1
2/* @(#)e_log10.c 1.3 95/01/18 */
3/*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14#include <sys/cdefs.h>
15__FBSDID("$FreeBSD$");
16
17/*
18 * Return the base 2 logarithm of x.  See e_log.c and k_log.h for most
19 * comments.
20 *
21 * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
22 * then does the combining and scaling steps
23 *    log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
24 * in not-quite-routine extra precision.
25 */
26
27#include <float.h>
28
29#include "math.h"
30#include "math_private.h"
31#include "k_log.h"
32
33static const double
34two54      =  1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
35ivln2hi    =  1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
36ivln2lo    =  1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */
37
38static const double zero   =  0.0;
39static volatile double vzero = 0.0;
40
41double
42__ieee754_log2(double x)
43{
44	double f,hfsq,hi,lo,r,val_hi,val_lo,w,y;
45	int32_t i,k,hx;
46	u_int32_t lx;
47
48	EXTRACT_WORDS(hx,lx,x);
49
50	k=0;
51	if (hx < 0x00100000) {			/* x < 2**-1022  */
52	    if (((hx&0x7fffffff)|lx)==0)
53		return -two54/vzero;		/* log(+-0)=-inf */
54	    if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */
55	    k -= 54; x *= two54; /* subnormal number, scale up x */
56	    GET_HIGH_WORD(hx,x);
57	}
58	if (hx >= 0x7ff00000) return x+x;
59	if (hx == 0x3ff00000 && lx == 0)
60	    return zero;			/* log(1) = +0 */
61	k += (hx>>20)-1023;
62	hx &= 0x000fffff;
63	i = (hx+0x95f64)&0x100000;
64	SET_HIGH_WORD(x,hx|(i^0x3ff00000));	/* normalize x or x/2 */
65	k += (i>>20);
66	y = (double)k;
67	f = x - 1.0;
68	hfsq = 0.5*f*f;
69	r = k_log1p(f);
70
71	/*
72	 * f-hfsq must (for args near 1) be evaluated in extra precision
73	 * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
74	 * This is fairly efficient since f-hfsq only depends on f, so can
75	 * be evaluated in parallel with R.  Not combining hfsq with R also
76	 * keeps R small (though not as small as a true `lo' term would be),
77	 * so that extra precision is not needed for terms involving R.
78	 *
79	 * Compiler bugs involving extra precision used to break Dekker's
80	 * theorem for spitting f-hfsq as hi+lo, unless double_t was used
81	 * or the multi-precision calculations were avoided when double_t
82	 * has extra precision.  These problems are now automatically
83	 * avoided as a side effect of the optimization of combining the
84	 * Dekker splitting step with the clear-low-bits step.
85	 *
86	 * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
87	 * precision to avoid a very large cancellation when x is very near
88	 * these values.  Unlike the above cancellations, this problem is
89	 * specific to base 2.  It is strange that adding +-1 is so much
90	 * harder than adding +-ln2 or +-log10_2.
91	 *
92	 * This uses Dekker's theorem to normalize y+val_hi, so the
93	 * compiler bugs are back in some configurations, sigh.  And I
94	 * don't want to used double_t to avoid them, since that gives a
95	 * pessimization and the support for avoiding the pessimization
96	 * is not yet available.
97	 *
98	 * The multi-precision calculations for the multiplications are
99	 * routine.
100	 */
101	hi = f - hfsq;
102	SET_LOW_WORD(hi,0);
103	lo = (f - hi) - hfsq + r;
104	val_hi = hi*ivln2hi;
105	val_lo = (lo+hi)*ivln2lo + lo*ivln2hi;
106
107	/* spadd(val_hi, val_lo, y), except for not using double_t: */
108	w = y + val_hi;
109	val_lo += (y - w) + val_hi;
110	val_hi = w;
111
112	return val_lo + val_hi;
113}
114
115#if (LDBL_MANT_DIG == 53)
116__weak_reference(log2, log2l);
117#endif
118