1 /* 2 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved. 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 4 * 5 * This code is free software; you can redistribute it and/or modify it 6 * under the terms of the GNU General Public License version 2 only, as 7 * published by the Free Software Foundation. Oracle designates this 8 * particular file as subject to the "Classpath" exception as provided 9 * by Oracle in the LICENSE file that accompanied this code. 10 * 11 * This code is distributed in the hope that it will be useful, but WITHOUT 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26 /* __ieee754_log(x) 27 * Return the logrithm of x 28 * 29 * Method : 30 * 1. Argument Reduction: find k and f such that 31 * x = 2^k * (1+f), 32 * where sqrt(2)/2 < 1+f < sqrt(2) . 33 * 34 * 2. Approximation of log(1+f). 35 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 36 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 37 * = 2s + s*R 38 * We use a special Reme algorithm on [0,0.1716] to generate 39 * a polynomial of degree 14 to approximate R The maximum error 40 * of this polynomial approximation is bounded by 2**-58.45. In 41 * other words, 42 * 2 4 6 8 10 12 14 43 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s 44 * (the values of Lg1 to Lg7 are listed in the program) 45 * and 46 * | 2 14 | -58.45 47 * | Lg1*s +...+Lg7*s - R(z) | <= 2 48 * | | 49 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 50 * In order to guarantee error in log below 1ulp, we compute log 51 * by 52 * log(1+f) = f - s*(f - R) (if f is not too large) 53 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) 54 * 55 * 3. Finally, log(x) = k*ln2 + log(1+f). 56 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 57 * Here ln2 is split into two floating point number: 58 * ln2_hi + ln2_lo, 59 * where n*ln2_hi is always exact for |n| < 2000. 60 * 61 * Special cases: 62 * log(x) is NaN with signal if x < 0 (including -INF) ; 63 * log(+INF) is +INF; log(0) is -INF with signal; 64 * log(NaN) is that NaN with no signal. 65 * 66 * Accuracy: 67 * according to an error analysis, the error is always less than 68 * 1 ulp (unit in the last place). 69 * 70 * Constants: 71 * The hexadecimal values are the intended ones for the following 72 * constants. The decimal values may be used, provided that the 73 * compiler will convert from decimal to binary accurately enough 74 * to produce the hexadecimal values shown. 75 */ 76 77 #include "fdlibm.h" 78 79 #ifdef __STDC__ 80 static const double 81 #else 82 static double 83 #endif 84 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 85 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 86 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 87 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 88 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 89 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 90 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 91 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 92 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 93 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 94 95 static double zero = 0.0; 96 97 #ifdef __STDC__ __ieee754_log(double x)98 double __ieee754_log(double x) 99 #else 100 double __ieee754_log(x) 101 double x; 102 #endif 103 { 104 double hfsq,f,s,z,R,w,t1,t2,dk; 105 int k,hx,i,j; 106 unsigned lx; 107 108 hx = __HI(x); /* high word of x */ 109 lx = __LO(x); /* low word of x */ 110 111 k=0; 112 if (hx < 0x00100000) { /* x < 2**-1022 */ 113 if (((hx&0x7fffffff)|lx)==0) 114 return -two54/zero; /* log(+-0)=-inf */ 115 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 116 k -= 54; x *= two54; /* subnormal number, scale up x */ 117 hx = __HI(x); /* high word of x */ 118 } 119 if (hx >= 0x7ff00000) return x+x; 120 k += (hx>>20)-1023; 121 hx &= 0x000fffff; 122 i = (hx+0x95f64)&0x100000; 123 __HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */ 124 k += (i>>20); 125 f = x-1.0; 126 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ 127 if(f==zero) { 128 if (k==0) return zero; 129 else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;} 130 } 131 R = f*f*(0.5-0.33333333333333333*f); 132 if(k==0) return f-R; else {dk=(double)k; 133 return dk*ln2_hi-((R-dk*ln2_lo)-f);} 134 } 135 s = f/(2.0+f); 136 dk = (double)k; 137 z = s*s; 138 i = hx-0x6147a; 139 w = z*z; 140 j = 0x6b851-hx; 141 t1= w*(Lg2+w*(Lg4+w*Lg6)); 142 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 143 i |= j; 144 R = t2+t1; 145 if(i>0) { 146 hfsq=0.5*f*f; 147 if(k==0) return f-(hfsq-s*(hfsq+R)); else 148 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); 149 } else { 150 if(k==0) return f-s*(f-R); else 151 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); 152 } 153 } 154