EXP(3) BSD Programmer's Manual EXP(3) NNAAMMEE eexxpp, eexxppmm11, lloogg, lloogg1100, lloogg11pp, ppooww - exponential, logarithm, power func- tions SSYYNNOOPPSSIISS ##iinncclluuddee <> _d_o_u_b_l_e eexxpp(_d_o_u_b_l_e _x); _d_o_u_b_l_e eexxppmm11(_d_o_u_b_l_e _x); _d_o_u_b_l_e lloogg(_d_o_u_b_l_e _x); _d_o_u_b_l_e lloogg1100(_d_o_u_b_l_e _x); _d_o_u_b_l_e lloogg11pp(_d_o_u_b_l_e _x); _d_o_u_b_l_e ppooww(_d_o_u_b_l_e _x, _d_o_u_b_l_e _y); DDEESSCCRRIIPPTTIIOONN The eexxpp() function computes the exponential value of the given argument _x. The eexxppmm11() function computes the value exp(x)-1 accurately even for tiny argument _x. The lloogg() function computes the value for the natural logarithm of the argument x. The lloogg1100() function computes the value for the logarithm of argument _x to base 10. The lloogg11pp() function computes the value of log(1+x) accurately even for tiny argument _x. The ppooww() computes the value of _x to the exponent _y. EERRRROORR ((dduuee ttoo RRoouunnddooffff eettcc..)) exp(x), log(x), expm1(x) and log1p(x) are accurate to within an _u_p, and log10(x) to within about 2 _u_p_s; an _u_p is one _U_n_i_t in the _L_a_s_t _P_l_a_c_e. The error in ppooww(_x, _y) is below about 2 _u_p_s when its magnitude is moderate, but increases as ppooww(_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_p_s for VAX D, 300 _u_p_s for IEEE 754 Double. Moderate values of ppooww() are accurate enough that ppooww(_i_n_t_e_g_e_r, _i_n_t_e_g_e_r) is exact until it is bigger than 2**56 on a VAX, 2**53 for IEEE 754. RREETTUURRNN VVAALLUUEESS These functions will return the approprate computation unless an error occurs or an argument is out of range. The functions eexxpp(), eexxppmm11() and ppooww() detect if the computed value will overflow, set the global variable _e_r_r_n_o _t_o RANGE and cause a reserved operand fault on a VAX or Tahoe. The function ppooww(_x, _y) checks to see if _x < 0 and _y is not an integer, in the event this is true, the global variable _e_r_r_n_o is set to EDOM and on the VAX and Tahoe generate a reserved operand fault. On a VAX and Tahoe, _e_r_r_n_o is set to EDOM and the reserved operand is returned by log unless _x > 0, by lloogg11pp() unless _x > -1. NNOOTTEESS 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 ppooww(_x, _0) returns x**0 = 1 for all x including x = 0, Infin- ity (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 al- ways: 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 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) 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., inde- pendently of x. SSEEEE AALLSSOO math(3), infnan(3) HHIISSTTOORRYY A eexxpp(), lloogg() and ppooww() function appeared in Version 6 AT&T UNIX. A lloogg1100() function appeared in Version 7 AT&T UNIX. The lloogg11pp() and eexxppmm11() functions appeared in 4.3BSD. 4th Berkeley Distribution June 4, 1993 2