118 lines
8.3 KiB
HTML
118 lines
8.3 KiB
HTML
<html>
|
||
<head>
|
||
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
|
||
<title>Floating-Point Representation Distance (ULP), and Finding Adjacent Floating-Point Values</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 3.0.0">
|
||
<link rel="up" href="../utils.html" title="Chapter 2. Floating Point Utilities">
|
||
<link rel="prev" href="fp_facets/rationale.html" title="Design Rationale">
|
||
<link rel="next" href="next_float/nextafter.html" title="Finding the Next Representable Value in a Specific Direction (nextafter)">
|
||
</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/rationale.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../utils.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="next_float/nextafter.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||
<a name="math_toolkit.next_float"></a><a class="link" href="next_float.html" title="Floating-Point Representation Distance (ULP), and Finding Adjacent Floating-Point Values">Floating-Point Representation
|
||
Distance (ULP), and Finding Adjacent Floating-Point Values</a>
|
||
</h2></div></div></div>
|
||
<div class="toc"><dl class="toc">
|
||
<dt><span class="section"><a href="next_float/nextafter.html">Finding the Next Representable
|
||
Value in a Specific Direction (nextafter)</a></span></dt>
|
||
<dt><span class="section"><a href="next_float/float_next.html">Finding the Next
|
||
Greater Representable Value (float_next)</a></span></dt>
|
||
<dt><span class="section"><a href="next_float/float_prior.html">Finding the Next
|
||
Smaller Representable Value (float_prior)</a></span></dt>
|
||
<dt><span class="section"><a href="next_float/float_distance.html">Calculating the
|
||
Representation Distance Between Two floating-point Values (ULP) float_distance</a></span></dt>
|
||
<dt><span class="section"><a href="next_float/float_advance.html">Advancing a floating-point
|
||
Value by a Specific Representation Distance (ULP) float_advance</a></span></dt>
|
||
<dt><span class="section"><a href="next_float/ulp.html">Obtaining the Size of a
|
||
Unit In the Last Place - ULP</a></span></dt>
|
||
</dl></div>
|
||
<p>
|
||
<a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">Unit of Least
|
||
Precision or Unit in the Last Place</a> is the gap between two different,
|
||
but as close as possible, floating-point numbers.
|
||
</p>
|
||
<p>
|
||
Most decimal values, for example 0.1, cannot be exactly represented as floating-point
|
||
values, but will be stored as the <a href="http://en.wikipedia.org/wiki/Floating_point#Representable_numbers.2C_conversion_and_rounding" target="_top">closest
|
||
representable floating-point</a>.
|
||
</p>
|
||
<p>
|
||
Functions are provided for finding adjacent greater and lesser floating-point
|
||
values, and estimating the number of gaps between any two floating-point values.
|
||
</p>
|
||
<p>
|
||
The floating-point type (FPT) must have has a fixed number of bits in the representation.
|
||
The number of bits may set at runtime, but must be the same for all numbers.
|
||
For example, <a href="http://shoup.net/ntl/doc/quad_float.txt" target="_top">NTL::quad_float</a>
|
||
type (fixed 128-bit representation), <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
|
||
type (arbitrary but fixed decimal digits, default 150) or <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
|
||
<a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a>
|
||
and<a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_bin_float.html" target="_top">cpp_bin_float</a>
|
||
are fixed at runtime, but <span class="bold"><strong>not</strong></span> a type that
|
||
extends the representation to provide an exact representation for any number,
|
||
for example <a href="http://keithbriggs.info/xrc.html" target="_top">XRC eXact Real in
|
||
C</a>.
|
||
</p>
|
||
<p>
|
||
The accuracy of mathematical functions can be assessed and displayed in terms
|
||
of <a href="https://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">Unit in
|
||
the Last Place</a>, often as a ulps plot or by binning the differences
|
||
as a histogram. Samples are evaluated using the implementation under test and
|
||
compared with 'known good' representation obtained using a more accurate method.
|
||
Other implementations, often using arbitrary precision arithmetic, for example
|
||
<a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> are one source
|
||
of references values. The other method, used widely in Boost.Math special functions,
|
||
it to carry out the same algorithm, but using a higher precision type, typically
|
||
using Boost.Multiprecision types like <code class="computeroutput"><span class="identifier">cpp_bin_float_quad</span></code>
|
||
for 128-bit (about 35 decimal digit precision), or <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>
|
||
(for 50 decimal digit precision).
|
||
</p>
|
||
<p>
|
||
When converted to a particular machine representation, say <code class="computeroutput"><span class="keyword">double</span></code>,
|
||
say using a <code class="computeroutput"><span class="keyword">static_cast</span></code>, the value
|
||
is the nearest representation possible for the <code class="computeroutput"><span class="keyword">double</span></code>
|
||
type. This value cannot be 'wrong' by more than half a <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">Unit
|
||
in the last place (ULP)</a>, and can be obtained using the Boost.Math function
|
||
<code class="computeroutput"><span class="identifier">ulp</span></code>. (Unless the algorithm
|
||
is fundamentally flawed, something that should be revealed by 'sanity' checks
|
||
using some independent sources).
|
||
</p>
|
||
<p>
|
||
See some discussion and example plots by Cleve Moler of Mathworks <a href="https://blogs.mathworks.com/cleve/2017/01/23/ulps-plots-reveal-math-function-accurary/" target="_top">ulps
|
||
plots reveal math-function accuracy</a>.
|
||
</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 © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||
Lalande, John Maddock, Evan Miller, 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/rationale.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../utils.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="next_float/nextafter.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
</div>
|
||
</body>
|
||
</html>
|