./_Man_NeXT_html/html3/exp.3m.html

Manual page for EXP(3M)

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

SYNOPSIS

#include <math.h>

double exp(double x);

double log(double x);

double log10(double x);

double pow(double x, double y);

(ALSO AVAILABLE IN BSD)

double expm1(double x);

double log1p(double x);

DESCRIPTION

Exp returns the exponential function of x.

Log returns the natural logarithm of x.

Log10 returns the logarithm of x to base 10.

Pow(x,y) returns xy.

Expm1 returns exp(x)-1 accurately even for tiny x.

Log1p returns log(1+x) accurately even for tiny x.

ERROR (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. No such drastic loss has been exposed by testing. 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.

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.

Pow(x,0) returns x**0 = 1 for all x including x = 0, \(if, 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 \(if**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.

AUTHOR

Kwok-Choi Ng, W. Kahan