382 lines
25 KiB
HTML
382 lines
25 KiB
HTML
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
|
<title>Introduction</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 2.11.0">
|
|
<link rel="up" href="../fp_facets.html" title="Facets for Floating-Point Infinities and NaNs">
|
|
<link rel="prev" href="../fp_facets.html" title="Facets for Floating-Point Infinities and NaNs">
|
|
<link rel="next" href="reference.html" title="Reference">
|
|
</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="../fp_facets.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../fp_facets.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="reference.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.fp_facets.facets_intro"></a><a class="link" href="facets_intro.html" title="Introduction">Introduction</a>
|
|
</h3></div></div></div>
|
|
<h6>
|
|
<a name="math_toolkit.fp_facets.facets_intro.h0"></a>
|
|
<span class="phrase"><a name="math_toolkit.fp_facets.facets_intro.the_problem"></a></span><a class="link" href="facets_intro.html#math_toolkit.fp_facets.facets_intro.the_problem">The
|
|
Problem</a>
|
|
</h6>
|
|
<p>
|
|
The C++98 standard does not specify how <span class="emphasis"><em>infinity</em></span> and
|
|
<span class="emphasis"><em>NaN</em></span> are represented in text streams. As a result, different
|
|
platforms use different string representations. This can cause undefined
|
|
behavior when text files are moved between different platforms. Some platforms
|
|
cannot even input parse their own output! So 'route-tripping' or loopback
|
|
of output to input is not possible. For instance, the following test fails
|
|
with MSVC:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">stringstream</span> <span class="identifier">ss</span><span class="special">;</span>
|
|
<span class="keyword">double</span> <span class="identifier">inf</span> <span class="special">=</span> <span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">();</span>
|
|
<span class="keyword">double</span> <span class="identifier">r</span><span class="special">;</span>
|
|
<span class="identifier">ss</span> <span class="special"><<</span> <span class="identifier">inf</span><span class="special">;</span> <span class="comment">// Write out.</span>
|
|
<span class="identifier">ss</span> <span class="special">>></span> <span class="identifier">r</span><span class="special">;</span> <span class="comment">// Read back in.</span>
|
|
|
|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"infinity output was "</span> <span class="special"><<</span> <span class="identifier">inf</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1.#INF</span>
|
|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"infinity input was "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1</span>
|
|
|
|
<span class="identifier">assert</span><span class="special">(</span><span class="identifier">inf</span> <span class="special">==</span> <span class="identifier">y</span><span class="special">);</span> <span class="comment">// Fails!</span>
|
|
</pre>
|
|
<h6>
|
|
<a name="math_toolkit.fp_facets.facets_intro.h1"></a>
|
|
<span class="phrase"><a name="math_toolkit.fp_facets.facets_intro.the_solution"></a></span><a class="link" href="facets_intro.html#math_toolkit.fp_facets.facets_intro.the_solution">The
|
|
Solution</a>
|
|
</h6>
|
|
<p>
|
|
The facets <code class="computeroutput"><span class="identifier">nonfinite_num_put</span></code>
|
|
and <code class="computeroutput"><span class="identifier">nonfinite_num_get</span></code> format
|
|
and parse all floating-point numbers, including <code class="computeroutput"><span class="identifier">infinity</span></code>
|
|
and <code class="computeroutput"><span class="identifier">NaN</span></code>, in a consistent
|
|
and portable manner.
|
|
</p>
|
|
<p>
|
|
The following test succeeds with MSVC.
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">locale</span> <span class="identifier">old_locale</span><span class="special">;</span>
|
|
<span class="identifier">locale</span> <span class="identifier">tmp_locale</span><span class="special">(</span><span class="identifier">old_locale</span><span class="special">,</span> <span class="keyword">new</span> <span class="identifier">nonfinite_num_put</span><span class="special"><</span><span class="keyword">char</span><span class="special">>);</span>
|
|
<span class="identifier">locale</span> <span class="identifier">new_locale</span><span class="special">(</span><span class="identifier">tmp_locale</span><span class="special">,</span> <span class="keyword">new</span> <span class="identifier">nonfinite_num_get</span><span class="special"><</span><span class="keyword">char</span><span class="special">>);</span>
|
|
</pre>
|
|
<div class="tip"><table border="0" summary="Tip">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td>
|
|
<th align="left">Tip</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top">
|
|
<p>
|
|
To add two facets, <code class="computeroutput"><span class="identifier">nonfinite_num_put</span></code>
|
|
and <code class="computeroutput"><span class="identifier">nonfinite_num_get</span></code>,
|
|
you may have to add one at a time, using a temporary locale.
