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/* @(#)s_log1p.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.
* ====================================================
*/
/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
for performance improvement on pipelined processors.
*/
/* double log1p(double x)
*
* Method :
* 1. Argument Reduction: find k and f such that
* 1+x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* Note. If k=0, then f=x is exact. However, if k!=0, then f
* may not be representable exactly. In that case, a correction
* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
* and add back the correction term c/u.
* (Note: when x > 2**53, one can simply return log(x))
*
* 2. Approximation of log1p(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) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
* (the values of Lp1 to Lp7 are listed in the program)
* and
* | 2 14 | -58.45
* | Lp1*s +...+Lp7*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
* log1p(f) = f - (hfsq - s*(hfsq+R)).
*
* 3. Finally, log1p(x) = k*ln2 + log1p(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:
* log1p(x) is NaN with signal if x < -1 (including -INF) ;
* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
* log1p(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.
*
* Note: Assuming log() return accurate answer, the following
* algorithm can be used to compute log1p(x) to within a few ULP:
*
* u = 1+x;
* if(u==1.0) return x ; else
* return log(u)*(x/(u-1.0));
*
* See HP-15C Advanced Functions Handbook, p.193.
*/
#include <math.h>
#include <math_private.h>
static const double
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
Lp[] = { 0.0, 6.666666666666735130e-01, /* 3FE55555 55555593 */
3.999999999940941908e-01, /* 3FD99999 9997FA04 */
2.857142874366239149e-01, /* 3FD24924 94229359 */
2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
1.818357216161805012e-01, /* 3FC74664 96CB03DE */
1.531383769920937332e-01, /* 3FC39A09 D078C69F */
1.479819860511658591e-01 }; /* 3FC2F112 DF3E5244 */
static const double zero = 0.0;
double
__log1p (double x)
{
double hfsq, f, c, s, z, R, u, z2, z4, z6, R1, R2, R3, R4;
int32_t k, hx, hu, ax;
GET_HIGH_WORD (hx, x);
ax = hx & 0x7fffffff;
k = 1;
if (hx < 0x3FDA827A) /* x < 0.41422 */
{
if (__builtin_expect (ax >= 0x3ff00000, 0)) /* x <= -1.0 */
{
if (x == -1.0)
return -two54 / (x - x); /* log1p(-1)=+inf */
else
return (x - x) / (x - x); /* log1p(x<-1)=NaN */
}
if (__builtin_expect (ax < 0x3e200000, 0)) /* |x| < 2**-29 */
{
math_force_eval (two54 + x); /* raise inexact */
if (ax < 0x3c900000) /* |x| < 2**-54 */
return x;
else
return x - x * x * 0.5;
}
if (hx > 0 || hx <= ((int32_t) 0xbfd2bec3))
{
k = 0; f = x; hu = 1;
} /* -0.2929<x<0.41422 */
}
else if (__builtin_expect (hx >= 0x7ff00000, 0))
return x + x;
if (k != 0)
{
if (hx < 0x43400000)
{
u = 1.0 + x;
GET_HIGH_WORD (hu, u);
k = (hu >> 20) - 1023;
c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */
c /= u;
}
else
{
u = x;
GET_HIGH_WORD (hu, u);
k = (hu >> 20) - 1023;
c = 0;
}
hu &= 0x000fffff;
if (hu < 0x6a09e)
{
SET_HIGH_WORD (u, hu | 0x3ff00000); /* normalize u */
}
else
{
k += 1;
SET_HIGH_WORD (u, hu | 0x3fe00000); /* normalize u/2 */
hu = (0x00100000 - hu) >> 2;
}
f = u - 1.0;
}
hfsq = 0.5 * f * f;
if (hu == 0) /* |f| < 2**-20 */
{
if (f == zero)
{
if (k == 0)
return zero;
else
{
c += k * ln2_lo; return k * ln2_hi + c;
}
}
R = hfsq * (1.0 - 0.66666666666666666 * f);
if (k == 0)
return f - R;
else
return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
}
s = f / (2.0 + f);
z = s * s;
R1 = z * Lp[1]; z2 = z * z;
R2 = Lp[2] + z * Lp[3]; z4 = z2 * z2;
R3 = Lp[4] + z * Lp[5]; z6 = z4 * z2;
R4 = Lp[6] + z * Lp[7];
R = R1 + z2 * R2 + z4 * R3 + z6 * R4;
if (k == 0)
return f - (hfsq - s * (hfsq + R));
else
return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
}
weak_alias (__log1p, log1p)
#ifdef NO_LONG_DOUBLE
strong_alias (__log1p, __log1pl)
weak_alias (__log1p, log1pl)
#endif
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