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| <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"><</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> |
| <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">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="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">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">&);</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> |
| </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> |
| </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="../../main_overview/error_handling.html#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="../../main_overview/error_handling.html#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="../../main_overview/error_handling.html#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="../../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> |
| <a name="math_toolkit.special.inv_hyper.atanh.accuracy"></a><h5> |
| <a name="id1155216"></a> |
| <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> |
| <a name="math_toolkit.special.inv_hyper.atanh.testing"></a><h5> |
| <a name="id1155233"></a> |
| <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> |
| <a name="math_toolkit.special.inv_hyper.atanh.implementation"></a><h5> |
| <a name="id1155277"></a> |
| <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 it's equivalent form: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/atanh3.png"></span> |
| </p> |
| <p> |
| is used. |
| </p> |
| </div> |
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| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
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