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```/* derived from /netlib/fdlibm */

/* @(#)s_log1p.c 1.3 95/01/18 */
/*
* ====================================================
*
* Developed at SunSoft, 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 "fdlibm.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 double zero = 0.0;

double log1p(double x)
{
double hfsq,f,c,s,z,R,u;
int k,hx,hu,ax;

hx = __HI(x);		/* high word of x */
ax = hx&0x7fffffff;

k = 1;
if (hx < 0x3FDA827A) {			/* x < 0.41422  */
if(ax>=0x3ff00000) {		/* x <= -1.0 */
if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
else return (x-x)/(x-x);	/* log1p(x<-1)=NaN */
}
if(ax<0x3e200000) {			/* |x| < 2**-29 */
if(two54+x>zero			/* raise inexact */
&&ax<0x3c900000) 		/* |x| < 2**-54 */
return x;
else
return x - x*x*0.5;
}
if(hx>0||hx<=((int)0xbfd2bec3)) {
k=0;f=x;hu=1;}	/* -0.2929<x<0.41422 */
}
if (hx >= 0x7ff00000) return x+x;
if(k!=0) {
if(hx<0x43400000) {
u  = 1.0+x;
hu = __HI(u);		/* high word of u */
k  = (hu>>20)-1023;
c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
c /= u;
} else {
u  = x;
hu = __HI(u);		/* high word of u */
k  = (hu>>20)-1023;
c  = 0;
}
hu &= 0x000fffff;
if(hu<0x6a09e) {
__HI(u) = hu|0x3ff00000;	/* normalize u */
} else {
k += 1;
__HI(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;
R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
if(k==0) return f-(hfsq-s*(hfsq+R)); else
return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
}
```