1/*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12/* __ieee754_log2(x)
13 * Return the logarithm to base 2 of x
14 *
15 * Method :
16 * 1. Argument Reduction: find k and f such that
17 * x = 2^k * (1+f),
18 * where sqrt(2)/2 < 1+f < sqrt(2) .
19 *
20 * 2. Approximation of log(1+f).
21 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
22 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
23 * = 2s + s*R
24 * We use a special Reme algorithm on [0,0.1716] to generate
25 * a polynomial of degree 14 to approximate R The maximum error
26 * of this polynomial approximation is bounded by 2**-58.45. In
27 * other words,
28 * 2 4 6 8 10 12 14
29 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
30 * (the values of Lg1 to Lg7 are listed in the program)
31 * and
32 * | 2 14 | -58.45
33 * | Lg1*s +...+Lg7*s - R(z) | <= 2
34 * | |
35 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
36 * In order to guarantee error in log below 1ulp, we compute log
37 * by
38 * log(1+f) = f - s*(f - R) (if f is not too large)
39 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
40 *
41 * 3. Finally, log(x) = k + log(1+f).
42 * = k+(f-(hfsq-(s*(hfsq+R))))
43 *
44 * Special cases:
45 * log2(x) is NaN with signal if x < 0 (including -INF) ;
46 * log2(+INF) is +INF; log(0) is -INF with signal;
47 * log2(NaN) is that NaN with no signal.
48 *
49 * Constants:
50 * The hexadecimal values are the intended ones for the following
51 * constants. The decimal values may be used, provided that the
52 * compiler will convert from decimal to binary accurately enough
53 * to produce the hexadecimal values shown.
54 */
55
56#include <math.h>
57#include <math_private.h>
58
59static const double ln2 = 0.69314718055994530942;
60static const double two54 = 1.80143985094819840000e+16; /* 4350000000000000 */
61static const double Lg1 = 6.666666666666735130e-01; /* 3FE5555555555593 */
62static const double Lg2 = 3.999999999940941908e-01; /* 3FD999999997FA04 */
63static const double Lg3 = 2.857142874366239149e-01; /* 3FD2492494229359 */
64static const double Lg4 = 2.222219843214978396e-01; /* 3FCC71C51D8E78AF */
65static const double Lg5 = 1.818357216161805012e-01; /* 3FC7466496CB03DE */
66static const double Lg6 = 1.531383769920937332e-01; /* 3FC39A09D078C69F */
67static const double Lg7 = 1.479819860511658591e-01; /* 3FC2F112DF3E5244 */
68
69static const double zero = 0.0;
70
71double
72__ieee754_log2 (double x)
73{
74 double hfsq, f, s, z, R, w, t1, t2, dk;
75 int64_t hx, i, j;
76 int32_t k;
77
78 EXTRACT_WORDS64 (hx, x);
79
80 k = 0;
81 if (hx < INT64_C(0x0010000000000000))
82 { /* x < 2**-1022 */
83 if (__glibc_unlikely ((hx & UINT64_C(0x7fffffffffffffff)) == 0))
84 return -two54 / (x - x); /* log(+-0)=-inf */
85 if (__glibc_unlikely (hx < 0))
86 return (x - x) / (x - x); /* log(-#) = NaN */
87 k -= 54;
88 x *= two54; /* subnormal number, scale up x */
89 EXTRACT_WORDS64 (hx, x);
90 }
91 if (__glibc_unlikely (hx >= UINT64_C(0x7ff0000000000000)))
92 return x + x;
93 k += (hx >> 52) - 1023;
94 hx &= UINT64_C(0x000fffffffffffff);
95 i = (hx + UINT64_C(0x95f6400000000)) & UINT64_C(0x10000000000000);
96 /* normalize x or x/2 */
97 INSERT_WORDS64 (x, hx | (i ^ UINT64_C(0x3ff0000000000000)));
98 k += (i >> 52);
99 dk = (double) k;
100 f = x - 1.0;
101 if ((UINT64_C(0x000fffffffffffff) & (2 + hx)) < 3)
102 { /* |f| < 2**-20 */
103 if (f == zero)
104 return dk;
105 R = f * f * (0.5 - 0.33333333333333333 * f);
106 return dk - (R - f) / ln2;
107 }
108 s = f / (2.0 + f);
109 z = s * s;
110 i = hx - UINT64_C(0x6147a00000000);
111 w = z * z;
112 j = UINT64_C(0x6b85100000000) - hx;
113 t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
114 t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
115 i |= j;
116 R = t2 + t1;
117 if (i > 0)
118 {
119 hfsq = 0.5 * f * f;
120 return dk - ((hfsq - (s * (hfsq + R))) - f) / ln2;
121 }
122 else
123 {
124 return dk - ((s * (f - R)) - f) / ln2;
125 }
126}
127
128strong_alias (__ieee754_log2, __log2_finite)
129