Quelle  xsimd_rem_pio2.hpp   Sprache: C

/***************************************************************************
 * Copyright (c) Johan Mabille, Sylvain Corlay, Wolf Vollprecht and         *
 * Martin Renou                                                             *
 * Copyright (c) QuantStack                                                 *
 * Copyright (c) Serge Guelton                                              *
 *                                                                          *
 * Distributed under the terms of the BSD 3-Clause License.                 *
 *                                                                          *
 * The full license is in the file LICENSE, distributed with this software. *
 ****************************************************************************/

#include <cmath>
#include <cstdint>
#include <cstring>

namespace xsimd
{
    namespace detail
    {

        /* origin: boost/simd/arch/common/scalar/function/rem_pio2.hpp */
        /*
         * ====================================================
         * copyright 2016 NumScale SAS
         *
         * Distributed under the Boost Software License, Version 1.0.
         * (See copy at http://boost.org/LICENSE_1_0.txt)
         * ====================================================
         */
#if defined(_MSC_VER)
#define ONCE0                                       \
    __pragma(warning(push))                         \
        __pragma(warning(disable : 4127)) while (0) \
            __pragma(warning(pop)) /**/
#else
#define ONCE0 while (0)
#endif

        /*
         * ====================================================
         * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
         *
         * Developed at SunPro, a Sun Microsystems, Inc. business.
         * Permission to use, copy, modify, and distribute this
         * software is freely granted, provided that this notice
         * is preserved.
         * ====================================================
         */

#if defined(__GNUC__) && defined(__BYTE_ORDER__)
#if __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__
#define XSIMD_LITTLE_ENDIAN
#endif
#elif defined(_WIN32)
        // We can safely assume that Windows is always little endian
#define XSIMD_LITTLE_ENDIAN
#elif defined(i386) || defined(i486) || defined(intel) || defined(x86) || defined(i86pc) || defined(__alpha) || defined(__osf__)
#define XSIMD_LITTLE_ENDIAN
#endif

#ifdef XSIMD_LITTLE_ENDIAN
#define LOW_WORD_IDX 0
#define HIGH_WORD_IDX sizeof(std::uint32_t)
#else
#define LOW_WORD_IDX sizeof(std::uint32_t)
#define HIGH_WORD_IDX 0
#endif

#define GET_HIGH_WORD(i, d)                                            \
    do                                                                 \
    {                                                                  \
        double f = (d);                                                \
        std::memcpy(&(i), reinterpret_cast<char*>(&f) + HIGH_WORD_IDX, \
                    sizeof(std::uint32_t));                            \
    }                                                                  \
    ONCE0                                                              \
    /**/

#define GET_LOW_WORD(i, d)                                            \
    do                                                                \
    {                                                                 \
        double f = (d);                                               \
        std::memcpy(&(i), reinterpret_cast<char*>(&f) + LOW_WORD_IDX, \
                    sizeof(std::uint32_t));                           \
    }                                                                 \
    ONCE0                                                             \
    /**/

#define SET_HIGH_WORD(d, v)                                      \
    do                                                           \
    {                                                            \
        double f = (d);                                          \
        std::uint32_t value = (v);                               \
        std::memcpy(reinterpret_cast<char*>(&f) + HIGH_WORD_IDX, \
                    &value, sizeof(std::uint32_t));              \
        (d) = f;                                                 \
    }                                                            \
    ONCE0                                                        \
    /**/

#define SET_LOW_WORD(d, v)                                      \
    do                                                          \
    {                                                           \
        double f = (d);                                         \
        std::uint32_t value = (v);                              \
        std::memcpy(reinterpret_cast<char*>(&f) + LOW_WORD_IDX, \
                    &value, sizeof(std::uint32_t));             \
        (d) = f;                                                \
    }                                                           \
    ONCE0                                                       \
    /**/

