code></pre>
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<a name="/s:12NumericAnnex4MathP"></a>
<a name="//apple_ref/swift/Protocol/Math" class="dashAnchor"></a>
<h4>
<code>Math</code>
</h4>
</div>
<div class="abstract">
<p>A signed numeric type that supports elementary functions.</p>
<p>The <code>Math</code> protocol provides a suitable basis for writing functions that
work on any real or complex floating-point type that supports the required
functions.</p>
<a href="Protocols/Math.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">protocol</span> <span class="kt">Math</span> <span class="p">:</span> <span class="kt">SignedNumeric</span></code></pre>
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</li>
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<a href="../Extensions/Int16.html">Int16</a>
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<a href="../Extensions/Int32.html">Int32</a>
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</li>
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<a href="../Extensions/UInt.html">UInt</a>
</li>
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<a href="../Extensions/UInt16.html">UInt16</a>
</li>
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<a href="../Extensions/UInt32.html">UInt32</a>
</li>
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<a href="../Extensions/UInt64.html">UInt64</a>
</li>
<li>
<a href="../Extensions/UInt8.html">UInt8</a>
</li>
<li>
<a href="../Extensions/UnsignedInteger.html">UnsignedInteger</a>
</li>
</ul>
</li>
<li>
<p><a href="../Functions.html">Functions</a></p>
<ul>
<li>
<a href="../Functions.html#/s:12NumericAnnex3absAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">abs(_:)</a>
</li>
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<a href="../Functions.html#/s:12NumericAnnex3absAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">abs(_:)</a>
</li>
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<a href="../Functions.html#/s:12NumericAnnex4ceilAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">ceil(_:)</a>
</li>
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<a href="../Functions.html#/s:12NumericAnnex5floorAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">floor(_:)</a>
</li>
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<a href="../Functions.html#/s:12NumericAnnex5roundAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">round(_:)</a>
</li>
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<a href="../Functions.html#/s:12NumericAnnex4sqrtAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">sqrt(_:)</a>
</li>
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<a href="../Functions.html#/s:12NumericAnnex5truncAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">trunc(_:)</a>
</li>
</ul>
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<li>
<p><a href="../Protocols.html">Protocols</a></p>
<ul>
<li>
<a href="../Protocols/Math.html">Math</a>
</li>
<li>
<a href="../Protocols/PRNG.html">PRNG</a>
</li>
<li>
<a href="../Protocols/Real.html">Real</a>
</li>
</ul>
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<p><a href="../Structs.html">Structures</a></p>
<ul>
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<a href="../Structs/Complex.html">Complex</a>
</li>
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<a href="../Structs/Rational.html">Rational</a>
</li>
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<p><a href="../Typealiases.html">Type Aliases</a></p>
<ul>
<li>
<a href="../Typealiases.html#/s:12NumericAnnex12RoundingRulea">RoundingRule</a>
</li>
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<a href="../Typealiases.html#/s:12NumericAnnex4Signa">Sign</a>
</li>
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</li>
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</nav>
<article>
<section>
<div class="wrapper">
<div class="overview">
<h1>PRNG</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">protocol</span> <span class="kt">PRNG</span> <span class="p">:</span> <span class="kd">class</span><span class="p">,</span> <span class="kt">IteratorProtocol</span><span class="p">,</span> <span class="kt">Sequence</span>
<span class="k">where</span> <span class="kt">Element</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="p">,</span>
<span class="kt">SubSequence</span> <span class="p">:</span> <span class="kt">Sequence</span><span class="p">,</span>
<span class="kt">Element</span> <span class="o">==</span> <span class="kt">SubSequence</span><span class="o">.</span><span class="kt">Element</span></code></pre>
</div>
</div>
<p>A pseudo-random number generator (PRNG).</p>
<p>Reference types that conform to <code>PRNG</code> are infinite sequences of
pseudo-random elements. Protocol extension methods iterate over such a
sequence as necessary to generate pseudo-random values from the desired
distribution.</p>
<h2 id='considerations-for-conforming-types' class='heading'>Considerations for Conforming Types</h2>
<p>For clarity to end users, custom PRNGs may be implemented in an extension to
<code><a href="../Classes/Random.html">Random</a></code>. For instance, the <code>xoroshiro128+</code> algorithm is implemented in a
final class named <code><a href="../Classes/Random/Xoroshiro.html">Random.Xoroshiro</a></code>.</p>
<p>The static methods <code>_entropy(_:)</code> and <code>_entropy(_:count:)</code> return
cryptographically secure random bytes that may be useful for seeding your
custom PRNG. However, these methods may return <code>nil</code> if the requested number
of random bytes is not available, and they are not recommended as a routine
source of random data.</p>
<h2 id='adding-other-probability-distributions' class='heading'>Adding Other Probability Distributions</h2>
<p>Many built-in protocol extension methods make use of the primitive,
overloaded method <code>_random(_:bitCount:)</code>. You may wish to use the same
method in new protocol extension methods that return pseudo-random
values from other probability distributions.</p>
<p>The method <code>_random(_:bitCount:)</code> generates uniformly distributed binary
floating-point values in the half-open range [0, 1) with a precision of
either <code>bitCount</code> or the significand bit count of the floating-point type,
whichever is less. Additionally, this method generates uniformly distributed
unsigned integers in the half-open range [0, 2 ** <em>x</em>), where ** is the
exponentiation operator and <em>x</em> is the lesser of <code>bitCount</code> and the bit
width of the integer type.</p>
<p>For end users, however, the recommended spelling for a uniformly distributed
numeric value is <code>uniform()</code>; that method is overloaded to permit custom
minimum and maximum values for the uniform distribution.</p>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex4PRNGP5State">State</a></li>
<li><a href="#/s:12NumericAnnex4PRNGP5state5StateQzv">state</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPx5StateQz5state_tcfc">init(state:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPxSgycfc">init()</a></li>
<li><a href="#/s:12NumericAnnex4PRNGP3max7ElementQzvZ">max</a></li>
