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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_private.h"

#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 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
#else
      int __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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