pow.c

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/*-
 * 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_pow.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.
 * =================================================================
 */

/**
 * @file  pow.c
 * @brief Expontation function
 */

#include "prim.h"

#ifdef __clang__
#pragma clang diagnostic ignored "-Wunused-variable"
#pragma clang diagnostic ignored "-Wstrict-aliasing"
#elif defined(__GNUC__) && __GNUC__ > 2
#pragma GCC diagnostic ignored "-Wunused-but-set-variable"
#pragma GCC diagnostic ignored "-Wstrict-aliasing"
#endif

static const double
    bp[] = {
        1.0, 1.5,
},
    dp_h[] = {
        0.0, 5.84962487220764160156e-01,
}, /* 0x3FE2B803, 0x40000000 */
    dp_l[] = {
        0.0, 1.35003920212974897128e-08,
}, /* 0x3E4CFDEB, 0x43CFD006 */
    zero = 0.0,
    one = 1.0, two = 2.0, two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
    huge = 1.0e300, tiny = 1.0e-300,
    /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
    L1 = 5.99999999999994648725e-01,      /* 0x3FE33333, 0x33333303 */
    L2 = 4.28571428578550184252e-01,      /* 0x3FDB6DB6, 0xDB6FABFF */
    L3 = 3.33333329818377432918e-01,      /* 0x3FD55555, 0x518F264D */
    L4 = 2.72728123808534006489e-01,      /* 0x3FD17460, 0xA91D4101 */
    L5 = 2.30660745775561754067e-01,      /* 0x3FCD864A, 0x93C9DB65 */
    L6 = 2.06975017800338417784e-01,      /* 0x3FCA7E28, 0x4A454EEF */
    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 */
    lg2 = 6.93147180559945286227e-01,     /* 0x3FE62E42, 0xFEFA39EF */
    lg2_h = 6.93147182464599609375e-01,   /* 0x3FE62E43, 0x00000000 */
    lg2_l = -1.90465429995776804525e-09,  /* 0xBE205C61, 0x0CA86C39 */
    ovt = 8.0085662595372944372e-0017,    /* -(1024-log2(ovfl+.5ulp)) */
    cp = 9.61796693925975554329e-01,      /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
    cp_h = 9.61796700954437255859e-01,    /* 0x3FEEC709, 0xE0000000 =(float)cp */
    cp_l = -7.02846165095275826516e-09,   /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
    ivln2 = 1.44269504088896338700e+00,   /* 0x3FF71547, 0x652B82FE =1/ln2 */
    ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
    ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/

/**
 * @brief   Expontation function
 * @details
 * <pre>
 * Method:  Let x =  2   * (1+f)
 *  1. Compute and return log2(x) in two pieces:
 *      log2(x) = w1 + w2,
 *     where w1 has 53-24 = 29 bit trailing zeros.
 *  2. Perform y*log2(x) = n+y' by simulating muti-precision
 *     arithmetic, where |y'|<=0.5.
 *  3. Return x**y = 2**n*exp(y'*log2)
 *
 * Special cases:
 *  1.  (anything) ** 0  is 1
 *  2.  (anything) ** 1  is itself
 *  3.  (anything) ** NAN is NAN
 *  4.  NAN ** (anything except 0) is NAN
 *  5.  +-(|x| > 1) **  +INF is +INF
 *  6.  +-(|x| > 1) **  -INF is +0
 *  7.  +-(|x| < 1) **  +INF is +0
 *  8.  +-(|x| < 1) **  -INF is +INF
 *  9.  +-1         ** +-INF is NAN
 *  10. +0 ** (+anything except 0, NAN)               is +0
 *  11. -0 ** (+anything except 0, NAN, odd integer)  is +0
 *  12. +0 ** (-anything except 0, NAN)               is +INF
 *  13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
 *  14. -0 ** (odd integer) = -( +0 ** (odd integer) )
 *  15. +INF ** (+anything except 0,NAN) is +INF
 *  16. +INF ** (-anything except 0,NAN) is +0
 *  17. -INF ** (anything)  = -0 ** (-anything)
 *  18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
 *  19. (-anything except 0 and inf) ** (non-integer) is NAN
 *
 * Accuracy:
 *  pow(x,y) returns x**y nearly rounded. In particular
 *          pow(integer,integer)
 *  always returns the correct integer provided it is
 *  representable.
 *
 * 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 pow(double x, double y)
{
    double z, ax, z_h, z_l, p_h, p_l;
    double y1, t1, t2, r, s, t, u, v, w;
    int32_t i, j, k, yisint, n;
    int32_t hx, hy, ix, iy;
    uint32_t lx, ly;