<li><a href="#/s:12NumericAnnex4PRNGP3min7ElementQzvZ">min</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4PRNGPAAE15_randomBitWidthSivZ">_randomBitWidth</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE8_entropyqd__Sgqd__ms17FixedWidthIntegerRd__s08UnsignedG0Rd__lFZ">_entropy(_:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE8_entropySayqd__GSgqd__m_Si5countts17FixedWidthIntegerRd__s08UnsignedH0Rd__lFZ">_entropy(_:count:)</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7_randomqd__qd__m_Si8bitCountts17FixedWidthIntegerRd__s08UnsignedI0Rd__lF">_random(_:bitCount:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__m_qd__1aqd__1bts17FixedWidthIntegerRd__s08UnsignedG0Rd__lF">uniform(_:a:b:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__ms17FixedWidthIntegerRd__s08UnsignedG0Rd__lF">uniform(_:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts17FixedWidthIntegerRd__s08UnsignedJ0Rd__lF">uniform(_:a:b:count:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts17FixedWidthIntegerRd__s08UnsignedJ0Rd__lF">uniform(_:count:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__m_qd__1aqd__1bts17FixedWidthIntegerRd__s06SignedG0Rd__sAG9Magnitudes0A0PRpd__s08UnsignedG0ALRQlF">uniform(_:a:b:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__ms17FixedWidthIntegerRd__s06SignedG0Rd__sAE9Magnitudes0A0PRpd__s08UnsignedG0AJRQlF">uniform(_:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts17FixedWidthIntegerRd__s06SignedJ0Rd__sAK9Magnitudes0A0PRpd__s08UnsignedJ0APRQlF">uniform(_:a:b:count:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts17FixedWidthIntegerRd__s06SignedJ0Rd__sAI9Magnitudes0A0PRpd__s08UnsignedJ0ANRQlF">uniform(_:count:)</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7_randomqd__qd__m_Si8bitCountts19BinaryFloatingPointRd__lF">_random(_:bitCount:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniformqd__qd__m_qd__1aqd__1bts19BinaryFloatingPointRd__lF">uniform(_:a:b:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniformqd__qd__ms19BinaryFloatingPointRd__lF">uniform(_:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts19BinaryFloatingPointRd__lF">uniform(_:a:b:count:)</a></li>
<li><a href="#/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts19BinaryFloatingPointRd__lF">uniform(_:count:)</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGP5State"></a>
<a name="//apple_ref/swift/Alias/State" class="dashAnchor"></a>
<h4>
<code>State</code>
</h4>
</div>
<div class="abstract">
<p>A type that can represent the internal state of the pseudo-random number
generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="k">associatedtype</span> <span class="kt">State</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGP5state5StateQzv"></a>
<a name="//apple_ref/swift/Property/state" class="dashAnchor"></a>
<h4>
<code>state</code>
</h4>
</div>
<div class="abstract">
<p>The internal state of the pseudo-random number generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="k">var</span> <span class="nv">state</span><span class="p">:</span> <span class="kt"><a href="../Protocols/PRNG.html#/s:12NumericAnnex4PRNGP5State">State</a></span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPx5StateQz5state_tcfc"></a>
<a name="//apple_ref/swift/Method/init(state:)" class="dashAnchor"></a>
<h4>
<code>init(state:)</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with the given internal state.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="nf">init</span><span class="p">(</span><span class="nv">state</span><span class="p">:</span> <span class="kt"><a href="../Protocols/PRNG.html#/s:12NumericAnnex4PRNGP5State">State</a></span><span class="p">)</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>state</code></dt>
<dd>
<p>The value to be used as the generator&rsquo;s internal state.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPxSgycfc"></a>
<a name="//apple_ref/swift/Method/init()" class="dashAnchor"></a>
<h4>
<code>init()</code>
</h4>
</div>
<div class="abstract">
<p>Creates a pseudo-random number generator with an internal state seeded
using cryptographically secure random bytes.</p>
<p>If cryptographically secure random bytes are unavailable, the result is
<code>nil</code>.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="nf">init</span><span class="p">?()</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGP3max7ElementQzvZ"></a>
<a name="//apple_ref/swift/Variable/max" class="dashAnchor"></a>
<h4>
<code>max</code>
<span class="declaration-note">Default implementation</span>
</h4>
</div>
<div class="abstract">
<p>The maximum value that may be generated by the pseudo-random number
generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">static</span> <span class="k">var</span> <span class="nv">max</span><span class="p">:</span> <span class="kt">Element</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGP3min7ElementQzvZ"></a>
<a name="//apple_ref/swift/Variable/min" class="dashAnchor"></a>
<h4>
<code>min</code>
<span class="declaration-note">Default implementation</span>
</h4>
</div>
<div class="abstract">
<p>The minimum value that may be generated by the pseudo-random number
generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">static</span> <span class="k">var</span> <span class="nv">min</span><span class="p">:</span> <span class="kt">Element</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE15_randomBitWidthSivZ"></a>
<a name="//apple_ref/swift/Variable/_randomBitWidth" class="dashAnchor"></a>
<h4>
<code>_randomBitWidth</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>The number of pseudo-random bits available from a value generated by the
pseudo-random number generator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="k">var</span> <span class="nv">_randomBitWidth</span><span class="p">:</span> <span class="kt">Int</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE8_entropyqd__Sgqd__ms17FixedWidthIntegerRd__s08UnsignedG0Rd__lFZ"></a>
<a name="//apple_ref/swift/Method/_entropy(_:)" class="dashAnchor"></a>
<h4>
<code>_entropy(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns a value filled with data from a source of cryptographically secure
random bytes, or <code>nil</code> if a sufficient number of cryptographically secure
random bytes is unavailable.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="n">_entropy</span><span class="o">&lt;</span>
<span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span>
<span class="o">&gt;</span><span class="p">(</span><span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span><span class="p">?</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE8_entropySayqd__GSgqd__m_Si5countts17FixedWidthIntegerRd__s08UnsignedH0Rd__lFZ"></a>
<a name="//apple_ref/swift/Method/_entropy(_:count:)" class="dashAnchor"></a>
<h4>
<code>_entropy(_:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns an array of <code>count</code> values filled with data from a source of
cryptographically secure random bytes, or <code>nil</code> if a sufficient number of
cryptographically secure random bytes is unavailable.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="n">_entropy</span><span class="o">&lt;</span>
<span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span>
<span class="o">&gt;</span><span class="p">(</span><span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="p">[</span><span class="kt">T</span><span class="p">]?</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7_randomqd__qd__m_Si8bitCountts17FixedWidthIntegerRd__s08UnsignedI0Rd__lF"></a>
<a name="//apple_ref/swift/Method/_random(_:bitCount:)" class="dashAnchor"></a>
<h4>
<code>_random(_:bitCount:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random unsigned integer of type <code>T</code> in the range from 0
to <code>2 ** min(bitCount, T.bitWidth)</code> (exclusive), where <code>**</code> is the
exponentiation operator.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">_random</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">bitCount</span><span class="p">:</span> <span class="kt">Int</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="n">bitWidth</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__m_qd__1aqd__1bts17FixedWidthIntegerRd__s08UnsignedG0Rd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random unsigned integer of type <code>T</code> in the range from
<code>a</code> through <code>b</code> (inclusive) from the discrete uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__ms17FixedWidthIntegerRd__s08UnsignedG0Rd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random unsigned integer of type <code>T</code> in the range from
<code>T.min</code> through <code>T.max</code> (inclusive) from the discrete uniform
distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts17FixedWidthIntegerRd__s08UnsignedJ0Rd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random unsigned integers of type
