/* 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)
*
* 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 "libm.h"
static const double
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
static const double zero = 0.0;
double log1p(double x)
{
double hfsq,f,c,s,z,R,u;
int32_t k,hx,hu,ax;
GET_HIGH_WORD(hx, x);
ax = hx & 0x7fffffff;
k = 1;
if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
if (ax >= 0x3ff00000) { /* x <= -1.0 */
if (x == -1.0)
return -two54/zero; /* log1p(-1)=+inf */
return (x-x)/(x-x); /* log1p(x<-1)=NaN */
}
if (ax < 0x3e200000) { /* |x| < 2**-29 */
/* raise inexact */
if (two54 + x > zero && ax < 0x3c900000) /* |x| < 2**-54 */
return x;
return x - x*x*0.5;
}
if (hx > 0 || hx <= (int32_t)0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
k = 0;
f = x;
hu = 1;
}
}
if (hx >= 0x7ff00000)
return x+x;
if (k != 0) {
if (hx < 0x43400000) {
STRICT_ASSIGN(double, 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;
/*
* The approximation to sqrt(2) used in thresholds is not
* critical. However, the ones used above must give less
* strict bounds than the one here so that the k==0 case is
* never reached from here, since here we have committed to
* using the correction term but don't use it if k==0.
*/
if (hu < 0x6a09e) { /* u ~< sqrt(2) */
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;
c += k*ln2_lo;
return k*ln2_hi + c;
}
R = hfsq*(1.0 - 0.66666666666666666*f);
if (k == 0)
return f - R;
return k*ln2_hi - ((R-(k*ln2_lo+c))-f);
}
s = f/(2.0+f);
z = s*s;
R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
if (k == 0)
return f - (hfsq-s*(hfsq+R));
return k*ln2_hi - ((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
}
