1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>acosh</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="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview"> 10<link rel="next" href="asinh.html" title="asinh"> 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="inv_hyper_over.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="asinh.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.acosh"></a><a class="link" href="acosh.html" title="acosh">acosh</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">acosh</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">acosh</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">acosh</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 (the restriction to the range of <code class="literal">[0;+∞[</code>) 39 <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the hyperbolic cosine 40 function</a>, at x. Values returned are positive. 41 </p> 42<p> 43 If x is in the range <code class="literal">]-∞;+1[</code> then returns the result of 44 <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 45 </p> 46<p> 47 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 48 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. 49 </p> 50<p> 51 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 52 be used to control the behaviour of the function: how it handles errors, 53 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 54 documentation for more details</a>. 55 </p> 56<div class="blockquote"><blockquote class="blockquote"><p> 57 <span class="inlinemediaobject"><img src="../../../graphs/acosh.svg" align="middle"></span> 58 59 </p></blockquote></div> 60<h5> 61<a name="math_toolkit.inv_hyper.acosh.h0"></a> 62 <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.accuracy"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.accuracy">Accuracy</a> 63 </h5> 64<p> 65 Generally accuracy is to within 1 or 2 <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine 66 epsilon</a> across all supported platforms. 67 </p> 68<h5> 69<a name="math_toolkit.inv_hyper.acosh.h1"></a> 70 <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.testing"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.testing">Testing</a> 71 </h5> 72<p> 73 This function is tested using a combination of random test values designed 74 to give full function coverage computed at high precision using the "naive" 75 formula: 76 </p> 77<div class="blockquote"><blockquote class="blockquote"><p> 78 <span class="inlinemediaobject"><img src="../../../equations/acosh1.svg"></span> 79 80 </p></blockquote></div> 81<p> 82 along with a selection of sanity check values computed using functions.wolfram.com 83 to at least 50 decimal digits. 84 </p> 85<h5> 86<a name="math_toolkit.inv_hyper.acosh.h2"></a> 87 <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.implementation"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.implementation">Implementation</a> 88 </h5> 89<p> 90 For sufficiently large x, we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/06/01/0001/" target="_top">approximation</a>: 91 </p> 92<div class="blockquote"><blockquote class="blockquote"><p> 93 <span class="inlinemediaobject"><img src="../../../equations/acosh2.svg"></span> 94 95 </p></blockquote></div> 96<p> 97 For x sufficiently close to 1 we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/04/01/0001/" target="_top">approximation</a>: 98 </p> 99<div class="blockquote"><blockquote class="blockquote"><p> 100 <span class="inlinemediaobject"><img src="../../../equations/acosh4.svg"></span> 101 102 </p></blockquote></div> 103<p> 104 Otherwise for x close to 1 we can use the following rearrangement of the 105 primary definition to preserve accuracy: 106 </p> 107<div class="blockquote"><blockquote class="blockquote"><p> 108 <span class="inlinemediaobject"><img src="../../../equations/acosh3.svg"></span> 109 110 </p></blockquote></div> 111<p> 112 Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/02/" target="_top">primary 113 definition</a> is used: 114 </p> 115<div class="blockquote"><blockquote class="blockquote"><p> 116 <span class="inlinemediaobject"><img src="../../../equations/acosh1.svg"></span> 117 118 </p></blockquote></div> 119</div> 120<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 121<td align="left"></td> 122<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 123 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 124 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 125 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 126 Daryle Walker and Xiaogang Zhang<p> 127 Distributed under the Boost Software License, Version 1.0. (See accompanying 128 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>) 129 </p> 130</div></td> 131</tr></table> 132<hr> 133<div class="spirit-nav"> 134<a accesskey="p" href="inv_hyper_over.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="asinh.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 135</div> 136</body> 137</html> 138
