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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>doubleexp(doublex);floatexpf(floatx);longdoubleexpl(longdoublex);doubleexp2(doublex);floatexp2f(floatx);longdoubleexp2l(longdoublex);doubleexpm1(doublex);floatexpm1f(floatx);longdoubleexpm1l(longdoublex);doublelog(doublex);floatlogf(floatx);longdoublelogl(longdoublex);doublelog2(doublex);floatlog2f(floatx);longdoublelog2l(longdoublex);doublelog10(doublex);floatlog10f(floatx);longdoublelog10l(longdoublex);doublelog1p(doublex);floatlog1pf(floatx);longdoublelog1pl(longdoublex);doublepow(doublex,doubley);floatpowf(floatx,floaty);longdoublepowl(longdoublex,longdoubley);## DESCRIPTION

Theexp() function computes the baseeexponential value of the given argumentx. Theexpf() function is a single precision version ofexp(). Theexpl() function is an extended precision version ofexp(). Theexp2() function computes the base 2 exponential of the given argumentx. Theexp2f() function is a single precision version ofexp2(). Theexp2l() function is an extended precision version ofexp2(). Theexpm1() function computes the value exp(x)-1 accurately even for tiny argumentx. Theexpm1f() function is a single precision version ofexpm1(). Theexpm1l() function is an extended precision version ofexpm1(). Thelog() function computes the value of the natural logarithm of argu- mentx. Thelogf() function is a single precision version oflog(). Thelogl() function is an extended precision version oflog(). Thelog2() function computes the value of the logarithm of argumentxto base 2. Thelog2f() function is a single precision version oflog2(). Thelog2l() function is an extended precision version oflog2(). Thelog10() function computes the value of the logarithm of argumentxto base 10. Thelog10f() function is a single precision version oflog10(). Thelog10l() function is an extended precision version oflog10(). Thelog1p() function computes the value of log(1+x) accurately even for tiny argumentx. Thelog1pf() function is a single precision version oflog1p(). Thelog1pl() function is an extended precision version oflog1p(). Thepow() function computes the value ofxto the exponenty. Thepowf() function is a single precision version ofpow(). Thepowl() function is an extended precision version ofpow().## RETURN VALUES

These functions will return the appropriate computation unless an error occurs or an argument is out of range. The functionsexp(),expm1() andpow() detect if the computed value will overflow, set the global variableerrnoto ERANGE and cause a reserved operand fault on a VAX or Tahoe. 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 and Tahoe generate a reserved operand fault. On a VAX and Tahoe,errnois set to EDOM and the reserved operand is returned by log unlessx> 0, bylog1p() unlessx> -1. ERRORS (due to Roundoff etc.) exp(x), log(x), expm1(x) and log1p(x) are accurate to within anulp, and log10(x) to within about 2ulps; anulpis oneUnitin theLastPlace. The error inpow(x,y) is below about 2ulpswhen its magnitude is moder- ate, but increases aspow(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 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.## 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 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). Pre- vious implementations ofpow() 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, 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) 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., inde- pendently of x.## SEE ALSO

ilogb(3), infnan(3)## HISTORY

Theexp() andlog() functions first appeared in Version 1 AT&T UNIX;pow() in Version 3 AT&T UNIX;log10() in Version 7 AT&T UNIX;log1p() andexpm1() in 4.3BSD. DragonFly 4.9 January 15, 2015 DragonFly 4.9

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