<code>T</code> in the range from <code>a</code> through <code>b</code> (inclusive) from the discrete
uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts17FixedWidthIntegerRd__s08UnsignedJ0Rd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random unsigned integers of type
<code>T</code> in the range from <code>T.min</code> through <code>T.max</code> (inclusive) from the
discrete uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__m_qd__1aqd__1bts17FixedWidthIntegerRd__s06SignedG0Rd__sAG9Magnitudes0A0PRpd__s08UnsignedG0ALRQlF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random signed integer of type <code>T</code> in the range from <code>a</code>
through <code>b</code> (inclusive) from the discrete uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">SignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span> <span class="k">where</span> <span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniformqd__qd__ms17FixedWidthIntegerRd__s06SignedG0Rd__sAE9Magnitudes0A0PRpd__s08UnsignedG0AJRQlF"></a>
<a name="//apple_ref/swift/Method/uniform(_:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random signed integer of type <code>T</code> in the range from
<code>T.min</code> through <code>T.max</code> (inclusive) from the discrete uniform
distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">SignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span> <span class="k">where</span> <span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts17FixedWidthIntegerRd__s06SignedJ0Rd__sAK9Magnitudes0A0PRpd__s08UnsignedJ0APRQlF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random signed integers of type <code>T</code>
in the range from <code>a</code> through <code>b</code> (inclusive) from the discrete uniform
distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">SignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span>
<span class="k">where</span> <span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPAAE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts17FixedWidthIntegerRd__s06SignedJ0Rd__sAI9Magnitudes0A0PRpd__s08UnsignedJ0ANRQlF"></a>
<a name="//apple_ref/swift/Method/uniform(_:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random signed integers of type <code>T</code>
in the range from <code>T.min</code> through <code>T.max</code> (inclusive) from the discrete
uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">SignedInteger</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span>
<span class="k">where</span> <span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">FixedWidthInteger</span> <span class="o">&amp;</span> <span class="kt">UnsignedInteger</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7_randomqd__qd__m_Si8bitCountts19BinaryFloatingPointRd__lF"></a>
<a name="//apple_ref/swift/Method/_random(_:bitCount:)" class="dashAnchor"></a>
<h4>
<code>_random(_:bitCount:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random binary floating-point value of type <code>T</code> in the
range from 0 to 1 (exclusive) with <code>min(bitCount, T.significandBitCount)</code>
bits of precision.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">_random</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">BinaryFloatingPoint</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">bitCount</span><span class="p">:</span> <span class="kt">Int</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="n">significandBitCount</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniformqd__qd__m_qd__1aqd__1bts19BinaryFloatingPointRd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random binary floating-point value of type <code>T</code> in the
range from <code>a</code> to <code>b</code> (exclusive) from the uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">BinaryFloatingPoint</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniformqd__qd__ms19BinaryFloatingPointRd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a pseudo-random binary floating-point value of type <code>T</code> in the
range from 0 to 1 (exclusive) from the uniform distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">BinaryFloatingPoint</span><span class="o">&gt;</span><span class="p">(</span><span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">T</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniforms14UnfoldSequenceVyqd__SiGqd__m_qd__1aqd__1bSi5countts19BinaryFloatingPointRd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:a:b:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:a:b:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random binary floating-point values
of type <code>T</code> in the range from <code>a</code> to <code>b</code> (exclusive) from the uniform
distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">BinaryFloatingPoint</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">a</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">b</span><span class="p">:</span> <span class="kt">T</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4PRNGPA2aBRzs6UInt64V7ElementRtzlE7uniforms14UnfoldSequenceVyqd__SiGqd__m_Si5countts19BinaryFloatingPointRd__lF"></a>
<a name="//apple_ref/swift/Method/uniform(_:count:)" class="dashAnchor"></a>
<h4>
<code>uniform(_:count:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Generates a sequence of <code>count</code> pseudo-random binary floating-point values
of type <code>T</code> in the range from 0 to 1 (exclusive) from the uniform
distribution.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="n">uniform</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">BinaryFloatingPoint</span><span class="o">&gt;</span><span class="p">(</span>
<span class="nv">_</span><span class="p">:</span> <span class="kt">T</span><span class="o">.</span><span class="k">Type</span> <span class="o">=</span> <span class="kt">T</span><span class="o">.</span><span class="k">self</span><span class="p">,</span> <span class="nv">count</span><span class="p">:</span> <span class="kt">Int</span>
<span class="p">)</span> <span class="o">-&gt;</span> <span class="kt">UnfoldSequence</span><span class="o">&lt;</span><span class="kt">T</span><span class="p">,</span> <span class="kt">Int</span><span class="o">&gt;</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
</div>
</section>
</article>
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<p>&copy; 2017 <a class="link" href="https://diploid.ca" target="_blank" rel="external">Xiaodi Wu</a>. All rights reserved. (Last updated: 2017-09-30)</p>
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<li>
<a href="../Classes/Random/Xoroshiro.html">– Xoroshiro</a>
</li>
</ul>
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<li>
<p><a href="../Extensions.html">Extensions</a></p>
<ul>
<li>
<a href="../Extensions/BinaryInteger.html">BinaryInteger</a>
</li>
<li>
<a href="../Extensions/Double.html">Double</a>
</li>
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<a href="../Extensions/DoubleWidth.html">DoubleWidth</a>
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<a href="../Extensions/Float.html">Float</a>
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<a href="../Extensions/FloatingPoint.html">FloatingPoint</a>
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<a href="../Extensions/Int.html">Int</a>
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<a href="../Extensions/Int16.html">Int16</a>
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<a href="../Extensions/Int32.html">Int32</a>
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<a href="../Extensions/Int64.html">Int64</a>
</li>
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<a href="../Extensions/Int8.html">Int8</a>
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<a href="../Extensions/UInt.html">UInt</a>
</li>
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<a href="../Extensions/UInt16.html">UInt16</a>
</li>