    EXTRACT_WORDS(hx, lx, x);
    EXTRACT_WORDS(hy, ly, y);
    ix = hx & 0x7fffffff;
    iy = hy & 0x7fffffff;

    /* y==zero: x**0 = 1 */
    if ((iy | ly) == 0)
        return one;

    /* +-NaN return x+y */
    if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
        iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
        return x + y;

    /* determine if y is an odd int when x < 0
     * yisint = 0   ... y is not an integer
     * yisint = 1   ... y is an odd int
     * yisint = 2   ... y is an even int
     */
    yisint = 0;
    if (hx < 0)
    {
        if (iy >= 0x43400000)
            yisint = 2; /* even integer y */
        else if (iy >= 0x3ff00000)
        {
            k = (iy >> 20) - 0x3ff; /* exponent */
            if (k > 20)
            {
                j = ly >> (52 - k);
                if ((uint32_t)(j << (52 - k)) == ly)
                    yisint = 2 - (j & 1);
            }
            else if (ly == 0)
            {
                j = iy >> (20 - k);
                if ((j << (20 - k)) == iy)
                    yisint = 2 - (j & 1);
            }
        }
    }

    /* special value of y */
    if (ly == 0)
    {
        if (iy == 0x7ff00000)
        { /* y is +-inf */
            if (((ix - 0x3ff00000) | lx) == 0)
                return y - y;          /* inf**+-1 is NaN */
            else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
                return (hy >= 0) ? y : zero;
            else /* (|x|<1)**-,+inf = inf,0 */
                return (hy < 0) ? -y : zero;
        }
        if (iy == 0x3ff00000)
        { /* y is  +-1 */
            if (hy < 0)
                return one / x;
            else
                return x;
        }
        if (hy == 0x40000000)
            return x * x; /* y is  2 */
        if (hy == 0x3fe00000)
        {                /* y is  0.5 */
            if (hx >= 0) /* x >= +0 */
                return sqrt(x);
        }
    }

    ax = fabs(x);
    /* special value of x */
    if (lx == 0)
    {
        if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000)
        {
            z = ax; /*x is +-0,+-inf,+-1*/
            if (hy < 0)
                z = one / z; /* z = (1/|x|) */
            if (hx < 0)
            {
                if (((ix - 0x3ff00000) | yisint) == 0)
                {
                    z = (z - z) / (z - z); /* (-1)**non-int is NaN */
                }
                else if (yisint == 1)
                    z = -z; /* (x<0)**odd = -(|x|**odd) */
            }
            return z;
        }
    }

    n = (hx >> 31) + 1;

    /* (x<0)**(non-int) is NaN */
    if ((n | yisint) == 0)
        return (x - x) / (x - x);

    s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
    if ((n | (yisint - 1)) == 0)
        s = -one; /* (-ve)**(odd int) */

    /* |y| is huge */
    if (iy > 0x41e00000)
    { /* if |y| > 2**31 */
        if (iy > 0x43f00000)
        { /* if |y| > 2**64, must o/uflow */
            if (ix <= 0x3fefffff)
                return (hy < 0) ? huge * huge : tiny * tiny;
            if (ix >= 0x3ff00000)
                return (hy > 0) ? huge * huge : tiny * tiny;
        }
        /* over/underflow if x is not close to one */
        if (ix < 0x3fefffff)
            return (hy < 0) ? s * huge * huge : s * tiny * tiny;
        if (ix > 0x3ff00000)
            return (hy > 0) ? s * huge * huge : s * tiny * tiny;
        /* now |1-x| is tiny <= 2**-20, suffice to compute
           log(x) by x-x^2/2+x^3/3-x^4/4 */
        t = ax - one; /* t has 20 trailing zeros */
        w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
        u = ivln2_h * t; /* ivln2_h has 21 sig. bits */
        v = t * ivln2_l - w * ivln2;
        t1 = u + v;
        SET_LOW_WORD(t1, 0);
        t2 = v - (t1 - u);
    }
    else
    {
        double ss, s2, s_h, s_l, t_h, t_l;
        n = 0;
        /* take care subnormal number */
        if (ix < 0x00100000)
        {
            ax *= two53;
            n -= 53;
            GET_HIGH_WORD(ix, ax);
        }
        n += ((ix) >> 20) - 0x3ff;
        j = ix & 0x000fffff;
        /* determine interval */
        ix = j | 0x3ff00000; /* normalize ix */
        if (j <= 0x3988E)
            k = 0; /* |x|<sqrt(3/2) */
        else if (j < 0xBB67A)
            k = 1; /* |x|<sqrt(3)   */
        else
        {
            k = 0;
            n += 1;
            ix -= 0x00100000;
        }
        SET_HIGH_WORD(ax, ix);