|
|
</p>
|
|
<p>
|
|
Or you can create a new locale in one step
|
|
</p>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">new_locale</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span><span class="special">(),</span> <span class="keyword">new</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">nonfinite_num_put</span><span class="special"><</span><span class="keyword">char</span><span class="special">>),</span> <span class="keyword">new</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">nonfinite_num_get</span><span class="special"><</span><span class="keyword">char</span><span class="special">>));</span></code>
|
|
</p>
|
|
<p>
|
|
and, for example, use it to imbue an input and output stringstream.
|
|
</p>
|
|
</td></tr>
|
|
</table></div>
|
|
<div class="tip"><table border="0" summary="Tip">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td>
|
|
<th align="left">Tip</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top"><p>
|
|
To just change an input or output stream, you can concisely write <code class="computeroutput"><span class="identifier">cout</span><span class="special">.</span><span class="identifier">imbue</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span><span class="special">(),</span> <span class="keyword">new</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">nonfinite_num_put</span><span class="special"><</span><span class="keyword">char</span><span class="special">>));</span></code> or <code class="computeroutput"><span class="identifier">cin</span><span class="special">.</span><span class="identifier">imbue</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span><span class="special">(),</span> <span class="keyword">new</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">nonfinite_num_get</span><span class="special"><</span><span class="keyword">char</span><span class="special">>));</span></code>
|
|
</p></td></tr>
|
|
</table></div>
|
|
<pre class="programlisting"><span class="identifier">stringstream</span> <span class="identifier">ss</span><span class="special">;</span>
|
|
<span class="identifier">ss</span><span class="special">.</span><span class="identifier">imbue</span><span class="special">(</span><span class="identifier">new_locale</span><span class="special">);</span>
|
|
<span class="keyword">double</span> <span class="identifier">inf</span> <span class="special">=</span> <span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">();</span>
|
|
<span class="identifier">ss</span> <span class="special"><<</span> <span class="identifier">inf</span><span class="special">;</span> <span class="comment">// Write out.</span>
|
|
<span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</span><span class="special">()</span> <span class="special">==</span> <span class="string">"inf"</span><span class="special">);</span>
|
|
<span class="keyword">double</span> <span class="identifier">r</span><span class="special">;</span>
|
|
<span class="identifier">ss</span> <span class="special">>></span> <span class="identifier">r</span><span class="special">;</span> <span class="comment">// Read back in.</span>
|
|
<span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">inf</span> <span class="special">==</span> <span class="identifier">r</span><span class="special">);</span> <span class="comment">// Confirms that the double values really are identical.</span>
|
|
|
|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"infinity output was "</span> <span class="special"><<</span> <span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"infinity input was "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="comment">// But the string representation of r displayed will be the native type</span>
|
|
<span class="comment">// because, when it was constructed, cout had NOT been imbued</span>
|
|
<span class="comment">// with the new locale containing the nonfinite_numput facet.</span>
|
|
<span class="comment">// So the cout output will be "1.#INF on MS platforms</span>
|
|
<span class="comment">// and may be "inf" or other string representation on other platforms.</span>
|
|
</pre>
|
|
<h5>
|
|
<a name="math_toolkit.fp_facets.facets_intro.h2"></a>
|
|
<span class="phrase"><a name="math_toolkit.fp_facets.facets_intro.c_0x_standard_for_output_of_infi"></a></span><a class="link" href="facets_intro.html#math_toolkit.fp_facets.facets_intro.c_0x_standard_for_output_of_infi">C++0X
|
|
standard for output of infinity and NaN</a>
|
|
</h5>
|
|
<p>
|
|
<a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2011/n3242.pdf" target="_top">C++0X
|
|
(final) draft standard</a> does not explicitly specify the representation
|
|
(and input) of nonfinite values, leaving it implementation-defined. So without
|
|
some specific action, input and output of nonfinite values is not portable.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.fp_facets.facets_intro.h3"></a>
|
|
<span class="phrase"><a name="math_toolkit.fp_facets.facets_intro.c99_standard_for_output_of_infin"></a></span><a class="link" href="facets_intro.html#math_toolkit.fp_facets.facets_intro.c99_standard_for_output_of_infin">C99
|
|
standard for output of infinity and NaN</a>
|
|
</h5>
|
|
<p>
|
|
The <a href="http://www.open-std.org/JTC1/SC22/WG14/www/docs/n1256.pdf" target="_top">C99
|
|
standard</a> <span class="bold"><strong>does</strong></span> specify how infinity
|
|
and NaN are formatted by printf and similar output functions, and parsed
|
|
by scanf and similar input functions.