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<a href="../Extensions/UInt32.html">UInt32</a>
</li>
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<a href="../Extensions/UInt64.html">UInt64</a>
</li>
<li>
<a href="../Extensions/UInt8.html">UInt8</a>
</li>
<li>
<a href="../Extensions/UnsignedInteger.html">UnsignedInteger</a>
</li>
</ul>
</li>
<li>
<p><a href="../Functions.html">Functions</a></p>
<ul>
<li>
<a href="../Functions.html#/s:12NumericAnnex3absAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">abs(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex3absAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">abs(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex4ceilAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">ceil(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex5floorAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">floor(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex5roundAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">round(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex4sqrtAA7ComplexVyxGAeA4RealRzs9DecodableRzs9EncodableRzs33_ExpressibleByBuiltinFloatLiteralRzlF">sqrt(_:)</a>
</li>
<li>
<a href="../Functions.html#/s:12NumericAnnex5truncAA8RationalVyxGAEs9DecodableRzs9EncodableRzs13SignedIntegerRzs021_ExpressibleByBuiltinH7LiteralRzs08UnsignedH09Magnitudes0A0PRpzAkM_AKQZANRSlF">trunc(_:)</a>
</li>
</ul>
</li>
<li>
<p><a href="../Protocols.html">Protocols</a></p>
<ul>
<li>
<a href="../Protocols/Math.html">Math</a>
</li>
<li>
<a href="../Protocols/PRNG.html">PRNG</a>
</li>
<li>
<a href="../Protocols/Real.html">Real</a>
</li>
</ul>
</li>
<li>
<p><a href="../Structs.html">Structures</a></p>
<ul>
<li>
<a href="../Structs/Complex.html">Complex</a>
</li>
<li>
<a href="../Structs/Rational.html">Rational</a>
</li>
</ul>
</li>
<li>
<p><a href="../Typealiases.html">Type Aliases</a></p>
<ul>
<li>
<a href="../Typealiases.html#/s:12NumericAnnex12RoundingRulea">RoundingRule</a>
</li>
<li>
<a href="../Typealiases.html#/s:12NumericAnnex4Signa">Sign</a>
</li>
</ul>
</li>
</ul>
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<article>
<section>
<div class="wrapper">
<div class="overview">
<h1>Real</h1>
<div class="declaration">
<div class="language">
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">protocol</span> <span class="kt">Real</span> <span class="p">:</span> <span class="kt"><a href="../Protocols/Math.html">Math</a></span><span class="p">,</span> <span class="kt">FloatingPoint</span></code></pre>
</div>
</div>
<p>A floating-point type that supports elementary functions and a selection of
special functions.</p>
<p>The <code>Real</code> protocol provides a suitable basis for writing functions that
work on any floating-point type that supports the required functions.</p>
</div>
</div>
</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex4RealP5hypotxx_xtFZ">hypot(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF">inverseTangent(dividingBy:)</a></li>
<li><a href="#/s:12NumericAnnex4RealP5errorxyF">error()</a></li>
<li><a href="#/s:12NumericAnnex4RealP18complementaryErrorxyF">complementaryError()</a></li>
<li><a href="#/s:12NumericAnnex4RealP5gammaxyF">gamma()</a></li>
<li><a href="#/s:12NumericAnnex4RealP16logarithmicGammaxyF">logarithmicGamma()</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4RealPAAE8cubeRootxyF">cubeRoot()</a></li>
</ul>
<ul>
<li><a href="#/s:12NumericAnnex4RealPAAE5atan2xx_xtFZ">atan2(_:_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealPAAE3erfxxFZ">erf(_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealPAAE4erfcxxFZ">erfc(_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealPAAE6tgammaxxFZ">tgamma(_:)</a></li>
<li><a href="#/s:12NumericAnnex4RealPAAE6lgammaxxFZ">lgamma(_:)</a></li>
</ul>
</div>
</details>
</section>
<section>
<div class="wrapper">
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4RealP5hypotxx_xtFZ"></a>
<a name="//apple_ref/swift/Method/hypot(_:_:)" class="dashAnchor"></a>
<h4>
<code>hypot(_:_:)</code>
<span class="declaration-note">Default implementation</span>
</h4>
</div>
<div class="abstract">
<p>Returns the hypotenuse of a right-angle triangle with legs (catheti) of
length <code>x</code> and <code>y</code>, preventing avoidable arithmetic overflow and
underflow. The return value is the square root of the sum of squares of
the arguments.</p>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">static</span> <span class="kd">func</span> <span class="nf">hypot</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">y</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>x</code></dt>
<dd>
<p>The length of one leg (cathetus) of a right-angle triangle.</p>
</dd>
<dt><code>y</code></dt>
<dd>
<p>The length of the other leg (cathetus) of a right-angle triangle.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF"></a>
<a name="//apple_ref/swift/Method/inverseTangent(dividingBy:)" class="dashAnchor"></a>
<h4>
<code>inverseTangent(dividingBy:)</code>
<span class="declaration-note">Default implementation</span>
</h4>
</div>
<div class="abstract">
<p>Returns the inverse tangent of <code>self / other</code>, using the signs of <code>self</code>
and <code>other</code> to determine the quadrant of the computed angle.</p>
<p>If <code>self == 0 &amp;&amp; other == 0</code>, the return value is still finite.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<p><code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealPAAE5atan2xx_xtFZ">atan2(_:_:)</a></code></p>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">func</span> <span class="nf">inverseTangent</span><span class="p">(</span><span class="n">dividingBy</span> <span class="nv">other</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>other</code></dt>
<dd>
<p>The divisor by which to divide <code>self</code>.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP5errorxyF"></a>
<a name="//apple_ref/swift/Method/error()" class="dashAnchor"></a>
<h4>
<code>error()</code>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/Erf.html">error function</a> of <code>self</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealPAAE3erfxxFZ">erf(_:)</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">func</span> <span class="nf">error</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP18complementaryErrorxyF"></a>
<a name="//apple_ref/swift/Method/complementaryError()" class="dashAnchor"></a>
<h4>
<code>complementaryError()</code>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/Erfc.html">complementary error function</a> of <code>self</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealPAAE4erfcxxFZ">erfc(_:)</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">func</span> <span class="nf">complementaryError</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP5gammaxyF"></a>
<a name="//apple_ref/swift/Method/gamma()" class="dashAnchor"></a>
<h4>
<code>gamma()</code>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/GammaFunction.html">gamma function</a> of <code>self</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealPAAE6tgammaxxFZ">tgamma(_:)</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">func</span> <span class="nf">gamma</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealP16logarithmicGammaxyF"></a>
<a name="//apple_ref/swift/Method/logarithmicGamma()" class="dashAnchor"></a>
<h4>
<code>logarithmicGamma()</code>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/LogGammaFunction.html">logarithmic gamma function</a> of <code>self</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealPAAE6lgammaxxFZ">lgamma(_:)</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">func</span> <span class="nf">logarithmicGamma</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE8cubeRootxyF"></a>
<a name="//apple_ref/swift/Method/cubeRoot()" class="dashAnchor"></a>
<h4>
<code>cubeRoot()</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the cube root of the value, rounded to a representable value
(default implementation).</p>
