k_rem_pio2.cs

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 * This code is distributed in the hope that it will be useful, but WITHOUT
 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
 * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
 * version 2 for more details (a copy is included in the LICENSE file that
 * accompanied this code).
 *
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 * 2 along with this work; if not, write to the Free Software Foundation,
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 *
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/*
 * __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.
 *
 */
 
 
/*
 * 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.
 */
 
namespace IKVM.Runtime.Util.Java.Lang
{
    static partial class fdlibm
    {
        static readonly int[] init_jk = { 2, 3, 4, 6 }; /* initial value for jk */
 
        static readonly 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 int __kernel_rem_pio2(double[] x, ref double y_0_, ref double y_1_, ref double y_2_, int e0, int nx, int prec, int[] ipio2)
        {
            const double
    zero = 0.0,
    one = 1.0,
    two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
    twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
 
            int jz, jx, jv, jp, jk, carry, n, i, j, k, m, q0, ih;
            int[] iq = new int[20];
            double z, fw;
            double[] f = new double[20];
            double[] fq = new double[20];
            double[] q = new double[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)((int)(twon24 * z));
                iq[i] = (int)(z - two24 * fw);
                z = q[j - 1] + fw;
            }
 
            /* compute n */
            z = scalbn(z, q0);              /* actual value of z */
            z -= 8.0 * floor(z * 0.125);                /* trim off integer >= 8 */
            n = (int)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 -= 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 = scalbn(z, -q0);
                if (z >= two24)
                {
                    fw = (double)((int)(twon24 * z));
                    iq[jz] = (int)(z - two24 * fw);
                    jz += 1; q0 += 24;
                    iq[jz] = (int)fw;
                }
                else iq[jz] = (int)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;
                    }
                    break;
            }
            return n & 7;
        }
    }
 
}