1 /* derived from /netlib/fdlibm */ 2 3 /* @(#)e_log.c 1.3 95/01/18 */ 4 /* 5 * ==================================================== 6 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 7 * 8 * Developed at SunSoft, a Sun Microsystems, Inc. business. 9 * Permission to use, copy, modify, and distribute this 10 * software is freely granted, provided that this notice 11 * is preserved. 12 * ==================================================== 13 */ 14 15 /* __ieee754_log(x) 16 * Return the logrithm of x 17 * 18 * Method : 19 * 1. Argument Reduction: find k and f such that 20 * x = 2^k * (1+f), 21 * where sqrt(2)/2 < 1+f < sqrt(2) . 22 * 23 * 2. Approximation of log(1+f). 24 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 25 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 26 * = 2s + s*R 27 * We use a special Reme algorithm on [0,0.1716] to generate 28 * a polynomial of degree 14 to approximate R The maximum error 29 * of this polynomial approximation is bounded by 2**-58.45. In 30 * other words, 31 * 2 4 6 8 10 12 14 32 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s 33 * (the values of Lg1 to Lg7 are listed in the program) 34 * and 35 * | 2 14 | -58.45 36 * | Lg1*s +...+Lg7*s - R(z) | <= 2 37 * | | 38 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 39 * In order to guarantee error in log below 1ulp, we compute log 40 * by 41 * log(1+f) = f - s*(f - R) (if f is not too large) 42 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) 43 * 44 * 3. Finally, log(x) = k*ln2 + log(1+f). 45 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 46 * Here ln2 is split into two floating point number: 47 * ln2_hi + ln2_lo, 48 * where n*ln2_hi is always exact for |n| < 2000. 49 * 50 * Special cases: 51 * log(x) is NaN with signal if x < 0 (including -INF) ; 52 * log(+INF) is +INF; log(0) is -INF with signal; 53 * log(NaN) is that NaN with no signal. 54 * 55 * Accuracy: 56 * according to an error analysis, the error is always less than 57 * 1 ulp (unit in the last place). 58 * 59 * Constants: 60 * The hexadecimal values are the intended ones for the following 61 * constants. The decimal values may be used, provided that the 62 * compiler will convert from decimal to binary accurately enough 63 * to produce the hexadecimal values shown. 64 */ 65 66 #include "fdlibm.h" 67 68 static const double 69 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 70 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 71 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 72 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 73 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 74 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 75 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 76 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 77 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 78 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 79 80 static double zero = 0.0; 81 82 double __ieee754_log(double x) 83 { 84 double hfsq,f,s,z,R,w,t1,t2,dk; 85 int k,hx,i,j; 86 unsigned lx; 87 88 hx = __HI(x); /* high word of x */ 89 lx = __LO(x); /* low word of x */ 90 91 k=0; 92 if (hx < 0x00100000) { /* x < 2**-1022 */ 93 if (((hx&0x7fffffff)|lx)==0) 94 return -two54/zero; /* log(+-0)=-inf */ 95 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ 96 k -= 54; x *= two54; /* subnormal number, scale up x */ 97 hx = __HI(x); /* high word of x */ 98 } 99 if (hx >= 0x7ff00000) return x+x; 100 k += (hx>>20)-1023; 101 hx &= 0x000fffff; 102 i = (hx+0x95f64)&0x100000; 103 __HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */ 104 k += (i>>20); 105 f = x-1.0; 106 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ 107 if(f==zero) if(k==0) return zero; else {dk=(double)k; 108 return dk*ln2_hi+dk*ln2_lo;} 109 R = f*f*(0.5-0.33333333333333333*f); 110 if(k==0) return f-R; else {dk=(double)k; 111 return dk*ln2_hi-((R-dk*ln2_lo)-f);} 112 } 113 s = f/(2.0+f); 114 dk = (double)k; 115 z = s*s; 116 i = hx-0x6147a; 117 w = z*z; 118 j = 0x6b851-hx; 119 t1= w*(Lg2+w*(Lg4+w*Lg6)); 120 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 121 i |= j; 122 R = t2+t1; 123 if(i>0) { 124 hfsq=0.5*f*f; 125 if(k==0) return f-(hfsq-s*(hfsq+R)); else 126 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); 127 } else { 128 if(k==0) return f-s*(f-R); else 129 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); 130 } 131 } 132