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<title>acosh</title>
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<div class="section math_toolkit_special_inv_hyper_acosh">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.inv_hyper.acosh"></a><a class="link" href="acosh.html" title="acosh">acosh</a>
</h4></div></div></div>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></span>
</pre>
<p>
</p>
<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../../main_overview/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">template</span><span class="special"><</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="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span>
</pre>
<p>
Computes the reciprocal of (the restriction to the range of <code class="literal">[0;+∞[</code>)
<a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the hyperbolic
cosine function</a>, at x. Values returned are positive.
</p>
<p>
If x is in the range <code class="literal">]-∞;+1[</code> then returns the result
of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>.
</p>
<p>
The return type of this function is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
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.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is
optional and can be used to control the behaviour of the function: how
it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>.
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/acosh.png" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.special.inv_hyper.acosh.h0"></a>
<span><a name="math_toolkit.special.inv_hyper.acosh.accuracy"></a></span><a class="link" href="acosh.html#math_toolkit.special.inv_hyper.acosh.accuracy">Accuracy</a>
</h5>
<p>
Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
</p>
<h5>
<a name="math_toolkit.special.inv_hyper.acosh.h1"></a>
<span><a name="math_toolkit.special.inv_hyper.acosh.testing"></a></span><a class="link" href="acosh.html#math_toolkit.special.inv_hyper.acosh.testing">Testing</a>
</h5>
<p>
This function is tested using a combination of random test values designed
to give full function coverage computed at high precision using the "naive"
formula:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/acosh1.png"></span>
</p>
<p>
along with a selection of sanity check values computed using functions.wolfram.com
to at least 50 decimal digits.
</p>
<h5>
<a name="math_toolkit.special.inv_hyper.acosh.h2"></a>
<span><a name="math_toolkit.special.inv_hyper.acosh.implementation"></a></span><a class="link" href="acosh.html#math_toolkit.special.inv_hyper.acosh.implementation">Implementation</a>
</h5>
<p>
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>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/acosh2.png"></span>
</p>
<p>
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>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/acosh4.png"></span>
</p>
<p>
Otherwise for x close to 1 we can use the following rearrangement of the
primary definition to preserve accuracy:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/acosh3.png"></span>
</p>
<p>
Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/02/" target="_top">primary
definition</a> is used:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/acosh1.png"></span>
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
Lalande, Johan Råde, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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>)
</p>
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