mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-10-25 01:50:30 -04:00
190 lines
13 KiB
HTML
190 lines
13 KiB
HTML
|
<html>
|
||
|
|
<head>
|
||
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||
|
|
<title>log1p</title>
|
||
|
|
<link rel="stylesheet" href="../../math.css" type="text/css">
|
||
|
|
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
||
|
|
<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
|
||
|
|
<link rel="up" href="../powers.html" title="Basic Functions">
|
||
|
|
<link rel="prev" href="cos_pi.html" title="cos_pi">
|
||
|
|
<link rel="next" href="expm1.html" title="expm1">
|
||
|
|
</head>
|
||
|
|
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||
|
|
<table cellpadding="2" width="100%"><tr>
|
||
|
|
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
|
||
|
|
<td align="center"><a href="../../../../../../index.html">Home</a></td>
|
||
|
|
<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
|
||
|
|
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
||
|
|
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
||
|
|
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
|
||
|
|
</tr></table>
|
||
|
|
<hr>
|
||
|
|
<div class="spirit-nav">
|
||
|
|
<a accesskey="p" href="cos_pi.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../powers.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="expm1.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
|
|
</div>
|
||
|
|
<div class="section">
|
||
|
|
<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"><</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>
|
||
|
|
</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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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>
|
||
|
|
<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||
|
|
|
||
|
|
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||
|
|
</pre>
|
||
|
|
<p>
|
||
|
|
Returns the natural logarithm of <code class="computeroutput"><span class="identifier">x</span><span class="special">+</span><span class="number">1</span></code>.
|
||
|
|
</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 15. 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 15. 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 <code class="computeroutput"><span class="identifier">x</span></code> then
|
||
|
|
<code class="computeroutput"><span class="identifier">x</span><span class="special">+</span><span class="number">1</span></code> suffers from catastrophic cancellation errors
|
||
|
|
so that <code class="computeroutput"><span class="identifier">x</span><span class="special">+</span><span class="number">1</span> <span class="special">==</span> <span class="number">1</span></code>
|
||
|
|
and <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> <span class="special">==</span> <span class="number">0</span></code>,
|
||
|
|
when in fact for very small x, the best approximation to <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> would be
|
||
|
|
<code class="computeroutput"><span class="identifier">x</span></code>. <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 2ɛ). Alternatively note that
|
||
|
|
there are faster methods available, for example using the equivalence:
|
||
|
|
</p>
|
||
|
|
<pre class="programlisting"><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> <span class="special">==</span> <span class="special">(</span><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> <span class="special">*</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">/</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><span class="special">)</span>
|
||
|
|
</pre>
|
||
|
|
<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 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>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../graphs/log1p.svg" align="middle"></span>
|
||
|
|
</p>
|
||
|
|
<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 epsilon accuracy.
|
||
|
|
</p>
|
||
|
|
<div class="table">
|
||
|
|
<a name="math_toolkit.powers.log1p.table_log1p"></a><p class="title"><b>Table 6.78. 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>
|
||
|
|
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
GNU C++ version 5.1.0<br> linux<br> double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
GNU C++ version 5.1.0<br> linux<br> long double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
</tr></thead>
|
||
|
|
<tbody><tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Random test data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0.509ε (Mean = 0.057ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.509ε (Mean = 0.057ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0.846ε (Mean = 0.153ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 0.846ε (Mean = 0.153ε))<br>
|
||
|
|
(<span class="emphasis"><em>Cephes:</em></span> Max = 0.799ε (Mean = 0.122ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0.818ε (Mean = 0.227ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 0.818ε (Mean = 0.227ε))<br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.818ε (Mean = 0.227ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.53ε (Mean = 0.627ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.818ε (Mean = 0.249ε))
|
||
|
|
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||
|
|
Benjamin Sobotta, 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>)
|
||
|
|
</p>
|
||
|
|
</div></td>
|
||
|
|
</tr></table>
|
||
|
|
<hr>
|
||
|
|
<div class="spirit-nav">
|
||
|
|
<a accesskey="p" href="cos_pi.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../powers.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="expm1.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
|
|
</div>
|
||
|
|
</body>
|
||
|
|
</html>
|