/* @(#)e_exp.c 1.6 04/04/22 */

/*

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

 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.

 *

 * 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 Remes 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 "fdlibm.h"

#ifdef __STDC__

static const double

#else

static double

#endif

one	= 1.0,

halF[2]	= {0.5,-0.5,},

huge	= 1.0e+300,

twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/

o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */

u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */

ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */

	     -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */

ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */

	     -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */

invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */

P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */

P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */

P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */

P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */

P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */

#ifdef __STDC__

	double __ieee754_exp(double x)	/* default IEEE double exp */

#else

	double __ieee754_exp(x)	/* default IEEE double exp */

	double x;

#endif

{

	double y,hi,lo,c,t;

	int k,xsb;

	unsigned hx;

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

		if(((hx&0xfffff)|__LO(x))!=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  = (int)(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) {

	    __HI(y) += (k<<20);	/* add k to y's exponent */

	    return y;

	} else {

	    __HI(y) += ((k+1000)<<20);/* add k to y's exponent */

	    return y*twom1000;

	}

}