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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.inv_hyper.acosh"></a><a class="link" href="acosh.html" title="acosh">acosh</a>
  28 </h3></div></div></div>
  29 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">acosh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  30 </pre>
  31 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  32 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">acosh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>
  33
  34 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  35 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">acosh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  36 </pre>
  37 <p>
  38         Computes the reciprocal of (the restriction to the range of <code class="literal">[0;+&#8734;[</code>)
  39         <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the hyperbolic cosine
  40         function</a>, at x. Values returned are positive.
  41       </p>
  42 <p>
  43         If x is in the range <code class="literal">]-&#8734;;+1[</code> then returns the result of
  44         <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
  45       </p>
  46 <p>
  47         The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
  48         type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and T otherwise.
  49       </p>
  50 <p>
  51         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  52         be used to control the behaviour of the function: how it handles errors,
  53         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  54         documentation for more details</a>.
  55       </p>
  56 <div class="blockquote"><blockquote class="blockquote"><p>
  57           <span class="inlinemediaobject"><img src="../../../graphs/acosh.svg" align="middle"></span>
  58
  59         </p></blockquote></div>
  60 <h5>
  61 <a name="math_toolkit.inv_hyper.acosh.h0"></a>
  62         <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.accuracy"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.accuracy">Accuracy</a>
  63       </h5>
  64 <p>
  65         Generally accuracy is to within 1 or 2 <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine
  66         epsilon</a> across all supported platforms.
  67       </p>
  68 <h5>
  69 <a name="math_toolkit.inv_hyper.acosh.h1"></a>
  70         <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.testing"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.testing">Testing</a>
  71       </h5>
  72 <p>
  73         This function is tested using a combination of random test values designed
  74         to give full function coverage computed at high precision using the "naive"
  75         formula:
  76       </p>
  77 <div class="blockquote"><blockquote class="blockquote"><p>
  78           <span class="inlinemediaobject"><img src="../../../equations/acosh1.svg"></span>
  79
  80         </p></blockquote></div>
  81 <p>
  82         along with a selection of sanity check values computed using functions.wolfram.com
  83         to at least 50 decimal digits.
  84       </p>
  85 <h5>
  86 <a name="math_toolkit.inv_hyper.acosh.h2"></a>
  87         <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.implementation"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.implementation">Implementation</a>
  88       </h5>
  89 <p>
  90         For sufficiently large x, we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/06/01/0001/" target="_top">approximation</a>:
  91       </p>
  92 <div class="blockquote"><blockquote class="blockquote"><p>
  93           <span class="inlinemediaobject"><img src="../../../equations/acosh2.svg"></span>
  94
  95         </p></blockquote></div>
  96 <p>
  97         For x sufficiently close to 1 we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/04/01/0001/" target="_top">approximation</a>:
  98       </p>
  99 <div class="blockquote"><blockquote class="blockquote"><p>
 100           <span class="inlinemediaobject"><img src="../../../equations/acosh4.svg"></span>
 101
 102         </p></blockquote></div>
 103 <p>
 104         Otherwise for x close to 1 we can use the following rearrangement of the
 105         primary definition to preserve accuracy:
 106       </p>
 107 <div class="blockquote"><blockquote class="blockquote"><p>
 108           <span class="inlinemediaobject"><img src="../../../equations/acosh3.svg"></span>
 109
 110         </p></blockquote></div>
 111 <p>
 112         Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/02/" target="_top">primary
 113         definition</a> is used:
 114       </p>
 115 <div class="blockquote"><blockquote class="blockquote"><p>
 116           <span class="inlinemediaobject"><img src="../../../equations/acosh1.svg"></span>
 117
 118         </p></blockquote></div>
 119 </div>
 120 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 121 <td align="left"></td>
 122 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 123       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 124       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 125       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 126       Daryle Walker and Xiaogang Zhang<p>
 127         Distributed under the Boost Software License, Version 1.0. (See accompanying
 128         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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