1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122

/* origin: FreeBSD /usr/src/lib/msun/src/e_log2.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
* Return the base 2 logarithm of x. See log.c for most comments.
*
* Reduce x to 2^k (1+f) and calculate r = log(1+f)  f + f*f/2
* as in log.c, then combine and scale in extra precision:
* log2(x) = (f  f*f/2 + r)/log(2) + k
*/
#include <math.h>
#include <stdint.h>
static const double
ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
ivln2lo = 1.67517131648865118353e10, /* 0x3de705fc, 0x2eefa200 */
Lg1 = 6.666666666666735130e01, /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e01, /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e01, /* 3FD24924 94229359 */
Lg4 = 2.222219843214978396e01, /* 3FCC71C5 1D8E78AF */
Lg5 = 1.818357216161805012e01, /* 3FC74664 96CB03DE */
Lg6 = 1.531383769920937332e01, /* 3FC39A09 D078C69F */
Lg7 = 1.479819860511658591e01; /* 3FC2F112 DF3E5244 */
double log2(double x)
{
union {double f; uint64_t i;} u = {x};
double_t hfsq,f,s,z,R,w,t1,t2,y,hi,lo,val_hi,val_lo;
uint32_t hx;
int k;
hx = u.i>>32;
k = 0;
if (hx < 0x00100000  hx>>31) {
if (u.i<<1 == 0)
return 1/(x*x); /* log(+0)=inf */
if (hx>>31)
return (xx)/0.0; /* log(#) = NaN */
/* subnormal number, scale x up */
k = 54;
x *= 0x1p54;
u.f = x;
hx = u.i>>32;
} else if (hx >= 0x7ff00000) {
return x;
} else if (hx == 0x3ff00000 && u.i<<32 == 0)
return 0;
/* reduce x into [sqrt(2)/2, sqrt(2)] */
hx += 0x3ff00000  0x3fe6a09e;
k += (int)(hx>>20)  0x3ff;
hx = (hx&0x000fffff) + 0x3fe6a09e;
u.i = (uint64_t)hx<<32  (u.i&0xffffffff);
x = u.f;
f = x  1.0;
hfsq = 0.5*f*f;
s = f/(2.0+f);
z = s*s;
w = z*z;
t1 = w*(Lg2+w*(Lg4+w*Lg6));
t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
R = t2 + t1;
/*
* fhfsq must (for args near 1) be evaluated in extra precision
* to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
* This is fairly efficient since fhfsq only depends on f, so can
* be evaluated in parallel with R. Not combining hfsq with R also
* keeps R small (though not as small as a true `lo' term would be),
* so that extra precision is not needed for terms involving R.
*
* Compiler bugs involving extra precision used to break Dekker's
* theorem for spitting fhfsq as hi+lo, unless double_t was used
* or the multiprecision calculations were avoided when double_t
* has extra precision. These problems are now automatically
* avoided as a side effect of the optimization of combining the
* Dekker splitting step with the clearlowbits step.
*
* y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
* precision to avoid a very large cancellation when x is very near
* these values. Unlike the above cancellations, this problem is
* specific to base 2. It is strange that adding +1 is so much
* harder than adding +ln2 or +log10_2.
*
* This uses Dekker's theorem to normalize y+val_hi, so the
* compiler bugs are back in some configurations, sigh. And I
* don't want to used double_t to avoid them, since that gives a
* pessimization and the support for avoiding the pessimization
* is not yet available.
*
* The multiprecision calculations for the multiplications are
* routine.
*/
/* hi+lo = f  hfsq + s*(hfsq+R) ~ log(1+f) */
hi = f  hfsq;
u.f = hi;
u.i &= (uint64_t)1<<32;
hi = u.f;
lo = f  hi  hfsq + s*(hfsq+R);
val_hi = hi*ivln2hi;
val_lo = (lo+hi)*ivln2lo + lo*ivln2hi;
/* spadd(val_hi, val_lo, y), except for not using double_t: */
y = k;
w = y + val_hi;
val_lo += (y  w) + val_hi;
val_hi = w;
return val_lo + val_hi;
}
