glibc/sysdeps/ieee754/dbl-64/k_rem_pio2.c

/* @(#)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 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
 *		precision, 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.h>
#include <math-narrow-eval.h>
#include <math_private.h>
#include <libc-diag.h>

static const int 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 */

int
__kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec,
                   const int32_t *ipio2)
{
  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 */
	{
	  /* On s390x gcc 6.1 -O3 produces the warning "array subscript is below
	     array bounds [-Werror=array-bounds]".  Only __ieee754_rem_pio2l
	     calls __kernel_rem_pio2 for normal numbers and |x| > pi/4 in case
	     of ldbl-96 and |x| > 3pi/4 in case of ldbl-128[ibm].
	     Thus x can't be zero and ipio2 is not zero, too.  Thus not all iq[]
	     values can't be zero.  */
	  DIAG_PUSH_NEEDS_COMMENT;
	  DIAG_IGNORE_NEEDS_COMMENT (6.1, "-Warray-bounds");
	  for (k = 1; iq[jk - k] == 0; k++)
	    ;                               /* k = no. of terms needed */
	  DIAG_POP_NEEDS_COMMENT;

	  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:;
      double fv = 0.0;
      for (i = jz; i >= 0; i--)
	fv = math_narrow_eval (fv + fq[i]);
      y[0] = (ih == 0) ? fv : -fv;
      fv = math_narrow_eval (fq[0] - fv);
      for (i = 1; i <= jz; i++)
	fv = math_narrow_eval (fv + fq[i]);
      y[1] = (ih == 0) ? fv : -fv;
      break;
    case 3:             /* painful */
      for (i = jz; i > 0; i--)
	{
	  double fv = math_narrow_eval (fq[i - 1] + fq[i]);
	  fq[i] += fq[i - 1] - fv;
	  fq[i - 1] = fv;
	}
      for (i = jz; i > 1; i--)
	{
	  double fv = math_narrow_eval (fq[i - 1] + fq[i]);
	  fq[i] += fq[i - 1] - fv;
	  fq[i - 1] = fv;
	}
      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;
}