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