|
|
</p>
|
|
<p>
|
|
The following string representations are used:
|
|
</p>
|
|
<div class="table">
|
|
<a name="math_toolkit.fp_facets.facets_intro.c99_representation_of_infinity_a"></a><p class="title"><b>Table 2.1. C99 Representation of Infinity and NaN</b></p>
|
|
<div class="table-contents"><table class="table" summary="C99 Representation of Infinity and NaN">
|
|
<colgroup>
|
|
<col>
|
|
<col>
|
|
</colgroup>
|
|
<thead><tr>
|
|
<th>
|
|
<p>
|
|
number
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
string
|
|
</p>
|
|
</th>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Positive infinity
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
"inf" or "infinity"
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Positive NaN
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
"nan" or "nan(...)"
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Negative infinity
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
"-inf" or "-infinity"
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Negative NaN
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
"-nan" or "-nan(...)"
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><p>
|
|
So following C99 provides a sensible 'standard' way of handling input and
|
|
output of nonfinites in C++, and this implementation follows most of these
|
|
formats.
|
|
</p>
|
|
<h6>
|
|
<a name="math_toolkit.fp_facets.facets_intro.h4"></a>
|
|
<span class="phrase"><a name="math_toolkit.fp_facets.facets_intro.signaling_nans"></a></span><a class="link" href="facets_intro.html#math_toolkit.fp_facets.facets_intro.signaling_nans">Signaling
|
|
NaNs</a>
|
|
</h6>
|
|
<p>
|
|
A particular type of NaN is the signaling NaN. The usual mechanism of signaling
|
|
is by raising a floating-point exception. Signaling NaNs are defined by
|
|
<a href="http://en.wikipedia.org/wiki/IEEE_floating-point_standard" target="_top">IEEE
|
|
754-2008</a>.
|
|
</p>
|
|
<p>
|
|
Floating-point values with layout <span class="emphasis"><em>s</em></span>111 1111 1<span class="emphasis"><em>a</em></span>xx
|
|
xxxx xxxx xxxx xxxx xxxx where <span class="emphasis"><em>s</em></span> is the sign, <span class="emphasis"><em>x</em></span>
|
|
is the payload, and bit <span class="emphasis"><em>a</em></span> determines the type of NaN.
|
|
</p>
|
|
<p>
|
|
If bit <span class="emphasis"><em>a</em></span> = 1, it is a quiet NaN.
|
|
</p>
|
|
<p>
|
|
If bit <span class="emphasis"><em>a</em></span> is zero and the payload <span class="emphasis"><em>x</em></span>
|
|
is nonzero, then it is a signaling NaN.
|
|
</p>
|
|
<p>
|
|
Although there has been theoretical interest in the ability of a signaling
|
|
NaN to raise an exception, for example to prevent use of an uninitialised
|
|
variable, in practice there appears to be no useful application of signaling
|
|
NaNs for most current processors. <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2011/n3242.pdf" target="_top">C++0X
|
|
18.3.2.2</a> still specifies a (implementation-defined) representation
|
|
for signaling NaN, and <code class="computeroutput"><span class="keyword">static</span> <span class="keyword">constexpr</span> <span class="keyword">bool</span> <span class="identifier">has_signaling_NaN</span></code> a method of checking
|
|
if a floating-point type has a representation for signaling NaN.
|
|
</p>
|
|
<p>
|
|
But in practice, most platforms treat signaling NaNs in the same as quiet
|
|
NaNs. So, for example, they are represented by "nan" on output
|
|
in <a href="http://www.open-std.org/JTC1/SC22/WG14/www/docs/n1256.pdf" target="_top">C99</a>
|
|
format, and output as <code class="computeroutput"><span class="number">1.</span><span class="special">#</span><span class="identifier">QNAN</span></code> by Microsoft compilers.
|
|
</p>
|
|
<div class="note"><table border="0" summary="Note">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
|
|
<th align="left">Note</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top">
|
|
<p>
|
|
The C99 standard does not distinguish between the quiet NaN and signaling
|
|
NaN values. A quiet NaN propagates through almost every arithmetic operation
|
|
without raising a floating-point exception; a signaling NaN generally raises
|
|
a floating-point exception when occurring as an arithmetic operand.
