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
k_log1p(double f)84 k_log1p(double f)
85 {
86 double hfsq,s,z,R,w,t1,t2;
87
88 s = f/(2.0+f);
89 z = s*s;
90 w = z*z;
91 t1= w*(Lg2+w*(Lg4+w*Lg6));
92 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
93 R = t2+t1;
94 hfsq=0.5*f*f;
95 return s*(hfsq+R);
96 }
97