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