EXP(3) BSD Programmer's Manual EXP(3)

exp,expf,exp2,exp2f,expm1,expm1f, - exponential functions

libm

#include <math.h>doubleexp(double x);floatexpf(float x);doubleexp2(double x);floatexp2f(float x);doubleexpm1(double x);floatexpm1f(float x);

Theexp() and theexpf() functions compute the baseeexponential value of the given argumentx. Theexp2() andexp2f() functions compute the base 2 exponential of the given argumentx. Theexpm1() and theexpm1f() functions computes the value exp(x)-1 accu- rately even for tiny argumentx.

These functions will return the appropriate computation unless an error occurs or an argument is out of range. The functionsexp() andexpm1() detect if the computed value will overflow, set the global variableerrnoto ERANGE and cause a reserved operand fault on a VAX. The functionpow(x,y) checks to see ifx< 0 andyis not an integer, in the event this is true, the global variableerrnois set to EDOM and on the VAX generate a reserved operand fault. On a VAX,errnois set to EDOM and the reserved operand is returned by log unlessx> 0, bylog1p() unlessx> -1.

exp(x), log(x), expm1(x) and log1p(x) are accurate to within anulp, and log10(x) to within about 2ulps; anulpis oneUnitin theLast Place. The error inpow(x,y) is below about 2ulpswhen its magnitude is moderate, but increases aspow(x,y) approaches the over/underflow thres- holds 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 test- ing; the worst errors observed have been below 20ulpsfor VAX D, 300ulpsfor IEEE 754 Double. Moderate values ofpow() are accurate enough thatpow(integer,integer) is exact until it is bigger than 2**56 on a VAX, 2**53 for IEEE 754.

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 functionpow(x,0) returns x**0 = 1 for all x including x = 0, Infin- ity (not found on a VAX), andNaN(the reserved operand on a VAX). Previ- ous 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 orNaN) 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, in- cluding x = 0. This is compatible with the convention that ac- cepts 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) areanyfunctions 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 thenNaN**0 = 1 too because x**0 = 1 for all finite and infinite x, i.e., in- dependently of x.

math(3)

Theexp() functions conform to ANSI X3.159-1989 ("ANSI C"). Theexp2(),exp2f(),expf(),expm1(), andexpm1f() functions conform to ISO/IEC 9899:1999 ("ISO C99").

Theexp() functions appeared in Version 6 AT&T UNIX. Theexpm1() function appeared in 4.3BSD. MirOS BSD #10-current February 9, 2014 1

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