/* @(#)s_log1p.c 1.3 95/01/18 */

/*

 * ====================================================

 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.

 *

 * 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"

#ifdef __STDC__

static const double

#else

static double

#endif

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;

#ifdef __STDC__

	double log1p(double x)

#else

	double log1p(x)

	double x;

#endif

{

	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);

}