ceph/src/boost/libs/math/doc/html/math_toolkit/powers/sqrt1pm1.html

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  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">&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">sqrt1pm1</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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&#160;15.&#160;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&#160;15.&#160;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&#160;6.81.&#160;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 = 1.36&#949; (Mean = 0.44&#949;)</span>
 120                 </p>
 121               </td>
 122 <td>
 123                 <p>
 124                   <span class="blue">Max = 1.3&#949; (Mean = 0.404&#949;)</span>
 125                 </p>
 126               </td>
 127 <td>
 128                 <p>
 129                   <span class="blue">Max = 1.33&#949; (Mean = 0.409&#949;)</span>
 130                 </p>
 131               </td>
 132 <td>
 133                 <p>
 134                   <span class="blue">Max = 1.54&#949; (Mean = 0.568&#949;)</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 &#169; 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 R&#229;de, 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>)
 157       </p>
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