1 2 /* @(#)e_log.c 1.3 95/01/18 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14 #include <sys/cdefs.h> 15 /* 16 * k_log1p(f): 17 * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)]. 18 * 19 * The following describes the overall strategy for computing 20 * logarithms in base e. The argument reduction and adding the final 21 * term of the polynomial are done by the caller for increased accuracy 22 * when different bases are used. 23 * 24 * Method : 25 * 1. Argument Reduction: find k and f such that 26 * x = 2^k * (1+f), 27 * where sqrt(2)/2 < 1+f < sqrt(2) . 28 * 29 * 2. Approximation of log(1+f). 30 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 31 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 32 * = 2s + s*R 33 * We use a special Reme algorithm on [0,0.1716] to generate 34 * a polynomial of degree 14 to approximate R The maximum error 35 * of this polynomial approximation is bounded by 2**-58.45. In 36 * other words, 37 * 2 4 6 8 10 12 14 38 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s 39 * (the values of Lg1 to Lg7 are listed in the program) 40 * and 41 * | 2 14 | -58.45 42 * | Lg1*s +...+Lg7*s - R(z) | <= 2 43 * | | 44 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 45 * In order to guarantee error in log below 1ulp, we compute log 46 * by 47 * log(1+f) = f - s*(f - R) (if f is not too large) 48 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) 49 * 50 * 3. Finally, log(x) = k*ln2 + log(1+f). 51 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 52 * Here ln2 is split into two floating point number: 53 * ln2_hi + ln2_lo, 54 * where n*ln2_hi is always exact for |n| < 2000. 55 * 56 * Special cases: 57 * log(x) is NaN with signal if x < 0 (including -INF) ; 58 * log(+INF) is +INF; log(0) is -INF with signal; 59 * log(NaN) is that NaN with no signal. 60 * 61 * Accuracy: 62 * according to an error analysis, the error is always less than 63 * 1 ulp (unit in the last place). 64 * 65 * Constants: 66 * The hexadecimal values are the intended ones for the following 67 * constants. The decimal values may be used, provided that the 68 * compiler will convert from decimal to binary accurately enough 69 * to produce the hexadecimal values shown. 70 */ 71 72 static const double 73 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 74 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 75 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 76 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 77 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 78 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 79 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 80 81 /* 82 * We always inline k_log1p(), since doing so produces a 83 * substantial performance improvement (~40% on amd64). 84 */ 85 static inline double 86 k_log1p(double f) 87 { 88 double hfsq,s,z,R,w,t1,t2; 89 90 s = f/(2.0+f); 91 z = s*s; 92 w = z*z; 93 t1= w*(Lg2+w*(Lg4+w*Lg6)); 94 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 95 R = t2+t1; 96 hfsq=0.5*f*f; 97 return s*(hfsq+R); 98 } 99