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 #include <sys/cdefs.h> 15 __FBSDID("$FreeBSD$"); 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 static const double 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 const double zero = 0.0; 84 85 double 86 __ieee754_log(double x) 87 { 88 double hfsq,f,s,z,R,w,t1,t2,dk; 89 int32_t k,hx,i,j; 90 u_int32_t lx; 91 92 EXTRACT_WORDS(hx,lx,x); 93 94 k=0; 95 if (hx < 0x00100000) { /* x < 2**-1022 */ 96 if (((hx&0x7fffffff)|lx)==0) 97 return -two54/zero; /* log(+-0)=-inf */ 98 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 99 k -= 54; x *= two54; /* subnormal number, scale up x */ 100 GET_HIGH_WORD(hx,x); 101 } 102 if (hx >= 0x7ff00000) return x+x; 103 k += (hx>>20)-1023; 104 hx &= 0x000fffff; 105 i = (hx+0x95f64)&0x100000; 106 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ 107 k += (i>>20); 108 f = x-1.0; 109 if((0x000fffff&(2+hx))<3) { /* -2**-20 <= f < 2**-20 */ 110 if(f==zero) { 111 if(k==0) { 112 return zero; 113 } else { 114 dk=(double)k; 115 return dk*ln2_hi+dk*ln2_lo; 116 } 117 } 118 R = f*f*(0.5-0.33333333333333333*f); 119 if(k==0) return f-R; else {dk=(double)k; 120 return dk*ln2_hi-((R-dk*ln2_lo)-f);} 121 } 122 s = f/(2.0+f); 123 dk = (double)k; 124 z = s*s; 125 i = hx-0x6147a; 126 w = z*z; 127 j = 0x6b851-hx; 128 t1= w*(Lg2+w*(Lg4+w*Lg6)); 129 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 130 i |= j; 131 R = t2+t1; 132 if(i>0) { 133 hfsq=0.5*f*f; 134 if(k==0) return f-(hfsq-s*(hfsq+R)); else 135 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); 136 } else { 137 if(k==0) return f-s*(f-R); else 138 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); 139 } 140 } 141