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<title>atanh</title>
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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.inv_hyper.atanh"></a><a class="link" href="atanh.html" title="atanh">atanh</a>
</h3></div></div></div>
<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">atanh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
</pre>
<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="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">atanh</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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<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">atanh</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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
</pre>
<p>
Computes the reciprocal of <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the
hyperbolic tangent function</a>, at x.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
If x is in the range <code class="literal">]-∞;-1[</code> or in the range <code class="literal">]+1;+∞[</code>
then returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<p>
If x is in the range <code class="literal">[-1;-1+ε[</code>, then the result of -<a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
is returned, with ε  
denoting numeric_limits<T>::epsilon().
</p>
<p>
If x is in the range <code class="literal">]+1-ε;+1]</code>, then the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
is returned, with ε  
denoting numeric_limits<T>::epsilon().
</p>
<p>
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
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>
<span class="inlinemediaobject"><img src="../../../graphs/atanh.svg" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.inv_hyper.atanh.h0"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.atanh.accuracy"></a></span><a class="link" href="atanh.html#math_toolkit.inv_hyper.atanh.accuracy">Accuracy</a>
</h5>
<p>
Generally accuracy is to within 1 or 2 epsilon across all supported platforms.
</p>
<h5>
<a name="math_toolkit.inv_hyper.atanh.h1"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.atanh.testing"></a></span><a class="link" href="atanh.html#math_toolkit.inv_hyper.atanh.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/atanh1.svg"></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.inv_hyper.atanh.h2"></a>
<span class="phrase"><a name="math_toolkit.inv_hyper.atanh.implementation"></a></span><a class="link" href="atanh.html#math_toolkit.inv_hyper.atanh.implementation">Implementation</a>
</h5>
<p>
For sufficiently small x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcTanh/06/01/03/01/" target="_top">approximation</a>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/atanh2.svg"></span>
</p>
<p>
Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcTanh/02/" target="_top">primary
definition</a>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/atanh1.svg"></span>
</p>
<p>
or its equivalent form:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/atanh3.svg"></span>
</p>
<p>
is used.
</p>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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