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