e_rem_pio2.c

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/* @(#)e_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.
 * ====================================================
 */

/* __ieee754_rem_pio2(x,y)
 * 
 * return the remainder of x rem pi/2 in y[0]+y[1] 
 * use __kernel_rem_pio2()
 */

#include "math.h"
#include "math_private.h"

static const int32_t 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
__ieee754_rem_pio2(double x, double *y)
{
    double z,w,t,r,fn;
    double tx[3];
    int32_t e0,i,j,nx,n,ix,hx;
    u_int32_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 {
            u_int32_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  = (ix>>20)-1046;    /* e0 = ilogb(z)-23; */
    SET_HIGH_WORD(z, ix - ((int32_t)(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);
    if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
    return n;
}