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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 <float.h>
69 :
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 : static volatile double vzero = 0.0;
86 :
87 : double
88 0 : __ieee754_log(double x)
89 : {
90 : double hfsq,f,s,z,R,w,t1,t2,dk;
91 : int32_t k,hx,i,j;
92 : u_int32_t lx;
93 :
94 0 : EXTRACT_WORDS(hx,lx,x);
95 :
96 0 : k=0;
97 0 : if (hx < 0x00100000) { /* x < 2**-1022 */
98 0 : if (((hx&0x7fffffff)|lx)==0)
99 0 : return -two54/vzero; /* log(+-0)=-inf */
100 0 : if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
101 0 : k -= 54; x *= two54; /* subnormal number, scale up x */
102 0 : GET_HIGH_WORD(hx,x);
103 : }
104 0 : if (hx >= 0x7ff00000) return x+x;
105 0 : k += (hx>>20)-1023;
106 0 : hx &= 0x000fffff;
107 0 : i = (hx+0x95f64)&0x100000;
108 0 : SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
109 0 : k += (i>>20);
110 0 : f = x-1.0;
111 0 : if((0x000fffff&(2+hx))<3) { /* -2**-20 <= f < 2**-20 */
112 0 : if(f==zero) {
113 0 : if(k==0) {
114 0 : return zero;
115 : } else {
116 0 : dk=(double)k;
117 0 : return dk*ln2_hi+dk*ln2_lo;
118 : }
119 : }
120 0 : R = f*f*(0.5-0.33333333333333333*f);
121 0 : if(k==0) return f-R; else {dk=(double)k;
122 0 : return dk*ln2_hi-((R-dk*ln2_lo)-f);}
123 : }
124 0 : s = f/(2.0+f);
125 0 : dk = (double)k;
126 0 : z = s*s;
127 0 : i = hx-0x6147a;
128 0 : w = z*z;
129 0 : j = 0x6b851-hx;
130 0 : t1= w*(Lg2+w*(Lg4+w*Lg6));
131 0 : t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
132 0 : i |= j;
133 0 : R = t2+t1;
134 0 : if(i>0) {
135 0 : hfsq=0.5*f*f;
136 0 : if(k==0) return f-(hfsq-s*(hfsq+R)); else
137 0 : return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
138 : } else {
139 0 : if(k==0) return f-s*(f-R); else
140 0 : return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
141 : }
142 : }
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