| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Why use a high-precision library rather than built-in floating-point types?</title> |
| <link rel="stylesheet" href="../../math.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> |
| <link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> |
| <link rel="up" href="../high_precision.html" title="Using Boost.Math with High-Precision Floating-Point Libraries"> |
| <link rel="prev" href="../high_precision.html" title="Using Boost.Math with High-Precision Floating-Point Libraries"> |
| <link rel="next" href="use_multiprecision.html" title="Using Boost.Multiprecision"> |
| </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="../high_precision.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../high_precision.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="use_multiprecision.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.high_precision.why_high_precision"></a><a class="link" href="why_high_precision.html" title="Why use a high-precision library rather than built-in floating-point types?">Why use |
| a high-precision library rather than built-in floating-point types?</a> |
| </h3></div></div></div> |
| <p> |
| For nearly all applications, the built-in floating-point types, <code class="computeroutput"><span class="keyword">double</span></code> (and <code class="computeroutput"><span class="keyword">long</span> |
| <span class="keyword">double</span></code> if this offers higher precision |
| than <code class="computeroutput"><span class="keyword">double</span></code>) offer enough precision, |
| typically a dozen decimal digits. |
| </p> |
| <p> |
| Some reasons why one would want to use a higher precision: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| A much more precise result (many more digits) is just a requirement. |
| </li> |
| <li class="listitem"> |
| The range of the computed value exceeds the range of the type: factorials |
| are the textbook example. |
| </li> |
| <li class="listitem"> |
| Using double is (or may be) too inaccurate. |
| </li> |
| <li class="listitem"> |
| Using long double (or may be) is too inaccurate. |
| </li> |
| <li class="listitem"> |
| Using an extended precision type implemented in software as <a href="http://en.wikipedia.org/wiki/Double-double_(arithmetic)#Double-double_arithmetic" target="_top">double-double</a> |
| (<a href="http://en.wikipedia.org/wiki/Darwin_(operating_system)" target="_top">Darwin</a>) |
| is sometimes unpredictably inaccurate. |
| </li> |
| <li class="listitem"> |
| Loss of precision or inaccuracy caused by extreme arguments or cancellation |
| error. |
| </li> |
| <li class="listitem"> |
| An accuracy as good as possible for a chosen built-in floating-point |
| type is required. |
| </li> |
| <li class="listitem"> |
| As a reference value, for example, to determine the inaccuracy of a value |
| computed with a built-in floating point type, (perhaps even using some |
| quick'n'dirty algorithm). The accuracy of many functions and distributions |
| in Boost.Math has been measured in this way from tables of very high |
| precision (up to 1000 decimal digits). |
| </li> |
| </ul></div> |
| <p> |
| Many functions and distributions have differences from exact values that |
| are only a few least significant bits - computation noise. Others, often |
| those for which analytical solutions are not available, require approximations |
| and iteration: these may lose several decimal digits of precision. |
| </p> |
| <p> |
| Much larger loss of precision can occur for <a href="http://en.wikipedia.org/wiki/Boundary_case" target="_top">boundary</a> |
| or <a href="http://en.wikipedia.org/wiki/Corner_case" target="_top">corner cases</a>, |
| often caused by <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation |
| errors</a>. |
| </p> |
| <p> |
| (Some of the worst and most common examples of <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">cancellation |
| error or loss of significance</a> can be avoided by using <a class="link" href="../stat_tut/overview/complements.html" title="Complements are supported too - and when to use them">complements</a>: |
| see <a class="link" href="../stat_tut/overview/complements.html#why_complements">why complements?</a>). |
| </p> |
| <p> |
| If you require a value which is as accurate as can be represented in the |
| floating-point type, and is thus the closest representable value and has |
| an error less than 1/2 a <a href="http://en.wikipedia.org/wiki/Least_significant_bit" target="_top">least |
| significant bit</a> or <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">ulp</a> |
| it may be useful to use a higher-precision type, for example, <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>, to generate this value. |
| Conversion of this value to a built-in floating-point type ('float', <code class="computeroutput"><span class="keyword">double</span></code> or <code class="computeroutput"><span class="keyword">long</span> |
| <span class="keyword">double</span></code>) will not cause any further |
| loss of precision. A decimal digit string will also be 'read' precisely by |
| the compiler into a built-in floating-point type to the nearest representable |
| value. |
| </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> |
| In contrast, reading a value from an <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">istream</span></code> |
| into a built-in floating-point type is <span class="bold"><strong>not guaranteed</strong></span> |
| by the C++ Standard to give the nearest representable value. |
| </p></td></tr> |
| </table></div> |
| <p> |
| William Kahan coined the term <a href="http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma" target="_top">Table-Maker's |
| Dilemma</a> for the problem of correctly rounding functions. Using a |
| much higher precision (50 or 100 decimal digits) is a practical way of generating |
| (almost always) correctly rounded values. |
| </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, 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="../high_precision.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../high_precision.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="use_multiprecision.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |
