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1 /* origin: FreeBSD /usr/src/lib/msun/src/e_log10.c */ | 1 /* origin: FreeBSD /usr/src/lib/msun/src/e_log10.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 10 logarithm of x. See log.c for most comments. | 13 * Return the base 10 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 * log10(x) = (f - f*f/2 + r)/log(10) + k*log10(2) | 17 * log10(x) = (f - f*f/2 + r)/log(10) + k*log10(2) |
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 ivln10hi = |
24 ivln10hi = 4.34294481878168880939e-01, /* 0x3fdbcb7b, 0x15200000 */ | 24 4.34294481878168880939e-01, /* 0x3fdbcb7b, 0x15200000 */ |
25 ivln10lo = 2.50829467116452752298e-11, /* 0x3dbb9438, 0xca9aadd5 */ | 25 ivln10lo = 2.50829467116452752298e-11, /* 0x3dbb9438, 0xca9aadd5 */ |
26 log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */ | 26 log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */ |
27 log10_2lo = 3.69423907715893078616e-13, /* 0x3D59FEF3, 0x11F12B36 */ | 27 log10_2lo = 3.69423907715893078616e-13, /* 0x3D59FEF3, 0x11F12B36 */ |
28 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ | 28 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ |
29 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ | 29 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ |
30 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ | 30 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ |
31 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ | 31 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ |
32 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ | 32 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ |
33 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ | 33 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ |
34 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ | 34 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
35 | 35 |
36 double log10(double x) | 36 double log10(double x) { |
37 { | 37 union { |
38 » union {double f; uint64_t i;} u = {x}; | 38 double f; |
39 » double_t hfsq,f,s,z,R,w,t1,t2,dk,y,hi,lo,val_hi,val_lo; | 39 uint64_t i; |
40 » uint32_t hx; | 40 } u = {x}; |
41 » int k; | 41 double_t hfsq, f, s, z, R, w, t1, t2, dk, y, hi, lo, val_hi, val_lo; |
| 42 uint32_t hx; |
| 43 int k; |
42 | 44 |
43 » hx = u.i>>32; | 45 hx = u.i >> 32; |
44 » k = 0; | 46 k = 0; |
45 » if (hx < 0x00100000 || hx>>31) { | 47 if (hx < 0x00100000 || hx >> 31) { |
46 » » if (u.i<<1 == 0) | 48 if (u.i << 1 == 0) |
47 » » » return -1/(x*x); /* log(+-0)=-inf */ | 49 return -1 / (x * x); /* log(+-0)=-inf */ |
48 » » if (hx>>31) | 50 if (hx >> 31) |
49 » » » return (x-x)/0.0; /* log(-#) = NaN */ | 51 return (x - x) / 0.0; /* log(-#) = NaN */ |
50 » » /* subnormal number, scale x up */ | 52 /* subnormal number, scale x up */ |
51 » » k -= 54; | 53 k -= 54; |
52 » » x *= 0x1p54; | 54 x *= 0x1p54; |
53 » » u.f = x; | 55 u.f = x; |
54 » » hx = u.i>>32; | 56 hx = u.i >> 32; |
55 » } else if (hx >= 0x7ff00000) { | 57 } else if (hx >= 0x7ff00000) { |
56 » » return x; | 58 return x; |
57 » } else if (hx == 0x3ff00000 && u.i<<32 == 0) | 59 } else if (hx == 0x3ff00000 && u.i << 32 == 0) |
58 » » return 0; | 60 return 0; |
59 | 61 |
60 » /* reduce x into [sqrt(2)/2, sqrt(2)] */ | 62 /* reduce x into [sqrt(2)/2, sqrt(2)] */ |
61 » hx += 0x3ff00000 - 0x3fe6a09e; | 63 hx += 0x3ff00000 - 0x3fe6a09e; |
62 » k += (int)(hx>>20) - 0x3ff; | 64 k += (int)(hx >> 20) - 0x3ff; |
63 » hx = (hx&0x000fffff) + 0x3fe6a09e; | 65 hx = (hx & 0x000fffff) + 0x3fe6a09e; |
64 » u.i = (uint64_t)hx<<32 | (u.i&0xffffffff); | 66 u.i = (uint64_t)hx << 32 | (u.i & 0xffffffff); |
65 » x = u.f; | 67 x = u.f; |
66 | 68 |
67 » f = x - 1.0; | 69 f = x - 1.0; |
68 » hfsq = 0.5*f*f; | 70 hfsq = 0.5 * f * f; |
69 » s = f/(2.0+f); | 71 s = f / (2.0 + f); |
70 » z = s*s; | 72 z = s * s; |
71 » w = z*z; | 73 w = z * z; |
72 » t1 = w*(Lg2+w*(Lg4+w*Lg6)); | 74 t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); |
73 » t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); | 75 t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); |
74 » R = t2 + t1; | 76 R = t2 + t1; |
75 | 77 |
76 » /* See log2.c for details. */ | 78 /* See log2.c for details. */ |
77 » /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */ | 79 /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */ |
78 » hi = f - hfsq; | 80 hi = f - hfsq; |
79 » u.f = hi; | 81 u.f = hi; |
80 » u.i &= (uint64_t)-1<<32; | 82 u.i &= (uint64_t)-1 << 32; |
81 » hi = u.f; | 83 hi = u.f; |
82 » lo = f - hi - hfsq + s*(hfsq+R); | 84 lo = f - hi - hfsq + s * (hfsq + R); |
83 | 85 |
84 » /* val_hi+val_lo ~ log10(1+f) + k*log10(2) */ | 86 /* val_hi+val_lo ~ log10(1+f) + k*log10(2) */ |
85 » val_hi = hi*ivln10hi; | 87 val_hi = hi * ivln10hi; |
86 » dk = k; | 88 dk = k; |
87 » y = dk*log10_2hi; | 89 y = dk * log10_2hi; |
88 » val_lo = dk*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi; | 90 val_lo = dk * log10_2lo + (lo + hi) * ivln10lo + lo * ivln10hi; |
89 | 91 |
90 » /* | 92 /* |
91 » * Extra precision in for adding y is not strictly needed | 93 * Extra precision in for adding y is not strictly needed |
92 » * since there is no very large cancellation near x = sqrt(2) or | 94 * since there is no very large cancellation near x = sqrt(2) or |
93 » * x = 1/sqrt(2), but we do it anyway since it costs little on CPUs | 95 * x = 1/sqrt(2), but we do it anyway since it costs little on CPUs |
94 » * with some parallelism and it reduces the error for many args. | 96 * with some parallelism and it reduces the error for many args. |
95 » */ | 97 */ |
96 » w = y + val_hi; | 98 w = y + val_hi; |
97 » val_lo += (y - w) + val_hi; | 99 val_lo += (y - w) + val_hi; |
98 » val_hi = w; | 100 val_hi = w; |
99 | 101 |
100 » return val_lo + val_hi; | 102 return val_lo + val_hi; |
101 } | 103 } |
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