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 __FBSDID("$FreeBSD$"); 16 17 /* __kernel_log(x) 18 * Return log(x) - (x-1) for x in ~[sqrt(2)/2, sqrt(2)]. 19 * 20 * The following describes the overall strategy for computing 21 * logarithms in base e. The argument reduction and adding the final 22 * term of the polynomial are done by the caller for increased accuracy 23 * when different bases are used. 24 * 25 * Method : 26 * 1. Argument Reduction: find k and f such that 27 * x = 2^k * (1+f), 28 * where sqrt(2)/2 < 1+f < sqrt(2) . 29 * 30 * 2. Approximation of log(1+f). 31 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 32 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 33 * = 2s + s*R 34 * We use a special Reme algorithm on [0,0.1716] to generate 35 * a polynomial of degree 14 to approximate R The maximum error 36 * of this polynomial approximation is bounded by 2**-58.45. In 37 * other words, 38 * 2 4 6 8 10 12 14 39 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s 40 * (the values of Lg1 to Lg7 are listed in the program) 41 * and 42 * | 2 14 | -58.45 43 * | Lg1*s +...+Lg7*s - R(z) | <= 2 44 * | | 45 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 46 * In order to guarantee error in log below 1ulp, we compute log 47 * by 48 * log(1+f) = f - s*(f - R) (if f is not too large) 49 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) 50 * 51 * 3. Finally, log(x) = k*ln2 + log(1+f). 52 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 53 * Here ln2 is split into two floating point number: 54 * ln2_hi + ln2_lo, 55 * where n*ln2_hi is always exact for |n| < 2000. 56 * 57 * Special cases: 58 * log(x) is NaN with signal if x < 0 (including -INF) ; 59 * log(+INF) is +INF; log(0) is -INF with signal; 60 * log(NaN) is that NaN with no signal. 61 * 62 * Accuracy: 63 * according to an error analysis, the error is always less than 64 * 1 ulp (unit in the last place). 65 * 66 * Constants: 67 * The hexadecimal values are the intended ones for the following 68 * constants. The decimal values may be used, provided that the 69 * compiler will convert from decimal to binary accurately enough 70 * to produce the hexadecimal values shown. 71 */ 72 73 static const double 74 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 75 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 76 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 77 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 78 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 79 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 80 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 81 82 /* 83 * We always inline __kernel_log(), since doing so produces a 84 * substantial performance improvement (~40% on amd64). 85 */ 86 static inline double 87 __kernel_log(double x) 88 { 89 double hfsq,f,s,z,R,w,t1,t2; 90 int32_t hx,i,j; 91 u_int32_t lx; 92 93 EXTRACT_WORDS(hx,lx,x); 94 95 f = x-1.0; 96 if((0x000fffff&(2+hx))<3) { /* -2**-20 <= f < 2**-20 */ 97 if(f==0.0) return 0.0; 98 return f*f*(0.33333333333333333*f-0.5); 99 } 100 s = f/(2.0+f); 101 z = s*s; 102 hx &= 0x000fffff; 103 i = hx-0x6147a; 104 w = z*z; 105 j = 0x6b851-hx; 106 t1= w*(Lg2+w*(Lg4+w*Lg6)); 107 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 108 i |= j; 109 R = t2+t1; 110 if (i>0) { 111 hfsq=0.5*f*f; 112 return s*(hfsq+R) - hfsq; 113 } else { 114 return s*(R-f); 115 } 116 } 117