<div class="aside aside-note">
<p class="aside-title">Note</p>
<p>This method is required by the protocol <code><a href="../Protocols/Math.html">Math</a></code>; it is not an
extension method.</p>
</div>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<p><code>cbrt(_:)</code></p>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">func</span> <span class="nf">cubeRoot</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
</div>
<div class="group">
<ul>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE5atan2xx_xtFZ"></a>
<a name="//apple_ref/swift/Method/atan2(_:_:)" class="dashAnchor"></a>
<h4>
<code>atan2(_:_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the inverse tangent of <code>y / x</code>, using the signs of <code>y</code> and <code>x</code> to
determine the quadrant of the computed angle.</p>
<p>If <code>x == 0 &amp;&amp; y == 0</code>, the return value is still finite.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<p><code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealP14inverseTangentxx10dividingBy_tF">inverseTangent(dividingBy:)</a></code></p>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">atan2</span><span class="p">(</span><span class="n">_</span> <span class="nv">y</span><span class="p">:</span> <span class="k">Self</span><span class="p">,</span> <span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
<div class="parameters">
<h5>Parameters</h5>
<dl>
<dt><code>y</code></dt>
<dd>
<p>The value to divide.</p>
</dd>
<dt><code>x</code></dt>
<dd>
<p>The divisor by which to divide <code>y</code>.</p>
</dd>
</dl>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE3erfxxFZ"></a>
<a name="//apple_ref/swift/Method/erf(_:)" class="dashAnchor"></a>
<h4>
<code>erf(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/Erf.html">error function</a> of <code>x</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealP5errorxyF">error()</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">erf</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE4erfcxxFZ"></a>
<a name="//apple_ref/swift/Method/erfc(_:)" class="dashAnchor"></a>
<h4>
<code>erfc(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/Erfc.html">complementary error function</a> of <code>x</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealP18complementaryErrorxyF">complementaryError()</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">erfc</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE6tgammaxxFZ"></a>
<a name="//apple_ref/swift/Method/tgamma(_:)" class="dashAnchor"></a>
<h4>
<code>tgamma(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/GammaFunction.html">gamma function</a> of <code>x</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealP5gammaxyF">gamma()</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">tgamma</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
<li>
<div>
<a name="/s:12NumericAnnex4RealPAAE6lgammaxxFZ"></a>
<a name="//apple_ref/swift/Method/lgamma(_:)" class="dashAnchor"></a>
<h4>
<code>lgamma(_:)</code>
<span class="declaration-note">Extension method</span>
</h4>
</div>
<div class="abstract">
<p>Returns the value of the <a href="http://mathworld.wolfram.com/LogGammaFunction.html">logarithmic gamma function</a> of <code>x</code>.</p>
<div class="aside aside-see-also">
<p class="aside-title">See also</p>
<code><a href="../Protocols/Real.html#/s:12NumericAnnex4RealP16logarithmicGammaxyF">logarithmicGamma()</a></code>
</div>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">static</span> <span class="kd">func</span> <span class="nf">lgamma</span><span class="p">(</span><span class="n">_</span> <span class="nv">x</span><span class="p">:</span> <span class="k">Self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="k">Self</span></code></pre>
</div>
</div>
</li>
</ul>
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</div>
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<div class="overview">
<h1>Structures</h1>
<p>The following structures are available globally.</p>
</div>
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</section>
<section class="toc">
<details open>
<summary>Topics</summary>
<div class="wrapper">
<ul>
<li><a href="#/s:12NumericAnnex7ComplexV">Complex</a></li>
</ul>
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<li><a href="#/s:12NumericAnnex8RationalV">Rational</a></li>
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<a name="/s:12NumericAnnex7ComplexV"></a>
<a name="//apple_ref/swift/Struct/Complex" class="dashAnchor"></a>
<h4>
<code>Complex</code>
</h4>
</div>
<div class="abstract">
<p>A type to represent a complex value in Cartesian form.</p>
<div class="aside aside-note">
<p class="aside-title">Note</p>
<code>Complex64</code> is a type alias for <code>Complex&lt;Float&gt;</code> and <code>Complex128</code> is
a type alias for <code>Complex&lt;Double&gt;</code>.
</div>
<p>Create new instances of <code>Complex&lt;T&gt;</code> using integer or floating-point
literals and the imaginary unit <code><a href="Structs/Complex.html#/s:12NumericAnnex7ComplexV1iACyxGvZ">Complex&lt;T&gt;.i</a></code>. For example:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span> <span class="o">*</span> <span class="o">.</span><span class="n">i</span> <span class="c1">// `x` is of type `Complex&lt;Double&gt;`</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="mf">3.5</span> <span class="o">+</span> <span class="mi">7</span> <span class="o">*</span> <span class="o">.</span><span class="n">i</span> <span class="c1">// `y` is of type `Complex&lt;Double&gt;`</span>
<span class="k">let</span> <span class="nv">z</span><span class="p">:</span> <span class="kt">Complex64</span> <span class="o">=</span> <span class="o">.</span><span class="n">e</span> <span class="o">+</span> <span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="o">.</span><span class="n">i</span> <span class="c1">// `z` is of type `Complex&lt;Float&gt;`</span>
</code></pre>
<h2 id='additional-considerations' class='heading'>Additional Considerations</h2>
<p>Floating-point types have special values that represent infinity or NaN
(<q>not a number</q>). Complex functions in different languages may return
different results when working with special values.</p>
<p>Many complex functions have <a href="http://mathworld.wolfram.com/BranchCut.html">branch cuts</a>, which are curves in the
complex plane across which a function is discontinuous. Different languages
may adopt different branch cut structures for the same complex function.</p>
<p>Implementations in <code>Complex&lt;T&gt;</code> adhere to the <a href="http://www.open-std.org/JTC1/SC22/WG14/www/standards.html#9899">C standard</a> (Annex G) as
closely as possible with respect to special values and branch cuts.</p>
<p>To users unfamiliar with complex functions, the principal value returned by
some complex functions may be unexpected. For example,
<code>Double.cbrt(-8) == -2</code>, which is the <strong>real root</strong>, while
<code>Complex.cbrt(-8) == 2 * Complex.exp(.i * .pi / 3)</code>, which is the
<strong>principal root</strong>.</p>
<a href="Structs/Complex.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">struct</span> <span class="kt">Complex</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt"><a href="Protocols/Real.html">Real</a></span><span class="o">&gt;</span> <span class="p">:</span> <span class="kt">Codable</span>
<span class="k">where</span> <span class="kt">T</span> <span class="p">:</span> <span class="kt">Codable</span> <span class="o">&amp;</span> <span class="n">_ExpressibleByBuiltinFloatLiteral</span></code></pre>
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<a name="/s:12NumericAnnex8RationalV"></a>
<a name="//apple_ref/swift/Struct/Rational" class="dashAnchor"></a>
<h4>
<code>Rational</code>
</h4>
</div>
<div class="abstract">
<p>A type to represent a rational value.</p>
<div class="aside aside-note">
<p class="aside-title">Note</p>
<code>Ratio</code> is a type alias for <code>Rational&lt;Int&gt;</code>.
