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 #include <sys/cdefs.h> 14 #if defined(LIBM_SCCS) && !defined(lint) 15 __RCSID("$NetBSD: e_log.c,v 1.12 2002/05/26 22:01:51 wiz 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.h" 70 #include "math_private.h" 71 72 static const double 73 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 74 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 75 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 76 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 77 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 78 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 79 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 80 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 81 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 82 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 83 84 static const double zero = 0.0; 85 86 double 87 __ieee754_log(double x) 88 { 89 double hfsq,f,s,z,R,w,t1,t2,dk; 90 int32_t k,hx,i,j; 91 u_int32_t lx; 92 93 EXTRACT_WORDS(hx,lx,x); 94 95 k=0; 96 if (hx < 0x00100000) { /* x < 2**-1022 */ 97 if (((hx&0x7fffffff)|lx)==0) 98 return -two54/zero; /* log(+-0)=-inf */ 99 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 100 k -= 54; x *= two54; /* subnormal number, scale up x */ 101 GET_HIGH_WORD(hx,x); 102 } 103 if (hx >= 0x7ff00000) return x+x; 104 k += (hx>>20)-1023; 105 hx &= 0x000fffff; 106 i = (hx+0x95f64)&0x100000; 107 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ 108 k += (i>>20); 109 f = x-1.0; 110 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ 111 if(f==zero) { if(k==0) return zero; else {dk=(double)k; 112 return dk*ln2_hi+dk*ln2_lo;} 113 } 114 R = f*f*(0.5-0.33333333333333333*f); 115 if(k==0) return f-R; else {dk=(double)k; 116 return dk*ln2_hi-((R-dk*ln2_lo)-f);} 117 } 118 s = f/(2.0+f); 119 dk = (double)k; 120 z = s*s; 121 i = hx-0x6147a; 122 w = z*z; 123 j = 0x6b851-hx; 124 t1= w*(Lg2+w*(Lg4+w*Lg6)); 125 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 126 i |= j; 127 R = t2+t1; 128 if(i>0) { 129 hfsq=0.5*f*f; 130 if(k==0) return f-(hfsq-s*(hfsq+R)); else 131 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); 132 } else { 133 if(k==0) return f-s*(f-R); else 134 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); 135 } 136 } 137