1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>asinh</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="acosh.html" title="acosh"> 10<link rel="next" href="atanh.html" title="atanh"> 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="acosh.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="atanh.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.asinh"></a><a class="link" href="asinh.html" title="asinh">asinh</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">asinh</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">asinh</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">asinh</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 sine function</a>. 40 </p> 41<p> 42 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 43 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. 44 </p> 45<div class="blockquote"><blockquote class="blockquote"><p> 46 <span class="inlinemediaobject"><img src="../../../graphs/asinh.svg" align="middle"></span> 47 48 </p></blockquote></div> 49<p> 50 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 51 be used to control the behaviour of the function: how it handles errors, 52 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 53 documentation for more details</a>. 54 </p> 55<h5> 56<a name="math_toolkit.inv_hyper.asinh.h0"></a> 57 <span class="phrase"><a name="math_toolkit.inv_hyper.asinh.accuracy"></a></span><a class="link" href="asinh.html#math_toolkit.inv_hyper.asinh.accuracy">Accuracy</a> 58 </h5> 59<p> 60 Generally accuracy is to within 1 or 2 <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine 61 epsilon</a> across all supported platforms. 62 </p> 63<h5> 64<a name="math_toolkit.inv_hyper.asinh.h1"></a> 65 <span class="phrase"><a name="math_toolkit.inv_hyper.asinh.testing"></a></span><a class="link" href="asinh.html#math_toolkit.inv_hyper.asinh.testing">Testing</a> 66 </h5> 67<p> 68 This function is tested using a combination of random test values designed 69 to give full function coverage computed at high precision using the "naive" 70 formula: 71 </p> 72<div class="blockquote"><blockquote class="blockquote"><p> 73 <span class="inlinemediaobject"><img src="../../../equations/asinh1.svg"></span> 74 75 </p></blockquote></div> 76<p> 77 along with a selection of sanity check values computed using functions.wolfram.com 78 to at least 50 decimal digits. 79 </p> 80<h5> 81<a name="math_toolkit.inv_hyper.asinh.h2"></a> 82 <span class="phrase"><a name="math_toolkit.inv_hyper.asinh.implementation"></a></span><a class="link" href="asinh.html#math_toolkit.inv_hyper.asinh.implementation">Implementation</a> 83 </h5> 84<p> 85 For sufficiently large x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/06/01/0001/" target="_top">approximation</a>: 86 </p> 87<div class="blockquote"><blockquote class="blockquote"><p> 88 <span class="inlinemediaobject"><img src="../../../equations/asinh2.svg"></span> 89 90 </p></blockquote></div> 91<p> 92 While for very small x we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/03/01/0001/" target="_top">approximation</a>: 93 </p> 94<div class="blockquote"><blockquote class="blockquote"><p> 95 <span class="inlinemediaobject"><img src="../../../equations/asinh3.svg"></span> 96 97 </p></blockquote></div> 98<p> 99 For 0.5 > x > ε the following rearrangement of the primary definition 100 is used: 101 </p> 102<div class="blockquote"><blockquote class="blockquote"><p> 103 <span class="inlinemediaobject"><img src="../../../equations/asinh4.svg"></span> 104 105 </p></blockquote></div> 106<p> 107 Otherwise evaluation is via the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcSinh/02/" target="_top">primary 108 definition</a>: 109 </p> 110<div class="blockquote"><blockquote class="blockquote"><p> 111 <span class="inlinemediaobject"><img src="../../../equations/asinh4.svg"></span> 112 113 </p></blockquote></div> 114</div> 115<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 116<td align="left"></td> 117<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 118 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 119 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 120 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 121 Daryle Walker and Xiaogang Zhang<p> 122 Distributed under the Boost Software License, Version 1.0. (See accompanying 123 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>) 124 </p> 125</div></td> 126</tr></table> 127<hr> 128<div class="spirit-nav"> 129<a accesskey="p" href="acosh.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="atanh.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 130</div> 131</body> 132</html> 133