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:

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.

SEE ALSO

fpclassify(3), ilogb(3)

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.