constexpr_math.inl

 
/*====================================================
* Copyright(C) 2004 by Sun Microsystems, Inc.All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
 
namespace detail
{
    inline constexpr auto nan_mix(auto x, auto y) noexcept { return ((x + 0.0L) + (y + 0)); }
 
    template <typename A, typename B>
    constexpr void extract_words(A& high, B& low, double d) noexcept
    {
        low = std::bit_cast<uint64_t>(d) & 0xFFFFFFFF;
        high = (std::bit_cast<uint64_t>(d) >> 32LL) & 0xFFFFFFFF;
    }
 
    template <typename A>
    constexpr void get_high_word(A& high, double d) noexcept
    {
        high = (std::bit_cast<uint64_t>(d) >> 32LL) & 0xFFFFFFFF;
    }
 
    template <typename A>
    constexpr void get_low_word(A& low, double d) noexcept
    {
        low = std::bit_cast<uint64_t>(d) & 0xFFFFFFFF;
    }
 
    template <typename A, typename B>
    constexpr void insert_words(double& d, A high, B low) noexcept
    {
        const auto bits = static_cast<uint64_t>(std::bit_cast<uint32_t>(low)) | (static_cast<uint64_t>(std::bit_cast<uint32_t>(high)) << 32ULL);
        d = std::bit_cast<double>(bits);
    }
 
    template <typename A>
    constexpr void set_high_word(double& d, A i) noexcept
    {
        uint32_t low = 0, high = 0;
        ::ghassanpl::cem::detail::extract_words(high, low, d);
        ::ghassanpl::cem::detail::insert_words(d, i, low);
    }
 
    template <typename A>
    constexpr void set_low_word(double& d, A i) noexcept
    {
        uint32_t low = 0, high = 0;
        ::ghassanpl::cem::detail::extract_words(high, low, d);
        ::ghassanpl::cem::detail::insert_words(d, high, i);
    }
 
    static constexpr double bp[] = { 1.0, 1.5 };
    static constexpr double dp_h[] = { 0.0, 5.84962487220764160156e-01 }; 
    static constexpr double dp_l[] = { 0.0, 1.35003920212974897128e-08 }; 
    static constexpr double zero = 0.0;
    static constexpr double half = 0.5;
    static constexpr double qrtr = 0.25;
    static constexpr double thrd = 3.3333333333333331e-01; 
    static constexpr double one = 1.0;
    static constexpr double two = 2.0;
    static constexpr double two53 = 9007199254740992.0; 
    static constexpr double huge = 1.0e300;
    static constexpr double tiny = 1.0e-300;
    static constexpr double L1 = 5.99999999999994648725e-01;      
    static constexpr double L2 = 4.28571428578550184252e-01;      
    static constexpr double L3 = 3.33333329818377432918e-01;      
    static constexpr double L4 = 2.72728123808534006489e-01;      
    static constexpr double L5 = 2.30660745775561754067e-01;      
    static constexpr double L6 = 2.06975017800338417784e-01;      
    static constexpr double P1 = 1.66666666666666019037e-01;      
    static constexpr double P2 = -2.77777777770155933842e-03;     
    static constexpr double P3 = 6.61375632143793436117e-05;      
    static constexpr double P4 = -1.65339022054652515390e-06;     
    static constexpr double P5 = 4.13813679705723846039e-08;      
    static constexpr double lg2 = 6.93147180559945286227e-01;     
    static constexpr double lg2_h = 6.93147182464599609375e-01;   
    static constexpr double lg2_l = -1.90465429995776804525e-09;  
    static constexpr double ovt = 8.0085662595372944372e-0017;    
    static constexpr double cp = 9.61796693925975554329e-01;      
    static constexpr double cp_h = 9.61796700954437255859e-01;    
    static constexpr double cp_l = -7.02846165095275826516e-09;   
    static constexpr double ivln2 = 1.44269504088896338700e+00;   
    static constexpr double ivln2_h = 1.44269502162933349609e+00; 
    static constexpr double ivln2_l = 1.92596299112661746887e-08; 
 
    constexpr double sqrt_impl(double x) noexcept
    {
        double z;
        int32_t sign = (int)0x80000000;
        int32_t ix0, s0, q, m, t, i;
        uint32_t r, t1, s1, ix1, q1;
 
        extract_words(ix0, ix1, x);
 
