.\" Copyright (c) 1985 Regents of the University of California. .\" All rights reserved. The Berkeley software License Agreement .\" specifies the terms and conditions for redistribution. .\" .\" @(#)exp.3m 6.9 (Berkeley) 5/27/86 .\" .TH EXP 3M "May 27, 1986" .UC 4 .ds nn \fINaN\fR .ds up \fIulp\fR .SH NAME exp, expm1, log, log10, log1p, pow \- exponential, logarithm, power .SH SYNOPSIS .nf .B #include <math.h> .PP .B double exp(x) .B double x; .PP .B double expm1(x) .B double x; .PP .B double log(x) .B double x; .PP .B double log10(x) .B double x; .PP .B double log1p(x) .B double x; .PP .B double pow(x,y) .B double x,y; .fi .SH DESCRIPTION .PP Exp returns the exponential function of x. .PP Expm1 returns exp(x)\-1 accurately even for tiny x. .PP Log returns the natural logarithm of x. .PP Log10 returns the logarithm of x to base 10. .PP Log1p returns log(1+x) accurately even for tiny x. .PP Pow(x,y) returns .if n \ x**y. .if t \ x\u\s8y\s10\d. .SH 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 \fIU\fRnit in the \fIL\fRast \fIP\fRlace. 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. .SH DIAGNOSTICS Exp, expm1 and pow return the reserved operand on a VAX when the correct value would overflow, and they set \fIerrno\fR to ERANGE. Pow(x,y) returns the reserved operand on a VAX and sets \fIerrno\fR to EDOM when x < 0 and y is not an integer. .PP On a VAX, \fIerrno\fR is set to EDOM and the reserved operand is returned by log unless x > 0, by log1p unless x > \-1. .SH 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. .PP Pow(x,0) returns x**0 = 1 for all x including x = 0, .if n \ Infinity .if t \ \(if (not found on a VAX), and \*(nn (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: .IP (1) \w'\0\0\0\0'u Any program that already tests whether x is zero (or infinite or \*(nn) 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. .IP (2) \w'\0\0\0\0'u 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 .ce p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n .IP at x = 0 rather than reject a[0]\(**0**0 as invalid. .IP (3) \w'\0\0\0\0'u 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: .IP If x(z) and y(z) are \fIany\fR 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. .IP (4) \w'\0\0\0\0'u If 0**0 = 1, then .if n \ infinity**0 = 1/0**0 = 1 too; and .if t \ \(if**0 = 1/0**0 = 1 too; and then \*(nn**0 = 1 too because x**0 = 1 for all finite and infinite x, i.e., independently of x. .SH SEE ALSO math(3M), infnan(3M) .SH AUTHOR Kwok\-Choi Ng, W. Kahan