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/* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */
/*
* ====================================================
* 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.
* ====================================================
*/
/* double log1p(double x)
* Return the natural logarithm of 1+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 log(1+f): See log.c
*
* 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
*
* 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 "libm.h"
static const double
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
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 log1p(double x)
{
union {double f; uint64_t i;} u = {x};
double_t hfsq,f,c,s,z,R,w,t1,t2,dk;
uint32_t hx,hu;
int k;
hx = u.i>>32;
k = 1;
if (hx < 0x3fda827a || hx>>31) { /* 1+x < sqrt(2)+ */
if (hx >= 0xbff00000) { /* x <= -1.0 */
if (x == -1)
return x/0.0; /* log1p(-1) = -inf */
return (x-x)/0.0; /* log1p(x<-1) = NaN */
}
if (hx<<1 < 0x3ca00000<<1) { /* |x| < 2**-53 */
/* underflow if subnormal */
if ((hx&0x7ff00000) == 0)
FORCE_EVAL((float)x);
return x;
}
if (hx <= 0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
k = 0;
c = 0;
f = x;
}
} else if (hx >= 0x7ff00000)
return x;
if (k) {
u.f = 1 + x;
hu = u.i>>32;
hu += 0x3ff00000 - 0x3fe6a09e;
k = (int)(hu>>20) - 0x3ff;
/* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
if (k < 54) {
c = k >= 2 ? 1-(u.f-x) : x-(u.f-1);
c /= u.f;
} else
c = 0;
/* reduce u into [sqrt(2)/2, sqrt(2)] */
hu = (hu&0x000fffff) + 0x3fe6a09e;
u.i = (uint64_t)hu<<32 | (u.i&0xffffffff);
f = u.f - 1;
}
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;
dk = k;
return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi;
}
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