|
|
</p>
|
|
<p>
|
|
C99 specification does not define the behavior of signaling NaNs. NaNs
|
|
created by IEC 60559 operations are always quiet. Therefore this implementation
|
|
follows C99, and treats the signaling NaN bit as just a part of the NaN
|
|
payload field. So this implementation does not distinguish between the
|
|
two classes of NaN.
|
|
</p>
|
|
</td></tr>
|
|
</table></div>
|
|
<div class="note"><table border="0" summary="Note">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
|
|
<th align="left">Note</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top">
|
|
<p>
|
|
An implementation may give zero and non-numeric values (such as infinities
|
|
and NaNs) a sign or may leave them unsigned. Wherever such values are unsigned,
|
|
any requirement in the C99 Standard to retrieve the sign shall produce
|
|
an unspecified sign, and any requirement to set the sign shall be ignored.
|
|
</p>
|
|
<p>
|
|
This might apply to user-defined types, but in practice built-in floating-point
|
|
types <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code> and <code class="computeroutput"><span class="keyword">long</span>
|
|
<span class="keyword">double</span></code> have well-behaved signs.
|
|
</p>
|
|
</td></tr>
|
|
</table></div>
|
|
<p>
|
|
The numbers can be of type <code class="computeroutput"><span class="keyword">float</span></code>,
|
|
<code class="computeroutput"><span class="keyword">double</span></code> and <code class="computeroutput"><span class="keyword">long</span>
|
|
<span class="keyword">double</span></code>. An optional + sign can be
|
|
used with positive numbers (controlled by ios manipulator <code class="computeroutput"><span class="identifier">showpos</span></code>).
|
|
The function <code class="computeroutput"><span class="identifier">printf</span></code> and similar
|
|
C++ functions use standard formatting flags to put all lower or all upper
|
|
case (controlled by <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">ios</span></code> manipulator <code class="computeroutput"><span class="identifier">uppercase</span></code>
|
|
and <code class="computeroutput"><span class="identifier">lowercase</span></code>).
|
|
</p>
|
|
<p>
|
|
The function <code class="computeroutput"><span class="identifier">scanf</span></code> and similar
|
|
input functions are case-insensitive.
|
|
</p>
|
|
<p>
|
|
The dots in <code class="computeroutput"><span class="identifier">nan</span><span class="special">(...)</span></code>
|
|
stand for an arbitrary string. The meaning of that string is implementation
|
|
dependent. It can be used to convey extra information about the NaN, from
|
|
the 'payload'. A particular value of the payload might be used to indicate
|
|
a <span class="emphasis"><em>missing value</em></span>, for example.
|
|
</p>
|
|
<p>
|
|
This library uses the string representations specified by the C99 standard.
|
|
</p>
|
|
<p>
|
|
An example of an implementation that optionally includes the NaN payload
|
|
information is at <a href="http://publib.boulder.ibm.com/infocenter/zos/v1r10/index.jsp?topic=/com.ibm.zos.r10.bpxbd00/fprints.htm" target="_top">AIX
|
|
NaN fprintf</a>. That implementation specifies for Binary Floating Point
|
|
NANs:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
A NaN ordinal sequence is a left-parenthesis character '(', followed
|
|
by a digit sequence representing an integer n, where 1 <= n <=
|
|
INT_MAX-1, followed by a right-parenthesis character ')'.
|
|
</li>
|
|
<li class="listitem">
|
|
The integer value, n, is determined by the fraction bits of the NaN argument
|
|
value as follows:
|
|
</li>
|
|
<li class="listitem">
|
|
For a signalling NaN value, NaN fraction bits are reversed (left to right)
|
|
to produce bits (right to left) of an even integer value, 2*n. Then formatted
|
|
output functions produce a (signalling) NaN ordinal sequence corresponding
|
|
to the integer value n.
|
|
</li>
|
|
<li class="listitem">
|
|
For a quiet NaN value, NaN fraction bits are reversed (left to right)
|
|
to produce bits (right to left) of an odd integer value, 2*n-1. Then
|
|
formatted output functions produce a (quiet) NaN ordinal sequence corresponding
|
|
to the integer value n.
|
|
</li>
|
|
</ul></div>
|
|
<div class="warning"><table border="0" summary="Warning">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../doc/src/images/warning.png"></td>
|
|
<th align="left">Warning</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top"><p>
|
|
This implementation does not (yet) provide output of, or access to, the
|
|
NaN payload.
|
|
</p></td></tr>
|
|
</table></div>
|
|
</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 © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="../fp_facets.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../fp_facets.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="reference.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
</body>
|
|
</html>
|