exp.c

Go to the documentation of this file.

/*-
 * Copyright (c) 2014-2018 Carsten Sonne Larsen <cs@innolan.net>
 * 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.
 *
 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``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 AUTHOR 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.
 *
 * Project homepage:
 * https://amath.innolan.net
 *
 * The original source code can be obtained from:
 * http://www.netlib.org/fdlibm/e_exp.c
 * 
 * =================================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * =================================================================
 */

#include "prim.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 */

/**
 * @brief   Returns the exponential of x
 * @details
 * <pre>
 * 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.
 * </pre>
 */
double exp(double x)
{
    double y, hi, lo, c, t;
    int32_t k, xsb;
    uint32_t hx, hy;

    lo = 0.0;
    hi = 0.0;

    GET_HIGH_WORD(hx, x); // high word of x
    xsb = (hx >> 31) & 1; // sign bit of x
    hx &= 0x7FFFFFFF;     // high word of |x|

    // filter out non-finite argument
    // if |x|>=709.78...
    if (hx >= 0x40862E42)
    {
        if (hx >= 0x7FF00000)
        {
            uint32_t lx;
            GET_LOW_WORD(lx, x);
            if (((hx & 0xFFFFF) | lx) != 0)
            {
                return NAN;
            }

            // exp(+-inf)={inf,0}
            return (xsb == 0) ? x : 0.0;
        }

        if (x > o_threshold)
        {
            // overflow
            return huge * huge;
        }

        if (x < u_threshold)
        {
            // underflow
            return twom1000 * twom1000;
        }
    }

    // argument reduction
    // |x| > 0.5 ln2
    if (hx > 0x3FD62E42)
    {
        // |x| < 1.5 ln2
        if (hx < 0x3FF0A2B2)
        {
            hi = x - ln2HI[xsb];
            lo = ln2LO[xsb];
            k = 1 - xsb - xsb;
        }
        else
        {
            k = (int32_t)(invln2 * x + halF[xsb]);
            t = k;
            hi = x - t * ln2HI[0]; // t*ln2HI is exact here
            lo = t * ln2LO[0];
        }
        x = hi - lo;
    }
    // when |x|<2**-28
    else if (hx < 0x3E300000)
    {
        if (huge + x > one)
        {
            return one + x; // trigger inexact
        }
        else
        {
            k = 0;
        }
    }
    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);
    }

    y = one - ((lo - (x * c) / (2.0 - c)) - hi);

    if (k >= -1021)
    {
        GET_HIGH_WORD(hy, y);
        SET_HIGH_WORD(y, hy + (k << 20)); // add k to y's exponent
        return y;
    }

    GET_HIGH_WORD(hy, y);
    SET_HIGH_WORD(y, hy + ((k + 1000) << 20)); // add k to y's exponent
    return y * twom1000;
}