DragonFly On-Line Manual Pages

EXP(3)		      DragonFly Library Functions Manual		EXP(3)

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 argu- ment 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, set the global variable errno to ERANGE 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. 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 moder- ate, but increases as pow(x, y) approaches the over/underflow thresholds until almost as many bits could be lost as are occupied by the float- ing-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 test- ing; the worst errors observed have been below 20 ulps for ``VAX D'', 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**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 Pas- cal, exp1 and log1 in C on APPLE Macintoshes, where they have been pro- vided 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, infin- ity (not found on a VAX), and NaN (the reserved operand on a VAX). Pre- vious 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 any- way since that expression's meaning and, if invalid, its conse- quences 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 any- thing 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., inde- pendently of x.

SEE ALSO

ilogb(3), infnan(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. DragonFly 4.3 January 15, 2015 DragonFly 4.3