EXP(3)

NAME
exp, expm1, log, log10, log1p, pow - exponential, logarithm, power functions

SYNOPSIS
#include <math.h>
double
exp(double x);
double
expm1(double x);
double
log(double x);
double
log10(double x);
double
log1p(double x);
double
pow(double x, double 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 for the natural logarithm of the argument x.

The log10() function computes the value for 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 pow() computes the value of x to the exponent y.

ERROR (due to Roundoff etc.)
exp(x), log(x), expm1(x) and log1p(x) are accurate to within an up, and log10(x) to within about 2 ups; an up is one Unit in the Last Place. The error in pow(x, y) is below about 2 ups 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 ups for VAX D, 300 ups 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.

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 RANGE 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.

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, 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) → 1 as z → 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), infnan(3)

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