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132 lines
6.9 KiB
HTML
132 lines
6.9 KiB
HTML
4 years ago
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<!-- This manual documents how to install and use the Multiple Precision
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Floating-Point Reliable Library, version 3.1.4.
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Copyright 1991, 1993-2016 Free Software Foundation, Inc.
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Permission is granted to copy, distribute and/or modify this document under
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the terms of the GNU Free Documentation License, Version 1.2 or any later
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version published by the Free Software Foundation; with no Invariant Sections,
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with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the
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license is included in GNU Free Documentation License. -->
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<!-- Created by GNU Texinfo 6.4, http://www.gnu.org/software/texinfo/ -->
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<head>
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<title>Rounding Modes (GNU MPFR 3.1.4)</title>
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<meta name="description" content="How to install and use GNU MPFR, a library for reliable multiple precision
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floating-point arithmetic, version 3.1.4.">
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<meta name="keywords" content="Rounding Modes (GNU MPFR 3.1.4)">
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<meta name="resource-type" content="document">
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<meta name="distribution" content="global">
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<link href="index.html#Top" rel="start" title="Top">
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<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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<link href="MPFR-Basics.html#MPFR-Basics" rel="up" title="MPFR Basics">
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<link href="Floating_002dPoint-Values-on-Special-Numbers.html#Floating_002dPoint-Values-on-Special-Numbers" rel="next" title="Floating-Point Values on Special Numbers">
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<link href="MPFR-Variable-Conventions.html#MPFR-Variable-Conventions" rel="prev" title="MPFR Variable Conventions">
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</style>
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</head>
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<body lang="en">
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<a name="Rounding-Modes"></a>
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<div class="header">
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<p>
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Next: <a href="Floating_002dPoint-Values-on-Special-Numbers.html#Floating_002dPoint-Values-on-Special-Numbers" accesskey="n" rel="next">Floating-Point Values on Special Numbers</a>, Previous: <a href="MPFR-Variable-Conventions.html#MPFR-Variable-Conventions" accesskey="p" rel="prev">MPFR Variable Conventions</a>, Up: <a href="MPFR-Basics.html#MPFR-Basics" accesskey="u" rel="up">MPFR Basics</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<a name="Rounding-Modes-1"></a>
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<h3 class="section">4.4 Rounding Modes</h3>
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<p>The following five rounding modes are supported:
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</p><ul>
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<li> <code>MPFR_RNDN</code>: round to nearest (roundTiesToEven in IEEE 754-2008),
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</li><li> <code>MPFR_RNDZ</code>: round toward zero (roundTowardZero in IEEE 754-2008),
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</li><li> <code>MPFR_RNDU</code>: round toward plus infinity (roundTowardPositive in IEEE 754-2008),
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</li><li> <code>MPFR_RNDD</code>: round toward minus infinity (roundTowardNegative in IEEE 754-2008),
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</li><li> <code>MPFR_RNDA</code>: round away from zero.
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</li></ul>
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<p>The ‘<samp>round to nearest</samp>’ mode works as in the IEEE 754 standard: in
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case the number to be rounded lies exactly in the middle of two representable
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numbers, it is rounded to the one with the least significant bit set to zero.
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For example, the number 2.5, which is represented by (10.1) in binary, is
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rounded to (10.0)=2 with a precision of two bits, and not to (11.0)=3.
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This rule avoids the <em>drift</em> phenomenon mentioned by Knuth in volume 2
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of The Art of Computer Programming (Section 4.2.2).
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</p>
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<a name="ternary-value"></a><a name="index-Ternary-value"></a>
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<p>Most MPFR functions take as first argument the destination variable, as
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second and following arguments the input variables, as last argument a
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rounding mode, and have a return value of type <code>int</code>, called the
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<em>ternary value</em>. The value stored in the destination variable is
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correctly rounded, i.e., MPFR behaves as if it computed the result with
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an infinite precision, then rounded it to the precision of this variable.
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The input variables are regarded as exact (in particular, their precision
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does not affect the result).
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</p>
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<p>As a consequence, in case of a non-zero real rounded result, the error
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on the result is less or equal to 1/2 ulp (unit in the last place) of
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that result in the rounding to nearest mode, and less than 1 ulp of that
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result in the directed rounding modes (a ulp is the weight of the least
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significant represented bit of the result after rounding).
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</p>
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<p>Unless documented otherwise, functions returning an <code>int</code> return
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a ternary value.
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If the ternary value is zero, it means that the value stored in the
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destination variable is the exact result of the corresponding mathematical
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function. If the ternary value is positive (resp. negative), it means
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the value stored in the destination variable is greater (resp. lower)
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than the exact result. For example with the <code>MPFR_RNDU</code> rounding mode,
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the ternary value is usually positive, except when the result is exact, in
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which case it is zero. In the case of an infinite result, it is considered
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as inexact when it was obtained by overflow, and exact otherwise. A NaN
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result (Not-a-Number) always corresponds to an exact return value.
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The opposite of a returned ternary value is guaranteed to be representable
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in an <code>int</code>.
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</p>
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<p>Unless documented otherwise, functions returning as result the value <code>1</code>
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(or any other value specified in this manual)
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for special cases (like <code>acos(0)</code>) yield an overflow or
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an underflow if that value is not representable in the current exponent range.
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</p>
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<hr>
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<div class="header">
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<p>
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Next: <a href="Floating_002dPoint-Values-on-Special-Numbers.html#Floating_002dPoint-Values-on-Special-Numbers" accesskey="n" rel="next">Floating-Point Values on Special Numbers</a>, Previous: <a href="MPFR-Variable-Conventions.html#MPFR-Variable-Conventions" accesskey="p" rel="prev">MPFR Variable Conventions</a>, Up: <a href="MPFR-Basics.html#MPFR-Basics" accesskey="u" rel="up">MPFR Basics</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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</body>
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</html>
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