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