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