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f671aeab | 1 | /* Adapted for log2 by Ulrich Drepper <drepper@cygnus.com>. */ |
e7fd8a39 UD |
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 | ||
601d2942 | 13 | /* __ieee754_log2(x) |
e7fd8a39 UD |
14 | * Return the logarithm to base 2 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 + ....., | |
0ac5ae23 | 24 | * = 2s + s*R |
e7fd8a39 | 25 | * We use a special Reme algorithm on [0,0.1716] to generate |
0ac5ae23 | 26 | * a polynomial of degree 14 to approximate R The maximum error |
e7fd8a39 UD |
27 | * of this polynomial approximation is bounded by 2**-58.45. In |
28 | * other words, | |
0ac5ae23 | 29 | * 2 4 6 8 10 12 14 |
e7fd8a39 | 30 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s |
0ac5ae23 | 31 | * (the values of Lg1 to Lg7 are listed in the program) |
e7fd8a39 UD |
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 + log(1+f). | |
43 | * = k+(f-(hfsq-(s*(hfsq+R)))) | |
44 | * | |
45 | * Special cases: | |
46 | * log2(x) is NaN with signal if x < 0 (including -INF) ; | |
47 | * log2(+INF) is +INF; log(0) is -INF with signal; | |
48 | * log2(NaN) is that NaN with no signal. | |
49 | * | |
50 | * Constants: | |
51 | * The hexadecimal values are the intended ones for the following | |
52 | * constants. The decimal values may be used, provided that the | |
53 | * compiler will convert from decimal to binary accurately enough | |
54 | * to produce the hexadecimal values shown. | |
55 | */ | |
56 | ||
1ed0291c RH |
57 | #include <math.h> |
58 | #include <math_private.h> | |
e7fd8a39 | 59 | |
9ea01d93 AZ |
60 | static const double ln2 = 0.69314718055994530942; |
61 | static const double two54 = 1.80143985094819840000e+16; /* 43500000 00000000 */ | |
62 | static const double Lg1 = 6.666666666666735130e-01; /* 3FE55555 55555593 */ | |
63 | static const double Lg2 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */ | |
64 | static const double Lg3 = 2.857142874366239149e-01; /* 3FD24924 94229359 */ | |
65 | static const double Lg4 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */ | |
66 | static const double Lg5 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */ | |
67 | static const double Lg6 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */ | |
68 | static const double Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ | |
e7fd8a39 | 69 | |
9ea01d93 | 70 | static const double zero = 0.0; |
e7fd8a39 | 71 | |
0ac5ae23 | 72 | double |
9ea01d93 | 73 | __ieee754_log2 (double x) |
e7fd8a39 | 74 | { |
9ea01d93 AZ |
75 | double hfsq, f, s, z, R, w, t1, t2, dk; |
76 | int32_t k, hx, i, j; | |
77 | u_int32_t lx; | |
e7fd8a39 | 78 | |
9ea01d93 | 79 | EXTRACT_WORDS (hx, lx, x); |
e7fd8a39 | 80 | |
9ea01d93 AZ |
81 | k = 0; |
82 | if (hx < 0x00100000) | |
83 | { /* x < 2**-1022 */ | |
84 | if (__builtin_expect (((hx & 0x7fffffff) | lx) == 0, 0)) | |
85 | return -two54 / (x - x); /* log(+-0)=-inf */ | |
86 | if (__builtin_expect (hx < 0, 0)) | |
87 | return (x - x) / (x - x); /* log(-#) = NaN */ | |
88 | k -= 54; | |
89 | x *= two54; /* subnormal number, scale up x */ | |
90 | GET_HIGH_WORD (hx, x); | |
91 | } | |
92 | if (__builtin_expect (hx >= 0x7ff00000, 0)) | |
93 | return x + x; | |
94 | k += (hx >> 20) - 1023; | |
95 | hx &= 0x000fffff; | |
96 | i = (hx + 0x95f64) & 0x100000; | |
97 | SET_HIGH_WORD (x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */ | |
98 | k += (i >> 20); | |
99 | dk = (double) k; | |
100 | f = x - 1.0; | |
101 | if ((0x000fffff & (2 + hx)) < 3) | |
102 | { /* |f| < 2**-20 */ | |
103 | if (f == zero) | |
104 | return dk; | |
105 | R = f * f * (0.5 - 0.33333333333333333 * f); | |
106 | return dk - (R - f) / ln2; | |
107 | } | |
108 | s = f / (2.0 + f); | |
109 | z = s * s; | |
110 | i = hx - 0x6147a; | |
111 | w = z * z; | |
112 | j = 0x6b851 - hx; | |
113 | t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); | |
114 | t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); | |
115 | i |= j; | |
116 | R = t2 + t1; | |
117 | if (i > 0) | |
118 | { | |
119 | hfsq = 0.5 * f * f; | |
120 | return dk - ((hfsq - (s * (hfsq + R))) - f) / ln2; | |
121 | } | |
122 | else | |
123 | { | |
124 | return dk - ((s * (f - R)) - f) / ln2; | |
125 | } | |
e7fd8a39 | 126 | } |
9ea01d93 | 127 | |
0ac5ae23 | 128 | strong_alias (__ieee754_log2, __log2_finite) |