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/* @(#)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

 *              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.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;

}