e_log.c (5278B)
1/* @(#)e_log.c 5.1 93/09/24 */ 2/* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, 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#if defined(LIBM_SCCS) && !defined(lint) 14static const char rcsid[] = 15 "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $"; 16#endif 17 18/* __ieee754_log(x) 19 * Return the logrithm of x 20 * 21 * Method : 22 * 1. Argument Reduction: find k and f such that 23 * x = 2^k * (1+f), 24 * where sqrt(2)/2 < 1+f < sqrt(2) . 25 * 26 * 2. Approximation of log(1+f). 27 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 28 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 29 * = 2s + s*R 30 * We use a special Reme algorithm on [0,0.1716] to generate 31 * a polynomial of degree 14 to approximate R The maximum error 32 * of this polynomial approximation is bounded by 2**-58.45. In 33 * other words, 34 * 2 4 6 8 10 12 14 35 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s 36 * (the values of Lg1 to Lg7 are listed in the program) 37 * and 38 * | 2 14 | -58.45 39 * | Lg1*s +...+Lg7*s - R(z) | <= 2 40 * | | 41 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 42 * In order to guarantee error in log below 1ulp, we compute log 43 * by 44 * log(1+f) = f - s*(f - R) (if f is not too large) 45 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) 46 * 47 * 3. Finally, log(x) = k*ln2 + log(1+f). 48 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 49 * Here ln2 is split into two floating point number: 50 * ln2_hi + ln2_lo, 51 * where n*ln2_hi is always exact for |n| < 2000. 52 * 53 * Special cases: 54 * log(x) is NaN with signal if x < 0 (including -INF) ; 55 * log(+INF) is +INF; log(0) is -INF with signal; 56 * log(NaN) is that NaN with no signal. 57 * 58 * Accuracy: 59 * according to an error analysis, the error is always less than 60 * 1 ulp (unit in the last place). 61 * 62 * Constants: 63 * The hexadecimal values are the intended ones for the following 64 * constants. The decimal values may be used, provided that the 65 * compiler will convert from decimal to binary accurately enough 66 * to produce the hexadecimal values shown. 67 */ 68 69#include "math_libm.h" 70#include "math_private.h" 71 72#ifdef __STDC__ 73static const double 74#else 75static double 76#endif 77 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 78 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 79 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 80 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 81 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 82 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 83 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 84 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 85 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 86 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 87 88#ifdef __STDC__ 89static const double zero = 0.0; 90#else 91static double zero = 0.0; 92#endif 93 94#ifdef __STDC__ 95double attribute_hidden 96__ieee754_log(double x) 97#else 98double attribute_hidden 99__ieee754_log(x) 100 double x; 101#endif 102{ 103 double hfsq, f, s, z, R, w, t1, t2, dk; 104 int32_t k, hx, i, j; 105 u_int32_t lx; 106 107 EXTRACT_WORDS(hx, lx, x); 108 109 k = 0; 110 if (hx < 0x00100000) { /* x < 2**-1022 */ 111 if (((hx & 0x7fffffff) | lx) == 0) 112 return -two54 / zero; /* log(+-0)=-inf */ 113 if (hx < 0) 114 return (x - x) / zero; /* log(-#) = NaN */ 115 k -= 54; 116 x *= two54; /* subnormal number, scale up x */ 117 GET_HIGH_WORD(hx, x); 118 } 119 if (hx >= 0x7ff00000) 120 return x + x; 121 k += (hx >> 20) - 1023; 122 hx &= 0x000fffff; 123 i = (hx + 0x95f64) & 0x100000; 124 SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */ 125 k += (i >> 20); 126 f = x - 1.0; 127 if ((0x000fffff & (2 + hx)) < 3) { /* |f| < 2**-20 */ 128 if (f == zero) { 129 if (k == 0) 130 return zero; 131 else { 132 dk = (double) k; 133 return dk * ln2_hi + dk * ln2_lo; 134 } 135 } 136 R = f * f * (0.5 - 0.33333333333333333 * f); 137 if (k == 0) 138 return f - R; 139 else { 140 dk = (double) k; 141 return dk * ln2_hi - ((R - dk * ln2_lo) - f); 142 } 143 } 144 s = f / (2.0 + f); 145 dk = (double) k; 146 z = s * s; 147 i = hx - 0x6147a; 148 w = z * z; 149 j = 0x6b851 - hx; 150 t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); 151 t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); 152 i |= j; 153 R = t2 + t1; 154 if (i > 0) { 155 hfsq = 0.5 * f * f; 156 if (k == 0) 157 return f - (hfsq - s * (hfsq + R)); 158 else 159 return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - 160 f); 161 } else { 162 if (k == 0) 163 return f - s * (f - R); 164 else 165 return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f); 166 } 167}