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<div class="section math_toolkit_special_inv_hyper_atanh">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.inv_hyper.atanh"></a><a class="link" href="atanh.html" title="atanh">atanh</a>
</h4></div></div></div>
<p>
</p>
<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">atanh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<p>
        </p>
<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>
<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">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">&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="Policies">Policy</a><span class="special">&gt;</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">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="Policies">Policy</a><span class="special">&amp;);</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="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>
          If x is in the range <code class="literal">]-&#8734;;-1[</code> or in the range <code class="literal">]+1;+&#8734;[</code>
          then returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>.
        </p>
<p>
          If x is in the range <code class="literal">[-1;-1+&#949;[</code>, then the result of -<a class="link" href="../../main_overview/error_handling.html#overflow_error">overflow_error</a> is returned, with &#949; &#160;
denoting
          numeric_limits&lt;T&gt;::epsilon().
        </p>
<p>
          If x is in the range <code class="literal">]+1-&#949;;+1]</code>, then the result of <a class="link" href="../../main_overview/error_handling.html#overflow_error">overflow_error</a> is returned, with &#949; &#160;
denoting
          numeric_limits&lt;T&gt;::epsilon().
        </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>
          <span class="inlinemediaobject"><img src="../../../../graphs/atanh.png" align="middle"></span>
        </p>
<h5>
<a name="math_toolkit.special.inv_hyper.atanh.h0"></a>
          <span><a name="math_toolkit.special.inv_hyper.atanh.accuracy"></a></span><a class="link" href="atanh.html#math_toolkit.special.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.special.inv_hyper.atanh.h1"></a>
          <span><a name="math_toolkit.special.inv_hyper.atanh.testing"></a></span><a class="link" href="atanh.html#math_toolkit.special.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.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.atanh.h2"></a>
          <span><a name="math_toolkit.special.inv_hyper.atanh.implementation"></a></span><a class="link" href="atanh.html#math_toolkit.special.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.png"></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.png"></span>
        </p>
<p>
          or its equivalent form:
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../equations/atanh3.png"></span>
        </p>
<p>
          is used.
        </p>
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      Lalande, Johan R&#229;de, Gautam Sewani and Thijs van den Berg<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>)
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