mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2026-06-02 05:54:41 -04:00
Squashed 'boost/' content from commit b4feb19f2
git-subtree-dir: boost git-subtree-split: b4feb19f287ee92d87a9624b5d36b7cf46aeadeb
This commit is contained in:
Bill Somerville
@@ -0,0 +1,165 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>cbrt</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="expm1.html" title="expm1">
|
||||
<link rel="next" href="sqrt1pm1.html" title="sqrt1pm1">
|
||||
</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="expm1.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="sqrt1pm1.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.cbrt"></a><a class="link" href="cbrt.html" title="cbrt">cbrt</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">cbrt</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">cbrt</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">cbrt</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 cubed root of x: x<sup>1/3</sup>.
|
||||
</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>
|
||||
Implemented using Halley iteration.
|
||||
</p>
|
||||
<p>
|
||||
The following graph illustrates the behaviour of cbrt:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../graphs/cbrt.svg" align="middle"></span>
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.cbrt.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.cbrt.accuracy"></a></span><a class="link" href="cbrt.html#math_toolkit.powers.cbrt.accuracy">Accuracy</a>
|
||||
</h5>
|
||||
<p>
|
||||
For built in floating-point types <code class="computeroutput"><span class="identifier">cbrt</span></code>
|
||||
should have approximately 2 epsilon accuracy.
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="math_toolkit.powers.cbrt.table_cbrt"></a><p class="title"><b>Table 6.80. Error rates for cbrt</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for cbrt">
|
||||
<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>
|
||||
cbrt Function
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.7ε (Mean = 0.565ε)</span><br> <br>
|
||||
(<span class="emphasis"><em><math.h>:</em></span> Max = 1.7ε (Mean = 0.565ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Cephes:</em></span>
|
||||
Max = 0ε (Mean = 0ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.34ε (Mean = 0.471ε)</span><br> <br>
|
||||
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.34ε (Mean = 0.471ε))<br>
|
||||
(<span class="emphasis"><em><math.h>:</em></span> Max = 1.34ε (Mean = 0.471ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.11ε (Mean = 0.424ε)</span><br> <br>
|
||||
(<span class="emphasis"><em><math.h>:</em></span> Max = 1.11ε (Mean = 0.424ε))
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><h5>
|
||||
<a name="math_toolkit.powers.cbrt.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.cbrt.testing"></a></span><a class="link" href="cbrt.html#math_toolkit.powers.cbrt.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="expm1.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="sqrt1pm1.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,169 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>cos_pi</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="sin_pi.html" title="sin_pi">
|
||||
<link rel="next" href="log1p.html" title="log1p">
|
||||
</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="sin_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="log1p.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.cos_pi"></a><a class="link" href="cos_pi.html" title="cos_pi">cos_pi</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">cos_pi</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">cos_pi</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">cos_pi</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 cosine of <span class="emphasis"><em>πx</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 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>
|
||||
This function performs exact all-integer arithmetic argument reduction before
|
||||
computing the cosine of <span class="emphasis"><em>πx</em></span>.
