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1 /* origin: FreeBSD /usr/src/lib/msun/src/e_log2.c */ | 1 /* origin: FreeBSD /usr/src/lib/msun/src/e_log2.c */ |
2 /* | 2 /* |
3 * ==================================================== | 3 * ==================================================== |
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. | 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
5 * | 5 * |
6 * Developed at SunSoft, a Sun Microsystems, Inc. business. | 6 * Developed at SunSoft, a Sun Microsystems, Inc. business. |
7 * Permission to use, copy, modify, and distribute this | 7 * Permission to use, copy, modify, and distribute this |
8 * software is freely granted, provided that this notice | 8 * software is freely granted, provided that this notice |
9 * is preserved. | 9 * is preserved. |
10 * ==================================================== | 10 * ==================================================== |
11 */ | 11 */ |
12 /* | 12 /* |
13 * Return the base 2 logarithm of x. See log.c for most comments. | 13 * Return the base 2 logarithm of x. See log.c for most comments. |
14 * | 14 * |
15 * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2 | 15 * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2 |
16 * as in log.c, then combine and scale in extra precision: | 16 * as in log.c, then combine and scale in extra precision: |
17 * log2(x) = (f - f*f/2 + r)/log(2) + k | 17 * log2(x) = (f - f*f/2 + r)/log(2) + k |
18 */ | 18 */ |
19 | 19 |
20 #include <math.h> | 20 #include <math.h> |
21 #include <stdint.h> | 21 #include <stdint.h> |
22 | 22 |
23 static const double | 23 static const double ivln2hi = |
24 ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */ | 24 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */ |
25 ivln2lo = 1.67517131648865118353e-10, /* 0x3de705fc, 0x2eefa200 */ | 25 ivln2lo = 1.67517131648865118353e-10, /* 0x3de705fc, 0x2eefa200 */ |
26 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ | 26 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ |
27 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ | 27 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ |
28 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ | 28 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ |
29 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ | 29 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ |
30 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ | 30 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ |
31 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ | 31 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ |
32 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ | 32 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
33 | 33 |
34 double log2(double x) | 34 double log2(double x) { |
35 { | 35 union { |
36 » union {double f; uint64_t i;} u = {x}; | 36 double f; |
37 » double_t hfsq,f,s,z,R,w,t1,t2,y,hi,lo,val_hi,val_lo; | 37 uint64_t i; |
38 » uint32_t hx; | 38 } u = {x}; |
39 » int k; | 39 double_t hfsq, f, s, z, R, w, t1, t2, y, hi, lo, val_hi, val_lo; |
| 40 uint32_t hx; |
| 41 int k; |
40 | 42 |
41 » hx = u.i>>32; | 43 hx = u.i >> 32; |
42 » k = 0; | 44 k = 0; |
43 » if (hx < 0x00100000 || hx>>31) { | 45 if (hx < 0x00100000 || hx >> 31) { |
44 » » if (u.i<<1 == 0) | 46 if (u.i << 1 == 0) |
45 » » » return -1/(x*x); /* log(+-0)=-inf */ | 47 return -1 / (x * x); /* log(+-0)=-inf */ |
46 » » if (hx>>31) | 48 if (hx >> 31) |
47 » » » return (x-x)/0.0; /* log(-#) = NaN */ | 49 return (x - x) / 0.0; /* log(-#) = NaN */ |
48 » » /* subnormal number, scale x up */ | 50 /* subnormal number, scale x up */ |
49 » » k -= 54; | 51 k -= 54; |
50 » » x *= 0x1p54; | 52 x *= 0x1p54; |
51 » » u.f = x; | 53 u.f = x; |
52 » » hx = u.i>>32; | 54 hx = u.i >> 32; |
53 » } else if (hx >= 0x7ff00000) { | 55 } else if (hx >= 0x7ff00000) { |
54 » » return x; | 56 return x; |
55 » } else if (hx == 0x3ff00000 && u.i<<32 == 0) | 57 } else if (hx == 0x3ff00000 && u.i << 32 == 0) |
56 » » return 0; | 58 return 0; |
57 | 59 |
58 » /* reduce x into [sqrt(2)/2, sqrt(2)] */ | 60 /* reduce x into [sqrt(2)/2, sqrt(2)] */ |
59 » hx += 0x3ff00000 - 0x3fe6a09e; | 61 hx += 0x3ff00000 - 0x3fe6a09e; |
60 » k += (int)(hx>>20) - 0x3ff; | 62 k += (int)(hx >> 20) - 0x3ff; |
61 » hx = (hx&0x000fffff) + 0x3fe6a09e; | 63 hx = (hx & 0x000fffff) + 0x3fe6a09e; |
62 » u.i = (uint64_t)hx<<32 | (u.i&0xffffffff); | 64 u.i = (uint64_t)hx << 32 | (u.i & 0xffffffff); |
63 » x = u.f; | 65 x = u.f; |
64 | 66 |
65 » f = x - 1.0; | 67 f = x - 1.0; |
66 » hfsq = 0.5*f*f; | 68 hfsq = 0.5 * f * f; |
