- change "the the" to "the" Approved by: lstewart Approved by: sahil (mentor) MFC after: 3 days
445 lines
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Groff
445 lines
13 KiB
Groff
.\" Copyright (c) 1985 Regents of the University of California.
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.\" All rights reserved.
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.\"
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.\" Redistribution and use in source and binary forms, with or without
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.\" modification, are permitted provided that the following conditions
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.\" are met:
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.\" 1. Redistributions of source code must retain the above copyright
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.\" notice, this list of conditions and the following disclaimer.
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.\" 2. Redistributions in binary form must reproduce the above copyright
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.\" notice, this list of conditions and the following disclaimer in the
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.\" documentation and/or other materials provided with the distribution.
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.\" 4. Neither the name of the University nor the names of its contributors
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.\" may be used to endorse or promote products derived from this software
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.\" without specific prior written permission.
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.\"
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.\" THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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.\" ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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.\" SUCH DAMAGE.
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.\"
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.\" from: @(#)ieee.3 6.4 (Berkeley) 5/6/91
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.\" $FreeBSD$
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.\"
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.Dd January 26, 2005
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.Dt IEEE 3
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.Os
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.Sh NAME
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.Nm ieee
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.Nd IEEE standard 754 for floating-point arithmetic
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.Sh DESCRIPTION
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The IEEE Standard 754 for Binary Floating-Point Arithmetic
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defines representations of floating-point numbers and abstract
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properties of arithmetic operations relating to precision,
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rounding, and exceptional cases, as described below.
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.Ss IEEE STANDARD 754 Floating-Point Arithmetic
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Radix: Binary.
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.Pp
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Overflow and underflow:
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.Bd -ragged -offset indent -compact
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Overflow goes by default to a signed \*(If.
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Underflow is
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.Em gradual .
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.Ed
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.Pp
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Zero is represented ambiguously as +0 or \-0.
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.Bd -ragged -offset indent -compact
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Its sign transforms correctly through multiplication or
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division, and is preserved by addition of zeros
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with like signs; but x\-x yields +0 for every
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finite x.
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The only operations that reveal zero's
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sign are division by zero and
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.Fn copysign x \(+-0 .
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In particular, comparison (x > y, x \(>= y, etc.)\&
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cannot be affected by the sign of zero; but if
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finite x = y then \*(If = 1/(x\-y) \(!= \-1/(y\-x) = \-\*(If.
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.Ed
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.Pp
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Infinity is signed.
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.Bd -ragged -offset indent -compact
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It persists when added to itself
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or to any finite number.
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Its sign transforms
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correctly through multiplication and division, and
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(finite)/\(+-\*(If\0=\0\(+-0
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(nonzero)/0 = \(+-\*(If.
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But
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\*(If\-\*(If, \*(If\(**0 and \*(If/\*(If
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are, like 0/0 and sqrt(\-3),
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invalid operations that produce \*(Na. ...
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.Ed
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.Pp
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Reserved operands (\*(Nas):
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.Bd -ragged -offset indent -compact
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An \*(Na is
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.Em ( N Ns ot Em a N Ns umber ) .
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Some \*(Nas, called Signaling \*(Nas, trap any floating-point operation
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performed upon them; they are used to mark missing
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or uninitialized values, or nonexistent elements
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of arrays.
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The rest are Quiet \*(Nas; they are
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the default results of Invalid Operations, and
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propagate through subsequent arithmetic operations.
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If x \(!= x then x is \*(Na; every other predicate
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(x > y, x = y, x < y, ...) is FALSE if \*(Na is involved.
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.Ed
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.Pp
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Rounding:
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.Bd -ragged -offset indent -compact
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Every algebraic operation (+, \-, \(**, /,
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\(sr)
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is rounded by default to within half an
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.Em ulp ,
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and when the rounding error is exactly half an
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.Em ulp
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then
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the rounded value's least significant bit is zero.
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(An
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.Em ulp
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is one
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.Em U Ns nit
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in the
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.Em L Ns ast
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.Em P Ns lace . )
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This kind of rounding is usually the best kind,
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sometimes provably so; for instance, for every
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x = 1.0, 2.0, 3.0, 4.0, ..., 2.0**52, we find
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(x/3.0)\(**3.0 == x and (x/10.0)\(**10.0 == x and ...
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despite that both the quotients and the products
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have been rounded.
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Only rounding like IEEE 754 can do that.
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But no single kind of rounding can be
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proved best for every circumstance, so IEEE 754
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provides rounding towards zero or towards
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+\*(If or towards \-\*(If
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at the programmer's option.
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.Ed
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.Pp
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Exceptions:
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.Bd -ragged -offset indent -compact
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IEEE 754 recognizes five kinds of floating-point exceptions,
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listed below in declining order of probable importance.
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.Bl -column -offset indent "Invalid Operation" "Gradual Underflow"
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.Em "Exception Default Result"
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Invalid Operation \*(Na, or FALSE
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Overflow \(+-\*(If
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Divide by Zero \(+-\*(If
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Underflow Gradual Underflow
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Inexact Rounded value
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.El
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.Pp
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NOTE: An Exception is not an Error unless handled
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badly.
