expm1.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/s_expm1.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"

/* expm1(x)
 * Returns exp(x)-1, the exponential of x minus 1.
 *
 * Method
 *   1. Argument reduction:
 *  Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
 *
 *      Here a correction term c will be computed to compensate
 *  the error in r when rounded to a floating-point number.
 *
 *   2. Approximating expm1(r) by a special rational function on
 *  the interval [0,0.34658]:
 *  Since
 *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
 *  we define R1(r*r) by
 *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
 *  That is,
 *      R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
 *           = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
 *           = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
 *      We use a special Remes algorithm on [0,0.347] to generate
 *  a polynomial of degree 5 in r*r to approximate R1. The
 *  maximum error of this polynomial approximation is bounded
 *  by 2**-61. In other words,
 *      R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
 *  where   Q1  =  -1.6666666666666567384E-2,
 *      Q2  =   3.9682539681370365873E-4,
 *      Q3  =  -9.9206344733435987357E-6,
 *      Q4  =   2.5051361420808517002E-7,
 *      Q5  =  -6.2843505682382617102E-9;
 *      (where z=r*r, and the values of Q1 to Q5 are listed below)
 *  with error bounded by
 *      |                  5           |     -61
 *      | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
 *      |                              |
 *
 *  expm1(r) = exp(r)-1 is then computed by the following
 *  specific way which minimize the accumulation rounding error:
 *                 2     3
 *                r     r    [ 3 - (R1 + R1*r/2)  ]
 *        expm1(r) = r + --- + --- * [--------------------]
 *                    2     2    [ 6 - r*(3 - R1*r/2) ]
 *
 *  To compensate the error in the argument reduction, we use
 *      expm1(r+c) = expm1(r) + c + expm1(r)*c
 *             ~ expm1(r) + c + r*c
 *  Thus c+r*c will be added in as the correction terms for
 *  expm1(r+c). Now rearrange the term to avoid optimization
 *  screw up:
 *              (      2                                    2 )
 *              ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
 *   expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
 *                  ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
 *                      (                                             )
 *
 *         = r - E
 *   3. Scale back to obtain expm1(x):
 *  From step 1, we have
 *     expm1(x) = either 2^k*[expm1(r)+1] - 1
 *          = or     2^k*[expm1(r) + (1-2^-k)]
 *   4. Implementation notes:
 *  (A). To save one multiplication, we scale the coefficient Qi
 *       to Qi*2^i, and replace z by (x^2)/2.
 *  (B). To achieve maximum accuracy, we compute expm1(x) by
 *    (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
 *    (ii)  if k=0, return r-E
 *    (iii) if k=-1, return 0.5*(r-E)-0.5
 *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
 *                 else      return  1.0+2.0*(r-E);
 *    (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
 *    (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
 *    (vii) return 2^k(1-((E+2^-k)-r))
 *
 * Special cases:
 *  expm1(INF) is INF, expm1(NaN) is NaN;
 *  expm1(-INF) is -1, and
 *  for finite argument, only expm1(0)=0 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 expm1(x) overflow
 *
 * 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.
 */

static const double
    one = 1.0,
    huge = 1.0e+300,
    tiny = 1.0e-300,
    o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
    ln2_hi = 6.93147180369123816490e-01,      /* 0x3fe62e42, 0xfee00000 */
    ln2_lo = 1.90821492927058770002e-10,      /* 0x3dea39ef, 0x35793c76 */
    invln2 = 1.44269504088896338700e+00,      /* 0x3ff71547, 0x652b82fe */
                                              /* scaled coefficients related to expm1 */
    Q1 = -3.33333333333331316428e-02,         /* BFA11111 111110F4 */
    Q2 = 1.58730158725481460165e-03,          /* 3F5A01A0 19FE5585 */
    Q3 = -7.93650757867487942473e-05,         /* BF14CE19 9EAADBB7 */
    Q4 = 4.00821782732936239552e-06,          /* 3ED0CFCA 86E65239 */
    Q5 = -2.01099218183624371326e-07;         /* BE8AFDB7 6E09C32D */

double expm1(double x)
{
    double y, hi, lo, c, t, e, hxs, hfx, r1;
    int32_t k, xsb;
    uint32_t hx;

    c = 0.0;

    GET_HIGH_WORD(hx, x);  /* high word of x */
    xsb = hx & 0x80000000; /* sign bit of x */
    hx &= 0x7fffffff; /* high word of |x| */

    /* filter out huge and non-finite argument */
    if (hx >= 0x4043687A)
    { /* if |x|>=56*ln2 */
        if (hx >= 0x40862E42)
        { /* if |x|>=709.78... */
            if (hx >= 0x7ff00000)
            {
                uint32_t low;
                GET_LOW_WORD(low, x);
                if (((hx & 0xfffff) | low) != 0)
                    return x + x; /* NaN */
                else
                    return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */
            }
            if (x > o_threshold)
                return huge * huge; /* overflow */
        }
        if (xsb != 0)
        {                          /* x < -56*ln2, return -1.0 with inexact */
            if (x + tiny < 0.0)    /* raise inexact */
                return tiny - one; /* return -1 */
        }
    }

    /* argument reduction */
    if (hx > 0x3fd62e42)
    { /* if  |x| > 0.5 ln2 */
        if (hx < 0x3FF0A2B2)
        { /* and |x| < 1.5 ln2 */
            if (xsb == 0)
            {
                hi = x - ln2_hi;
                lo = ln2_lo;
                k = 1;
            }
            else
            {
                hi = x + ln2_hi;
                lo = -ln2_lo;
                k = -1;
            }
        }
        else
        {
            k = (int32_t)(invln2 * x + ((xsb == 0) ? 0.5 : -0.5));
            t = k;
            hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
            lo = t * ln2_lo;
        }
        x = hi - lo;
        c = (hi - x) - lo;
    }
    else if (hx < 0x3c900000)
    {                 /* when |x|<2**-54, return x */
        t = huge + x; /* return x with inexact flags when x!=0 */
        return x - (t - (huge + x));
    }
    else
        k = 0;

    /* x is now in primary range */
    hfx = 0.5 * x;
    hxs = x * hfx;
    r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
    t = 3.0 - r1 * hfx;
    e = hxs * ((r1 - t) / (6.0 - x * t));
    if (k == 0)
        return x - (x * e - hxs); /* c is 0 */
    else
    {
        e = (x * (e - c) - c);
        e -= hxs;
        if (k == -1)
            return 0.5 * (x - e) - 0.5;
        if (k == 1)
        {
            if (x < -0.25)
                return -2.0 * (e - (x + 0.5));
            else
                return one + 2.0 * (x - e);
        }
        if (k <= -2 || k > 56)
        { /* suffice to return exp(x)-1 */
            uint32_t hy;

            y = one - (e - x);
            GET_HIGH_WORD(hy, y);
            SET_HIGH_WORD(y, hy + (k << 20)); /* add k to y's exponent */
            return y - one;
        }
        t = one;
        if (k < 20)
        {
            uint32_t hy;

            SET_HIGH_WORD(t, 0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */
            y = t - (e - x);
            GET_HIGH_WORD(hy, y);
            SET_HIGH_WORD(y, hy + (k << 20)); /* add k to y's exponent */
        }
        else
        {
            uint32_t hy;

            SET_HIGH_WORD(t, (0x3ff - k) << 20); /* 2^-k */
            y = x - (e + t);
            y += one;
            GET_HIGH_WORD(hy, y);
            SET_HIGH_WORD(y, hy + (k << 20)); /* add k to y's exponent */
        }
    }
    return y;
}