/* @(#)e_log.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.

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

 */

/* __ieee754_log(x)

 * Return the logrithm of x

 *

 * Method :                  

 *   1. Argument Reduction: find k and f such that 

 *			x = 2^k * (1+f), 

 *	   where  sqrt(2)/2 < 1+f < sqrt(2) .

 *

 *   2. Approximation of log(1+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) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s

 *  	(the values of Lg1 to Lg7 are listed in the program)

 *	and

 *	    |      2          14          |     -58.45

 *	    | Lg1*s +...+Lg7*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

 *		log(1+f) = f - s*(f - R)	(if f is not too large)

 *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)

 *	

 *	3. Finally,  log(x) = k*ln2 + log(1+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:

 *	log(x) is NaN with signal if x < 0 (including -INF) ; 

 *	log(+INF) is +INF; log(0) is -INF with signal;

 *	log(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.

 */

#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 */

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 */

static double zero   =  0.0;

#ifdef __STDC__

	double __ieee754_log(double x)

#else

	double __ieee754_log(x)

	double x;

#endif

{

	double hfsq,f,s,z,R,w,t1,t2,dk;

	int k,hx,i,j;

	unsigned lx;

	hx = __HI(x);		/* high word of x */

	lx = __LO(x);		/* low  word of x */

	k=0;

	if (hx < 0x00100000) {			/* x < 2**-1022  */

	    if (((hx&0x7fffffff)|lx)==0) 

		return -two54/zero;		/* log(+-0)=-inf */

	    if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */

	    k -= 54; x *= two54; /* subnormal number, scale up x */

	    hx = __HI(x);		/* high word of x */

	} 

	if (hx >= 0x7ff00000) return x+x;

	k += (hx>>20)-1023;

	hx &= 0x000fffff;

	i = (hx+0x95f64)&0x100000;

	__HI(x) = hx|(i^0x3ff00000);	/* normalize x or x/2 */

	k += (i>>20);

	f = x-1.0;

	if((0x000fffff&(2+hx))<3) {	/* |f| < 2**-20 */

	    if(f==zero) if(k==0) return zero;  else {dk=(double)k;

				 return dk*ln2_hi+dk*ln2_lo;}

	    R = f*f*(0.5-0.33333333333333333*f);

	    if(k==0) return f-R; else {dk=(double)k;

	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}

	}

 	s = f/(2.0+f); 

	dk = (double)k;

	z = s*s;

	i = hx-0x6147a;

	w = z*z;

	j = 0x6b851-hx;

	t1= w*(Lg2+w*(Lg4+w*Lg6)); 

	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 

	i |= j;

	R = t2+t1;

	if(i>0) {

	    hfsq=0.5*f*f;

	    if(k==0) return f-(hfsq-s*(hfsq+R)); else

		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);

	} else {

	    if(k==0) return f-s*(f-R); else

		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);

	}

}