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| 24 | </div> | |
| 25 | <div class="section"> | |
| 26 | <div class="titlepage"><div><div><h3 class="title"> | |
| 27 | <a name="math_toolkit.powers.sqrt1pm1"></a><a class="link" href="sqrt1pm1.html" title="sqrt1pm1">sqrt1pm1</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">sqrt1pm1</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
| 30 | </pre> | |
| 31 | <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> | |
| 32 | ||
| 33 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
| 34 | <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">sqrt1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> | |
| 35 | ||
| 36 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 37 | <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">sqrt1pm1</span><span class="special">(</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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 38 | ||
| 39 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
| 40 | </pre> | |
| 41 | <p> | |
| 42 | Returns <code class="computeroutput"><span class="identifier">sqrt</span><span class="special">(</span><span class="number">1</span><span class="special">+</span><span class="identifier">x</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span></code>. | |
| 43 | </p> | |
| 44 | <p> | |
| 45 | 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 | |
| 46 | type calculation rules</em></span></a>: the return is <code class="computeroutput"><span class="keyword">double</span></code> | |
| 47 | when <span class="emphasis"><em>x</em></span> is an integer type and T otherwise. | |
| 48 | </p> | |
| 49 | <p> | |
| 50 | The final <a class="link" href="../../policy.html" title="Chapter 15. 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 15. Policies: Controlling Precision, Error Handling etc">policy | |
| 53 | documentation for more details</a>. | |
| 54 | </p> | |
| 55 | <p> | |
| 56 | This function is useful when you need the difference between sqrt(x) and | |
| 57 | 1, when x is itself close to 1. | |
| 58 | </p> | |
| 59 | <p> | |
| 60 | Implemented in terms of <code class="computeroutput"><span class="identifier">log1p</span></code> | |
| 61 | and <code class="computeroutput"><span class="identifier">expm1</span></code>. | |
| 62 | </p> | |
| 63 | <p> | |
| 64 | The following graph illustrates the behaviour of sqrt1pm1: | |
| 65 | </p> | |
| 66 | <p> | |
| 67 | <span class="inlinemediaobject"><img src="../../../graphs/sqrt1pm1.svg" align="middle"></span> | |
| 68 | </p> | |
| 69 | <h5> | |
| 70 | <a name="math_toolkit.powers.sqrt1pm1.h0"></a> | |
| 71 | <span class="phrase"><a name="math_toolkit.powers.sqrt1pm1.accuracy"></a></span><a class="link" href="sqrt1pm1.html#math_toolkit.powers.sqrt1pm1.accuracy">Accuracy</a> | |
| 72 | </h5> | |
| 73 | <p> | |
| 74 | For built in floating-point types <code class="computeroutput"><span class="identifier">sqrt1pm1</span></code> | |
| 75 | should have approximately 3 epsilon accuracy. | |
| 76 | </p> | |
| 77 | <div class="table"> | |
| 78 | <a name="math_toolkit.powers.sqrt1pm1.table_sqrt1pm1"></a><p class="title"><b>Table 6.81. Error rates for sqrt1pm1</b></p> | |
| 79 | <div class="table-contents"><table class="table" summary="Error rates for sqrt1pm1"> | |
| 80 | <colgroup> | |
| 81 | <col> | |
| 82 | <col> | |
| 83 | <col> | |
| 84 | <col> | |
| 85 | <col> | |
| 86 | </colgroup> | |
| 87 | <thead><tr> | |
| 88 | <th> | |
| 89 | </th> | |
| 90 | <th> | |
| 91 | <p> | |
| 92 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
| 93 | </p> | |
| 94 | </th> | |
| 95 | <th> | |
| 96 | <p> | |
| 97 | GNU C++ version 5.1.0<br> linux<br> double | |
| 98 | </p> | |
| 99 | </th> | |
| 100 | <th> | |
| 101 | <p> | |
| 102 | GNU C++ version 5.1.0<br> linux<br> long double | |
| 103 | </p> | |
| 104 | </th> | |
| 105 | <th> | |
| 106 | <p> | |
| 107 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
| 108 | </p> | |
| 109 | </th> | |
| 110 | </tr></thead> | |
| 111 | <tbody><tr> | |
| 112 | <td> | |
| 113 | <p> | |
| 114 | sqrt1pm1 | |
| 115 | </p> | |
| 116 | </td> | |
| 117 | <td> | |
| 118 | <p> | |
| 119 | <span class="blue">Max = 1.36ε (Mean = 0.44ε)</span> | |
| 120 | </p> | |
| 121 | </td> | |
| 122 | <td> | |
| 123 | <p> | |
| 124 | <span class="blue">Max = 1.3ε (Mean = 0.404ε)</span> | |
| 125 | </p> | |
| 126 | </td> | |
| 127 | <td> | |
| 128 | <p> | |
| 129 | <span class="blue">Max = 1.33ε (Mean = 0.409ε)</span> | |
| 130 | </p> | |
| 131 | </td> | |
| 132 | <td> | |
| 133 | <p> | |
| 134 | <span class="blue">Max = 1.54ε (Mean = 0.568ε)</span> | |
| 135 | </p> | |
| 136 | </td> | |
| 137 | </tr></tbody> | |
| 138 | </table></div> | |
| 139 | </div> | |
| 140 | <br class="table-break"><h5> | |
| 141 | <a name="math_toolkit.powers.sqrt1pm1.h1"></a> | |
| 142 | <span class="phrase"><a name="math_toolkit.powers.sqrt1pm1.testing"></a></span><a class="link" href="sqrt1pm1.html#math_toolkit.powers.sqrt1pm1.testing">Testing</a> | |
| 143 | </h5> | |
| 144 | <p> | |
| 145 | A selection of random high precision test values calculated using NTL::RR | |
| 146 | at 1000-bit precision. | |
| 147 | </p> | |
| 148 | </div> | |
| 149 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 150 | <td align="left"></td> | |
| 151 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 152 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 153 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 154 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 155 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 156 | 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>) | |
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