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e_log.c (4400B)
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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 #ifndef lint
14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_log.c,v 1.9 2003/07/23 04:53:46 peter Exp $";
15 #endif
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) { /* |f| < 2**-20 */
110 if(f==zero) if(k==0) return zero; else {dk=(double)k;
111 return dk*ln2_hi+dk*ln2_lo;}
112 R = f*f*(0.5-0.33333333333333333*f);
113 if(k==0) return f-R; else {dk=(double)k;
114 return dk*ln2_hi-((R-dk*ln2_lo)-f);}
115 }
116 s = f/(2.0+f);
117 dk = (double)k;
118 z = s*s;
119 i = hx-0x6147a;
120 w = z*z;
121 j = 0x6b851-hx;
122 t1= w*(Lg2+w*(Lg4+w*Lg6));
123 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
124 i |= j;
125 R = t2+t1;
126 if(i>0) {
127 hfsq=0.5*f*f;
128 if(k==0) return f-(hfsq-s*(hfsq+R)); else
129 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
130 } else {
131 if(k==0) return f-s*(f-R); else
132 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
133 }
134 }