EXP(3)

NAME
exp, expf, exp2, exp2f, 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
exp2(double x);
float
exp2f(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() and the expf() functions compute the base e exponential value of the given argument x.

The exp2(), and exp2f() functions compute the base 2 exponential of the given argument x.

The expm1() and the expm1f() functions computes the value exp(x)-1 accurately even for tiny argument x.

The log() function computes the value of the natural logarithm of argument 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() and powf() functions compute 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.

ERRORS
The values of exp(x), expm1(x), exp2(x), log(x), and log1p(x), are exact provided that they are representable. Otherwise the error in these functions is generally below one ulp. The values of log10(x) are 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(x, y) 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(x) and logp1(x) in BASIC on the Hewlett-Packard HP-71B and APPLE Macintosh, EXP1 and LN1 in Pascal, exp1(x) and log1(x) 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 ) 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) *[Gt] 1 as z *[Gt] 0.

4.  If 0**0 = 1, then infinity**0 = 1/0**0 = 1 too; and then **0 = 1 too because x**0 = 1 for all finite and infinite x, i.e., independently of x.

SEE ALSO
math(3)

STANDARDS
The exp(), log(), log10() and pow() functions conform to ansiC. The exp2(), exp2f(), expf(), expm1(), expm1f(), log1p(), log1pf(), log2(), log2f(), log10f(), logf(), and powf() functions conform to isoC-99.

HISTORY
A exp(), log() and pow() functions appeared in AT&T UNIX. A log10() function appeared in AT&T UNIX. The log1p() and expm1() functions appeared in 4.3BSD.