        /*
         * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
         * double x[],y[]; int e0,nx,prec; int ipio2[];
         *
         * __kernel_rem_pio2 return the last three digits of N with
         * y = x - N*pi/2
         * so that |y| < pi/2.
         *
         * The method is to compute the integer (mod 8) and fraction parts of
         * (2/pi)*x without doing the full multiplication. In general we
         * skip the part of the product that are known to be a huge integer (
         * more accurately, = 0 mod 8 ). Thus the number of operations are
         * independent of the exponent of the input.
         *
         * (2/pi) is represented by an array of 24-bit integers in ipio2[].
         *
         * Input parameters:
         *  x[] The input value (must be positive) is broken into nx
         * pieces of 24-bit integers in double precision format.
         * x[i] will be the i-th 24 bit of x. The scaled exponent
         * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
         * match x's up to 24 bits.
         *
         * Example of breaking a double positive z into x[0]+x[1]+x[2]:
         * e0 = ilogb(z)-23
         * z  = scalbn(z,-e0)
         * for i = 0,1,2
         * x[i] = floor(z)
         * z    = (z-x[i])*2**24
         *
         *
         * y[] ouput result in an array of double precision numbers.
         * The dimension of y[] is:
         * 24-bit  precision 1
         * 53-bit  precision 2
         * 64-bit  precision 2
         * 113-bit precision 3
         * The actual value is the sum of them. Thus for 113-bit
         * precison, one may have to do something like:
         *
         * long double t,w,r_head, r_tail;
         * t = (long double)y[2] + (long double)y[1];
         * w = (long double)y[0];
         * r_head = t+w;
         * r_tail = w - (r_head - t);
         *
         * e0 The exponent of x[0]
         *
         * nx dimension of x[]
         *
         *   prec an integer indicating the precision:
         * 0 24  bits (single)
         * 1 53  bits (double)
         * 2 64  bits (extended)
         * 3 113 bits (quad)
         *
         * ipio2[]
         * integer array, contains the (24*i)-th to (24*i+23)-th
         * bit of 2/pi after binary point. The corresponding
         * floating value is
         *
         * ipio2[i] * 2^(-24(i+1)).
         *
         * External function:
         * double scalbn(), floor();
         *
         *
         * Here is the description of some local variables:
         *
         *  jk jk+1 is the initial number of terms of ipio2[] needed
         * in the computation. The recommended value is 2,3,4,
         * 6 for single, double, extended,and quad.
         *
         *  jz local integer variable indicating the number of
         * terms of ipio2[] used.
         *
         * jx nx - 1
         *
         * jv index for pointing to the suitable ipio2[] for the
         * computation. In general, we want
         * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
         * is an integer. Thus
         * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
         * Hence jv = max(0,(e0-3)/24).
         *
         * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
         *
         *  q[] double array with integral value, representing the
         * 24-bits chunk of the product of x and 2/pi.
         *
         * q0 the corresponding exponent of q[0]. Note that the
         * exponent for q[i] would be q0-24*i.
         *
         * PIo2[] double precision array, obtained by cutting pi/2
         * into 24 bits chunks.
         *
         * f[] ipio2[] in floating point
         *
         * iq[] integer array by breaking up q[] in 24-bits chunk.
         *
         * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
         *
         * ih integer. If >0 it indicates q[] is >= 0.5, hence
         * it also indicates the *sign* of the result.
         *
         */

        XSIMD_INLINE int32_t __kernel_rem_pio2(double* x, double* y, int32_t e0, int32_t nx, int32_t prec, const int32_t* ipio2) noexcept
        {
            static const int32_t init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */

            static const double PIo2[] = {
                1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
                7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
                5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
                3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
                1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
                1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
                2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
                2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
            };

            static const double
                zero
                = 0.0,
                one = 1.0,
                two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
                twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */

            int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
            double z, fw, f[20], fq[20], q[20];

            /* initialize jk*/
            jk = init_jk[prec];
            jp = jk;

            /* determine jx,jv,q0, note that 3>q0 */
            jx = nx - 1;
            jv = (e0 - 3) / 24;
            if (jv < 0)
                jv = 0;
            q0 = e0 - 24 * (jv + 1);

            /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
            j = jv - jx;
            m = jx + jk;
            for (i = 0; i <= m; i++, j++)
                f[i] = (j < 0) ? zero : (double)ipio2[j];

