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| | /* 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 = 1.67517131648865118353e-10, /* 0x3de705fc, 0x2eefa200 */
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
Lg7 = 1.479819860511658591e-01; /* 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 (x-x)/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;
/*
* f-hfsq 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 f-hfsq 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 f-hfsq as hi+lo, unless double_t was used
* or the multi-precision 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 clear-low-bits 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 multi-precision 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;
}
|