NAME
exp, expf,
expl, exp2,
exp2f, exp2l,
expm1, expm1f,
expm1l, log,
logf, logl,
log2, log2f,
log2l, log10,
log10f, log10l,
log1p, log1pf,
log1pl, pow,
powf, powl —
exponential, logarithm, power
functions
SYNOPSIS
#include
<math.h>
double
exp(double
x);
float
expf(float
x);
long double
expl(long
double x);
double
exp2(double
x);
float
exp2f(float
x);
long double
exp2l(long
double x);
double
expm1(double
x);
float
expm1f(float
x);
long double
expm1l(long
double x);
double
log(double
x);
float
logf(float
x);
long double
logl(long
double x);
double
log2(double
x);
float
log2f(float
x);
long double
log2l(long
double x);
double
log10(double
x);
float
log10f(float
x);
long double
log10l(long
double x);
double
log1p(double
x);
float
log1pf(float
x);
long double
log1pl(long
double x);
double
pow(double
x, double y);
float
powf(float
x, float y);
long double
powl(long
double x, long double
y);
DESCRIPTION
The
exp()
function computes the base e exponential value of
the given argument x. The
expf()
function is a single precision version of exp(). The
expl()
function is an extended precision version of
exp().
The
exp2()
function computes the base 2 exponential of the given argument
x. The
exp2f()
function is a single precision version of exp2().
The
exp2l()
function is an extended precision version of
exp2().
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
expm1l()
function is an extended 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
logl()
function is an extended precision version of
log().
The
log2()
function computes the value of the logarithm of argument
x to base 2. The
log2f()
function is a single precision version of log2().
The
log2l()
function is an extended precision version of
log2().
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
log10l()
function is an extended 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
log1pl()
function is an extended 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(). The
powl()
function is an extended precision version of
pow().
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 and
set the global variable errno to
ERANGE. 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.
ERRORS (due to Roundoff etc.)
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 11 bits for IEEE 754 Double. No such
drastic loss has been exposed by testing; the worst errors observed have
been below 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**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 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 and
infinity. 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:
- 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.
- 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.
- 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.
- 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.
SEE ALSO
HISTORY
The exp() and
log() functions first appeared in
Version 1 AT&T UNIX;
pow() in Version 3 AT&T
UNIX; log10() in
Version 7 AT&T UNIX;
log1p() and expm1() in
4.3BSD.