remp2.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/e_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.
 * =================================================================
 */

#include "prim.h"

/**
 * @file  remp2.c
 * @brief
 */

/* rem_pio2(x,y)
 *
 * return the remainder of x rem pi/2 in y[0]+y[1]
 * use __kernel_rem_pio2()
 */


/*
 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
 */

static const int two_over_pi[] = {
    0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
    0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
    0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
    0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
    0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
    0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
    0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
    0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
    0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
    0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
    0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
};

static const int npio2_hw[] = {
    0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
    0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
    0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
    0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
    0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
    0x404858EB, 0x404921FB,
};

/*
 * invpio2:  53 bits of 2/pi
 * pio2_1:   first  33 bit of pi/2
 * pio2_1t:  pi/2 - pio2_1
 * pio2_2:   second 33 bit of pi/2
 * pio2_2t:  pi/2 - (pio2_1+pio2_2)
 * pio2_3:   third  33 bit of pi/2
 * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
 */

static const double
zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
half =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
invpio2 =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
pio2_1  =  1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
pio2_1t =  6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
pio2_2  =  6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
pio2_2t =  2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
pio2_3  =  2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
pio2_3t =  8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */

int32_t rempio2(double x, double *y)
{
    double z = 0.,w,t,r,fn;
    double tx[3];
    int32_t i,j,n,ix,hx;
    int e0,nx;
    uint32_t low;

    GET_HIGH_WORD(hx,x);        /* high word of x */
    ix = hx&0x7fffffff;
    if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */
    {
        y[0] = x;
        y[1] = 0;
        return 0;
    }
    if(ix<0x4002d97c) {  /* |x| < 3pi/4, special case with n=+-1 */
        if(hx>0) {
            z = x - pio2_1;
            if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
                y[0] = z - pio2_1t;
                y[1] = (z-y[0])-pio2_1t;
            } else {        /* near pi/2, use 33+33+53 bit pi */
                z -= pio2_2;
                y[0] = z - pio2_2t;
                y[1] = (z-y[0])-pio2_2t;
            }
            return 1;
        } else {    /* negative x */
            z = x + pio2_1;
            if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
                y[0] = z + pio2_1t;
                y[1] = (z-y[0])+pio2_1t;
            } else {        /* near pi/2, use 33+33+53 bit pi */
                z += pio2_2;
                y[0] = z + pio2_2t;
                y[1] = (z-y[0])+pio2_2t;
            }
            return -1;
        }
    }
    if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
        t  = fabs(x);
        n  = (int32_t) (t*invpio2+half);
        fn = (double)n;
        r  = t-fn*pio2_1;
        w  = fn*pio2_1t;    /* 1st round good to 85 bit */
        if(n<32&&ix!=npio2_hw[n-1]) {
            y[0] = r-w; /* quick check no cancellation */
        } else {
            uint32_t high;

            j  = ix>>20;
            y[0] = r-w;
            GET_HIGH_WORD(high, y[0]);
            i = j-((high>>20)&0x7ff);
            if(i>16) {  /* 2nd iteration needed, good to 118 */
                t  = r;
                w  = fn*pio2_2;
                r  = t-w;
                w  = fn*pio2_2t-((t-r)-w);
                y[0] = r-w;
                GET_HIGH_WORD(high,y[0]);
                i = j-((high>>20)&0x7ff);
                if(i>49)  { /* 3rd iteration need, 151 bits acc */
                    t  = r; /* will cover all possible cases */
                    w  = fn*pio2_3;
                    r  = t-w;
                    w  = fn*pio2_3t-((t-r)-w);
                    y[0] = r-w;
                }
            }
        }
        y[1] = (r-y[0])-w;
        if(hx<0)    {
            y[0] = -y[0];
            y[1] = -y[1];
            return -n;
        }
        else     return n;
    }
    /*
     * all other (large) arguments
     */
    if(ix>=0x7ff00000) {        /* x is inf or NaN */
        y[0]=y[1]=x-x;
        return 0;
    }
    /* set z = scalbn(|x|,ilogb(x)-23) */
    GET_LOW_WORD(low,x);
    SET_LOW_WORD(z,low);
    e0  = (int32_t)(ix>>20)-1046;   /* e0 = ilogb(z)-23; */
    SET_HIGH_WORD(z,ix - (e0<<20));
    for(i=0; i<2; i++) {
        tx[i] = (double)((int32_t)(z));
        z     = (z-tx[i])*two24;
    }
    tx[2] = z;
    nx = 3;
    while(tx[nx-1]==zero) nx--; /* skip zero term */
    n  =  __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
    if(hx<0) {
        y[0] = -y[0];
        y[1] = -y[1];
        return -n;
    }
    return n;
}