3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
10 * ====================================================
13 #include <sys/cdefs.h>
15 * Return the base 2 logarithm of x. See e_log.c and k_log.h for most
18 * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
19 * then does the combining and scaling steps
20 * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
21 * in not-quite-routine extra precision.
27 #include "math_private.h"
31 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
32 ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
33 ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */
35 static const double zero = 0.0;
36 static volatile double vzero = 0.0;
41 double f,hfsq,hi,lo,r,val_hi,val_lo,w,y;
45 EXTRACT_WORDS(hx,lx,x);
48 if (hx < 0x00100000) { /* x < 2**-1022 */
49 if (((hx&0x7fffffff)|lx)==0)
50 return -two54/vzero; /* log(+-0)=-inf */
51 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
52 k -= 54; x *= two54; /* subnormal number, scale up x */
55 if (hx >= 0x7ff00000) return x+x;
56 if (hx == 0x3ff00000 && lx == 0)
57 return zero; /* log(1) = +0 */
60 i = (hx+0x95f64)&0x100000;
61 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
69 * f-hfsq must (for args near 1) be evaluated in extra precision
70 * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
71 * This is fairly efficient since f-hfsq only depends on f, so can
72 * be evaluated in parallel with R. Not combining hfsq with R also
73 * keeps R small (though not as small as a true `lo' term would be),
74 * so that extra precision is not needed for terms involving R.
76 * Compiler bugs involving extra precision used to break Dekker's
77 * theorem for spitting f-hfsq as hi+lo, unless double_t was used
78 * or the multi-precision calculations were avoided when double_t
79 * has extra precision. These problems are now automatically
80 * avoided as a side effect of the optimization of combining the
81 * Dekker splitting step with the clear-low-bits step.
83 * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
84 * precision to avoid a very large cancellation when x is very near
85 * these values. Unlike the above cancellations, this problem is
86 * specific to base 2. It is strange that adding +-1 is so much
87 * harder than adding +-ln2 or +-log10_2.
89 * This uses Dekker's theorem to normalize y+val_hi, so the
90 * compiler bugs are back in some configurations, sigh. And I
91 * don't want to used double_t to avoid them, since that gives a
92 * pessimization and the support for avoiding the pessimization
93 * is not yet available.
95 * The multi-precision calculations for the multiplications are
100 lo = (f - hi) - hfsq + r;
102 val_lo = (lo+hi)*ivln2lo + lo*ivln2hi;
104 /* spadd(val_hi, val_lo, y), except for not using double_t: */
106 val_lo += (y - w) + val_hi;
109 return val_lo + val_hi;
112 #if (LDBL_MANT_DIG == 53)
113 __weak_reference(log2, log2l);