2 /* @(#)e_log10.c 1.3 95/01/18 */
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
11 * ====================================================
14 #include <sys/cdefs.h>
16 * Return the base 2 logarithm of x. See e_log.c and k_log.h for most
19 * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
20 * then does the combining and scaling steps
21 * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
22 * in not-quite-routine extra precision.
28 #include "math_private.h"
32 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
33 ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
34 ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */
36 static const double zero = 0.0;
37 static volatile double vzero = 0.0;
42 double f,hfsq,hi,lo,r,val_hi,val_lo,w,y;
46 EXTRACT_WORDS(hx,lx,x);
49 if (hx < 0x00100000) { /* x < 2**-1022 */
50 if (((hx&0x7fffffff)|lx)==0)
51 return -two54/vzero; /* log(+-0)=-inf */
52 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
53 k -= 54; x *= two54; /* subnormal number, scale up x */
56 if (hx >= 0x7ff00000) return x+x;
57 if (hx == 0x3ff00000 && lx == 0)
58 return zero; /* log(1) = +0 */
61 i = (hx+0x95f64)&0x100000;
62 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
70 * f-hfsq must (for args near 1) be evaluated in extra precision
71 * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
72 * This is fairly efficient since f-hfsq only depends on f, so can
73 * be evaluated in parallel with R. Not combining hfsq with R also
74 * keeps R small (though not as small as a true `lo' term would be),
75 * so that extra precision is not needed for terms involving R.
77 * Compiler bugs involving extra precision used to break Dekker's
78 * theorem for spitting f-hfsq as hi+lo, unless double_t was used
79 * or the multi-precision calculations were avoided when double_t
80 * has extra precision. These problems are now automatically
81 * avoided as a side effect of the optimization of combining the
82 * Dekker splitting step with the clear-low-bits step.
84 * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
85 * precision to avoid a very large cancellation when x is very near
86 * these values. Unlike the above cancellations, this problem is
87 * specific to base 2. It is strange that adding +-1 is so much
88 * harder than adding +-ln2 or +-log10_2.
90 * This uses Dekker's theorem to normalize y+val_hi, so the
91 * compiler bugs are back in some configurations, sigh. And I
92 * don't want to used double_t to avoid them, since that gives a
93 * pessimization and the support for avoiding the pessimization
94 * is not yet available.
96 * The multi-precision calculations for the multiplications are
101 lo = (f - hi) - hfsq + r;
103 val_lo = (lo+hi)*ivln2lo + lo*ivln2hi;
105 /* spadd(val_hi, val_lo, y), except for not using double_t: */
107 val_lo += (y - w) + val_hi;
110 return val_lo + val_hi;
113 #if (LDBL_MANT_DIG == 53)
114 __weak_reference(log2, log2l);