</div>
<p>Create new instances of <code>Rational&lt;T&gt;</code> by using integer literals and the
division (<code>/</code>) operator. For example:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">x</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="mi">3</span> <span class="k">as</span> <span class="kt">Ratio</span> <span class="c1">// `x` is of type `Rational&lt;Int&gt;`</span>
<span class="k">let</span> <span class="nv">y</span> <span class="o">=</span> <span class="mi">2</span> <span class="k">as</span> <span class="kt">Ratio</span> <span class="c1">// `y` is of type `Rational&lt;Int&gt;`</span>
<span class="k">let</span> <span class="nv">z</span><span class="p">:</span> <span class="kt">Ratio</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">/</span> <span class="mi">3</span> <span class="c1">// `z` is also of type `Rational&lt;Int&gt;`</span>
<span class="nf">print</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span> <span class="c1">// Prints "3"</span>
</code></pre>
<p>You can create an unreduced fraction by using the initializer
<code><a href="Structs/Rational.html#/s:12NumericAnnex8RationalVACyxGx9numerator_x11denominatortcfc">Rational&lt;T&gt;.init(numerator:denominator:)</a></code>. For example:</p>
<pre class="highlight swift"><code><span class="k">let</span> <span class="nv">a</span> <span class="o">=</span> <span class="kt">Ratio</span><span class="p">(</span><span class="nv">numerator</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span> <span class="nv">denominator</span><span class="p">:</span> <span class="mi">3</span><span class="p">)</span>
<span class="nf">print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="c1">// Prints "3/3"</span>
</code></pre>
<p>All arithmetic operations on values in canonical form (i.e. reduced to
lowest terms) return results in canonical form. However, operations on
values not in canonical form may or may not return results that are
themselves in canonical form. The property <code>canonicalized</code> is the canonical
form of any value.</p>
<h2 id='additional-considerations' class='heading'>Additional Considerations</h2>
<h3 id='special-values' class='heading'>Special Values</h3>
<p><code>Rational&lt;T&gt;</code> does not prohibit zero as a denominator. Any instance with a
positive numerator and zero denominator represents (positive) infinity; any
instance with a negative numerator and zero denominator represents negative
infinity; and any instance with zero numerator and zero denominator
represents NaN (<q>not a number</q>).</p>
<p>As with floating-point types, <code><a href="Structs/Rational.html#/s:12NumericAnnex8RationalV8infinityACyxGvZ">Rational&lt;T&gt;.infinity</a></code> compares greater than
every finite value and negative infinity, and <code>-Rational&lt;T&gt;.infinity</code>
compares less than every finite value and positive infinity. Infinite values
of the same sign compare equal to each other.</p>
<p>As with floating-point types, <code><a href="Structs/Rational.html#/s:12NumericAnnex8RationalV3nanACyxGvZ">Rational&lt;T&gt;.nan</a></code> does not compare equal to
any other value, including another NaN. Use the property <code>isNaN</code> to test if
a value is NaN. <code>Rational&lt;T&gt;</code> arithmetic operations are intended to
propagate NaN in the same manner as analogous floating-point operations.</p>
<h3 id='numerical-limits' class='heading'>Numerical Limits</h3>
<p>When a value of type <code>Rational&lt;T&gt;</code> is in canonical form, the sign of the
numerator is the sign of the value; that is, in canonical form, the sign of
the denominator is always positive. Therefore, <code>-1 / T.min</code> cannot be
represented as a value of type <code>Rational&lt;T&gt;</code> because <code>abs(T.min)</code> cannot be
represented as a value of type <code>T</code>.</p>
<p>To ensure that every representable value of type <code>Rational&lt;T&gt;</code> has a
representable magnitude and reciprocal of the same type, an overflow trap
occurs when the division (<code>/</code>) operator is used to create a value of type
<code>Rational&lt;T&gt;</code> with numerator <code>T.min</code>.</p>
<a href="Structs/Rational.html">See more…</a>
</div>
<div class="declaration">
<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">struct</span> <span class="kt">Rational</span><span class="o">&lt;</span><span class="kt">T</span> <span class="p">:</span> <span class="kt">SignedInteger</span><span class="o">&gt;</span> <span class="p">:</span> <span class="kt">Codable</span>
<span class="k">where</span> <span class="kt">T</span> <span class="p">:</span> <span class="kt">Codable</span> <span class="o">&amp;</span> <span class="n">_ExpressibleByBuiltinIntegerLiteral</span><span class="p">,</span>
<span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="p">:</span> <span class="kt">UnsignedInteger</span><span class="p">,</span>
<span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span><span class="o">.</span><span class="kt">Magnitude</span> <span class="o">==</span> <span class="kt">T</span><span class="o">.</span><span class="kt">Magnitude</span></code></pre>
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<code>Sign</code>
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<p>The mathematical sign of a number.</p>
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<h5>Declaration</h5>
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<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">typealias</span> <span class="kt">Sign</span> <span class="o">=</span> <span class="kt">FloatingPointSign</span></code></pre>
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<h4>
<code>RoundingRule</code>
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<p>A rule for rounding a number.</p>
</div>
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<h5>Declaration</h5>
<div class="language">
<p class="aside-title">Swift</p>
<pre class="highlight"><code><span class="kd">public</span> <span class="kd">typealias</span> <span class="kt">RoundingRule</span> <span class="o">=</span> <span class="kt">FloatingPointRoundingRule</span></code></pre>
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<h1 id='img-src-https-xwu-github-io-numericannex-img-numericannex-2017-09-02-svg-alt-numericannex-height-72-br-numericannex' class='heading'><img src="https://xwu.github.io/NumericAnnex/img/NumericAnnex-2017-09-02.svg" alt="NumericAnnex" height="72"><br>NumericAnnex</h1>
<p>NumericAnnex supplements the numeric facilities provided in the Swift standard
library.</p>
<p><a href="https://travis-ci.org/xwu/NumericAnnex"><img src="https://travis-ci.org/xwu/NumericAnnex.svg?branch=master" alt="Build Status"></a>
<a href="https://codecov.io/gh/xwu/NumericAnnex"><img src="https://codecov.io/gh/xwu/NumericAnnex/branch/master/graph/badge.svg" alt="codecov"></a></p>
<h2 id='features' class='heading'>Features</h2>
<ul>
<li>The exponentiation operator <code>**</code> and the compound assignment operator <code>**=</code>.</li>
<li>Extension methods for <code>BinaryInteger</code> exponentiation, square root, cube root,
greatest common divisor, and least common multiple.</li>
<li><code><a href="Protocols/Math.html">Math</a></code>, a protocol for signed numeric types that support elementary functions.</li>
<li><code><a href="Protocols/Real.html">Real</a></code>, a protocol for floating-point types that support elementary functions
and a selection of special functions.</li>
