EXP(3M) EXP(3M) NAME exp, expm1, log, log10, log1p, pow - exponential, logarithm, power SYNOPSIS #include double exp(x) double x; double expm1(x) double x; double log(x) double x; double log10(x) double x; double log1p(x) double x; double pow(x,y) double x,y; DESCRIPTION Exp returns the exponential function of x. Expm1 returns exp(x)-1 accurately even for tiny x. Log returns the natural logarithm of x. Log10 returns the logarithm of x to base 10. Log1p returns log(1+x) accurately even for tiny x. Pow(x,y) returns x**y. ERROR (due to Roundoff etc.) exp(x), log(x), expm1(x) and log1p(x) are accurate to within an _u_l_p, and log10(x) to within about 2 _u_l_ps; an _u_l_p is one _Unit in the _Last _Place. The error in pow(x,y) is below about 2 _u_l_ps 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 _u_l_ps for VAX D, 300 _u_l_ps 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. DIAGNOSTICS Exp, expm1 and pow return the reserved operand on a VAX when the cor‐ rect value would overflow, and they set _e_r_r_n_o to ERANGE. Pow(x,y) returns the reserved operand on a VAX and sets _e_r_r_n_o to EDOM when x < 0 and y is not an integer. On a VAX, _e_r_r_n_o is set to EDOM and the reserved operand is returned by log unless x > 0, by log1p unless x > -1. 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 pro‐ vide accurate inverse hyperbolic functions. Pow(x,0) returns x**0 = 1 for all x including x = 0, Infinity (not found on a VAX), and _N_a_N (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: (1) Any program that already tests whether x is zero (or infinite or _N_a_N) 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. (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 _a_n_y 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 _N_a_N**0 = 1 too because x**0 = 1 for all finite and infinite x, i.e., indepen‐ dently of x. SEE ALSO math(3M), infnan(3M) AUTHOR Kwok-Choi Ng, W. Kahan 4th Berkeley Distribution May 27, 1986 EXP(3M)