b_exp__D.c

Go to the documentation of this file.

/*  $OpenBSD: b_exp__D.c,v 1.5 2009/10/27 23:59:29 deraadt Exp $    */
/*
 * Copyright (c) 1985, 1993
 *  The Regents of the University of California.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 * 3. Neither the name of the University nor the names of its contributors
 *    may be used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 */

/* EXP(X)
 * RETURN THE EXPONENTIAL OF X
 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
 * CODED IN C BY K.C. NG, 1/19/85;
 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
 *
 * Required system supported functions:
 *  scalb(x,n)
 *  copysign(x,y)
 *  finite(x)
 *
 * Method:
 *  1. Argument Reduction: given the input x, find r and integer k such
 *     that
 *                     x = k*ln2 + r,  |r| <= 0.5*ln2 .
 *     r will be represented as r := z+c for better accuracy.
 *
 *  2. Compute exp(r) by
 *
 *      exp(r) = 1 + r + r*R1/(2-R1),
 *     where
 *      R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
 *
 *  3. exp(x) = 2^k * exp(r) .
 *
 * Special cases:
 *  exp(INF) is INF, exp(NaN) is NaN;
 *  exp(-INF)=  0;
 *  for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *  exp(x) returns the exponential of x nearly rounded. In a test run
 *  with 1,156,000 random arguments on a VAX, the maximum observed
 *  error was 0.869 ulps (units in the last place).
 */

#include "math.h"
#include "math_private.h"

static const double p1 = 0x1.555555555553ep-3;
static const double p2 = -0x1.6c16c16bebd93p-9;
static const double p3 = 0x1.1566aaf25de2cp-14;
static const double p4 = -0x1.bbd41c5d26bf1p-20;
static const double p5 = 0x1.6376972bea4d0p-25;
static const double ln2hi = 0x1.62e42fee00000p-1;
static const double ln2lo = 0x1.a39ef35793c76p-33;
static const double lnhuge = 0x1.6602b15b7ecf2p9;
static const double lntiny = -0x1.77af8ebeae354p9;
static const double invln2 = 0x1.71547652b82fep0;

/* returns exp(r = x + c) for |c| < |x| with no overlap.  */

double
__exp__D(double x, double c)
{
    double z, hi, lo;
    int k;

    if (isnan(x))   /* x is NaN */
        return(x);
    if ( x <= lnhuge ) {
        if ( x >= lntiny ) {

            /* argument reduction : x --> x - k*ln2 */
            z = invln2*x;
            k = z + copysign(.5, x);

            /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */

            hi=(x-k*ln2hi);         /* Exact. */
            x= hi - (lo = k*ln2lo-c);
            /* return 2^k*[1+x+x*c/(2+c)]  */
            z=x*x;
            c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
            c = (x*c)/(2.0-c);

            return  scalb(1.+(hi-(lo - c)), k);
        }
        /* end of x > lntiny */

        else
             /* exp(-big#) underflows to zero */
             if(finite(x))  return(scalb(1.0,-5000));

             /* exp(-INF) is zero */
             else return(0.0);
    }
    /* end of x < lnhuge */

    else
    /* exp(INF) is INF, exp(+big#) overflows to INF */
        return( finite(x) ?  scalb(1.0,5000)  : x);
}