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| Rev | Author | Line No. | Line |
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| 2 | pj | 1 | /* @(#)e_log.c 5.1 93/09/24 */ |
| 2 | /* |
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| 3 | * ==================================================== |
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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| 5 | * |
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| 6 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
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| 7 | * Permission to use, copy, modify, and distribute this |
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| 8 | * software is freely granted, provided that this notice |
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| 9 | * is preserved. |
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| 10 | * ==================================================== |
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| 11 | */ |
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| 12 | |||
| 13 | #ifndef lint |
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| 14 | static char rcsid[] = "$\Id: e_log.c,v 1.2.6.1 1997/02/23 11:03:05 joerg Exp $"; |
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| 15 | #endif |
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| 16 | |||
| 17 | /* __ieee754_log(x) |
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| 18 | * Return the logrithm of x |
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| 19 | * |
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| 20 | * Method : |
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| 21 | * 1. Argument Reduction: find k and f such that |
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| 22 | * x = 2^k * (1+f), |
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| 23 | * where sqrt(2)/2 < 1+f < sqrt(2) . |
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| 24 | * |
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| 25 | * 2. Approximation of log(1+f). |
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| 26 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) |
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| 27 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., |
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| 28 | * = 2s + s*R |
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| 29 | * We use a special Reme algorithm on [0,0.1716] to generate |
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| 30 | * a polynomial of degree 14 to approximate R The maximum error |
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| 31 | * of this polynomial approximation is bounded by 2**-58.45. In |
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| 32 | * other words, |
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| 33 | * 2 4 6 8 10 12 14 |
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| 34 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s |
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| 35 | * (the values of Lg1 to Lg7 are listed in the program) |
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| 36 | * and |
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| 37 | * | 2 14 | -58.45 |
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| 38 | * | Lg1*s +...+Lg7*s - R(z) | <= 2 |
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| 39 | * | | |
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| 40 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. |
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| 41 | * In order to guarantee error in log below 1ulp, we compute log |
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| 42 | * by |
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| 43 | * log(1+f) = f - s*(f - R) (if f is not too large) |
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| 44 | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) |
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| 45 | * |
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| 46 | * 3. Finally, log(x) = k*ln2 + log(1+f). |
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| 47 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
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| 48 | * Here ln2 is split into two floating point number: |
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| 49 | * ln2_hi + ln2_lo, |
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| 50 | * where n*ln2_hi is always exact for |n| < 2000. |
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| 51 | * |
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| 52 | * Special cases: |
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| 53 | * log(x) is NaN with signal if x < 0 (including -INF) ; |
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| 54 | * log(+INF) is +INF; log(0) is -INF with signal; |
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| 55 | * log(NaN) is that NaN with no signal. |
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| 56 | * |
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| 57 | * Accuracy: |
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| 58 | * according to an error analysis, the error is always less than |
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| 59 | * 1 ulp (unit in the last place). |
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| 60 | * |
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| 61 | * Constants: |
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| 62 | * The hexadecimal values are the intended ones for the following |
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| 63 | * constants. The decimal values may be used, provided that the |
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| 64 | * compiler will convert from decimal to binary accurately enough |
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| 65 | * to produce the hexadecimal values shown. |
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| 66 | */ |
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| 67 | |||
| 68 | #include "math.h" |
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| 69 | #include "math_private.h" |
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| 70 | |||
| 71 | #ifdef __STDC__ |
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| 72 | static const double |
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| 73 | #else |
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| 74 | static double |
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| 75 | #endif |
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| 76 | ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ |
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| 77 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ |
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| 78 | two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ |
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| 79 | Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ |
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| 80 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ |
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| 81 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ |
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| 82 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ |
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| 83 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ |
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| 84 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ |
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| 85 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
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| 86 | |||
| 87 | #ifdef __STDC__ |
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| 88 | static const double zero = 0.0; |
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| 89 | #else |
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| 90 | static double zero = 0.0; |
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| 91 | #endif |
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| 92 | |||
| 93 | #ifdef __STDC__ |
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| 94 | double __generic___ieee754_log(double x) |
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| 95 | #else |
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| 96 | double __generic___ieee754_log(x) |
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| 97 | double x; |
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| 98 | #endif |
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| 99 | { |
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| 100 | double hfsq,f,s,z,R,w,t1,t2,dk; |
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| 101 | int32_t k,hx,i,j; |
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| 102 | u_int32_t lx; |
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| 103 | |||
| 104 | EXTRACT_WORDS(hx,lx,x); |
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| 105 | |||
| 106 | k=0; |
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| 107 | if (hx < 0x00100000) { /* x < 2**-1022 */ |
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| 108 | if (((hx&0x7fffffff)|lx)==0) |
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| 109 | return -two54/zero; /* log(+-0)=-inf */ |
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| 110 | if (hx<0) return (x-x)/zero; /* log(-#) = NaN */ |
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| 111 | k -= 54; x *= two54; /* subnormal number, scale up x */ |
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| 112 | GET_HIGH_WORD(hx,x); |
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| 113 | } |
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| 114 | if (hx >= 0x7ff00000) return x+x; |
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| 115 | k += (hx>>20)-1023; |
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| 116 | hx &= 0x000fffff; |
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| 117 | i = (hx+0x95f64)&0x100000; |
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| 118 | SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */ |
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| 119 | k += (i>>20); |
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| 120 | f = x-1.0; |
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| 121 | if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */ |
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| 122 | if(f==zero) if(k==0) return zero; else {dk=(double)k; |
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| 123 | return dk*ln2_hi+dk*ln2_lo;} |
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| 124 | R = f*f*(0.5-0.33333333333333333*f); |
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| 125 | if(k==0) return f-R; else {dk=(double)k; |
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| 126 | return dk*ln2_hi-((R-dk*ln2_lo)-f);} |
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| 127 | } |
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| 128 | s = f/(2.0+f); |
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| 129 | dk = (double)k; |
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| 130 | z = s*s; |
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| 131 | i = hx-0x6147a; |
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| 132 | w = z*z; |
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| 133 | j = 0x6b851-hx; |
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| 134 | t1= w*(Lg2+w*(Lg4+w*Lg6)); |
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| 135 | t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); |
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| 136 | i |= j; |
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| 137 | R = t2+t1; |
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| 138 | if(i>0) { |
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| 139 | hfsq=0.5*f*f; |
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| 140 | if(k==0) return f-(hfsq-s*(hfsq+R)); else |
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| 141 | return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f); |
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| 142 | } else { |
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| 143 | if(k==0) return f-s*(f-R); else |
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| 144 | return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f); |
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| 145 | } |
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| 146 | } |