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e_log.c
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/* @(#)e_log.c 5.1 93/09/24 */
/*
* ====================================================
* 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.
* ====================================================
*/
#if defined(LIBM_SCCS) && !defined(lint)
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static const char rcsid[] =
"$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
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#endif
/* __ieee754_log(x)
* Return the logrithm 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*ln2 + log(1+f).
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
* Here ln2 is split into two floating point number:
* ln2_hi + ln2_lo,
* where n*ln2_hi is always exact for |n| < 2000.
*
* Special cases:
* log(x) is NaN with signal if x < 0 (including -INF) ;
* log(+INF) is +INF; log(0) is -INF with signal;
* log(NaN) is that NaN with no signal.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* 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.
*/
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#include "math.h"
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#include "math_private.h"
#ifdef __STDC__
static const double
#else
static double
#endif
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ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
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two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
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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 */
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#ifdef __STDC__
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static const double zero = 0.0;
#else
static double zero = 0.0;
#endif
#ifdef __STDC__
double attribute_hidden
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__ieee754_log(double x)
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double attribute_hidden
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__ieee754_log(x)
double x;
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double hfsq, f, s, z, R, w, t1, t2, dk;
int32_t k, hx, i, j;
u_int32_t lx;
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EXTRACT_WORDS(hx, lx, x);
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k = 0;
if (hx < 0x00100000) { /* x < 2**-1022 */
if (((hx & 0x7fffffff) | lx) == 0)
return -two54 / zero; /* log(+-0)=-inf */
if (hx < 0)
return (x - x) / zero; /* log(-#) = NaN */
k -= 54;
x *= two54; /* subnormal number, scale up x */
GET_HIGH_WORD(hx, x);
}
if (hx >= 0x7ff00000)
return x + x;
k += (hx >> 20) - 1023;
hx &= 0x000fffff;
i = (hx + 0x95f64) & 0x100000;
SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
k += (i >> 20);
f = x - 1.0;
if ((0x000fffff & (2 + hx)) < 3) { /* |f| < 2**-20 */
if (f == zero) {
if (k == 0)
return zero;
else {
dk = (double) k;
return dk * ln2_hi + dk * ln2_lo;
}
}
R = f * f * (0.5 - 0.33333333333333333 * f);
if (k == 0)
return f - R;
else {
dk = (double) k;
return dk * ln2_hi - ((R - dk * ln2_lo) - f);
}
}
s = f / (2.0 + f);
dk = (double) k;
z = s * s;
i = hx - 0x6147a;
w = z * z;
j = 0x6b851 - 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;
if (k == 0)
return f - (hfsq - s * (hfsq + R));
else
return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -
f);
} else {
if (k == 0)
return f - s * (f - R);
else
return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
}