k_rem_pio2.c

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/* @(#)k_rem_pio2.c 5.1 93/09/24 */
/*
 * ====================================================
 * 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(LIBM_SCCS) && !defined(lint)
static const char rcsid[] =
    "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
#endif

/*
 * __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.
 */

#include "math_libm.h"
#include "math_private.h"

libm_hidden_proto(scalbn)
    libm_hidden_proto(floor)
#ifdef __STDC__
     static const int init_jk[] = { 2, 3, 4, 6 };       /* initial value for jk */
#else
     static int init_jk[] = { 2, 3, 4, 6 };
#endif

#ifdef __STDC__
static const double PIo2[] = {
#else
static double PIo2[] = {
#endif
    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 */
};

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

#ifdef __STDC__
int attribute_hidden
__kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
                  const int32_t * ipio2)
#else
int attribute_hidden
__kernel_rem_pio2(x, y, e0, nx, prec, ipio2)
     double x[], y[];
     int e0, nx, prec;
     int32_t ipio2[];
#endif
{
    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] = (int32_t) (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 = (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 -= 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) ((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;
}