exp, expf, expm1, expm1f, log, logf, log10, log10f, log1p,
log1pf, pow,
powf - exponential, logarithm, power functions
#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
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);
The exp() function computes the exponential value of the
given argument
x. The expf() function is a single precision version of
exp().
The expm1() function computes the value exp(x)-1 accurately
even for tiny
argument x. The expm1f() function is a single precision
version of
expm1().
The log() function computes the value of the natural logarithm of argument
x. The logf() function is a single precision version
of log().
The log10() function computes the value of the logarithm of
argument x to
base 10. The log10f() function is a single precision version of log10().
The log1p() function computes the value of log(1+x) accurately even for
tiny argument x. The log1pf() function is a single precision version of
log1p().
The pow() function computes the value of x to the exponent
y. The powf()
function is a single precision version of pow().
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
or Tahoe.
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 and Tahoe generate a reserved operand fault. On a
VAX and Tahoe,
errno is set to EDOM and the reserved operand is returned by
log unless x
> 0, by log1p() unless x > -1.
ERRORS (due to Roundoff etc.) [Toc] [Back] exp(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 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 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.
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.
The function pow(x, 0) returns x**0 = 1 for all x including
x = 0, Infinity
(not found on a VAX), and NaN (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
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 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 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., independently
of x.
infnan(3), math(3)
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.
OpenBSD 3.6 July 31, 1991
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