29bf6af890
manpages. They are not very related, so separating them makes it easier to add meaningful cross-references and extend some of the descriptions. - Move the part of math(3) that discusses IEEE 754 to the ieee(3) manpage.
451 lines
13 KiB
Groff
451 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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.\" 3. All advertising materials mentioning features or use of this software
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.\" must display the following acknowledgement:
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.\" This product includes software developed by the University of
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.\" California, Berkeley and its contributors.
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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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.Bl -column "" -compact
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Overflow and underflow:
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.El
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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 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 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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1.0e\-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**16451 = 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 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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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
|
|
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.
|
|
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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