libs/math/doc/html/math_toolkit/inv_hyper/atanh.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   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 2.11.0">
   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">&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>
  30 </pre>
  31 <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>
  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">&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="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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&#160;20.&#160;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&#160;20.&#160;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">]-&#8734;;-1[</code> or in the range <code class="literal">]+1;+&#8734;[</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+&#949;[</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 &#949;
  54 denoting <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</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-&#949;;+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 &#949;
  59 denoting <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</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 &#169; 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       R&#229;de, 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>