            /* compute q[0],q[1],...q[jk] */
            for (i = 0; i <= jk; i++)
            {
                for (j = 0, fw = 0.0; j <= jx; j++)
                    fw += x[j] * f[jx + i - j];
                q[i] = fw;
            }

            jz = jk;

        recompute:
            /* distill q[] into iq[] reversingly */
            for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
            {
                fw = (double)((int32_t)(twon24 * z));
                iq[i] = (int)(z - two24 * fw);
                z = q[j - 1] + fw;
            }

            /* compute n */
            z = std::scalbn(z, q0); /* actual value of z */
            z -= 8.0 * std::floor(z * 0.125); /* trim off integer >= 8 */
            n = (int32_t)z;
            z -= (double)n;
            ih = 0;
            if (q0 > 0)
            { /* need iq[jz-1] to determine n */
                i = (iq[jz - 1] >> (24 - q0));
                n += i;
                iq[jz - 1] -= i << (24 - q0);
                ih = iq[jz - 1] >> (23 - q0);
            }
            else if (q0 == 0)
                ih = iq[jz - 1] >> 23;
            else if (z >= 0.5)
                ih = 2;

            if (ih > 0)
            { /* q > 0.5 */
                n += 1;
                carry = 0;
                for (i = 0; i < jz; i++)
                { /* compute 1-q */
                    j = iq[i];
                    if (carry == 0)
                    {
                        if (j != 0)
                        {
                            carry = 1;
                            iq[i] = 0x1000000 - j;
                        }
                    }
                    else
                        iq[i] = 0xffffff - j;
                }
                if (q0 > 0)
                { /* rare case: chance is 1 in 12 */
                    switch (q0)
                    {
                    case 1:
                        iq[jz - 1] &= 0x7fffff;
                        break;
                    case 2:
                        iq[jz - 1] &= 0x3fffff;
                        break;
                    }
                }
                if (ih == 2)
                {
                    z = one - z;
                    if (carry != 0)
                        z -= std::scalbn(one, q0);
                }
            }

            /* check if recomputation is needed */
            if (z == zero)
            {
                j = 0;
                for (i = jz - 1; i >= jk; i--)
                    j |= iq[i];
                if (j == 0)
                { /* need recomputation */
                    for (k = 1; iq[jk - k] == 0; k++)
                        ; /* k = no. of terms needed */

                    for (i = jz + 1; i <= jz + k; i++)
                    { /* add q[jz+1] to q[jz+k] */
                        f[jx + i] = (double)ipio2[jv + i];
                        for (j = 0, fw = 0.0; j <= jx; j++)
                            fw += x[j] * f[jx + i - j];
                        q[i] = fw;
                    }
                    jz += k;
                    goto recompute;
                }
            }

            /* chop off zero terms */
            if (z == 0.0)
            {
                jz -= 1;
                q0 -= 24;
                while (iq[jz] == 0)
                {
                    jz--;
                    q0 -= 24;
                }
            }
            else
            { /* break z into 24-bit if necessary */
                z = std::scalbn(z, -q0);
                if (z >= two24)
                {
                    fw = (double)((int32_t)(twon24 * z));
                    iq[jz] = (int32_t)(z - two24 * fw);
                    jz += 1;
                    q0 += 24;
                    iq[jz] = (int32_t)fw;
                }
                else
                    iq[jz] = (int32_t)z;
            }

            /* convert integer "bit" chunk to floating-point value */
            fw = scalbn(one, q0);
            for (i = jz; i >= 0; i--)
            {
                q[i] = fw * (double)iq[i];
                fw *= twon24;
            }

            /* compute PIo2[0,...,jp]*q[jz,...,0] */
            for (i = jz; i >= 0; i--)
            {
                for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
                    fw += PIo2[k] * q[i + k];
                fq[jz - i] = fw;
            }

