kremp2.c

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/*-
 * Copyright (c) 2014-2018 Carsten Sonne Larsen <cs@innolan.net>
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 *
 * Project homepage:
 * https://amath.innolan.net
 *
 * The original source code can be obtained from:
 * http://www.netlib.org/fdlibm/k_rem_pio2.c
 * 
 * =================================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * =================================================================
 */

/**
 * @file  kremp2.c
 * @brief Kernel reduction function
 */

#include "prim.h"

/*
 * 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.
 */

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 */

/**
 * @brief   Kernel reduction function
 * @details
 * <pre>
 * __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.
 * </pre>
 */
int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2)
{
    int 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)((int)(twon24* z));
        iq[i] =  (int)(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  = (int) 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)((int)(twon24*z));
            iq[jz] = (int)(z-two24*fw);
            jz += 1;
            q0 += 24;
            iq[jz] = (int) fw;
        } else iq[jz] = (int) 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;
}