67 » s = f/(2.0+f); | 69 s = f / (2.0 + f); |
68 » z = s*s; | 70 z = s * s; |
69 » w = z*z; | 71 w = z * z; |
70 » t1 = w*(Lg2+w*(Lg4+w*Lg6)); | 72 t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); |
71 » t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); | 73 t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); |
72 » R = t2 + t1; | 74 R = t2 + t1; |
73 | 75 |
74 » /* | 76 /* |
75 » * f-hfsq must (for args near 1) be evaluated in extra precision | 77 * f-hfsq must (for args near 1) be evaluated in extra precision |
76 » * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). | 78 * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). |
77 » * This is fairly efficient since f-hfsq only depends on f, so can | 79 * This is fairly efficient since f-hfsq only depends on f, so can |
78 » * be evaluated in parallel with R. Not combining hfsq with R also | 80 * be evaluated in parallel with R. Not combining hfsq with R also |
79 » * keeps R small (though not as small as a true `lo' term would be), | 81 * keeps R small (though not as small as a true `lo' term would be), |
80 » * so that extra precision is not needed for terms involving R. | 82 * so that extra precision is not needed for terms involving R. |
81 » * | 83 * |
82 » * Compiler bugs involving extra precision used to break Dekker's | 84 * Compiler bugs involving extra precision used to break Dekker's |
83 » * theorem for spitting f-hfsq as hi+lo, unless double_t was used | 85 * theorem for spitting f-hfsq as hi+lo, unless double_t was used |
84 » * or the multi-precision calculations were avoided when double_t | 86 * or the multi-precision calculations were avoided when double_t |
85 » * has extra precision. These problems are now automatically | 87 * has extra precision. These problems are now automatically |
86 » * avoided as a side effect of the optimization of combining the | 88 * avoided as a side effect of the optimization of combining the |
87 » * Dekker splitting step with the clear-low-bits step. | 89 * Dekker splitting step with the clear-low-bits step. |
88 » * | 90 * |
89 » * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra | 91 * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra |
90 » * precision to avoid a very large cancellation when x is very near | 92 * precision to avoid a very large cancellation when x is very near |
91 » * these values. Unlike the above cancellations, this problem is | 93 * these values. Unlike the above cancellations, this problem is |
92 » * specific to base 2. It is strange that adding +-1 is so much | 94 * specific to base 2. It is strange that adding +-1 is so much |
93 » * harder than adding +-ln2 or +-log10_2. | 95 * harder than adding +-ln2 or +-log10_2. |
94 » * | 96 * |
95 » * This uses Dekker's theorem to normalize y+val_hi, so the | 97 * This uses Dekker's theorem to normalize y+val_hi, so the |
96 » * compiler bugs are back in some configurations, sigh. And I | 98 * compiler bugs are back in some configurations, sigh. And I |
97 » * don't want to used double_t to avoid them, since that gives a | 99 * don't want to used double_t to avoid them, since that gives a |
98 » * pessimization and the support for avoiding the pessimization | 100 * pessimization and the support for avoiding the pessimization |
99 » * is not yet available. | 101 * is not yet available. |
100 » * | 102 * |
101 » * The multi-precision calculations for the multiplications are | 103 * The multi-precision calculations for the multiplications are |
102 » * routine. | 104 * routine. |
103 » */ | 105 */ |
104 | 106 |
105 » /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */ | 107 /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */ |
106 » hi = f - hfsq; | 108 hi = f - hfsq; |
107 » u.f = hi; | 109 u.f = hi; |
108 » u.i &= (uint64_t)-1<<32; | 110 u.i &= (uint64_t)-1 << 32; |
109 » hi = u.f; | 111 hi = u.f; |
110 » lo = f - hi - hfsq + s*(hfsq+R); | 112 lo = f - hi - hfsq + s * (hfsq + R); |
111 | 113 |
112 » val_hi = hi*ivln2hi; | 114 val_hi = hi * ivln2hi; |
113 » val_lo = (lo+hi)*ivln2lo + lo*ivln2hi; | 115 val_lo = (lo + hi) * ivln2lo + lo * ivln2hi; |
114 | 116 |
115 » /* spadd(val_hi, val_lo, y), except for not using double_t: */ | 117 /* spadd(val_hi, val_lo, y), except for not using double_t: */ |
116 » y = k; | 118 y = k; |
117 » w = y + val_hi; | 119 w = y + val_hi; |
118 » val_lo += (y - w) + val_hi; | 120 val_lo += (y - w) + val_hi; |
119 » val_hi = w; | 121 val_hi = w; |
120 | 122 |
121 » return val_lo + val_hi; | 123 return val_lo + val_hi; |
122 } | 124 } |
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