# FreeBSD Manual Pages

EXP(3) FreeBSD Library Functions Manual EXP(3)NAMEexp,expf,exp10,exp10f,expm1,expm1f,log,logf,log10,log10f,log1p,log1pf,pow,powf-- exponential, logarithm, power functionsLIBRARYMath Library (libm, -lm)SYNOPSIS#include<math.h>doubleexp(doublex);floatexpf(floatx);doubleexpm1(doublex);floatexpm1f(floatx);doublelog(doublex);floatlogf(floatx);doublelog10(doublex);floatlog10f(floatx);doublelog1p(doublex);floatlog1pf(floatx);doublepow(doublex,doubley);floatpowf(floatx,floaty);DESCRIPTIONTheexp() and theexpf() functions compute the exponential value of the given argumentx. Theexpm1() and theexpm1f() functions compute the value exp(x)-1 accu- rately even for tiny argumentx. Thelog() and thelogf() functions compute the value of the natural loga- rithm of argumentx. Thelog10() and thelog10f() functions compute the value of the logarithm of argumentxto base 10. Thelog1p() and thelog1pf() functions compute the value of log(1+x) accurately even for tiny argumentx. Thepow() and thepowf() functions compute the value ofxto the exponenty.ERROR (due to Roundoff etc.)exp(x),log(x),expm1(x) andlog1p(x) are accurate to within anulp, andlog10(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 testing; 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.RETURN VALUESThese functions will return the appropriate computation unless an error occurs or an argument is out of range. The functionsexp(),expm1(),pow() detect if the computed value will overflow, set the global variableerrnoto ERANGE and cause a reserved operand fault on a VAX or Tahoe. The functionspow(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.NOTESThe 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 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, 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 ALSOmath(3)HISTORYAexp(),log() andpow() functions appeared in Version 6 AT&T UNIX. Alog10() function appeared in Version 7 AT&T UNIX. Thelog1p() andexpm1() functions appeared in 4.3BSD. FreeBSD 11.1 July 31, 1991 FreeBSD 11.1

NAME | LIBRARY | SYNOPSIS | DESCRIPTION | ERROR (due to Roundoff etc.) | RETURN VALUES | NOTES | SEE ALSO | HISTORY

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