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What makes a class of exceptions exceptional
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is that no single default response can be satisfactory
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in every instance.
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On the other hand, if a default
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response will serve most instances satisfactorily,
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the unsatisfactory instances cannot justify aborting
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computation every time the exception occurs.
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.Ed
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.Ss Data Formats
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Single-precision:
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.Bd -ragged -offset indent -compact
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Type name:
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.Vt float
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.Pp
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Wordsize: 32 bits.
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.Pp
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Precision: 24 significant bits,
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roughly like 7 significant decimals.
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.Bd -ragged -offset indent -compact
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If x and x' are consecutive positive single-precision
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numbers (they differ by 1
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.Em ulp ) ,
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then
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.Bd -ragged -compact
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5.9e\-08 < 0.5**24 < (x'\-x)/x \(<= 0.5**23 < 1.2e\-07.
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.Ed
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.Ed
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.Pp
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.Bl -column "XXX" -compact
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Range: Overflow threshold = 2.0**128 = 3.4e38
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Underflow threshold = 0.5**126 = 1.2e\-38
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.El
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.Bd -ragged -offset indent -compact
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Underflowed results round to the nearest
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integer multiple of 0.5**149 = 1.4e\-45.
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.Ed
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.Ed
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.Pp
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Double-precision:
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.Bd -ragged -offset indent -compact
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Type name:
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.Vt double
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.Bd -ragged -offset indent -compact
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On some architectures,
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.Vt long double
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is the same as
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.Vt double .
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.Ed
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.Pp
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Wordsize: 64 bits.
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.Pp
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Precision: 53 significant bits,
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roughly like 16 significant decimals.
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.Bd -ragged -offset indent -compact
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If x and x' are consecutive positive double-precision
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numbers (they differ by 1
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.Em ulp ) ,
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then
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.Bd -ragged -compact
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1.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16.
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.Ed
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.Ed
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.Pp
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.Bl -column "XXX" -compact
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Range: Overflow threshold = 2.0**1024 = 1.8e308
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Underflow threshold = 0.5**1022 = 2.2e\-308
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.El
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.Bd -ragged -offset indent -compact
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Underflowed results round to the nearest
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integer multiple of 0.5**1074 = 4.9e\-324.
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.Ed
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.Ed
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.Pp
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Extended-precision:
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.Bd -ragged -offset indent -compact
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Type name:
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.Vt long double
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(when supported by the hardware)
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.Pp
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Wordsize: 96 bits.
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.Pp
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Precision: 64 significant bits,
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roughly like 19 significant decimals.
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.Bd -ragged -offset indent -compact
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If x and x' are consecutive positive extended-precision
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numbers (they differ by 1
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.Em ulp ) ,
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then
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.Bd -ragged -compact
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1.0e\-19 < 0.5**63 < (x'\-x)/x \(<= 0.5**62 < 2.2e\-19.
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.Ed
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.Ed
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.Pp
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.Bl -column "XXX" -compact
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Range: Overflow threshold = 2.0**16384 = 1.2e4932
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Underflow threshold = 0.5**16382 = 3.4e\-4932
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.El
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.Bd -ragged -offset indent -compact
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Underflowed results round to the nearest
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integer multiple of 0.5**16445 = 5.7e\-4953.
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.Ed
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.Ed
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.Pp
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Quad-extended-precision:
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.Bd -ragged -offset indent -compact
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Type name:
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.Vt long double
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(when supported by the hardware)
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.Pp
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Wordsize: 128 bits.
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.Pp
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Precision: 113 significant bits,
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roughly like 34 significant decimals.
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.Bd -ragged -offset indent -compact
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If x and x' are consecutive positive quad-extended-precision
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numbers (they differ by 1
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.Em ulp ) ,
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then
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.Bd -ragged -compact
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9.6e\-35 < 0.5**113 < (x'\-x)/x \(<= 0.5**112 < 2.0e\-34.
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.Ed
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.Ed
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.Pp
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.Bl -column "XXX" -compact
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Range: Overflow threshold = 2.0**16384 = 1.2e4932
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Underflow threshold = 0.5**16382 = 3.4e\-4932
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.El
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.Bd -ragged -offset indent -compact
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Underflowed results round to the nearest
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integer multiple of 0.5**16494 = 6.5e\-4966.
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.Ed
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.Ed
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.Ss Additional Information Regarding Exceptions
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.Pp
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For each kind of floating-point exception, IEEE 754
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provides a Flag that is raised each time its exception
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is signaled, and stays raised until the program resets
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it.
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Programs may also test, save and restore a flag.
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Thus, IEEE 754 provides three ways by which programs
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may cope with exceptions for which the default result
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might be unsatisfactory:
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.Bl -enum
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.It
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Test for a condition that might cause an exception
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later, and branch to avoid the exception.
