kc3-lang/SDL/src/libm/k_rem_pio2.c

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  Date : 2013-11-29T10:06:08
  

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• src/libm/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 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;
  }
  

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