<li><code><a href="Protocols/PRNG.html">PRNG</a></code>, a protocol for pseudo-random number generators.</li>
<li><code><a href="Structs/Rational.html">Rational</a></code>, a value type to represent rational values which supports division
by zero.</li>
<li><code><a href="Structs/Complex.html">Complex</a></code>, a value type to represent complex values in Cartesian form.</li>
<li><code><a href="Classes/Random.html">Random</a></code> and <code><a href="Classes/Random/Xoroshiro.html">Random.Xoroshiro</a></code>, two reference types implementing efficient
pseudo-random number generators.</li>
</ul>
<blockquote>
<p>Note: This project is in the early stages of development and is not
production-ready at this time.</p>
</blockquote>
<h2 id='requirements' class='heading'>Requirements</h2>
<p>NumericAnnex requires Swift 4.0. On Apple platforms, it also requires the
Security framework for cryptographically secure random bytes.</p>
<h2 id='installation' class='heading'>Installation</h2>
<p>After NumericAnnex has been cloned or downloaded locally, build the library
using the command <code>swift build</code> (macOS) or <code>swift build -Xcc -D_GNU_SOURCE</code>
(Linux). Run tests with the command <code>swift test</code> (macOS) or
<code>swift test -Xcc -D_GNU_SOURCE</code> (Linux). An Xcode project can be generated with
the command <code>swift package generate-xcodeproj</code>.</p>
<p>To add the package as a dependency using <a href="https://cocoapods.org">CocoaPods</a>,
insert the following line in your <code>Podfile</code>:</p>
<pre class="highlight ruby"><code><span class="n">pod</span> <span class="s1">'NumericAnnex'</span><span class="p">,</span> <span class="s1">'~&gt; 0.1.19'</span>
</code></pre>
<p><a href="https://swift.org/package-manager/">Swift Package Manager</a> can also be used to
add the package as a dependency. See Swift documentation for details.</p>
<h2 id='basic-usage' class='heading'>Basic Usage</h2>
<pre class="highlight swift"><code><span class="kd">import</span> <span class="kt">NumericAnnex</span>
<span class="nf">print</span><span class="p">(</span><span class="mi">2</span> <span class="o">**</span> <span class="mi">3</span><span class="p">)</span>
<span class="c1">// Prints "8".</span>
<span class="nf">print</span><span class="p">(</span><span class="mf">4.0</span> <span class="o">**</span> <span class="mf">5.0</span><span class="p">)</span>
<span class="c1">// Prints "1024.0".</span>
<span class="nf">print</span><span class="p">(</span><span class="kt">Int</span><span class="o">.</span><span class="nf">cbrt</span><span class="p">(</span><span class="mi">8</span><span class="p">))</span>
<span class="c1">// Prints "2".</span>
<span class="nf">print</span><span class="p">(</span><span class="kt">Double</span><span class="o">.</span><span class="nf">cbrt</span><span class="p">(</span><span class="mf">27.0</span><span class="p">))</span>
<span class="c1">// Prints "3.0".</span>
<span class="k">var</span> <span class="nv">x</span><span class="p">:</span> <span class="kt">Ratio</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="mi">4</span>
<span class="c1">// Ratio is a type alias for Rational&lt;Int&gt;.</span>
<span class="nf">print</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="nf">reciprocal</span><span class="p">())</span>
<span class="c1">// Prints "4".</span>
<span class="n">x</span> <span class="o">*=</span> <span class="mi">8</span>
<span class="nf">print</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span>
<span class="c1">// Prints "4".</span>
<span class="n">x</span> <span class="o">=</span> <span class="kt">Ratio</span><span class="p">(</span><span class="kt">Float</span><span class="o">.</span><span class="n">phi</span><span class="p">)</span> <span class="c1">// Golden ratio.</span>
<span class="nf">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="c1">// Prints "13573053/8388608".</span>
<span class="k">var</span> <span class="nv">z</span><span class="p">:</span> <span class="kt">Complex64</span> <span class="o">=</span> <span class="mi">42</span> <span class="o">*</span> <span class="o">.</span><span class="n">i</span>
<span class="c1">// Complex64 is a type alias for Complex&lt;Float&gt;.</span>
<span class="nf">print</span><span class="p">(</span><span class="kt">Complex</span><span class="o">.</span><span class="nf">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
<span class="c1">// Prints "4.58258 + 4.58258i".</span>
<span class="n">z</span> <span class="o">=</span> <span class="o">.</span><span class="n">pi</span> <span class="o">+</span> <span class="o">.</span><span class="n">i</span> <span class="o">*</span> <span class="o">.</span><span class="nf">log</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="o">.</span><span class="nf">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="nf">print</span><span class="p">(</span><span class="kt">Complex</span><span class="o">.</span><span class="nf">cos</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">real</span><span class="p">)</span>
<span class="c1">// Prints "-2.0".</span>
</code></pre>
<h2 id='documentation' class='heading'>Documentation</h2>
<p>All public protocols, types, and functions have been carefully documented in the
code. See the <a href="https://xwu.github.io/NumericAnnex/">formatted reference</a> for
details.</p>
<p>The project adheres to many design patterns found in the Swift standard library.
For example, <code><a href="Protocols/Math.html">Math</a></code> types provide methods such as <code>cubeRoot()</code> and <code>tangent()</code>
just as <code>FloatingPoint</code> types provide methods such as <code>squareRoot()</code>.</p>
<p>No free functions are declared in this library unless they overload existing
ones in the Swift standard library. Instead, functions such as <code>cbrt(_:)</code> and
<code>tan(_:)</code> are provided as static members. This avoids collisions with C standard
library functions that you may wish to use. It also promotes clarity at the call
site when the result of a complex operation differs from that of its real
counterpart (e.g., <code>Complex128.cbrt(-8) != -2</code>).</p>
<h2 id='future-directions' class='heading'>Future Directions</h2>
<ul>
<li>Add more tests, including performance tests</li>
<li>Design and implement additional methods on <code><a href="Protocols/PRNG.html">PRNG</a></code></li>
</ul>
<h2 id='license' class='heading'>License</h2>
<p>All original work is released under the MIT license. See
<a href="https://github.com/xwu/NumericAnnex/blob/master/LICENSE">LICENSE</a> for details.</p>
<p>Portions of the complex square root and elementary transcendental functions use
checks for special values adapted from libc++. Code in libc++ is dual-licensed
under the MIT and UIUC/NCSA licenses.</p>
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<h1 id='img-src-https-xwu-github-io-numericannex-img-numericannex-2017-09-02-svg-alt-numericannex-height-72-br-numericannex' class='heading'><img src="https://xwu.github.io/NumericAnnex/img/NumericAnnex-2017-09-02.svg" alt="NumericAnnex" height="72"><br>NumericAnnex</h1>
<p>NumericAnnex supplements the numeric facilities provided in the Swift standard
library.</p>
<p><a href="https://travis-ci.org/xwu/NumericAnnex"><img src="https://travis-ci.org/xwu/NumericAnnex.svg?branch=master" alt="Build Status"></a>
<a href="https://codecov.io/gh/xwu/NumericAnnex"><img src="https://codecov.io/gh/xwu/NumericAnnex/branch/master/graph/badge.svg" alt="codecov"></a></p>
