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/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __ieee754_log2(x)
* Return the logarithm to base 2 of x
*
* Method :
* 1. Argument Reduction: find k and f such that
* x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* 2. Approximation of log(1+f).
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
* = 2s + s*R
* We use a special Reme algorithm on [0,0.1716] to generate
* a polynomial of degree 14 to approximate R The maximum error
* of this polynomial approximation is bounded by 2**-58.45. In
* other words,
* 2 4 6 8 10 12 14
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
* (the values of Lg1 to Lg7 are listed in the program)
* and
* | 2 14 | -58.45
* | Lg1*s +...+Lg7*s - R(z) | <= 2
* | |
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
* In order to guarantee error in log below 1ulp, we compute log
* by
* log(1+f) = f - s*(f - R) (if f is not too large)
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
*
* 3. Finally, log(x) = k + log(1+f).
* = k+(f-(hfsq-(s*(hfsq+R))))
*
* Special cases:
* log2(x) is NaN with signal if x < 0 (including -INF) ;
* log2(+INF) is +INF; log(0) is -INF with signal;
* log2(NaN) is that NaN with no signal.
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
#include <math.h>
#include <math_private.h>
static const double ln2 = 0.69314718055994530942;
static const double two54 = 1.80143985094819840000e+16; /* 4350000000000000 */
static const double Lg1 = 6.666666666666735130e-01; /* 3FE5555555555593 */
static const double Lg2 = 3.999999999940941908e-01; /* 3FD999999997FA04 */
static const double Lg3 = 2.857142874366239149e-01; /* 3FD2492494229359 */
static const double Lg4 = 2.222219843214978396e-01; /* 3FCC71C51D8E78AF */
static const double Lg5 = 1.818357216161805012e-01; /* 3FC7466496CB03DE */
static const double Lg6 = 1.531383769920937332e-01; /* 3FC39A09D078C69F */
static const double Lg7 = 1.479819860511658591e-01; /* 3FC2F112DF3E5244 */
static const double zero = 0.0;
double
__ieee754_log2 (double x)
{
double hfsq, f, s, z, R, w, t1, t2, dk;
int64_t hx, i, j;
int32_t k;
EXTRACT_WORDS64 (hx, x);
k = 0;
if (hx < INT64_C(0x0010000000000000))
{ /* x < 2**-1022 */
if (__glibc_unlikely ((hx & UINT64_C(0x7fffffffffffffff)) == 0))
return -two54 / (x - x); /* log(+-0)=-inf */
if (__glibc_unlikely (hx < 0))
return (x - x) / (x - x); /* log(-#) = NaN */
k -= 54;
x *= two54; /* subnormal number, scale up x */
EXTRACT_WORDS64 (hx, x);
}
if (__glibc_unlikely (hx >= UINT64_C(0x7ff0000000000000)))
return x + x;
k += (hx >> 52) - 1023;
hx &= UINT64_C(0x000fffffffffffff);
i = (hx + UINT64_C(0x95f6400000000)) & UINT64_C(0x10000000000000);
/* normalize x or x/2 */
INSERT_WORDS64 (x, hx | (i ^ UINT64_C(0x3ff0000000000000)));
k += (i >> 52);
dk = (double) k;
f = x - 1.0;
if ((UINT64_C(0x000fffffffffffff) & (2 + hx)) < 3)
{ /* |f| < 2**-20 */
if (f == zero)
return dk;
R = f * f * (0.5 - 0.33333333333333333 * f);
return dk - (R - f) / ln2;
}
s = f / (2.0 + f);
z = s * s;
i = hx - UINT64_C(0x6147a00000000);
w = z * z;
j = UINT64_C(0x6b85100000000) - hx;
t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
i |= j;
R = t2 + t1;
if (i > 0)
{
hfsq = 0.5 * f * f;
return dk - ((hfsq - (s * (hfsq + R))) - f) / ln2;
}
else
{
return dk - ((s * (f - R)) - f) / ln2;
}
}
strong_alias (__ieee754_log2, __log2_finite)
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