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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.powers.log1p"></a><a class="link" href="log1p.html" title="log1p">log1p</a>
</h3></div></div></div>
<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>
</pre>
<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>
<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>
<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">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>
<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>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
Returns the natural logarithm of <span class="emphasis"><em>x+1</em></span>.
</p>
<p>
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
type calculation rules</em></span></a>: the return is <code class="computeroutput"><span class="keyword">double</span></code>
when <span class="emphasis"><em>x</em></span> is an integer type and T otherwise.
</p>
<p>
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
be used to control the behaviour of the function: how it handles errors,
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
documentation for more details</a>.
</p>
<p>
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>.
However, for small <span class="emphasis"><em>x</em></span> then <span class="emphasis"><em>x+1</em></span> suffers
from catastrophic cancellation errors so that <span class="emphasis"><em>x+1 == 1</em></span>
and <span class="emphasis"><em>log(x+1) == 0</em></span>, when in fact for very small x, the
best approximation to <span class="emphasis"><em>log(x+1)</em></span> would be <span class="emphasis"><em>x</em></span>.
<code class="computeroutput"><span class="identifier">log1p</span></code> calculates the best
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
(less than 2&#603;). Alternatively note that there are faster methods available,
for example using the equivalence:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="emphasis"><em>log(1+x) == (log(1+x) * x) / ((1+x) - 1)</em></span>
</p></blockquote></div>
<p>
However, experience has shown that these methods tend to fail quite spectacularly
once the compiler's optimizations are turned on, consequently they are used
only when known not to break with a particular compiler. In contrast, the
series expansion method seems to be reasonably immune to optimizer-induced
errors.
</p>
<p>
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>
specializations of this template simply forward to the platform's native
(POSIX) implementation of this function.
</p>
<p>
The following graph illustrates the behaviour of log1p:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/log1p.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.powers.log1p.h0"></a>
<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>
</h5>
<p>
For built in floating point types <code class="computeroutput"><span class="identifier">log1p</span></code>
should have approximately 1 <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine
epsilon</a> accuracy.
</p>
<div class="table">
<a name="math_toolkit.powers.log1p.table_log1p"></a><p class="title"><b>Table&#160;8.81.&#160;Error rates for log1p</b></p>
<div class="table-contents"><table class="table" summary="Error rates for log1p">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody><tr>
<td>
<p>
Random test data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.818&#949; (Mean = 0.227&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.818&#949; (Mean = 0.227&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.818&#949; (Mean = 0.227&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.846&#949; (Mean = 0.153&#949;)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 0.846&#949; (Mean = 0.153&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.3&#949; (Mean = 0.66&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.818&#949; (Mean = 0.249&#949;))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.509&#949; (Mean = 0.057&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.509&#949; (Mean = 0.057&#949;))
</p>
</td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><h5>
<a name="math_toolkit.powers.log1p.h1"></a>
<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>
</h5>
<p>
A mixture of spot test sanity checks, and random high precision test values
calculated using NTL::RR at 1000-bit precision.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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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