<h2 id='features' class='heading'>Features</h2>
<ul>
<li>The exponentiation operator <code>**</code> and the compound assignment operator <code>**=</code>.</li>
<li>Extension methods for <code>BinaryInteger</code> exponentiation, square root, cube root,
greatest common divisor, and least common multiple.</li>
<li><code><a href="Protocols/Math.html">Math</a></code>, a protocol for signed numeric types that support elementary functions.</li>
<li><code><a href="Protocols/Real.html">Real</a></code>, a protocol for floating-point types that support elementary functions
and a selection of special functions.</li>
<li><code><a href="Protocols/PRNG.html">PRNG</a></code>, a protocol for pseudo-random number generators.</li>
<li><code><a href="Structs/Rational.html">Rational</a></code>, a value type to represent rational values which supports division
by zero.</li>
<li><code><a href="Structs/Complex.html">Complex</a></code>, a value type to represent complex values in Cartesian form.</li>
<li><code><a href="Classes/Random.html">Random</a></code> and <code><a href="Classes/Random/Xoroshiro.html">Random.Xoroshiro</a></code>, two reference types implementing efficient
pseudo-random number generators.</li>
</ul>
<blockquote>
<p>Note: This project is in the early stages of development and is not
production-ready at this time.</p>
</blockquote>
<h2 id='requirements' class='heading'>Requirements</h2>
<p>NumericAnnex requires Swift 4.0. On Apple platforms, it also requires the
Security framework for cryptographically secure random bytes.</p>
<h2 id='installation' class='heading'>Installation</h2>
<p>After NumericAnnex has been cloned or downloaded locally, build the library
using the command <code>swift build</code> (macOS) or <code>swift build -Xcc -D_GNU_SOURCE</code>
(Linux). Run tests with the command <code>swift test</code> (macOS) or
<code>swift test -Xcc -D_GNU_SOURCE</code> (Linux). An Xcode project can be generated with
the command <code>swift package generate-xcodeproj</code>.</p>
<p>To add the package as a dependency using <a href="https://cocoapods.org">CocoaPods</a>,
insert the following line in your <code>Podfile</code>:</p>
<pre class="highlight ruby"><code><span class="n">pod</span> <span class="s1">'NumericAnnex'</span><span class="p">,</span> <span class="s1">'~&gt; 0.1.19'</span>
</code></pre>
<p><a href="https://swift.org/package-manager/">Swift Package Manager</a> can also be used to
add the package as a dependency. See Swift documentation for details.</p>
<h2 id='basic-usage' class='heading'>Basic Usage</h2>
<pre class="highlight swift"><code><span class="kd">import</span> <span class="kt">NumericAnnex</span>
<span class="nf">print</span><span class="p">(</span><span class="mi">2</span> <span class="o">**</span> <span class="mi">3</span><span class="p">)</span>
<span class="c1">// Prints "8".</span>
<span class="nf">print</span><span class="p">(</span><span class="mf">4.0</span> <span class="o">**</span> <span class="mf">5.0</span><span class="p">)</span>
<span class="c1">// Prints "1024.0".</span>
<span class="nf">print</span><span class="p">(</span><span class="kt">Int</span><span class="o">.</span><span class="nf">cbrt</span><span class="p">(</span><span class="mi">8</span><span class="p">))</span>
<span class="c1">// Prints "2".</span>
<span class="nf">print</span><span class="p">(</span><span class="kt">Double</span><span class="o">.</span><span class="nf">cbrt</span><span class="p">(</span><span class="mf">27.0</span><span class="p">))</span>
<span class="c1">// Prints "3.0".</span>
<span class="k">var</span> <span class="nv">x</span><span class="p">:</span> <span class="kt">Ratio</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="mi">4</span>
<span class="c1">// Ratio is a type alias for Rational&lt;Int&gt;.</span>
<span class="nf">print</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="nf">reciprocal</span><span class="p">())</span>
<span class="c1">// Prints "4".</span>
<span class="n">x</span> <span class="o">*=</span> <span class="mi">8</span>
<span class="nf">print</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span>
<span class="c1">// Prints "4".</span>
<span class="n">x</span> <span class="o">=</span> <span class="kt">Ratio</span><span class="p">(</span><span class="kt">Float</span><span class="o">.</span><span class="n">phi</span><span class="p">)</span> <span class="c1">// Golden ratio.</span>
<span class="nf">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="c1">// Prints "13573053/8388608".</span>
<span class="k">var</span> <span class="nv">z</span><span class="p">:</span> <span class="kt">Complex64</span> <span class="o">=</span> <span class="mi">42</span> <span class="o">*</span> <span class="o">.</span><span class="n">i</span>
<span class="c1">// Complex64 is a type alias for Complex&lt;Float&gt;.</span>
<span class="nf">print</span><span class="p">(</span><span class="kt">Complex</span><span class="o">.</span><span class="nf">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
<span class="c1">// Prints "4.58258 + 4.58258i".</span>
<span class="n">z</span> <span class="o">=</span> <span class="o">.</span><span class="n">pi</span> <span class="o">+</span> <span class="o">.</span><span class="n">i</span> <span class="o">*</span> <span class="o">.</span><span class="nf">log</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="o">.</span><span class="nf">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="nf">print</span><span class="p">(</span><span class="kt">Complex</span><span class="o">.</span><span class="nf">cos</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">real</span><span class="p">)</span>
<span class="c1">// Prints "-2.0".</span>
</code></pre>
<h2 id='documentation' class='heading'>Documentation</h2>
<p>All public protocols, types, and functions have been carefully documented in the
code. See the <a href="https://xwu.github.io/NumericAnnex/">formatted reference</a> for
details.</p>
<p>The project adheres to many design patterns found in the Swift standard library.
For example, <code><a href="Protocols/Math.html">Math</a></code> types provide methods such as <code>cubeRoot()</code> and <code>tangent()</code>
just as <code>FloatingPoint</code> types provide methods such as <code>squareRoot()</code>.</p>
<p>No free functions are declared in this library unless they overload existing
ones in the Swift standard library. Instead, functions such as <code>cbrt(_:)</code> and
<code>tan(_:)</code> are provided as static members. This avoids collisions with C standard
library functions that you may wish to use. It also promotes clarity at the call
site when the result of a complex operation differs from that of its real
counterpart (e.g., <code>Complex128.cbrt(-8) != -2</code>).</p>
<h2 id='future-directions' class='heading'>Future Directions</h2>
<ul>
<li>Add more tests, including performance tests</li>
<li>Design and implement additional methods on <code><a href="Protocols/PRNG.html">PRNG</a></code></li>
</ul>
<h2 id='license' class='heading'>License</h2>
<p>All original work is released under the MIT license. See
<a href="https://github.com/xwu/NumericAnnex/blob/master/LICENSE">LICENSE</a> for details.</p>
<p>Portions of the complex square root and elementary transcendental functions use
checks for special values adapted from libc++. Code in libc++ is dual-licensed
under the MIT and UIUC/NCSA licenses.</p>
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