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 <sys/cdefs.h>
15 __FBSDID("$FreeBSD$");
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
k_log1p(double f)88 k_log1p(double f)
89 {
90 double hfsq,s,z,R,w,t1,t2;
91
92 s = f/(2.0+f);
93 z = s*s;
94 w = z*z;
95 t1= w*(Lg2+w*(Lg4+w*Lg6));
96 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
97 R = t2+t1;
98 hfsq=0.5*f*f;
99 return s*(hfsq+R);
100 }
101