prime/src/Runtime/k_rem_pio2.c

/* @(#)k_rem_pio2.c 1.2 95/01/04 */
/*
 * ====================================================
 * 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.
 * ====================================================
 */

/*
 * __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 "fdlibm.h"

#ifdef __STDC__
static const _INT32 init_jk[] = {2, 3, 4, 6}; /* initial value for jk */
#else
static _INT32 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__
_INT32 __kernel_rem_pio2(double* x, double* y, _INT32 e0, _INT32 nx, _INT32 prec,
                         const _INT32* ipio2)
#else
_INT32 __kernel_rem_pio2(x, y, e0, nx, prec, ipio2)
double x[], y[];
_INT32 e0, nx, prec;
_INT32 ipio2[];
#endif
{
  _INT32 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)(twon24 * z));
    iq[i] = (_INT32)(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)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)(twon24 * z));
      iq[jz] = (_INT32)(z - two24 * fw);
      jz += 1;
      q0 += 24;
      iq[jz] = (_INT32)fw;
    } else
      iq[jz] = (_INT32)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;
}