e_exp.c

Go to the documentation of this file.

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

/* 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 "math.h"
#include "math_private.h"

static const double
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 */


double
exp(double x)   /* default IEEE double exp */
{
    double y,hi,lo,c,t;
    int32_t k,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;
    }
}