            /* compress fq[] into y[] */
            switch (prec)
            {
            case 0:
                fw = 0.0;
                for (i = jz; i >= 0; i--)
                    fw += fq[i];
                y[0] = (ih == 0) ? fw : -fw;
                break;
            case 1:
            case 2:
                fw = 0.0;
                for (i = jz; i >= 0; i--)
                    fw += fq[i];
                y[0] = (ih == 0) ? fw : -fw;
                fw = fq[0] - fw;
                for (i = 1; i <= jz; i++)
                    fw += fq[i];
                y[1] = (ih == 0) ? fw : -fw;
                break;
            case 3: /* painful */
                for (i = jz; i > 0; i--)
                {
                    fw = fq[i - 1] + fq[i];
                    fq[i] += fq[i - 1] - fw;
                    fq[i - 1] = fw;
                }
                for (i = jz; i > 1; i--)
                {
                    fw = fq[i - 1] + fq[i];
                    fq[i] += fq[i - 1] - fw;
                    fq[i - 1] = fw;
                }
                for (fw = 0.0, i = jz; i >= 2; i--)
                    fw += fq[i];
                if (ih == 0)
                {
                    y[0] = fq[0];
                    y[1] = fq[1];
                    y[2] = fw;
                }
                else
                {
                    y[0] = -fq[0];
                    y[1] = -fq[1];
                    y[2] = -fw;
                }
            }
            return n & 7;
        }

        XSIMD_INLINE std::int32_t __ieee754_rem_pio2(double x, double* y) noexcept
        {
            static const std::int32_t two_over_pi[] = {
                0xA2F983,
                0x6E4E44,
                0x1529FC,
                0x2757D1,
                0xF534DD,
                0xC0DB62,
                0x95993C,
                0x439041,
                0xFE5163,
                0xABDEBB,
                0xC561B7,
                0x246E3A,
                0x424DD2,
                0xE00649,
                0x2EEA09,
                0xD1921C,
                0xFE1DEB,
                0x1CB129,
                0xA73EE8,
                0x8235F5,
                0x2EBB44,
                0x84E99C,
                0x7026B4,
                0x5F7E41,
                0x3991D6,
                0x398353,
                0x39F49C,
                0x845F8B,
                0xBDF928,
                0x3B1FF8,
                0x97FFDE,
                0x05980F,
                0xEF2F11,
                0x8B5A0A,
                0x6D1F6D,
                0x367ECF,
                0x27CB09,
                0xB74F46,
                0x3F669E,
                0x5FEA2D,
                0x7527BA,
                0xC7EBE5,
                0xF17B3D,
                0x0739F7,
                0x8A5292,
                0xEA6BFB,
                0x5FB11F,
                0x8D5D08,
                0x560330,
                0x46FC7B,
                0x6BABF0,
                0xCFBC20,
                0x9AF436,
                0x1DA9E3,
                0x91615E,
                0xE61B08,
                0x659985,
                0x5F14A0,
                0x68408D,
                0xFFD880,
                0x4D7327,
                0x310606,
                0x1556CA,
                0x73A8C9,
                0x60E27B,
                0xC08C6B,
            };

            static const std::int32_t npio2_hw[] = {
                0x3FF921FB,
                0x400921FB,
                0x4012D97C,
                0x401921FB,
                0x401F6A7A,
                0x4022D97C,
                0x4025FDBB,
                0x402921FB,
                0x402C463A,
                0x402F6A7A,
                0x4031475C,
                0x4032D97C,
                0x40346B9C,
                0x4035FDBB,
                0x40378FDB,
                0x403921FB,
                0x403AB41B,
                0x403C463A,
                0x403DD85A,
                0x403F6A7A,
                0x40407E4C,
                0x4041475C,
                0x4042106C,
                0x4042D97C,
                0x4043A28C,
                0x40446B9C,
                0x404534AC,
                0x4045FDBB,
                0x4046C6CB,
                0x40478FDB,
                0x404858EB,
                0x404921FB,
            };

            /*
             * invpio2:  53 bits of 2/pi
             * pio2_1:   first  33 bit of pi/2
             * pio2_1t:  pi/2 - pio2_1
             * pio2_2:   second 33 bit of pi/2
             * pio2_2t:  pi/2 - (pio2_1+pio2_2)
             * pio2_3:   third  33 bit of pi/2
             * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
             */

            static const double
                zero
                = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
                half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
                two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
                invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
                pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
                pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
                pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
                pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
                pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
                pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */

            double z = 0., w, t, r, fn;
            double tx[3];
            std::int32_t e0, i, j, nx, n, ix, hx;
            std::uint32_t low;

