From Prof. Kahan at UC at Berkeley
.\" Copyright (c) 1985, 1991 Regents of the University of California.
.\" All rights reserved.
.\"
.\" %sccs.include.redist.man%
.\"
.\"     @(#)ieee.3	6.3 (Berkeley) %G%
.\"
.Dd 
.Dt IEEE 3
.Os BSD 4.3
.Sh NAME
.Nm copysign ,
.Nm drem ,
.Nm finite ,
.Nm logb ,
.Nm scalb
.Nm copysign ,
.Nm remainder,
.Nd exponent manipulations
.Sh SYNOPSIS
.Fd #include <math.h>
.Ft double 
.Fn copysign "double x" "double y"
.Ft double 
.Fn drem "double x" "double y"
.Ft int 
.Fn finite "double x"
.Ft double 
.Fn logb "double x"
.Ft double 
.Fn scalb "double x" "int n"
.Sh DESCRIPTION
These functions are required for, or recommended by the
.Tn IEEE
standard
754 for floating\-point arithmetic.
.Pp
The
.Fn copysign
function
returns
.Fa x
with its sign changed to
.Fa y Ns 's.
.Pp
The
.Fn drem
function
returns the remainder
.Fa r
:=
.Fa x
\-
.Fa n\(**y
where
.Fa n
is the integer nearest the exact value of
.Bk -words
.Fa x Ns / Ns Fa y ;
.Ek
moreover if
.Pf \\*(Ba Fa n
\-
.Sm off
.Fa x No / Fa y No \\*(Ba
.Sm on
=
1/2
then
.Fa n
is even.  Consequently
the remainder is computed exactly and
.Sm off
.Pf \\*(Ba Fa r No \\*(Ba
.Sm on
\*(Le
.Sm off
.Pf \\*(Ba Fa y No \\*(Ba/2.
.Sm on
But
.Fn drem x 0
is exceptional.
(See below under
.Sx DIAGNOSTICS . )
.Pp
The
.Fn finite
function returns the value 1 just when
\-\*(If \*(Lt
.Fa x
\*(Lt +\*(If;
otherwise a
zero is returned
(when
.Pf \\*(Ba Ns Fa x Ns \\*(Ba
= \*(If or
.Fa x
is \*(Na or
is the
.Tn VAX Ns 's
reserved operand).
.Pp
The
.Fn logb
function returns
.Fa x Ns 's exponent
.Fa n ,
a signed integer converted to double\-precision floating\-point and so
chosen that
1 (<=
.Pf \\*(Ba Ns Fa x Ns \\*(Ba2** Ns Fa n
< 2
unless
.Fa x
= 0 or
(only on machines that conform to
.Tn IEEE
754)
.Pf \\*(Ba Fa x Ns \\*(Ba
= \*(If
or
.Fa x
lies between 0 and the Underflow Threshold.
(See below under
.Sx BUGS . )
.Pp
The
Fn calb
returns
.Fa x Ns \(**(2** Ns Fa n )
computed, for integer n, without first computing
.Pf 2\(** Fa n .
.Sh RETURN VALUES
The
.Tn IEEE
standard
754 defines
.Fn drem x 0
and
.Fn drem \\*(If y
to be invalid operations that produce a \*(Na.
On the
.Tn VAX , 
.Fn drem x 0
generates a reserved operand fault.  No \*(If
exists on a
.Tn VAX . 
.Pp
.Tn IEEE
754 defines
.if n \
.Fn logb \(+-\\*(If
= \*(If and
.Fn logb 0
= \-\*(If, and
requires the latter to signal Division\-by\-Zero.
But on a
.Tn VAX , 
.Fn logb 0
= 1.0 \- 2.0**31 = \-2,147,483,647.0.
And if the correct value of
.Fn scalb
would overflow on a
.Tn VAX , 
it generates a reserved operand fault and sets the global variable
.Va errno
to
.Dv ERANGE . 
.Sh SEE ALSO
.Xr floor 3 ,
.Xr math 3 ,
.Xr infnan 3
.Sh AUTHOR
Kwok\-Choi Ng
.Sh HISTORY
The
.Nm ieee
functions appeared in 
.Bx 4.3 .
.Sh BUGS
Should
.Fn drem x 0
and
.Fn logb 0
on a
.Tn VAX
signal invalidity 
by setting
.Va errno No = Dv EDOM ?
Should
.Fn logb 0
return  \-1.7e38?
.Pp
.Tn IEEE
754 currently specifies that
.Fn logb "denormalized no."
=
.Fn logb "tiniest normalized no. > 0"
but the consensus has changed to the specification in the new 
proposed
.Tn IEEE
standard p854, namely that
.Fn logb x
satisfy 
.Bd -filled -offset indent
1 \(<=
.Fn scalb \\*(Bax\\*(Ba \-logb(x)
<
Radix\0 ... = 2
for
.Tn IEEE
754
.Ed
.Pp
for every x except 0, 
\*(If
and \*(Na.
Almost every program that assumes 754's specification will work
correctly if
.Fn logb
follows 854's specification instead.
.Pp
.Tn IEEE
754 requires
.Fn copysign x \\*(Na)
=
.Pf \(+- Ns Fa x
but says nothing
else about the sign of a \*(Na.  A \*(Na
.Em N Ns ot
.Em a
.Em N Ns umber )
is
similar in spirit to the
.Tn VAX Ns 's
reserved operand, but very
different in important details.  Since the sign bit of a
reserved operand makes it look negative,  
.Bd -filled -offset indent
.Fn copysign x "reserved operand"
=
.Pf \- Fa x ;
.Ed
.Pp
should this return the reserved operand instead?