        /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
        u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
        v = one / (ax + bp[k]);
        ss = u * v;
        s_h = ss;
        SET_LOW_WORD(s_h, 0);
        /* t_h=ax+bp[k] High */
        t_h = zero;
        SET_HIGH_WORD(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
        t_l = ax - (t_h - bp[k]);
        s_l = v * ((u - s_h * t_h) - s_h * t_l);
        /* compute log(ax) */
        s2 = ss * ss;
        r = s2 * s2 * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
        r += s_l * (s_h + ss);
        s2 = s_h * s_h;
        t_h = 3.0 + s2 + r;
        SET_LOW_WORD(t_h, 0);
        t_l = r - ((t_h - 3.0) - s2);
        /* u+v = ss*(1+...) */
        u = s_h * t_h;
        v = s_l * t_h + t_l * ss;
        /* 2/(3log2)*(ss+...) */
        p_h = u + v;
        SET_LOW_WORD(p_h, 0);
        p_l = v - (p_h - u);
        z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
        z_l = cp_l * p_h + p_l * cp + dp_l[k];
        /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
        t = (double)n;
        t1 = (((z_h + z_l) + dp_h[k]) + t);
        SET_LOW_WORD(t1, 0);
        t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
    }

    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
    y1 = y;
    SET_LOW_WORD(y1, 0);
    p_l = (y - y1) * t1 + y * t2;
    p_h = y1 * t1;
    z = p_l + p_h;
    EXTRACT_WORDS(j, i, z);
    if (j >= 0x40900000)
    {                                    /* z >= 1024 */
        if (((j - 0x40900000) | i) != 0) /* if z > 1024 */
            return s * huge * huge;      /* overflow */
        else
        {
            if (p_l + ovt > z - p_h)
                return s * huge * huge; /* overflow */
        }
    }
    else if ((j & 0x7fffffff) >= 0x4090cc00)
    {                                    /* z <= -1075 */
        if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */
            return s * tiny * tiny;      /* underflow */
        else
        {
            if (p_l <= z - p_h)
                return s * tiny * tiny; /* underflow */
        }
    }
    /*
     * compute 2**(p_h+p_l)
     */
    i = j & 0x7fffffff;
    k = (i >> 20) - 0x3ff;
    n = 0;
    if (i > 0x3fe00000)
    { /* if |z| > 0.5, set n = [z+0.5] */
        n = j + (0x00100000 >> (k + 1));
        k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */
        t = zero;
        SET_HIGH_WORD(t, (n & ~(0x000fffff >> k)));
        n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
        if (j < 0)
            n = -n;
        p_h -= t;
    }
    t = p_l + p_h;
    SET_LOW_WORD(t, 0);
    u = t * lg2_h;
    v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
    z = u + v;
    w = v - (z - u);
    t = z * z;
    t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
    r = (z * t1) / (t1 - two) - (w + z * w);
    z = one - (r - z);
    GET_HIGH_WORD(j, z);
    j += (n << 20);
    if ((j >> 20) <= 0)
        z = scalbn(z, n); /* subnormal output */
    else
    {
        uint32_t hz;
        GET_HIGH_WORD(hz, z);
        SET_HIGH_WORD(z, hz + (n << 20));
    }
    return s * z;
}