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[]	output 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;
      /* GCC mainline (to be GCC 9), as of 2018-05-22 on i686, warns
	 that fq[0] may be used uninitialized.  This is not possible
	 because jz is always nonnegative when the above loop
	 initializing fq is executed, because the result is never zero
	 to full precision (this function is not called for zero
	 arguments).  */
      DIAG_PUSH_NEEDS_COMMENT;
      DIAG_IGNORE_NEEDS_COMMENT (9, "-Wmaybe-uninitialized");
      fv = math_narrow_eval (fq[0] - fv);
      DIAG_POP_NEEDS_COMMENT;
      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;
}