            GET_HIGH_WORD(hx, x); /* high word of x */
            ix = hx & 0x7fffffff;
            if (ix <= 0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
            {
                y[0] = x;
                y[1] = 0;
                return 0;
            }
            if (ix < 0x4002d97c)
            { /* |x| < 3pi/4, special case with n=+-1 */
                if (hx > 0)
                {
                    z = x - pio2_1;
                    if (ix != 0x3ff921fb)
                    { /* 33+53 bit pi is good enough */
                        y[0] = z - pio2_1t;
                        y[1] = (z - y[0]) - pio2_1t;
                    }
                    else
                    { /* near pi/2, use 33+33+53 bit pi */
                        z -= pio2_2;
                        y[0] = z - pio2_2t;
                        y[1] = (z - y[0]) - pio2_2t;
                    }
                    return 1;
                }
                else
                { /* negative x */
                    z = x + pio2_1;
                    if (ix != 0x3ff921fb)
                    { /* 33+53 bit pi is good enough */
                        y[0] = z + pio2_1t;
                        y[1] = (z - y[0]) + pio2_1t;
                    }
                    else
                    { /* near pi/2, use 33+33+53 bit pi */
                        z += pio2_2;
                        y[0] = z + pio2_2t;
                        y[1] = (z - y[0]) + pio2_2t;
                    }

                    return -1;
                }
            }
            if (ix <= 0x413921fb)
            { /* |x| ~<= 2^19*(pi/2), medium_ size */
                t = std::fabs(x);
                n = (std::int32_t)(t * invpio2 + half);
                fn = (double)n;
                r = t - fn * pio2_1;
                w = fn * pio2_1t; /* 1st round good to 85 bit */
                if ((n < 32) && (n > 0) && (ix != npio2_hw[n - 1]))
                {
                    y[0] = r - w; /* quick check no cancellation */
                }
                else
                {
                    std::uint32_t high;
                    j = ix >> 20;
                    y[0] = r - w;
                    GET_HIGH_WORD(high, y[0]);
                    i = j - static_cast<int32_t>((high >> 20) & 0x7ff);
                    if (i > 16)
                    { /* 2nd iteration needed, good to 118 */
                        t = r;
                        w = fn * pio2_2;
                        r = t - w;
                        w = fn * pio2_2t - ((t - r) - w);
                        y[0] = r - w;
                        GET_HIGH_WORD(high, y[0]);
                        i = j - static_cast<int32_t>((high >> 20) & 0x7ff);
                        if (i > 49)
                        { /* 3rd iteration need, 151 bits acc */
                            t = r; /* will cover all possible cases */
                            w = fn * pio2_3;
                            r = t - w;
                            w = fn * pio2_3t - ((t - r) - w);
                            y[0] = r - w;
                        }
                    }
                }
                y[1] = (r - y[0]) - w;
                if (hx < 0)
                {
                    y[0] = -y[0];
                    y[1] = -y[1];
                    return -n;
                }
                else
                    return n;
            }
            /*
             * all other (large) arguments
             */
            if (ix >= 0x7ff00000)
            { /* x is inf or NaN */
                y[0] = y[1] = x - x;
                return 0;
            }
            /* set z = scalbn(|x|,ilogb(x)-23) */
            GET_LOW_WORD(low, x);
            SET_LOW_WORD(z, low);
            e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
            SET_HIGH_WORD(z, static_cast<uint32_t>(ix - (e0 << 20)));
            for (i = 0; i < 2; i++)
            {
                tx[i] = (double)((std::int32_t)(z));
                z = (z - tx[i]) * two24;
            }
            tx[2] = z;
            nx = 3;
            while (tx[nx - 1] == zero)
                nx--; /* skip zero term */
            n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
            if (hx < 0)
            {
                y[0] = -y[0];
                y[1] = -y[1];
                return -n;
            }
            return n;
        }
    }

#undef XSIMD_LITTLE_ENDIAN
#undef SET_LOW_WORD
#undef SET_HIGH_WORD
#undef GET_LOW_WORD
#undef GET_HIGH_WORD
#undef HIGH_WORD_IDX
#undef LOW_WORD_IDX
#undef ONCE0
}