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/* @(#)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.

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

 */

#if defined(LIBM_SCCS) && !defined(lint)

static char rcsid[] = "$NetBSD: e_exp.c,v 1.8 1995/05/10 20:45:03 jtc Exp $";

#endif

/* __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"

#include "math_private.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;

        double hi = 0.0;

        double lo = 0.0;

        double c;

        double t;

        int32_t k=0;

        int32_t 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  = 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) {

            u_int32_t hy;

            GET_HIGH_WORD(hy,y);

            SET_HIGH_WORD(y,hy+(k<<20));        /* add k to y's exponent */

            return y;

        } else {

            u_int32_t hy;

            GET_HIGH_WORD(hy,y);

            SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */

            return y*twom1000;

        }

}