<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>atanh</title> <link rel="stylesheet" href="../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../index.html" title="Math Toolkit"> <link rel="up" href="../inv_hyper.html" title="Inverse Hyperbolic Functions"> <link rel="prev" href="asinh.html" title="asinh"> <link rel="next" href="../../utils.html" title="Floating Point Utilities"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="asinh.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../inv_hyper.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../../utils.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section" lang="en"> <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="id1158169"></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="id1158185"></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="id1158230"></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> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, 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 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> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="asinh.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../inv_hyper.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../../utils.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>