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="math_toolkit.powers.cos_pi.table_cos_pi"></a><p class="title"><b>Table 6.77. Error rates for cos_pi</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for cos_pi">
|
||||
<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>
|
||||
sin_pi and cos_pi
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.996ε (Mean = 0.281ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.991ε (Mean = 0.302ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.991ε (Mean = 0.302ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
sin_pi and cos_pi near integers and half integers
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.996ε (Mean = 0.298ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.976ε (Mean = 0.28ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.976ε (Mean = 0.28ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break">
|
||||
</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="sin_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="log1p.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,265 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Compile Time Power of a Runtime Base</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="hypot.html" title="hypot">
|
||||
<link rel="next" href="../sinc.html" title="Sinus Cardinal and Hyperbolic Sinus Cardinal Functions">
|
||||
</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="hypot.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="../sinc.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.ct_pow"></a><a class="link" href="ct_pow.html" title="Compile Time Power of a Runtime Base">Compile Time Power of a Runtime
|
||||
Base</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
The <code class="computeroutput"><span class="identifier">pow</span></code> function effectively
|
||||
computes the compile-time integral power of a run-time base.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.ct_pow.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.ct_pow.synopsis"></a></span><a class="link" href="ct_pow.html#math_toolkit.powers.ct_pow.synopsis">Synopsis</a>
|
||||
</h5>
|
||||
<p>
|
||||
<a href="../../../../../../boost/math/special_functions/pow.hpp" target="_top"><code class="computeroutput"><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">pow</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code></a>
|
||||
</p>
|
||||
<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">int</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">typename</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">pow</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">base</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">int</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</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">pow</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">base</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">policy</span><span class="special">);</span>
|
||||
|
||||
<span class="special">}}</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.ct_pow.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.ct_pow.rationale_and_usage"></a></span><a class="link" href="ct_pow.html#math_toolkit.powers.ct_pow.rationale_and_usage">Rationale
|
||||
and Usage</a>
|
||||
</h5>
|
||||
<p>
|
||||
Computing the power of a number with an exponent that is known at compile
|
||||
time is a common need for programmers. In such cases, the usual method is
|
||||
to avoid the overhead implied by the <code class="computeroutput"><span class="identifier">pow</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">powf</span></code> and <code class="computeroutput"><span class="identifier">powl</span></code>
|
||||
C functions by hardcoding an expression such as:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">// Hand-written 8th power of a 'base' variable</span>
|
||||
<span class="keyword">double</span> <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
However, this kind of expression is not really readable (knowing the value
|
||||
of the exponent involves counting the number of occurrences of <span class="emphasis"><em>base</em></span>),
|
||||
error-prone (it's easy to forget an occurrence), syntactically bulky, and
|
||||
non-optimal in terms of performance.
|
||||
</p>
|
||||
<p>
|
||||
The pow function of Boost.Math helps writing this kind expression along with
|
||||
solving all the problems listed above:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">// 8th power of a 'base' variable using math::pow</span>
|
||||
<span class="keyword">double</span> <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">pow</span><span class="special"><</span><span class="number">8</span><span class="special">>(</span><span class="identifier">base</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
The expression is now shorter, easier to read, safer, and even faster. Indeed,
|
||||
<code class="computeroutput"><span class="identifier">pow</span></code> will compute the expression
|
||||
such that only log2(N) products are made for a power of N. For instance in
|
||||
the example above, the resulting expression will be the same as if we had
|
||||
written this, with only one computation of each identical subexpression:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">// Internal effect of pow<8>(base)</span>
|
||||
<span class="keyword">double</span> <span class="identifier">result</span> <span class="special">=</span> <span class="special">((</span><span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">)*(</span><span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">))*((</span><span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">)*(</span><span class="identifier">base</span><span class="special">*</span><span class="identifier">base</span><span class="special">));</span>
|
||||
</pre>
|
||||
<p>
|
||||
Only 3 different products were actually computed.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.ct_pow.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.ct_pow.return_type"></a></span><a class="link" href="ct_pow.html#math_toolkit.powers.ct_pow.return_type">Return
|
||||
Type</a>
|
||||
</h5>
|
||||
<p>
|
||||
The return type of these functions 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>. For example:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
If T is a <code class="computeroutput"><span class="keyword">float</span></code>, the return
|
||||
type is a <code class="computeroutput"><span class="keyword">float</span></code>.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
If T is a <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>,
|
||||
the return type is a <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
Otherwise, the return type is a <code class="computeroutput"><span class="keyword">double</span></code>.
