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.\" Copyright (c) 1985 Regents of the University of California.
.\" All rights reserved.  The Berkeley software License Agreement
.\" specifies the terms and conditions for redistribution.
.\"
.\"	@(#)exp.3	6.9 (Berkeley) 5/27/86
.\"
.TH EXP 3M  "May 27, 1986"
.UC 4
.ds nn \fINaN\fR
.ds up \fIulp\fR
.SH NAME
exp, expm1, log, log10, log1p, pow \- exponential, logarithm, power
.SH SYNOPSIS
.nf
.B #include <math.h>
.PP
.B double exp(x)
.B double x;
.PP
.B double expm1(x)
.B double x;
.PP
.B double log(x)
.B double x;
.PP
.B double log10(x)
.B double x;
.PP
.B double log1p(x)
.B double x;
.PP
.B double pow(x,y)
.B double x,y;
.fi
.SH DESCRIPTION
.PP
Exp returns the exponential function of x.
.PP
Expm1 returns exp(x)\-1 accurately even for tiny x.
.PP
Log returns the natural logarithm of x.
.PP
Log10 returns the logarithm of x to base 10.
.PP
Log1p returns log(1+x) accurately even for tiny x.
.PP
Pow(x,y) returns
.if n \
x**y.
.if t \
x\u\s8y\s10\d.
.SH ERROR (due to Roundoff etc.)
exp(x), log(x), expm1(x) and log1p(x) are accurate to within 
an \*(up, and log10(x) to within about 2 \*(ups;
an \*(up is one \fIU\fRnit in the \fIL\fRast \fIP\fRlace.
The error in pow(x,y) is below about 2 \*(ups when its
magnitude is moderate, but increases as pow(x,y) approaches
the over/underflow thresholds until almost as many bits could be
lost as are occupied by the floating\-point format's exponent
field; that is 8 bits for VAX D and 11 bits for IEEE 754 Double.
No such drastic loss has been exposed by testing; the worst
errors observed have been below 20 \*(ups for VAX D,
300 \*(ups for IEEE 754 Double.
Moderate values of pow are accurate enough that pow(integer,integer)
is exact until it is bigger than 2**56 on a VAX, 2**53 for IEEE 754.
.SH DIAGNOSTICS
Exp, expm1 and pow return the reserved operand on a VAX when the correct
value would overflow, and they set \fIerrno\fR to ERANGE.
Pow(x,y) returns the reserved operand on a VAX and sets \fIerrno\fR
to EDOM when x < 0 and y is not an integer.
.PP
On a VAX, \fIerrno\fR is set to EDOM and the reserved operand is returned
by log unless x > 0, by log1p unless x > \-1.
.SH NOTES
The functions exp(x)\-1 and log(1+x) are called expm1
and logp1 in BASIC on the Hewlett\-Packard HP\-71B and APPLE
Macintosh, EXP1 and LN1 in Pascal, exp1 and log1 in C
on APPLE Macintoshes, where they have been provided to make
sure financial calculations of ((1+x)**n\-1)/x, namely
expm1(n\(**log1p(x))/x, will be accurate when x is tiny.
They also provide accurate inverse hyperbolic functions.
.PP
Pow(x,0) returns x**0 = 1 for all x including x = 0,
.if n \
Infinity
.if t \
\(if
(not found on a VAX), and \*(nn (the reserved
operand on a VAX).  Previous implementations of pow may
have defined x**0 to be undefined in some or all of these
cases.  Here are reasons for returning x**0 = 1 always:
.IP (1) \w'\0\0\0\0'u
Any program that already tests whether x is zero (or
infinite or \*(nn) before computing x**0 cannot care
whether 0**0 = 1 or not. Any program that depends
upon 0**0 to be invalid is dubious anyway since that
expression's meaning and, if invalid, its consequences 
vary from one computer system to another.
.IP (2) \w'\0\0\0\0'u
Some Algebra texts (e.g. Sigler's) define x**0 = 1 for 
all x, including x = 0.
This is compatible with the convention that accepts a[0]
as the value of polynomial
.ce
p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n
.IP
at x = 0 rather than reject a[0]\(**0**0 as invalid.
.IP (3) \w'\0\0\0\0'u
Analysts will accept 0**0 = 1 despite that x**y can
approach anything or nothing as x and y approach 0
independently.
The reason for setting 0**0 = 1 anyway is this:
.IP
If x(z) and y(z) are \fIany\fR functions analytic (expandable
in power series) in z around z = 0, and if there 
x(0) = y(0) = 0, then x(z)**y(z) \(-> 1 as z \(-> 0.
.IP (4) \w'\0\0\0\0'u
If 0**0 = 1, then
.if n \
infinity**0 = 1/0**0 = 1 too; and
.if t \
\(if**0 = 1/0**0 = 1 too; and
then \*(nn**0 = 1 too because x**0 = 1 for all finite
and infinite x, i.e., independently of x.
.SH SEE ALSO
math(3M), infnan(3M)
.SH AUTHOR
Kwok\-Choi Ng, W. Kahan

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