/src/moddable/xs/tools/fdlibm/k_log.h
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1 | | |
2 | | /* |
3 | | * ==================================================== |
4 | | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
5 | | * |
6 | | * Developed at SunSoft, 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 | | /* |
14 | | * k_log1p(f): |
15 | | * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)]. |
16 | | * |
17 | | * The following describes the overall strategy for computing |
18 | | * logarithms in base e. The argument reduction and adding the final |
19 | | * term of the polynomial are done by the caller for increased accuracy |
20 | | * when different bases are used. |
21 | | * |
22 | | * Method : |
23 | | * 1. Argument Reduction: find k and f such that |
24 | | * x = 2^k * (1+f), |
25 | | * where sqrt(2)/2 < 1+f < sqrt(2) . |
26 | | * |
27 | | * 2. Approximation of log(1+f). |
28 | | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) |
29 | | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., |
30 | | * = 2s + s*R |
31 | | * We use a special Reme algorithm on [0,0.1716] to generate |
32 | | * a polynomial of degree 14 to approximate R The maximum error |
33 | | * of this polynomial approximation is bounded by 2**-58.45. In |
34 | | * other words, |
35 | | * 2 4 6 8 10 12 14 |
36 | | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s |
37 | | * (the values of Lg1 to Lg7 are listed in the program) |
38 | | * and |
39 | | * | 2 14 | -58.45 |
40 | | * | Lg1*s +...+Lg7*s - R(z) | <= 2 |
41 | | * | | |
42 | | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. |
43 | | * In order to guarantee error in log below 1ulp, we compute log |
44 | | * by |
45 | | * log(1+f) = f - s*(f - R) (if f is not too large) |
46 | | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) |
47 | | * |
48 | | * 3. Finally, log(x) = k*ln2 + log(1+f). |
49 | | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
50 | | * Here ln2 is split into two floating point number: |
51 | | * ln2_hi + ln2_lo, |
52 | | * where n*ln2_hi is always exact for |n| < 2000. |
53 | | * |
54 | | * Special cases: |
55 | | * log(x) is NaN with signal if x < 0 (including -INF) ; |
56 | | * log(+INF) is +INF; log(0) is -INF with signal; |
57 | | * log(NaN) is that NaN with no signal. |
58 | | * |
59 | | * Accuracy: |
60 | | * according to an error analysis, the error is always less than |
61 | | * 1 ulp (unit in the last place). |
62 | | * |
63 | | * Constants: |
64 | | * The hexadecimal values are the intended ones for the following |
65 | | * constants. The decimal values may be used, provided that the |
66 | | * compiler will convert from decimal to binary accurately enough |
67 | | * to produce the hexadecimal values shown. |
68 | | */ |
69 | | |
70 | | static const double |
71 | | Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ |
72 | | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ |
73 | | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ |
74 | | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ |
75 | | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ |
76 | | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ |
77 | | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
78 | | |
79 | | /* |
80 | | * We always inline k_log1p(), since doing so produces a |
81 | | * substantial performance improvement (~40% on amd64). |
82 | | */ |
83 | | static inline double |
84 | | k_log1p(double f) |
85 | 17 | { |
86 | 17 | double hfsq,s,z,R,w,t1,t2; |
87 | | |
88 | 17 | s = f/(2.0+f); |
89 | 17 | z = s*s; |
90 | 17 | w = z*z; |
91 | 17 | t1= w*(Lg2+w*(Lg4+w*Lg6)); |
92 | 17 | t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); |
93 | 17 | R = t2+t1; |
94 | 17 | hfsq=0.5*f*f; |
95 | 17 | return s*(hfsq+R); |
96 | 17 | } |