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.ct_pow.h3"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.ct_pow.policies"></a></span><a class="link" href="ct_pow.html#math_toolkit.powers.ct_pow.policies">Policies</a>
|
||||
</h5>
|
||||
<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>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.ct_pow.h4"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.ct_pow.error_handling"></a></span><a class="link" href="ct_pow.html#math_toolkit.powers.ct_pow.error_handling">Error
|
||||
Handling</a>
|
||||
</h5>
|
||||
<p>
|
||||
Two cases of errors can occur when using <code class="computeroutput"><span class="identifier">pow</span></code>:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
In case of null base and negative exponent, an <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
|
||||
occurs since this operation is a division by 0 (it equals to 1/0).
|
||||
</li>
|
||||
<li class="listitem">
|
||||
In case of null base and null exponent, an <a class="link" href="../error_handling.html#math_toolkit.error_handling.indeterminate_result_error">indeterminate_result_error</a>
|
||||
occurs since the result of this operation is indeterminate. Those errors
|
||||
follow the <a class="link" href="../error_handling.html" title="Error Handling">general policies
|
||||
of error handling in Boost.Math</a>.
|
||||
</li>
|
||||
</ul></div>
|
||||
<p>
|
||||
The default overflow error policy is <code class="computeroutput"><span class="identifier">throw_on_error</span></code>.
|
||||
A call like <code class="computeroutput"><span class="identifier">pow</span><span class="special"><-</span><span class="number">2</span><span class="special">>(</span><span class="number">0</span><span class="special">)</span></code> will thus throw a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">overflow_error</span></code>
|
||||
exception. As shown in the link given above, other error handling policies
|
||||
can be used:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">errno_on_error</span></code>: Sets
|
||||
<code class="computeroutput"><span class="special">::</span><span class="identifier">errno</span></code>
|
||||
to <code class="computeroutput"><span class="identifier">ERANGE</span></code> and returns
|
||||
<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">()</span></code>.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">ignore_error</span></code>: Returns
|
||||
<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">()</span></code>.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">user_error</span></code>: Returns the
|
||||
result of <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">user_overflow_error</span></code>: this function
|
||||
must be defined by the user.
|
||||
</li>
|
||||
</ul></div>
|
||||
<p>
|
||||
The default indeterminate result error policy is <code class="computeroutput"><span class="identifier">ignore_error</span></code>,
|
||||
which for this function returns 1 since it's the most commonly chosen result
|
||||
for a power of 0. Here again, other error handling policies can be used:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">throw_on_error</span></code>: Throws
|
||||
<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">domain_error</span></code>
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">errno_on_error</span></code>: Sets
|
||||
<code class="computeroutput"><span class="special">::</span><span class="identifier">errno</span></code>
|
||||
to <code class="computeroutput"><span class="identifier">EDOM</span></code> and returns 1.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">user_error</span></code>: Returns the
|
||||
result of <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">user_indeterminate_result_error</span></code>: this
|
||||
function must be defined by the user.
|
||||
</li>
|
||||
</ul></div>
|
||||
<p>
|
||||
Here is an example of error handling customization where we want to specify
|
||||
the result that has to be returned in case of error. We will thus use the
|
||||
<code class="computeroutput"><span class="identifier">user_error</span></code> policy, by passing
|
||||
as second argument an instance of an overflow_error policy templated with
|
||||
<code class="computeroutput"><span class="identifier">user_error</span></code>:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">// First we open the boost::math::policies namespace and define the `user_overflow_error`</span>
|
||||
<span class="comment">// by making it return the value we want in case of error (-1 here)</span>
|
||||
|
||||
<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">namespace</span> <span class="identifier">policies</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="identifier">T</span> <span class="identifier">user_overflow_error</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">char</span><span class="special">*,</span> <span class="keyword">const</span> <span class="keyword">char</span><span class="special">*,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&)</span>
|
||||
<span class="special">{</span> <span class="keyword">return</span> <span class="special">-</span><span class="number">1</span><span class="special">;</span> <span class="special">}</span>
|
||||
<span class="special">}}}</span>
|
||||
|
