1/* origin: FreeBSD /usr/src/lib/msun/src/e_log10.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/*
13 * Return the base 10 logarithm of x.  See log.c for most comments.
14 *
15 * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2
16 * as in log.c, then combine and scale in extra precision:
17 *    log10(x) = (f - f*f/2 + r)/log(10) + k*log10(2)
18 */
19
20#include <math.h>
21#include <stdint.h>
22
23static const double
24ivln10hi  = 4.34294481878168880939e-01, /* 0x3fdbcb7b, 0x15200000 */
25ivln10lo  = 2.50829467116452752298e-11, /* 0x3dbb9438, 0xca9aadd5 */
26log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
27log10_2lo = 3.69423907715893078616e-13, /* 0x3D59FEF3, 0x11F12B36 */
28Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
29Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
30Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
31Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
32Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
33Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
34Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
35
36double log10(double x)
37{
38	union {double f; uint64_t i;} u = {x};
39	double_t hfsq,f,s,z,R,w,t1,t2,dk,y,hi,lo,val_hi,val_lo;
40	uint32_t hx;
41	int k;
42
43	hx = u.i>>32;
44	k = 0;
45	if (hx < 0x00100000 || hx>>31) {
46		if (u.i<<1 == 0)
47			return -1/(x*x);  /* log(+-0)=-inf */
48		if (hx>>31)
49			return (x-x)/0.0; /* log(-#) = NaN */
50		/* subnormal number, scale x up */
51		k -= 54;
52		x *= 0x1p54;
53		u.f = x;
54		hx = u.i>>32;
55	} else if (hx >= 0x7ff00000) {
56		return x;
57	} else if (hx == 0x3ff00000 && u.i<<32 == 0)
58		return 0;
59
60	/* reduce x into [sqrt(2)/2, sqrt(2)] */
61	hx += 0x3ff00000 - 0x3fe6a09e;
62	k += (int)(hx>>20) - 0x3ff;
63	hx = (hx&0x000fffff) + 0x3fe6a09e;
64	u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
65	x = u.f;
66
67	f = x - 1.0;
68	hfsq = 0.5*f*f;
69	s = f/(2.0+f);
70	z = s*s;
71	w = z*z;
72	t1 = w*(Lg2+w*(Lg4+w*Lg6));
73	t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
74	R = t2 + t1;
75
76	/* See log2.c for details. */
77	/* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */
78	hi = f - hfsq;
79	u.f = hi;
80	u.i &= (uint64_t)-1<<32;
81	hi = u.f;
82	lo = f - hi - hfsq + s*(hfsq+R);
83
84	/* val_hi+val_lo ~ log10(1+f) + k*log10(2) */
85	val_hi = hi*ivln10hi;
86	dk = k;
87	y = dk*log10_2hi;
88	val_lo = dk*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi;
89
90	/*
91	 * Extra precision in for adding y is not strictly needed
92	 * since there is no very large cancellation near x = sqrt(2) or
93	 * x = 1/sqrt(2), but we do it anyway since it costs little on CPUs
94	 * with some parallelism and it reduces the error for many args.
95	 */
96	w = y + val_hi;
97	val_lo += (y - w) + val_hi;
98	val_hi = w;
99
100	return val_lo + val_hi;
101}
102