EXP(3) BSD Programmer's Manual EXP(3)NAME
exp, expf, expm1, expm1f, log, logf, log2, log2f, log10, log10f, log1p,
log1pf, pow, powf - exponential, logarithm, power functions
LIBRARY
libm
SYNOPSIS
#include <math.h>
double
exp(double x);
float
expf(float x);
double
expm1(double x);
float
expm1f(float x);
double
log(double x);
float
logf(float x);
double
log2(double x);
float
log2f(float x);
double
log10(double x);
float
log10f(float x);
double
log1p(double x);
float
log1pf(float x);
double
pow(double x, double y);
float
powf(float x, float y);
DESCRIPTION
The exp() function computes the exponential value of the given argument
x.
The expm1() function computes the value exp(x)-1 accurately even for tiny
argument x.
The log() function computes the value of the natural logarithm of argu-
ment x.
The log10() function computes the value of the logarithm of argument x to
base 10.
The log1p() function computes the value of log(1+x) accurately even for
tiny argument x.
The log2() and the log2f() functions compute the value of the logarithm
of argument x to base 2.
The pow() computes the value of x to the exponent y.
RETURN VALUES
These functions will return the appropriate computation unless an error
occurs or an argument is out of range. The functions exp(), expm1() and
pow() detect if the computed value will overflow, set the global variable
errno to ERANGE and cause a reserved operand fault on a VAX. The function
pow(x, y) checks to see if x < 0 and y is not an integer, in the event
this is true, the global variable errno is set to EDOM and on the VAX
generate a reserved operand fault. On a VAX, errno is set to EDOM and the
reserved operand is returned by log unless x > 0, by log1p() unless x >
-1.
ERRORSexp(x), log(x), expm1(x) and log1p(x) are accurate to within an ulp, and
log10(x) to within about 2 ulps; an ulp is one Unit in the Last Place.
The error in pow(x, y) is below about 2 ulps when its magnitude is
moderate, but increases as pow(x, y) approaches the over/underflow thres-
holds 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 test-
ing; the worst errors observed have been below 20 ulps for VAX D, 300
ulps 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.
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 Pas-
cal, exp1 and log1 in C on APPLE Macintoshes, where they have been pro-
vided 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.
The function pow(x, 0) returns x**0 = 1 for all x including x = 0, Infin-
ity (not found on a VAX), and NaN (the reserved operand on a VAX). Previ-
ous 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
NaN) before computing x**0 cannot care whether 0**0 = 1 or not.
Any program that depends upon 0**0 to be invalid is dubious any-
way since that expression's meaning and, if invalid, its conse-
quences vary from one computer system to another.
2. Some Algebra texts (e.g. Sigler's) define x**0 = 1 for all x, in-
cluding x = 0. This is compatible with the convention that ac-
cepts 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 any-
thing 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 any 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 NaN**0 =
1 too because x**0 = 1 for all finite and infinite x, i.e., in-
dependently of x.
SEE ALSOmath(3)STANDARDS
The exp(), log(), log10() and pow() functions conform to ANSI X3.159-1989
("ANSI C").
HISTORY
A exp(), log() and pow() functions appeared in Version 6 AT&T UNIX. A
log10() function appeared in Version 7 AT&T UNIX. The log1p() and expm1()
functions appeared in 4.3BSD.
MirOS BSD #10-current July 21, 2005 2
