relibc_openlibm/src/e_exp.c

/* @(#)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.
 * ====================================================
 */

#include "cdefs-compat.h"
//__FBSDID("$FreeBSD: src/lib/msun/src/e_exp.c,v 1.14 2011/10/21 06:26:38 das Exp $");

/* __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 <float.h>
#include <openlibm.h>

#include "math_private.h"

static const double
one	= 1.0,
halF[2]	= {0.5,-0.5,},
huge	= 1.0e+300,
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 */

static volatile double
twom1000= 9.33263618503218878990e-302;     /* 2**-1000=0x01700000,0*/

DLLEXPORT double
__ieee754_exp(double x)	/* default IEEE double exp */
{
	double y,hi=0.0,lo=0.0,c,t,twopk;
	int32_t k=0,xsb;
	u_int32_t hx;

	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) {
	        u_int32_t lx;
		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 */
	}

        /* this implementation gives 2.7182818284590455 for exp(1.0),
           which is well within the allowable error. however,
           2.718281828459045 is closer to the true value so we prefer that
           answer, given that 1.0 is such an important argument value. */
        if (x == 1.0)
            return 2.718281828459045235360;

    /* 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];
	    }
	    STRICT_ASSIGN(double, 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;
	if(k >= -1021)
	    INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0);
	else
	    INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0);
	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) {
	    if (k==1024) return y*2.0*0x1p1023;
	    return y*twopk;
	} else {
	    return y*twopk*twom1000;
	}
}