1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>atanh</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../inv_hyper.html" title="Inverse Hyperbolic Functions"> 9<link rel="prev" href="asinh.html" title="asinh"> 10<link rel="next" href="../owens_t.html" title="Owen's T function"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<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="../owens_t.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.inv_hyper.atanh"></a><a class="link" href="atanh.html" title="atanh">atanh</a> 28</h3></div></div></div> 29<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> 30</pre> 31<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> 32<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> 33 34<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 35<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 36</pre> 37<p> 38 Computes the reciprocal of <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the 39 hyperbolic tangent function</a>, at x. 40 </p> 41<p> 42 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 43 be used to control the behaviour of the function: how it handles errors, 44 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 45 documentation for more details</a>. 46 </p> 47<p> 48 If x is in the range <code class="literal">]-∞;-1[</code> or in the range <code class="literal">]+1;+∞[</code> 49 then returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 50 </p> 51<p> 52 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> 53 is returned, with ε 54denoting <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code>. 55 </p> 56<p> 57 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> 58 is returned, with ε 59denoting <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code>. 60 </p> 61<p> 62 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 63 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. 64 </p> 65<div class="blockquote"><blockquote class="blockquote"><p> 66 <span class="inlinemediaobject"><img src="../../../graphs/atanh.svg" align="middle"></span> 67 68 </p></blockquote></div> 69<h5> 70<a name="math_toolkit.inv_hyper.atanh.h0"></a> 71 <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> 72 </h5> 73<p> 74 Generally accuracy is to within 1 or 2 <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine 75 epsilon</a> across all supported platforms. 76 </p> 77<h5> 78<a name="math_toolkit.inv_hyper.atanh.h1"></a> 79 <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> 80 </h5> 81<p> 82 This function is tested using a combination of random test values designed 83 to give full function coverage computed at high precision using the "naive" 84 formula: 85 </p> 86<div class="blockquote"><blockquote class="blockquote"><p> 87 <span class="inlinemediaobject"><img src="../../../equations/atanh1.svg"></span> 88 89 </p></blockquote></div> 90<p> 91 along with a selection of sanity check values computed using functions.wolfram.com 92 to at least 50 decimal digits. 93 </p> 94<h5> 95<a name="math_toolkit.inv_hyper.atanh.h2"></a> 96 <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> 97 </h5> 98<p> 99 For sufficiently small x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcTanh/06/01/03/01/" target="_top">approximation</a>: 100 </p> 101<div class="blockquote"><blockquote class="blockquote"><p> 102 <span class="inlinemediaobject"><img src="../../../equations/atanh2.svg"></span> 103 104 </p></blockquote></div> 105<p> 106 Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcTanh/02/" target="_top">primary 107 definition</a>: 108 </p> 109<div class="blockquote"><blockquote class="blockquote"><p> 110 <span class="inlinemediaobject"><img src="../../../equations/atanh1.svg"></span> 111 112 </p></blockquote></div> 113<p> 114 or its equivalent form: 115 </p> 116<div class="blockquote"><blockquote class="blockquote"><p> 117 <span class="inlinemediaobject"><img src="../../../equations/atanh3.svg"></span> 118 119 </p></blockquote></div> 120<p> 121 is used. 122 </p> 123</div> 124<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 125<td align="left"></td> 126<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 127 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 128 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 129 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 130 Daryle Walker and Xiaogang Zhang<p> 131 Distributed under the Boost Software License, Version 1.0. (See accompanying 132 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>) 133 </p> 134</div></td> 135</tr></table> 136<hr> 137<div class="spirit-nav"> 138<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="../owens_t.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 139</div> 140</body> 141</html> 142