        /* take care of Inf and NaN */
        if ((ix0 & 0x7ff00000) == 0x7ff00000)
        {
            return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
                         sqrt(-inf)=sNaN */
        }
        /* take care of zero */
        if (ix0 <= 0)
        {
            if (((ix0 & (~sign)) | ix1) == 0)
                return x; /* sqrt(+-0) = +-0 */
            else if (ix0 < 0)
                return std::numeric_limits<double>::quiet_NaN(); /* sqrt(-ve) = sNaN */
        }
        /* normalize x */
        m = (ix0 >> 20);
        if (m == 0)
        { /* subnormal x */
            while (ix0 == 0)
            {
                m -= 21;
                ix0 |= (ix1 >> 11);
                ix1 <<= 21;
            }
            for (i = 0; (ix0 & 0x00100000) == 0; i++)
                ix0 <<= 1;
            m -= i - 1;
            ix0 |= (ix1 >> (32 - i));
            ix1 <<= i;
        }
        m -= 1023; /* unbias exponent */
        ix0 = (ix0 & 0x000fffff) | 0x00100000;
        if (m & 1)
        { /* odd m, double x to make it even */
            ix0 += ix0 + ((ix1 & sign) >> 31);
            ix1 += ix1;
        }
        m >>= 1; /* m = [m/2] */
 
        /* generate sqrt(x) bit by bit */
        ix0 += ix0 + ((ix1 & sign) >> 31);
        ix1 += ix1;
        q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
        r = 0x00200000;       /* r = moving bit from right to left */
 
        while (r != 0)
        {
            t = s0 + r;
            if (t <= ix0)
            {
                s0 = t + r;
                ix0 -= t;
                q += r;
            }
            ix0 += ix0 + ((ix1 & sign) >> 31);
            ix1 += ix1;
            r >>= 1;
        }
 
        r = sign;
        while (r != 0)
        {
            t1 = s1 + r;
            t = s0;
            if ((t < ix0) || ((t == ix0) && (t1 <= ix1)))
            {
                s1 = t1 + r;
                if (((t1 & sign) == sign) && (s1 & sign) == 0)
                    s0 += 1;
                ix0 -= t;
                if (ix1 < t1)
                    ix0 -= 1;
                ix1 -= t1;
                q1 += r;
            }
            ix0 += ix0 + ((ix1 & sign) >> 31);
            ix1 += ix1;
            r >>= 1;
        }
 
        /* use floating add to find out rounding direction */
        if ((ix0 | ix1) != 0)
        {
            z = one - tiny; /* trigger inexact flag */
            if (z >= one)
            {
                z = one + tiny;
                if (q1 == (uint32_t)0xffffffff)
                {
                    q1 = 0;
                    q += 1;
                }
                else if (z > one)
                {
                    if (q1 == (uint32_t)0xfffffffe)
                        q += 1;
                    q1 += 2;
                }
                else
                    q1 += (q1 & 1);
            }
        }
        ix0 = (q >> 1) + 0x3fe00000;
        ix1 = q1 >> 1;
        if ((q & 1) == 1)
            ix1 |= sign;
        ix0 += (m << 20);
        insert_words(z, ix0, ix1);
        return z;
    }
 
    constexpr double scalbn(double x, int n) noexcept
    {
        double y = x;
 
        if (n > 1023)
        {
            y *= 0x1p1023;
            n -= 1023;
            if (n > 1023)
            {
                y *= 0x1p1023;
                n -= 1023;
                if (n > 1023)
                    n = 1023;
            }
        }
        else if (n < -1022)
        {
            /* make sure final n < -53 to avoid double
            rounding in the subnormal range */
            y *= 0x1p-1022 * 0x1p53;
            n += 1022 - 53;
            if (n < -1022)
            {
                y *= 0x1p-1022 * 0x1p53;
                n += 1022 - 53;
                if (n < -1022)
                    n = -1022;
            }
        }
        x = y * std::bit_cast<double>((uint64_t)(0x3ff + n) << 52);
        return x;
    }
 
    constexpr double pow_impl(double x, double y) noexcept
    {
        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;
 
        /* x==1: 1**y = 1, even if y is NaN */
        if (hx == 0x3ff00000 && lx == 0)
            return one;
 
        /* y!=zero: result is NaN if either arg is NaN */
        if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
            iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
            return nan_mix(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 one;            /* (-1)**+-inf is 1 */
                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_impl(x);
            }
        }
 
        ax = cem::abs(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;
            }
        }
 
        /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
        n = (hx>>31)+1;
           but ANSI C says a right shift of a signed negative quantity is
           implementation defined.  */
        n = ((uint32_t)hx >> 31) - 1;
 
        /* (x<0)**(non-int) is NaN */
        if ((n | yisint) == 0)
            return std::numeric_limits<double>::quiet_NaN();
 
        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) * (half - t * (thrd - t * qrtr));
            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 + s2 + r;
            set_low_word(t_h, 0);
            t_l = r - ((t_h - 3) - 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 = 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
            set_high_word(z, j);
        return s * z;
    }
}