1/* origin: FreeBSD /usr/src/lib/msun/src/e_log.c */
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/* log(x)
13 * Return the logarithm 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 Remez 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*ln2 + log(1+f).
42 *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
43 *         Here ln2 is split into two floating point number:
44 *                      ln2_hi + ln2_lo,
45 *         where n*ln2_hi is always exact for |n| < 2000.
46 *
47 * Special cases:
48 *      log(x) is NaN with signal if x < 0 (including -INF) ;
49 *      log(+INF) is +INF; log(0) is -INF with signal;
50 *      log(NaN) is that NaN with no signal.
51 *
52 * Accuracy:
53 *      according to an error analysis, the error is always less than
54 *      1 ulp (unit in the last place).
55 *
56 * Constants:
57 * The hexadecimal values are the intended ones for the following
58 * constants. The decimal values may be used, provided that the
59 * compiler will convert from decimal to binary accurately enough
60 * to produce the hexadecimal values shown.
61 */
62
63#include <math.h>
64#include <stdint.h>
65
66static const double
67ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
68ln2_lo = 1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
69Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
70Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
71Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
72Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
73Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
74Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
75Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
76
77double log(double x)
78{
79	union {double f; uint64_t i;} u = {x};
80	double_t hfsq,f,s,z,R,w,t1,t2,dk;
81	uint32_t hx;
82	int k;
83
84	hx = u.i>>32;
85	k = 0;
86	if (hx < 0x00100000 || hx>>31) {
87		if (u.i<<1 == 0)
88			return -1/(x*x);  /* log(+-0)=-inf */
89		if (hx>>31)
90			return (x-x)/0.0; /* log(-#) = NaN */
91		/* subnormal number, scale x up */
92		k -= 54;
93		x *= 0x1p54;
94		u.f = x;
95		hx = u.i>>32;
96	} else if (hx >= 0x7ff00000) {
97		return x;
98	} else if (hx == 0x3ff00000 && u.i<<32 == 0)
99		return 0;
100
101	/* reduce x into [sqrt(2)/2, sqrt(2)] */
102	hx += 0x3ff00000 - 0x3fe6a09e;
103	k += (int)(hx>>20) - 0x3ff;
104	hx = (hx&0x000fffff) + 0x3fe6a09e;
105	u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
106	x = u.f;
107
108	f = x - 1.0;
109	hfsq = 0.5*f*f;
110	s = f/(2.0+f);
111	z = s*s;
112	w = z*z;
113	t1 = w*(Lg2+w*(Lg4+w*Lg6));
114	t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
115	R = t2 + t1;
116	dk = k;
117	return s*(hfsq+R) + dk*ln2_lo - hfsq + f + dk*ln2_hi;
118}
119