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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.powers.log1p"></a><a class="link" href="log1p.html" title="log1p">log1p</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">log1p</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">log1p</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;20.&#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">log1p</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;20.&#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 the natural logarithm of <span class="emphasis"><em>x+1</em></span>.
  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;20.&#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;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  53         documentation for more details</a>.
  54       </p>
  55 <p>
  56         There are many situations where it is desirable to compute <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="identifier">x</span><span class="special">+</span><span class="number">1</span><span class="special">)</span></code>.
  57         However, for small <span class="emphasis"><em>x</em></span> then <span class="emphasis"><em>x+1</em></span> suffers
  58         from catastrophic cancellation errors so that <span class="emphasis"><em>x+1 == 1</em></span>
  59         and <span class="emphasis"><em>log(x+1) == 0</em></span>, when in fact for very small x, the
  60         best approximation to <span class="emphasis"><em>log(x+1)</em></span> would be <span class="emphasis"><em>x</em></span>.
  61         <code class="computeroutput"><span class="identifier">log1p</span></code> calculates the best
  62         approximation to <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="number">1</span><span class="special">+</span><span class="identifier">x</span><span class="special">)</span></code> using a Taylor series expansion for accuracy
  63         (less than 2&#603;). Alternatively note that there are faster methods available,
  64         for example using the equivalence:
  65       </p>
  66 <div class="blockquote"><blockquote class="blockquote"><p>
  67           <span class="emphasis"><em>log(1+x) == (log(1+x) * x) / ((1+x) - 1)</em></span>
  68         </p></blockquote></div>
  69 <p>
  70         However, experience has shown that these methods tend to fail quite spectacularly
  71         once the compiler's optimizations are turned on, consequently they are used
  72         only when known not to break with a particular compiler. In contrast, the
  73         series expansion method seems to be reasonably immune to optimizer-induced
  74         errors.
  75       </p>
  76 <p>
  77         Finally when macro BOOST_HAS_LOG1P is defined then the <code class="computeroutput"><span class="keyword">float</span><span class="special">/</span><span class="keyword">double</span><span class="special">/</span><span class="keyword">long</span> <span class="keyword">double</span></code>
  78         specializations of this template simply forward to the platform's native
  79         (POSIX) implementation of this function.
  80       </p>
  81 <p>
  82         The following graph illustrates the behaviour of log1p:
  83       </p>
  84 <div class="blockquote"><blockquote class="blockquote"><p>
  85           <span class="inlinemediaobject"><img src="../../../graphs/log1p.svg" align="middle"></span>
  86
  87         </p></blockquote></div>
  88 <h5>
  89 <a name="math_toolkit.powers.log1p.h0"></a>
  90         <span class="phrase"><a name="math_toolkit.powers.log1p.accuracy"></a></span><a class="link" href="log1p.html#math_toolkit.powers.log1p.accuracy">Accuracy</a>
  91       </h5>
  92 <p>
  93         For built in floating point types <code class="computeroutput"><span class="identifier">log1p</span></code>
  94         should have approximately 1 <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine
  95         epsilon</a> accuracy.
  96       </p>
  97 <div class="table">
  98 <a name="math_toolkit.powers.log1p.table_log1p"></a><p class="title"><b>Table&#160;8.81.&#160;Error rates for log1p</b></p>
  99 <div class="table-contents"><table class="table" summary="Error rates for log1p">
 100 <colgroup>
 101 <col>
 102 <col>
 103 <col>
 104 <col>
 105 <col>
 106 </colgroup>
 107 <thead><tr>
 108 <th>
 109               </th>
 110 <th>
 111                 <p>
 112                   GNU C++ version 7.1.0<br> linux<br> long double
 113                 </p>
 114               </th>
 115 <th>
 116                 <p>
 117                   GNU C++ version 7.1.0<br> linux<br> double
 118                 </p>
 119               </th>
 120 <th>
 121                 <p>
 122                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 123                 </p>
 124               </th>
 125 <th>
 126                 <p>
 127                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 128                 </p>
 129               </th>
 130 </tr></thead>
 131 <tbody><tr>
 132 <td>
 133                 <p>
 134                   Random test data
 135                 </p>
 136               </td>
 137 <td>
 138                 <p>
 139                   <span class="blue">Max = 0.818&#949; (Mean = 0.227&#949;)</span><br> <br>
 140                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.818&#949; (Mean = 0.227&#949;))<br>
 141                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.818&#949; (Mean = 0.227&#949;))
 142                 </p>
 143               </td>
 144 <td>
 145                 <p>
 146                   <span class="blue">Max = 0.846&#949; (Mean = 0.153&#949;)</span><br> <br>
 147                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 0.846&#949; (Mean = 0.153&#949;))
 148                 </p>
 149               </td>
 150 <td>
 151                 <p>
 152                   <span class="blue">Max = 2.3&#949; (Mean = 0.66&#949;)</span><br> <br>
 153                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.818&#949; (Mean = 0.249&#949;))
 154                 </p>
 155               </td>
 156 <td>
 157                 <p>
 158                   <span class="blue">Max = 0.509&#949; (Mean = 0.057&#949;)</span><br> <br>
 159                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.509&#949; (Mean = 0.057&#949;))
 160                 </p>
 161               </td>
 162 </tr></tbody>
 163 </table></div>
 164 </div>
 165 <br class="table-break"><h5>
 166 <a name="math_toolkit.powers.log1p.h1"></a>
 167         <span class="phrase"><a name="math_toolkit.powers.log1p.testing"></a></span><a class="link" href="log1p.html#math_toolkit.powers.log1p.testing">Testing</a>
 168       </h5>
 169 <p>
 170         A mixture of spot test sanity checks, and random high precision test values
 171         calculated using NTL::RR at 1000-bit precision.
 172       </p>
 173 </div>
 174 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 175 <td align="left"></td>
 176 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 177       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 178       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 179       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 180       Daryle Walker and Xiaogang Zhang<p>
 181         Distributed under the Boost Software License, Version 1.0. (See accompanying
 182         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>)
 183       </p>
 184 </div></td>
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