// 15 August 1997, Rick Huebner:  Small changes made to adapt for MathLib

/* @(#)e_exp.c 5.1 93/09/24 */
/*
 * ====================================================
 * 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.
 * ====================================================
 */


/* __ieee754_exp(x)
 * Returns the exponential of x.
 *
 * Method
 *   1. Argument reduction:
 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
 *	Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2.  
 *
 *      Here r will be represented as r = hi-lo for better 
 *	accuracy.
 *
 *   2. Approximation of exp(r) by a special rational function on
 *	the interval [0,0.34658]:
 *	Write
 *	    R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
 *      We use a special Reme algorithm on [0,0.34658] to generate 
 * 	a polynomial of degree 5 to approximate R. The maximum error 
 *	of this polynomial approximation is bounded by 2**-59. In
 *	other words,
 *	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
 *  	(where z=r*r, and the values of P1 to P5 are listed below)
 *	and
 *	    |                  5          |     -59
 *	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2 
 *	    |                             |
 *	The computation of exp(r) thus becomes
 *                             2*r
 *		exp(r) = 1 + -------
 *		              R - r
 *                                 r*R1(r)	
 *		       = 1 + r + ----------- (for better accuracy)
 *		                  2 - R1(r)
 *	where
 *			         2       4             10
 *		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
 *	
 *   3. Scale back to obtain exp(x):
 *	From step 1, we have
 *	   exp(x) = 2^k * exp(r)
 *
 * Special cases:
 *	exp(INF) is INF, exp(NaN) is NaN;
 *	exp(-INF) is 0, and
 *	for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *	according to an error analysis, the error is always less than
 *	1 ulp (unit in the last place).
 *
 * Misc. info.
 *	For IEEE double 
 *	    if x >  7.09782712893383973096e+02 then exp(x) overflow
 *	    if x < -7.45133219101941108420e+02 then exp(x) underflow
 *
 * 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 "..\Math.h"


#define ONE						 1.0
#define HUGE					 1.0e+300
#define TWOM1000			 9.33263618503218878990e-302	/* 2**-1000=0x01700000,0  */
#define O_THRESHOLD		 7.09782712893383973096e+02   /* 0x40862E42, 0xFEFA39EF */
#define U_THRESHOLD	  -7.45133219101941108420e+02   /* 0xc0874910, 0xD52D3051 */
#define INVLN2				 1.44269504088896338700e+00   /* 0x3ff71547, 0x652b82fe */
#define P1						 1.66666666666666019037e-01   /* 0x3FC55555, 0x5555553E */
#define P2						-2.77777777770155933842e-03   /* 0xBF66C16C, 0x16BEBD93 */
#define P3						 6.61375632143793436117e-05   /* 0x3F11566A, 0xAF25DE2C */
#define P4						-1.65339022054652515390e-06   /* 0xBEBBBD41, 0xC5D26BF1 */
#define P5						 4.13813679705723846039e-08   /* 0x3E663769, 0x72BEA4D0 */


tDouble math_exp(tDouble x)
/***********************************************************************
 *
 *	exp
 *
 ***********************************************************************/
{
tDouble halF[2], ln2HI[2], ln2LO[2];
tDouble y,hi,lo,c,t;
tULong hx;
tULong lx;
tLong hy;
tLong k,xsb;

	halF[0] = 0.5; halF[1] = -0.5;
	ln2HI[0] =  6.93147180369123816490e-01;  /* 0x3fe62e42, 0xfee00000 */
	ln2HI[1] = -6.93147180369123816490e-01;  /* 0xbfe62e42, 0xfee00000 */
	ln2LO[0] =  1.90821492927058770002e-10;  /* 0x3dea39ef, 0x35793c76 */
	ln2LO[1] = -1.90821492927058770002e-10;  /* 0xbdea39ef, 0x35793c76 */
	
	GET_HIGH_WORD(hx,x);
	xsb = (hx>>31)&1;		/* sign bit of x */
	hx &= 0x7fffffff;		/* high word of |x| */

  /* filter out non-finite argument */
	if (hx >= 0x40862E42) 			/* if |x|>=709.78... */
	{
  	if(hx>=0x7ff00000)
  	{
			GET_LOW_WORD(lx,x);
			
			if(((hx & 0xfffff) | lx)!=0) 
		     return x+x; 		/* NaN */
			else 
				return (xsb==0) ? x : 0.0;	/* exp(+-inf)={inf,0} */
	  }
	  
	  if (x > O_THRESHOLD) return (HUGE * HUGE); /* overflow */
	  if (x < U_THRESHOLD) return (TWOM1000 * TWOM1000); /* underflow */
	}

  /* argument reduction */
	if (hx > 0x3fd62e42) 		/* if  |x| > 0.5 ln2 */ 
	{
	  if (hx < 0x3FF0A2B2) 	/* and |x| < 1.5 ln2 */
	  {
			hi = x - ln2HI[xsb]; 
			lo=ln2LO[xsb]; 
			k = 1 - xsb - xsb;
	  } 
	  else
	  {
			k  = INVLN2 * x + halF[xsb];
			t  = k;
			hi = x - t*ln2HI[0];	/* t*ln2HI is exact here */
			lo = t*ln2LO[0];
	  }
	  x  = hi - lo;
	} 
	else if (hx < 0x3e300000)  /* when |x|<2**-28 */
	{
	  if ((HUGE + x) > ONE) 
	  	return (ONE + x);/* trigger inexact */
	}
	else 
		k = 0;

  /* x is now in primary range */
	t  = x*x;
	c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
	
	if (k == 0) 	
		return (ONE - ((x*c)/(c-2.0)-x)); 
	else
		y = (ONE - ((lo-(x*c)/(2.0-c))-hi));
		
	if (k >= -1021)
	{
	  GET_HIGH_WORD(hy,y);
	  SET_HIGH_WORD(y,hy+(k<<20));	/* add k to y's exponent */
	  return (y);
	}
	else
	{
	  GET_HIGH_WORD(hy,y);
	  SET_HIGH_WORD(y,hy+((k+1000)<<20));	/* add k to y's exponent */
	  return (y * TWOM1000);
	}
}
