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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.powers.log1p"></a><a class="link" href="log1p.html" title="log1p">log1p</a>
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</h3></div></div></div>
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<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">log1p</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<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>
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<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>
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<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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Returns the natural logarithm of <span class="emphasis"><em>x+1</em></span>.
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</p>
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<p>
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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
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type calculation rules</em></span></a>: the return is <code class="computeroutput"><span class="keyword">double</span></code>
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when <span class="emphasis"><em>x</em></span> is an integer type and T otherwise.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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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
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documentation for more details</a>.
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</p>
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<p>
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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>.
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However, for small <span class="emphasis"><em>x</em></span> then <span class="emphasis"><em>x+1</em></span> suffers
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from catastrophic cancellation errors so that <span class="emphasis"><em>x+1 == 1</em></span>
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and <span class="emphasis"><em>log(x+1) == 0</em></span>, when in fact for very small x, the
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best approximation to <span class="emphasis"><em>log(x+1)</em></span> would be <span class="emphasis"><em>x</em></span>.
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<code class="computeroutput"><span class="identifier">log1p</span></code> calculates the best
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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
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(less than 2ɛ). Alternatively note that there are faster methods available,
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for example using the equivalence:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="emphasis"><em>log(1+x) == (log(1+x) * x) / ((1+x) - 1)</em></span>
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</p></blockquote></div>
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<p>
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However, experience has shown that these methods tend to fail quite spectacularly
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once the compiler's optimizations are turned on, consequently they are used
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only when known not to break with a particular compiler. In contrast, the
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series expansion method seems to be reasonably immune to optimizer-induced
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errors.
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</p>
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<p>
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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>
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specializations of this template simply forward to the platform's native
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(POSIX) implementation of this function.
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</p>
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<p>
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The following graph illustrates the behaviour of log1p:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/log1p.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.powers.log1p.h0"></a>
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<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>
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</h5>
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<p>
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For built in floating point types <code class="computeroutput"><span class="identifier">log1p</span></code>
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should have approximately 1 <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine
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epsilon</a> accuracy.
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</p>
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<div class="table">
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<a name="math_toolkit.powers.log1p.table_log1p"></a><p class="title"><b>Table 8.81. Error rates for log1p</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for log1p">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
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</p>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 14.1<br> Win32<br> double
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</p>
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</th>
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</tr></thead>
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<tbody><tr>
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<td>
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<p>
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Random test data
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.818ε (Mean = 0.227ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 0.818ε (Mean = 0.227ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.818ε (Mean = 0.227ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.846ε (Mean = 0.153ε)</span><br> <br>
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(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 0.846ε (Mean = 0.153ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.3ε (Mean = 0.66ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.818ε (Mean = 0.249ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.509ε (Mean = 0.057ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.509ε (Mean = 0.057ε))
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</p>
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</td>
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</tr></tbody>
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</table></div>
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</div>
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<br class="table-break"><h5>
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<a name="math_toolkit.powers.log1p.h1"></a>
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<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>
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</h5>
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<p>
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A mixture of spot test sanity checks, and random high precision test values
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calculated using NTL::RR at 1000-bit precision.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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