1EXP(3) 386BSD Programmer's Manual EXP(3) 2 3NNAAMMEE 4 eexxpp, eexxppmm11, lloogg, lloogg1100, lloogg11pp, ppooww - exponential, logarithm, power 5 functions 6 7SSYYNNOOPPSSIISS 8 ##iinncclluuddee <<mmaatthh..hh>> 9 10 _d_o_u_b_l_e 11 eexxpp(_d_o_u_b_l_e _x) 12 13 _d_o_u_b_l_e 14 eexxppmm11(_d_o_u_b_l_e _x) 15 16 _d_o_u_b_l_e 17 lloogg(_d_o_u_b_l_e _x) 18 19 _d_o_u_b_l_e 20 lloogg1100(_d_o_u_b_l_e _x) 21 22 _d_o_u_b_l_e 23 lloogg11pp(_d_o_u_b_l_e _x) 24 25 _d_o_u_b_l_e 26 ppooww(_d_o_u_b_l_e _x, _d_o_u_b_l_e _y) 27 28DDEESSCCRRIIPPTTIIOONN 29 The eexxpp() function computes the exponential value of the given argument 30 _x. 31 32 The eexxppmm11() function computes the value exp(x)-1 accurately even for tiny 33 argument _x. 34 35 The lloogg() function computes the value for the natural logarithm of the 36 argument x. 37 38 The lloogg1100() function computes the value for the logarithm of argument _x 39 to base 10. 40 41 The lloogg11pp() function computes the value of log(1+x) accurately even for 42 tiny argument _x. 43 44 The ppooww() computes the value of _x to the exponent _y. 45 46EERRRROORR ((dduuee ttoo RRoouunnddooffff eettcc..)) 47 exp(x), log(x), expm1(x) and log1p(x) are accurate to within an _u_p, and 48 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 49 error in ppooww(_x, _y) is below about 2 _u_p_s when its magnitude is moderate, 50 but increases as ppooww(_x, _y) approaches the over/underflow thresholds until 51 almost as many bits could be lost as are occupied by the floating-point 52 format's exponent field; that is 8 bits for VAX D and 11 bits for IEEE 53 754 Double. No such drastic loss has been exposed by testing; the worst 54 errors observed have been below 20 _u_p_s for VAX D, 300 _u_p_s for IEEE 754 55 Double. Moderate values of ppooww() are accurate enough that ppooww(_i_n_t_e_g_e_r, 56 _i_n_t_e_g_e_r) is exact until it is bigger than 2**56 on a VAX, 2**53 for IEEE 57 754. 58 59RREETTUURRNN VVAALLUUEESS 60 These functions will return the approprate computation unless an error 61 occurs or an argument is out of range. The functions eexxpp(), eexxppmm11() and 62 ppooww() detect if the computed value will overflow, set the global variable 63 _e_r_r_n_o _t_o RANGE and cause a reserved operand fault on a VAX or Tahoe. The 64 function ppooww(_x, _y) checks to see if _x < 0 and _y is not an integer, in the 65 event this is true, the global variable _e_r_r_n_o is set to EDOM and on the 66 VAX and Tahoe generate a reserved operand fault. On a VAX and Tahoe, 67 _e_r_r_n_o is set to EDOM and the reserved operand is returned by log unless _x 68 > 0, by lloogg11pp() unless _x > -1. 69 70NNOOTTEESS 71 The functions exp(x)-1 and log(1+x) are called expm1 and logp1 in BASIC 72 on the Hewlett-Packard HP-71B and APPLE Macintosh, EXP1 and LN1 in 73 Pascal, exp1 and log1 in C on APPLE Macintoshes, where they have been 74 provided to make sure financial calculations of ((1+x)**n-1)/x, namely 75 expm1(n*log1p(x))/x, will be accurate when x is tiny. They also provide 76 accurate inverse hyperbolic functions. 77 78 The function ppooww(_x, _0) returns x**0 = 1 for all x including x = 0, 79 Infinity (not found on a VAX), and _N_a_N (the reserved operand on a VAX). 80 Previous implementations of pow may have defined x**0 to be undefined in 81 some or all of these cases. Here are reasons for returning x**0 = 1 82 always: 83 84 1. Any program that already tests whether x is zero (or infinite or 85 _N_a_N) before computing x**0 cannot care whether 0**0 = 1 or not. 86 Any program that depends upon 0**0 to be invalid is dubious 87 anyway since that expression's meaning and, if invalid, its 88 consequences vary from one computer system to another. 89 90 2. Some Algebra texts (e.g. Sigler's) define x**0 = 1 for all x, 91 including x = 0. This is compatible with the convention that 92 accepts a[0] as the value of polynomial 93 94 p(x) = a[0]*x**0 + a[1]*x**1 + a[2]*x**2 +...+ a[n]*x**n 95 96 at x = 0 rather than reject a[0]*0**0 as invalid. 97 98 3. Analysts will accept 0**0 = 1 despite that x**y can approach 99 anything or nothing as x and y approach 0 independently. The 100 reason for setting 0**0 = 1 anyway is this: 101 102 If x(z) and y(z) are _a_n_y functions analytic (expandable in 103 power series) in z around z = 0, and if there x(0) = y(0) = 104 0, then x(z)**y(z) -> 1 as z -> 0. 105 106 4. If 0**0 = 1, then infinity**0 = 1/0**0 = 1 too; and then _N_a_N**0 = 107 1 too because x**0 = 1 for all finite and infinite x, i.e., 108 independently of x. 109 110SSEEEE AALLSSOO 111 math(3), infnan(3) 112 113HHIISSTTOORRYY 114 A eexxpp(), lloogg() and ppooww() function appeared in Version 6 AT&T UNIX. A 115 lloogg1100() function appeared in Version 7 AT&T UNIX. The lloogg11pp() and 116 eexxppmm11() functions appeared in 4.3BSD. 117 1184th Berkeley Distribution July 31, 1991 2 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133