||||
|
||||
<span class="comment">// Then we invoke pow and indicate that we want to use the user_error policy</span>
|
||||
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">;</span>
|
||||
<span class="keyword">double</span> <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">pow</span><span class="special"><-</span><span class="number">5</span><span class="special">>(</span><span class="identifier">base</span><span class="special">,</span> <span class="identifier">policy</span><span class="special"><</span><span class="identifier">overflow_error</span><span class="special"><</span><span class="identifier">user_error</span><span class="special">></span> <span class="special">>());</span>
|
||||
|
||||
<span class="comment">// We can now test the returned value and treat the special case if needed:</span>
|
||||
<span class="keyword">if</span> <span class="special">(</span><span class="identifier">result</span> <span class="special">==</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="comment">// there was an error, do something...</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
Another way is to redefine the default <code class="computeroutput"><span class="identifier">overflow_error</span></code>
|
||||
policy by using the BOOST_MATH_OVERFLOW_ERROR_POLICY macro. Once the <code class="computeroutput"><span class="identifier">user_overflow_error</span></code> function is defined
|
||||
as above, we can achieve the same result like this:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">// Redefine the default error_overflow policy</span>
|
||||
<span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_OVERFLOW_ERROR_POLICY</span> <span class="identifier">user_error</span>
|
||||
<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">pow</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
|
||||
<span class="comment">// From this point, passing a policy in argument is no longer needed, a call like this one</span>
|
||||
<span class="comment">// will return -1 in case of error:</span>
|
||||
|
||||
<span class="keyword">double</span> <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">pow</span><span class="special"><-</span><span class="number">5</span><span class="special">>(</span><span class="identifier">base</span><span class="special">);</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.ct_pow.h5"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.ct_pow.acknowledgements"></a></span><a class="link" href="ct_pow.html#math_toolkit.powers.ct_pow.acknowledgements">Acknowledgements</a>
|
||||
</h5>
|
||||
<p>
|
||||
Bruno Lalande submitted this addition to Boost.Math.
|
||||
</p>
|
||||
<p>
|
||||
Thanks to Joaquín López Muñoz and Scott McMurray for their help in
|
||||
improving the implementation.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.ct_pow.h6"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.ct_pow.references"></a></span><a class="link" href="ct_pow.html#math_toolkit.powers.ct_pow.references">References</a>
|
||||
</h5>
|
||||
<p>
|
||||
D.E. Knuth, <span class="emphasis"><em>The Art of Computer Programming, Vol. 2: Seminumerical
|
||||
Algorithms</em></span>, 2nd ed., Addison-Wesley, Reading, MA, 1981
|
||||
</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="hypot.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="../sinc.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,177 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>expm1</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="log1p.html" title="log1p">
|
||||
<link rel="next" href="cbrt.html" title="cbrt">
|
||||
</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="log1p.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="cbrt.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.expm1"></a><a class="link" href="expm1.html" title="expm1">expm1</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">expm1</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">expm1</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">expm1</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 e<sup>x</sup> - 1.
|
||||
</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>
|
||||
For small x, then <code class="computeroutput">e<sup>x</sup></code> is very close to 1, as a result calculating <code class="computeroutput">e<sup>x</sup> - 1</code> results in
|
||||
catastrophic cancellation errors when x is small. <code class="computeroutput"><span class="identifier">expm1</span></code>
|
||||
calculates <code class="computeroutput">e<sup>x</sup> - 1</code> using rational approximations (for up to 128-bit long doubles),
|
||||
otherwise via a series expansion when x is small (giving an accuracy of less
|
||||
than 2ɛ).
|
||||
</p>
|
||||
<p>
|
||||
Finally when BOOST_HAS_EXPM1 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 expm1:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../graphs/expm1.svg" align="middle"></span>
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.expm1.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.expm1.accuracy"></a></span><a class="link" href="expm1.html#math_toolkit.powers.expm1.accuracy">Accuracy</a>
|
||||
</h5>
|
||||
<p>
|
||||
For built in floating point types <code class="computeroutput"><span class="identifier">expm1</span></code>
|
||||
should have approximately 1 epsilon accuracy.
