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1 : :
2 : : /* @(#)e_log.c 1.3 95/01/18 */
3 : : /*
4 : : * ====================================================
5 : : * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 : : *
7 : : * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 : : * Permission to use, copy, modify, and distribute this
9 : : * software is freely granted, provided that this notice
10 : : * is preserved.
11 : : * ====================================================
12 : : */
13 : :
14 : : #include "cdefs-compat.h"
15 : : //__FBSDID("$FreeBSD: src/lib/msun/src/k_log.h,v 1.2 2011/10/15 05:23:28 das Exp $");
16 : :
17 : : /*
18 : : * k_log1p(f):
19 : : * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)].
20 : : *
21 : : * The following describes the overall strategy for computing
22 : : * logarithms in base e. The argument reduction and adding the final
23 : : * term of the polynomial are done by the caller for increased accuracy
24 : : * when different bases are used.
25 : : *
26 : : * Method :
27 : : * 1. Argument Reduction: find k and f such that
28 : : * x = 2^k * (1+f),
29 : : * where sqrt(2)/2 < 1+f < sqrt(2) .
30 : : *
31 : : * 2. Approximation of log(1+f).
32 : : * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
33 : : * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
34 : : * = 2s + s*R
35 : : * We use a special Reme algorithm on [0,0.1716] to generate
36 : : * a polynomial of degree 14 to approximate R The maximum error
37 : : * of this polynomial approximation is bounded by 2**-58.45. In
38 : : * other words,
39 : : * 2 4 6 8 10 12 14
40 : : * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
41 : : * (the values of Lg1 to Lg7 are listed in the program)
42 : : * and
43 : : * | 2 14 | -58.45
44 : : * | Lg1*s +...+Lg7*s - R(z) | <= 2
45 : : * | |
46 : : * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
47 : : * In order to guarantee error in log below 1ulp, we compute log
48 : : * by
49 : : * log(1+f) = f - s*(f - R) (if f is not too large)
50 : : * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
51 : : *
52 : : * 3. Finally, log(x) = k*ln2 + log(1+f).
53 : : * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
54 : : * Here ln2 is split into two floating point number:
55 : : * ln2_hi + ln2_lo,
56 : : * where n*ln2_hi is always exact for |n| < 2000.
57 : : *
58 : : * Special cases:
59 : : * log(x) is NaN with signal if x < 0 (including -INF) ;
60 : : * log(+INF) is +INF; log(0) is -INF with signal;
61 : : * log(NaN) is that NaN with no signal.
62 : : *
63 : : * Accuracy:
64 : : * according to an error analysis, the error is always less than
65 : : * 1 ulp (unit in the last place).
66 : : *
67 : : * Constants:
68 : : * The hexadecimal values are the intended ones for the following
69 : : * constants. The decimal values may be used, provided that the
70 : : * compiler will convert from decimal to binary accurately enough
71 : : * to produce the hexadecimal values shown.
72 : : */
73 : :
74 : : static const double
75 : : Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
76 : : Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
77 : : Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
78 : : Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
79 : : Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
80 : : Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
81 : : Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
82 : :
83 : : /*
84 : : * We always inline k_log1p(), since doing so produces a
85 : : * substantial performance improvement (~40% on amd64).
86 : : */
87 : : static inline double
88 : 11 : k_log1p(double f)
89 : : {
90 : : double hfsq,s,z,R,w,t1,t2;
91 : :
92 : 11 : s = f/(2.0+f);
93 : 11 : z = s*s;
94 : 11 : w = z*z;
95 : 11 : t1= w*(Lg2+w*(Lg4+w*Lg6));
96 : 11 : t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
97 : 11 : R = t2+t1;
98 : 11 : hfsq=0.5*f*f;
99 : 11 : return s*(hfsq+R);
100 : : }
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