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``````
``````/* @(#)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 <sys/cdefs.h>
``````//__FBSDID("\$FreeBSD\$");
``````
``````/* __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 "math_private.h"
``````
``````static const double
``````one	= 1.0,
``````halF	= {0.5,-0.5,},
``````o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
``````u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
``````ln2HI   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
``````	     -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
``````ln2LO   ={ 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
``````huge	= 1.0e+300,
``````twom1000= 9.33263618503218878990e-302;     /* 2**-1000=0x01700000,0*/
``````
``````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 */
``````	}
``````
``````    /* 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;	/* t*ln2HI is exact here */
``````		lo = t*ln2LO;
``````	    }
``````	    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) {
``````	        double const_0x1p1023 = pow(2, 1023);
``````	        return y*2.0*const_0x1p1023;
``````	    }
``````	    return y*twopk;
``````	} else {
``````	    return y*twopk*twom1000;
``````	}
``````}
``````