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="math_toolkit.powers.expm1.table_expm1"></a><p class="title"><b>Table 6.79. Error rates for expm1</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for expm1">
|
||||
<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 = 0.996ε (Mean = 0.283ε)</span><br> <br>
|
||||
(<span class="emphasis"><em><math.h>:</em></span> Max = 1.31ε (Mean = 0.496ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.793ε (Mean = 0.126ε)</span><br> <br>
|
||||
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 0.793ε (Mean = 0.126ε))<br>
|
||||
(<span class="emphasis"><em>Cephes:</em></span> Max = 1.53ε (Mean = 0.535ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.992ε (Mean = 0.402ε)</span><br> <br>
|
||||
(<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max
|
||||
= 1.26e+19ε (Mean = 4.89e+18ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_expm1__tr1_cmath__Random_test_data">And
|
||||
other failures.</a>)</span><br> (<span class="emphasis"><em><math.h>:</em></span>
|
||||
Max = 0.992ε (Mean = 0.402ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.31ε (Mean = 0.406ε)</span><br> <br>
|
||||
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.996ε (Mean = 0.426ε))
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><h5>
|
||||
<a name="math_toolkit.powers.expm1.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.expm1.testing"></a></span><a class="link" href="expm1.html#math_toolkit.powers.expm1.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="log1p.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="cbrt.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,91 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>hypot</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="powm1.html" title="powm1">
|
||||
<link rel="next" href="ct_pow.html" title="Compile Time Power of a Runtime Base">
|
||||
</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="powm1.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="ct_pow.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.hypot"></a><a class="link" href="hypot.html" title="hypot">hypot</a>
|
||||
</h3></div></div></div>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">hypot</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">hypot</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</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>
|
||||
</pre>
|
||||
<p>
|
||||
<span class="bold"><strong>Effects: </strong></span> computes <span class="inlinemediaobject"><img src="../../../equations/hypot.svg"></span>
|
||||
in such a way
|
||||
as to avoid undue underflow and overflow.
|
||||
</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> when T1 and T2 are of different
|
||||
types.
|
||||
</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>
|
||||
When calculating <span class="inlinemediaobject"><img src="../../../equations/hypot.svg"></span> it's quite easy for the intermediate terms to
|
||||
either overflow or underflow, even though the result is in fact perfectly
|
||||
representable.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.hypot.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.hypot.implementation"></a></span><a class="link" href="hypot.html#math_toolkit.powers.hypot.implementation">Implementation</a>
|
||||
</h5>
|
||||
<p>
|
||||
The function is even and symmetric in x and y, so first take assume <span class="emphasis"><em>x,y
|
||||
> 0</em></span> and <span class="emphasis"><em>x > y</em></span> (we can permute the arguments
|
||||
if this is not the case).
|
||||
</p>
|
||||
<p>
|
||||
Then if <span class="emphasis"><em>x * ε   >= y</em></span> we can simply return <span class="emphasis"><em>x</em></span>.
|
||||
</p>
|
||||
<p>
|
||||
Otherwise the result is given by:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/hypot2.svg"></span>
|
||||
</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="powm1.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="ct_pow.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,189 @@
|
||||
<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>
|
||||
@@ -0,0 +1,162 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>powm1</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="sqrt1pm1.html" title="sqrt1pm1">
|
||||
<link rel="next" href="hypot.html" title="hypot">
|
||||
</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="sqrt1pm1.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="hypot.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.powm1"></a><a class="link" href="powm1.html" title="powm1">powm1</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">powm1</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">powm1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">powm1</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">y</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 x<sup>y </sup> - 1.
|
||||
</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> when T1 and T2 are dufferent types.
|
||||
</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 two domains where this is useful: when y is very small, or when
|
||||
x is close to 1.
|
||||
</p>
|
||||
<p>
|
||||
Implemented in terms of <code class="computeroutput"><span class="identifier">expm1</span></code>.
|
||||
</p>
|
||||
<p>
|
||||
The following graph illustrates the behaviour of powm1:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../graphs/powm1.svg" align="middle"></span>
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.powm1.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.powm1.accuracy"></a></span><a class="link" href="powm1.html#math_toolkit.powers.powm1.accuracy">Accuracy</a>
|
||||
</h5>
|
||||
<p>
|
||||
Should have approximately 2-3 epsilon accuracy.
