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 * Return the base 2 logarithm of x. See e_log.c and k_log.h for most
15 * comments.
16 *
17 * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
18 * then does the combining and scaling steps
19 * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
20 * in not-quite-routine extra precision.
21 */
22
23 #include <float.h>
24
25 #include "math.h"
26 #include "math_private.h"
27 #include "k_log.h"
28
29 static const double
30 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
31 ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
32 ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */
33
34 static const double zero = 0.0;
35 static volatile double vzero = 0.0;
36
37 double
log2(double x)38 log2(double x)
39 {
40 double f,hfsq,hi,lo,r,val_hi,val_lo,w,y;
41 int32_t i,k,hx;
42 u_int32_t lx;
43
44 EXTRACT_WORDS(hx,lx,x);
45
46 k=0;
47 if (hx < 0x00100000) { /* x < 2**-1022 */
48 if (((hx&0x7fffffff)|lx)==0)
49 return -two54/vzero; /* log(+-0)=-inf */
50 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
51 k -= 54; x *= two54; /* subnormal number, scale up x */
52 GET_HIGH_WORD(hx,x);
53 }
54 if (hx >= 0x7ff00000) return x+x;
55 if (hx == 0x3ff00000 && lx == 0)
56 return zero; /* log(1) = +0 */
57 k += (hx>>20)-1023;
58 hx &= 0x000fffff;
59 i = (hx+0x95f64)&0x100000;
60 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
61 k += (i>>20);
62 y = (double)k;
63 f = x - 1.0;
64 hfsq = 0.5*f*f;
65 r = k_log1p(f);
66
67 /*
68 * f-hfsq must (for args near 1) be evaluated in extra precision
69 * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
70 * This is fairly efficient since f-hfsq only depends on f, so can
71 * be evaluated in parallel with R. Not combining hfsq with R also
72 * keeps R small (though not as small as a true `lo' term would be),
73 * so that extra precision is not needed for terms involving R.
74 *
75 * Compiler bugs involving extra precision used to break Dekker's
76 * theorem for spitting f-hfsq as hi+lo, unless double_t was used
77 * or the multi-precision calculations were avoided when double_t
78 * has extra precision. These problems are now automatically
79 * avoided as a side effect of the optimization of combining the
80 * Dekker splitting step with the clear-low-bits step.
81 *
82 * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
83 * precision to avoid a very large cancellation when x is very near
84 * these values. Unlike the above cancellations, this problem is
85 * specific to base 2. It is strange that adding +-1 is so much
86 * harder than adding +-ln2 or +-log10_2.
87 *
88 * This uses Dekker's theorem to normalize y+val_hi, so the
89 * compiler bugs are back in some configurations, sigh. And I
90 * don't want to used double_t to avoid them, since that gives a
91 * pessimization and the support for avoiding the pessimization
92 * is not yet available.
93 *
94 * The multi-precision calculations for the multiplications are
95 * routine.
96 */
97 hi = f - hfsq;
98 SET_LOW_WORD(hi,0);
99 lo = (f - hi) - hfsq + r;
100 val_hi = hi*ivln2hi;
101 val_lo = (lo+hi)*ivln2lo + lo*ivln2hi;
102
103 /* spadd(val_hi, val_lo, y), except for not using double_t: */
104 w = y + val_hi;
105 val_lo += (y - w) + val_hi;
106 val_hi = w;
107
108 return val_lo + val_hi;
109 }
110
111 #if (LDBL_MANT_DIG == 53)
112 __weak_reference(log2, log2l);
113 #endif
114