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.It
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Test a flag to see whether an exception has occurred
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since the program last reset its flag.
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.It
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Test a result to see whether it is a value that only
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an exception could have produced.
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.Pp
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CAUTION: The only reliable ways to discover
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whether Underflow has occurred are to test whether
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products or quotients lie closer to zero than the
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underflow threshold, or to test the Underflow
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flag.
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(Sums and differences cannot underflow in
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IEEE 754; if x \(!= y then x\-y is correct to
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full precision and certainly nonzero regardless of
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how tiny it may be.)
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Products and quotients that
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underflow gradually can lose accuracy gradually
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without vanishing, so comparing them with zero
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(as one might on a VAX) will not reveal the loss.
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Fortunately, if a gradually underflowed value is
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destined to be added to something bigger than the
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underflow threshold, as is almost always the case,
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digits lost to gradual underflow will not be missed
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because they would have been rounded off anyway.
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So gradual underflows are usually
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.Em provably
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ignorable.
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The same cannot be said of underflows flushed to 0.
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.El
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.Pp
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At the option of an implementor conforming to IEEE 754,
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other ways to cope with exceptions may be provided:
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.Bl -enum
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.It
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ABORT.
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This mechanism classifies an exception in
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advance as an incident to be handled by means
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traditionally associated with error-handling
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statements like "ON ERROR GO TO ...".
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Different
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languages offer different forms of this statement,
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but most share the following characteristics:
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.Bl -dash
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.It
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No means is provided to substitute a value for
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the offending operation's result and resume
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computation from what may be the middle of an
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expression.
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An exceptional result is abandoned.
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.It
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In a subprogram that lacks an error-handling
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statement, an exception causes the subprogram to
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abort within whatever program called it, and so
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on back up the chain of calling subprograms until
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an error-handling statement is encountered or the
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whole task is aborted and memory is dumped.
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.El
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.It
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STOP.
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This mechanism, requiring an interactive
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debugging environment, is more for the programmer
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than the program.
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It classifies an exception in
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advance as a symptom of a programmer's error; the
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exception suspends execution as near as it can to
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the offending operation so that the programmer can
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look around to see how it happened.
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Quite often
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the first several exceptions turn out to be quite
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unexceptionable, so the programmer ought ideally
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to be able to resume execution after each one as if
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execution had not been stopped.
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.It
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\&... Other ways lie beyond the scope of this document.
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.El
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.Pp
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Ideally, each
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elementary function should act as if it were indivisible, or
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atomic, in the sense that ...
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.Bl -enum
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.It
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No exception should be signaled that is not deserved by
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the data supplied to that function.
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.It
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Any exception signaled should be identified with that
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function rather than with one of its subroutines.
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.It
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The internal behavior of an atomic function should not
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be disrupted when a calling program changes from
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one to another of the five or so ways of handling
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exceptions listed above, although the definition
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of the function may be correlated intentionally
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with exception handling.
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.El
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.Pp
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The functions in
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.Nm libm
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are only approximately atomic.
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They signal no inappropriate exception except possibly ...
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.Bl -tag -width indent -offset indent -compact
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.It Xo
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Over/Underflow
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.Xc
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when a result, if properly computed, might have lain barely within range, and
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.It Xo
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Inexact in
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.Fn cabs ,
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.Fn cbrt ,
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.Fn hypot ,
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.Fn log10
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and
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.Fn pow
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.Xc
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when it happens to be exact, thanks to fortuitous cancellation of errors.
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.El
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Otherwise, ...
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.Bl -tag -width indent -offset indent -compact
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.It Xo
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Invalid Operation is signaled only when
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.Xc
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any result but \*(Na would probably be misleading.
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.It Xo
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Overflow is signaled only when
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.Xc
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the exact result would be finite but beyond the overflow threshold.
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.It Xo
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Divide-by-Zero is signaled only when
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.Xc
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a function takes exactly infinite values at finite operands.
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.It Xo
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Underflow is signaled only when
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.Xc
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the exact result would be nonzero but tinier than the underflow threshold.
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.It Xo
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Inexact is signaled only when
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.Xc
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greater range or precision would be needed to represent the exact result.
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.El
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.Sh SEE ALSO
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.Xr fenv 3 ,
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.Xr ieee_test 3 ,
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.Xr math 3
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.Pp
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An explanation of IEEE 754 and its proposed extension p854
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was published in the IEEE magazine MICRO in August 1984 under
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the title "A Proposed Radix- and Word-length-independent
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Standard for Floating-point Arithmetic" by
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.An "W. J. Cody"
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et al.
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The manuals for Pascal, C and BASIC on the Apple Macintosh
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document the features of IEEE 754 pretty well.
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Articles in the IEEE magazine COMPUTER vol.\& 14 no.\& 3 (Mar.\&
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1981), and in the ACM SIGNUM Newsletter Special Issue of
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Oct.\& 1979, may be helpful although they pertain to
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superseded drafts of the standard.
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.Sh STANDARDS
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.St -ieee754
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