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 #ifndef lint 14 static char rcsid[] = "$FreeBSD$"; 15 #endif 16 17 /* __ieee754_log(x) 18 * Return the logrithm of x 19 * 20 * Method : 21 * 1. Argument Reduction: find k and f such that 22 * x = 2^k * (1+f), 23 * where sqrt(2)/2 < 1+f < sqrt(2) . 24 * 25 * 2. Approximation of log(1+f). 26 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 27 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 28 * = 2s + s*R 29 * We use a special Reme algorithm on [0,0.1716] to generate 30 * a polynomial of degree 14 to approximate R The maximum error 31 * of this polynomial approximation is bounded by 2**-58.45. In 32 * other words, 33 * 2 4 6 8 10 12 14 34 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s 35 * (the values of Lg1 to Lg7 are listed in the program) 36 * and 37 * | 2 14 | -58.45 38 * | Lg1*s +...+Lg7*s - R(z) | <= 2 39 * | | 40 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 41 * In order to guarantee error in log below 1ulp, we compute log 42 * by 43 * log(1+f) = f - s*(f - R) (if f is not too large) 44 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) 45 * 46 * 3. Finally, log(x) = k*ln2 + log(1+f). 47 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 48 * Here ln2 is split into two floating point number: 49 * ln2_hi + ln2_lo, 50 * where n*ln2_hi is always exact for |n| < 2000. 51 * 52 * Special cases: 53 * log(x) is NaN with signal if x < 0 (including -INF) ; 54 * log(+INF) is +INF; log(0) is -INF with signal; 55 * log(NaN) is that NaN with no signal. 56 * 57 * Accuracy: 58 * according to an error analysis, the error is always less than 59 * 1 ulp (unit in the last place). 60 * 61 * Constants: 62 * The hexadecimal values are the intended ones for the following 63 * constants. The decimal values may be used, provided that the 64 * compiler will convert from decimal to binary accurately enough 65 * to produce the hexadecimal values shown. 66 */ 67 68 #include "math.h" 69 #include "math_private.h" 70 71 #ifdef __STDC__ 72 static const double 73 #else 74 static double 75 #endif 76 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 77 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 78 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 79 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 80 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 81 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 82 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 83 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 84 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 85 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 86 87 #ifdef __STDC__ 88 static const double zero = 0.0; 89 #else 90 static double zero = 0.0; 91 #endif 92 93 #ifdef __STDC__ 94 double __generic___ieee754_log(double x) 95 #else 96 double __generic___ieee754_log(x) 97 double x; 98 #endif 99 { 100 double hfsq,f,s,z,R,w,t1,t2,dk; 101 int32_t k,hx,i,j; 102 u_int32_t lx; 103 104 EXTRACT_WORDS(hx,lx,x); 105 106 k=0; 107 if (hx < 0x00100000) { /* x < 2**-1022 */ 108 if (((hx&0x7fffffff)|lx)==0) 109 return -two54/zero; /* log(+-0)=-inf */ 110 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 111 k -= 54; x *= two54; /* subnormal number, scale up x */ 112 GET_HIGH_WORD(hx,x); 113 } 114 if (hx >= 0x7ff00000) return x+x; 115 k += (hx>>20)-1023; 116 hx &= 0x000fffff; 117 i = (hx+0x95f64)&0x100000; 118 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ 119 k += (i>>20); 120 f = x-1.0; 121 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ 122 if(f==zero) if(k==0) return zero; else {dk=(double)k; 123 return dk*ln2_hi+dk*ln2_lo;} 124 R = f*f*(0.5-0.33333333333333333*f); 125 if(k==0) return f-R; else {dk=(double)k; 126 return dk*ln2_hi-((R-dk*ln2_lo)-f);} 127 } 128 s = f/(2.0+f); 129 dk = (double)k; 130 z = s*s; 131 i = hx-0x6147a; 132 w = z*z; 133 j = 0x6b851-hx; 134 t1= w*(Lg2+w*(Lg4+w*Lg6)); 135 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 136 i |= j; 137 R = t2+t1; 138 if(i>0) { 139 hfsq=0.5*f*f; 140 if(k==0) return f-(hfsq-s*(hfsq+R)); else 141 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); 142 } else { 143 if(k==0) return f-s*(f-R); else 144 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); 145 } 146 } 147