usr/src/lib/libm/common_source/ieee.3

From Prof. Kahan at UC at Berkeley
.\" Copyright (c) 1985 Regents of the University of California.
.\" All rights reserved.  The Berkeley software License Agreement
.\" specifies the terms and conditions for redistribution.
.\"
.\"     @(#)ieee.3      6.2 (Berkeley) %G%
.\"
.TH IEEE 3M  ""
.UC 6
.ds nn \fINaN\fR
.SH NAME
copysign, drem, finite, logb, scalb \- copysign, remainder,
exponent manipulations
.SH SYNOPSIS
.nf
.B #include <math.h>
.PP
.B double copysign(x,y)
.B double x,y;
.PP
.B double drem(x,y)
.B double x,y;
.PP
.B int finite(x)
.B double x;
.PP
.B double logb(x)
.B double x;
.PP
.B double scalb(x,n)
.B double x;
.B int n;
.fi
.SH DESCRIPTION
These functions are required for, or recommended by the IEEE standard
754 for floating\-point arithmetic.
.PP
Copysign(x,y)
returns x with its sign changed to y's.
.PP
Drem(x,y) returns the remainder r := x \- n\(**y
where n is the integer nearest the exact value of x/y;
moreover if |n\|\-\|x/y|\|=\|1/2 then n is even.  Consequently
the remainder is computed exactly and |r| \(<= |y|/2.  But
drem(x,0) is exceptional; see below under DIAGNOSTICS.
.PP
.nf
.ta \w'Finite(x)'u+1n +\w'= 0 otherwise'u+1n
.if n \
Finite(x)       = 1 just when \-infinity < x < +infinity,
.if t \
Finite(x)       = 1 just when \-\(if < x < +\(if,
.if n \
        = 0 otherwise   (when |x| = infinity or x is \*(nn or
.if t \
        = 0 otherwise   (when |x| = \(if or x is \*(nn or
                \0x is the VAX's reserved operand.)
.ta
.fi
.PP
Logb(x) returns x's exponent n,
a signed integer converted to double\-precision floating\-point and so
chosen that 1\0\(<=\0|x|/2**n\0<\02 unless x = 0 or
(only on machines that conform to IEEE 754)
.if n \
|x| = infinity
.if t \
|x| = \(if
or x lies between 0 and the Underflow Threshold; see below under "BUGS".
.PP
Scalb(x,n) = x\(**(2**n) computed, for integer n, without first computing
2**n.
.SH DIAGNOSTICS
IEEE 754 defines drem(x,0) and
.if n \
drem(infinity,y)
.if t \
drem(\(if,y)
to be invalid operations that produce a \*(nn.
On a VAX, drem(x,0) returns the reserved operand.  No
.if n \
infinity
.if t \
\(if
exists on a VAX.
.PP
IEEE 754 defines
.if n \
logb(\(+-infinity) = +infinity and logb(0) = \-infinity,
.if t \
logb(\(+-\(if) = +\(if and logb(0) = \-\(if, and
requires the latter to signal Division\-by\-Zero.
But on a VAX, logb(0) = 1.0 \- 2.0**31 = \-2,147,483,647.0.
And if the correct value of scalb(x,n) would overflow on a VAX,
it returns the reserved operand and sets \fIerrno\fR to ERANGE.
.SH SEE ALSO
floor(3M), math(3M), infnan(3M)
.SH AUTHOR
Kwok\-Choi Ng
.SH BUGS
Should drem(x,0) and logb(0) on a VAX signal invalidity
by setting \fIerrno\fR = EDOM?  Should  logb(0) return  \-1.7e38?
.PP
IEEE 754 currently specifies that
logb(denormalized no.) = logb(tiniest normalized no. > 0)
but the consensus has changed to the specification in the new
proposed IEEE standard p854, namely that logb(x) satisfy
.RS
1 \(<= scalb(|x|,\-logb(x)) < Radix\0\0\0... = 2 for IEEE 754
.RE
for every x except 0,
.if n \
infinity
.if t \
\(if
and \*(nn.
Almost every program that assumes 754's specification will work
correctly if logb follows 854's specification instead.
.PP
IEEE 754 requires copysign(x,\*(nn) = \(+-x  but says nothing
else about the sign of a \*(nn.  A \*(nn (\fIN\fRot \fIa\fR \fIN\fRumber) is
similar in spirit to the VAX's reserved operand, but very
different in important details.  Since the sign bit of a
reserved operand makes it look negative,
.RS
copysign(x,reserved operand) = \-x;
.RE
should this return the reserved operand instead?