1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*- 2 * 3 * ***** BEGIN LICENSE BLOCK ***** 4 * Version: MPL 1.1/GPL 2.0/LGPL 2.1 5 * 6 * The contents of this file are subject to the Mozilla Public License Version 7 * 1.1 (the "License"); you may not use this file except in compliance with 8 * the License. You may obtain a copy of the License at 9 * http://www.mozilla.org/MPL/ 10 * 11 * Software distributed under the License is distributed on an "AS IS" basis, 12 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License 13 * for the specific language governing rights and limitations under the 14 * License. 15 * 16 * The Original Code is Mozilla Communicator client code, released 17 * March 31, 1998. 18 * 19 * The Initial Developer of the Original Code is 20 * Sun Microsystems, Inc. 21 * Portions created by the Initial Developer are Copyright (C) 1998 22 * the Initial Developer. All Rights Reserved. 23 * 24 * Contributor(s): 25 * 26 * Alternatively, the contents of this file may be used under the terms of 27 * either of the GNU General Public License Version 2 or later (the "GPL"), 28 * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"), 29 * in which case the provisions of the GPL or the LGPL are applicable instead 30 * of those above. If you wish to allow use of your version of this file only 31 * under the terms of either the GPL or the LGPL, and not to allow others to 32 * use your version of this file under the terms of the MPL, indicate your 33 * decision by deleting the provisions above and replace them with the notice 34 * and other provisions required by the GPL or the LGPL. If you do not delete 35 * the provisions above, a recipient may use your version of this file under 36 * the terms of any one of the MPL, the GPL or the LGPL. 37 * 38 * ***** END LICENSE BLOCK ***** */ 39 40 /* @(#)e_log.c 1.3 95/01/18 */ 41 /* 42 * ==================================================== 43 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 44 * 45 * Developed at SunSoft, a Sun Microsystems, Inc. business. 46 * Permission to use, copy, modify, and distribute this 47 * software is freely granted, provided that this notice 48 * is preserved. 49 * ==================================================== 50 */ 51 52 /* __ieee754_log(x) 53 * Return the logrithm of x 54 * 55 * Method : 56 * 1. Argument Reduction: find k and f such that 57 * x = 2^k * (1+f), 58 * where sqrt(2)/2 < 1+f < sqrt(2) . 59 * 60 * 2. Approximation of log(1+f). 61 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 62 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 63 * = 2s + s*R 64 * We use a special Reme algorithm on [0,0.1716] to generate 65 * a polynomial of degree 14 to approximate R The maximum error 66 * of this polynomial approximation is bounded by 2**-58.45. In 67 * other words, 68 * 2 4 6 8 10 12 14 69 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s 70 * (the values of Lg1 to Lg7 are listed in the program) 71 * and 72 * | 2 14 | -58.45 73 * | Lg1*s +...+Lg7*s - R(z) | <= 2 74 * | | 75 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 76 * In order to guarantee error in log below 1ulp, we compute log 77 * by 78 * log(1+f) = f - s*(f - R) (if f is not too large) 79 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) 80 * 81 * 3. Finally, log(x) = k*ln2 + log(1+f). 82 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 83 * Here ln2 is split into two floating point number: 84 * ln2_hi + ln2_lo, 85 * where n*ln2_hi is always exact for |n| < 2000. 86 * 87 * Special cases: 88 * log(x) is NaN with signal if x < 0 (including -INF) ; 89 * log(+INF) is +INF; log(0) is -INF with signal; 90 * log(NaN) is that NaN with no signal. 91 * 92 * Accuracy: 93 * according to an error analysis, the error is always less than 94 * 1 ulp (unit in the last place). 95 * 96 * Constants: 97 * The hexadecimal values are the intended ones for the following 98 * constants. The decimal values may be used, provided that the 99 * compiler will convert from decimal to binary accurately enough 100 * to produce the hexadecimal values shown. 101 */ 102 103 #include "fdlibm.h" 104 105 #ifdef __STDC__ 106 static const double 107 #else 108 static double 109 #endif 110 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 111 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 112 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 113 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 114 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 115 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 116 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 117 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 118 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 119 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 120 121 static double zero = 0.0; 122 123 #ifdef __STDC__ __ieee754_log(double x)124 double __ieee754_log(double x) 125 #else 126 double __ieee754_log(x) 127 double x; 128 #endif 129 { 130 fd_twoints u; 131 double hfsq,f,s,z,R,w,t1,t2,dk; 132 int k,hx,i,j; 133 unsigned lx; 134 135 u.d = x; 136 hx = __HI(u); /* high word of x */ 137 lx = __LO(u); /* low word of x */ 138 139 k=0; 140 if (hx < 0x00100000) { /* x < 2**-1022 */ 141 if (((hx&0x7fffffff)|lx)==0) 142 return -two54/zero; /* log(+-0)=-inf */ 143 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 144 k -= 54; x *= two54; /* subnormal number, scale up x */ 145 u.d = x; 146 hx = __HI(u); /* high word of x */ 147 } 148 if (hx >= 0x7ff00000) return x+x; 149 k += (hx>>20)-1023; 150 hx &= 0x000fffff; 151 i = (hx+0x95f64)&0x100000; 152 u.d = x; 153 __HI(u) = hx|(i^0x3ff00000); /* normalize x or x/2 */ 154 x = u.d; 155 k += (i>>20); 156 f = x-1.0; 157 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ 158 if(f==zero) { 159 if(k==0) return zero; else {dk=(double)k; 160 return dk*ln2_hi+dk*ln2_lo;} 161 } 162 R = f*f*(0.5-0.33333333333333333*f); 163 if(k==0) return f-R; else {dk=(double)k; 164 return dk*ln2_hi-((R-dk*ln2_lo)-f);} 165 } 166 s = f/(2.0+f); 167 dk = (double)k; 168 z = s*s; 169 i = hx-0x6147a; 170 w = z*z; 171 j = 0x6b851-hx; 172 t1= w*(Lg2+w*(Lg4+w*Lg6)); 173 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 174 i |= j; 175 R = t2+t1; 176 if(i>0) { 177 hfsq=0.5*f*f; 178 if(k==0) return f-(hfsq-s*(hfsq+R)); else 179 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); 180 } else { 181 if(k==0) return f-s*(f-R); else 182 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); 183 } 184 } 185