4.3BSD-UWisc/man/cat3/exp.3m
EXP(3M) UNIX Programmer's Manual EXP(3M)
NAME
exp, expm1, log, log10, log1p, pow - exponential, logarithm,
power
SYNOPSIS
#include <math.h>
double exp(x)
double x;
double expm1(x)
double x;
double log(x)
double x;
double log10(x)
double x;
double log1p(x)
double x;
double pow(x,y)
double x,y;
DESCRIPTION
Exp returns the exponential function of x.
Expm1 returns exp(x)-1 accurately even for tiny x.
Log returns the natural logarithm of x.
Log10 returns the logarithm of x to base 10.
Log1p returns log(1+x) accurately even for tiny x.
Pow(x,y) returns x**y.
ERROR (due to Roundoff etc.)
exp(x), log(x), expm1(x) and log1p(x) are accurate to within
an _u_l_p, and log10(x) to within about 2 _u_l_ps; an _u_l_p is one
_Unit in the _Last _Place. The error in pow(x,y) is below
about 2 _u_l_ps 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 _u_l_ps for VAX D, 300 _u_l_ps 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.
Printed 12/27/86 May 27, 1986 1
EXP(3M) UNIX Programmer's Manual EXP(3M)
DIAGNOSTICS
Exp, expm1 and pow return the reserved operand on a VAX when
the correct value would overflow, and they set _e_r_r_n_o to
ERANGE. Pow(x,y) returns the reserved operand on a VAX and
sets _e_r_r_n_o to EDOM when x < 0 and y is not an integer.
On a VAX, _e_r_r_n_o is set to EDOM and the reserved operand is
returned by log unless x > 0, by log1p unless x > -1.
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.
Pow(x,0) returns x**0 = 1 for all x including x = 0, Infin-
ity (not found on a VAX), and _N_a_N (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:
(1) Any program that already tests whether x is zero (or
infinite or _N_a_N) 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.
(2) 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
p(x) = a[0]*x**0 + a[1]*x**1 + a[2]*x**2 +...+ a[n]*x**n
at x = 0 rather than reject a[0]*0**0 as invalid.
(3) 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:
If x(z) and y(z) are _a_n_y 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.
(4) If 0**0 = 1, then infinity**0 = 1/0**0 = 1 too; and then
_N_a_N**0 = 1 too because x**0 = 1 for all finite and
infinite x, i.e., independently of x.
Printed 12/27/86 May 27, 1986 2
EXP(3M) UNIX Programmer's Manual EXP(3M)
SEE ALSO
math(3M), infnan(3M)
AUTHOR
Kwok-Choi Ng, W. Kahan
Printed 12/27/86 May 27, 1986 3