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comparison src/libm/e_log.c @ 2756:a98604b691c8

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Expanded the libm support and put it into a separate directory.
author Sam Lantinga <slouken@libsdl.org>
date Mon, 15 Sep 2008 06:33:23 +0000
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children dc1eb82ffdaa
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2755:2a3ec308d995 2756:a98604b691c8
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 #if defined(LIBM_SCCS) && !defined(lint)
14 static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc 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 #ifdef __STDC__
72 static const double
73 #else
74 static double
75 #endif
76 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
77 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
78 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
79 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
80 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
81 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
82 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
83 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
84 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
85 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
86
87 #ifdef __STDC__
88 static const double zero = 0.0;
89 #else
90 static double zero = 0.0;
91 #endif
92
93 #ifdef __STDC__
94 double attribute_hidden
95 __ieee754_log(double x)
96 #else
97 double attribute_hidden
98 __ieee754_log(x)
99 double x;
100 #endif
101 {
102 double hfsq, f, s, z, R, w, t1, t2, dk;
103 int32_t k, hx, i, j;
104 u_int32_t lx;
105
106 EXTRACT_WORDS(hx, lx, x);
107
108 k = 0;
109 if (hx < 0x00100000) { /* x < 2**-1022 */
110 if (((hx & 0x7fffffff) | lx) == 0)
111 return -two54 / zero; /* log(+-0)=-inf */
112 if (hx < 0)
113 return (x - x) / zero; /* log(-#) = NaN */
114 k -= 54;
115 x *= two54; /* subnormal number, scale up x */
116 GET_HIGH_WORD(hx, x);
117 }
118 if (hx >= 0x7ff00000)
119 return x + x;
120 k += (hx >> 20) - 1023;
121 hx &= 0x000fffff;
122 i = (hx + 0x95f64) & 0x100000;
123 SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
124 k += (i >> 20);
125 f = x - 1.0;
126 if ((0x000fffff & (2 + hx)) < 3) { /* |f| < 2**-20 */
127 if (f == zero) {
128 if (k == 0)
129 return zero;
130 else {
131 dk = (double) k;
132 return dk * ln2_hi + dk * ln2_lo;
133 }
134 }
135 R = f * f * (0.5 - 0.33333333333333333 * f);
136 if (k == 0)
137 return f - R;
138 else {
139 dk = (double) k;
140 return dk * ln2_hi - ((R - dk * ln2_lo) - f);
141 }
142 }
143 s = f / (2.0 + f);
144 dk = (double) k;
145 z = s * s;
146 i = hx - 0x6147a;
147 w = z * z;
148 j = 0x6b851 - hx;
149 t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
150 t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
151 i |= j;
152 R = t2 + t1;
153 if (i > 0) {
154 hfsq = 0.5 * f * f;
155 if (k == 0)
156 return f - (hfsq - s * (hfsq + R));
157 else
158 return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -
159 f);
160 } else {
161 if (k == 0)
162 return f - s * (f - R);
163 else
164 return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
165 }
166 }