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="math_toolkit.powers.powm1.table_powm1"></a><p class="title"><b>Table 6.82. Error rates for powm1</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for powm1">
|
||||
<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>
|
||||
powm1
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.99ε (Mean = 0.461ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.26ε (Mean = 0.428ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 2.04ε (Mean = 0.521ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 2.14ε (Mean = 0.498ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><h5>
|
||||
<a name="math_toolkit.powers.powm1.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.powers.powm1.testing"></a></span><a class="link" href="powm1.html#math_toolkit.powers.powm1.testing">Testing</a>
|
||||
</h5>
|
||||
<p>
|
||||
A selection of 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="sqrt1pm1.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="hypot.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,169 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>sin_pi</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="../powers.html" title="Basic Functions">
|
||||
<link rel="next" href="cos_pi.html" title="cos_pi">
|
||||
</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="../powers.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="cos_pi.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.sin_pi"></a><a class="link" href="sin_pi.html" title="sin_pi">sin_pi</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">sin_pi</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">sin_pi</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">sin_pi</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 sine of <span class="emphasis"><em>πx</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 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>
|
||||
This function performs exact all-integer arithmetic argument reduction before
|
||||
computing the sine of <span class="emphasis"><em>πx</em></span>.
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="math_toolkit.powers.sin_pi.table_sin_pi"></a><p class="title"><b>Table 6.76. Error rates for sin_pi</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for sin_pi">
|
||||
<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>
|
||||
sin_pi and cos_pi
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.99ε (Mean = 0.328ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.996ε (Mean = 0.335ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.996ε (Mean = 0.336ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
sin_pi and cos_pi near integers and half integers
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.996ε (Mean = 0.343ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.976ε (Mean = 0.293ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.976ε (Mean = 0.293ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break">
|
||||
</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="../powers.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="cos_pi.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,165 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>sqrt1pm1</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="cbrt.html" title="cbrt">
|
||||
<link rel="next" href="powm1.html" title="powm1">
|
||||
</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="cbrt.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="powm1.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.sqrt1pm1"></a><a class="link" href="sqrt1pm1.html" title="sqrt1pm1">sqrt1pm1</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">sqrt1pm1</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">sqrt1pm1</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">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 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 <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>.
|
||||
</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>
|
||||
This function is useful when you need the difference between sqrt(x) and
|
||||
1, when x is itself close to 1.
|
||||
</p>
|
||||
<p>
|
||||
Implemented in terms of <code class="computeroutput"><span class="identifier">log1p</span></code>
|
||||
and <code class="computeroutput"><span class="identifier">expm1</span></code>.
|
||||
</p>
|
||||
<p>
|
||||
The following graph illustrates the behaviour of sqrt1pm1:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../graphs/sqrt1pm1.svg" align="middle"></span>
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.powers.sqrt1pm1.h0"></a>
|
||||
<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>
|
||||
</h5>
|
||||
<p>
|
||||
For built in floating-point types <code class="computeroutput"><span class="identifier">sqrt1pm1</span></code>
|
||||
should have approximately 3 epsilon accuracy.
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="math_toolkit.powers.sqrt1pm1.table_sqrt1pm1"></a><p class="title"><b>Table 6.81. Error rates for sqrt1pm1</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for sqrt1pm1">
|
||||
<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>
|
||||
sqrt1pm1
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.36ε (Mean = 0.44ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.3ε (Mean = 0.404ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.33ε (Mean = 0.409ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.54ε (Mean = 0.568ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><h5>
|
||||
<a name="math_toolkit.powers.sqrt1pm1.h1"></a>
|
||||
<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>
|
||||
</h5>
|
||||
<p>
|
||||
A selection of 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="cbrt.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="powm1.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
Reference in New Issue
Block a user