third_party/ulib/musl/third_party/math/__rem_pio2.c

/* origin: FreeBSD /usr/src/lib/msun/src/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.
 * ====================================================
 *
 * Optimized by Bruce D. Evans.
 */
/* __rem_pio2(x,y)
 *
 * return the remainder of x rem pi/2 in y[0]+y[1]
 * use __rem_pio2_large() for large x
 */

#include "libm.h"

#if FLT_EVAL_METHOD == 0 || FLT_EVAL_METHOD == 1
#define EPS DBL_EPSILON
#elif FLT_EVAL_METHOD == 2
#define EPS LDBL_EPSILON
#endif

/*
 * 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 toint = 1.5 / EPS,
                    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 */

/* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
int __rem_pio2(double x, double* y) {
    union {
        double f;
        uint64_t i;
    } u = {x};
    double_t z, w, t, r, fn;
    double tx[3], ty[2];
    uint32_t ix;
    int sign, n, ex, ey, i;

    sign = u.i >> 63;
    ix = u.i >> 32 & 0x7fffffff;
    if (ix <= 0x400f6a7a) {            /* |x| ~<= 5pi/4 */
        if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
            goto medium;               /* cancellation -- use medium case */
        if (ix <= 0x4002d97c) {        /* |x| ~<= 3pi/4 */
            if (!sign) {
                z = x - pio2_1; /* one round good to 85 bits */
                y[0] = z - pio2_1t;
                y[1] = (z - y[0]) - pio2_1t;
                return 1;
            } else {
                z = x + pio2_1;
                y[0] = z + pio2_1t;
                y[1] = (z - y[0]) + pio2_1t;
                return -1;
            }
        } else {
            if (!sign) {
                z = x - 2 * pio2_1;
                y[0] = z - 2 * pio2_1t;
                y[1] = (z - y[0]) - 2 * pio2_1t;
                return 2;
            } else {
                z = x + 2 * pio2_1;
                y[0] = z + 2 * pio2_1t;
                y[1] = (z - y[0]) + 2 * pio2_1t;
                return -2;
            }
        }
    }
    if (ix <= 0x401c463b) {       /* |x| ~<= 9pi/4 */
        if (ix <= 0x4015fdbc) {   /* |x| ~<= 7pi/4 */
            if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
                goto medium;
            if (!sign) {
                z = x - 3 * pio2_1;
                y[0] = z - 3 * pio2_1t;
                y[1] = (z - y[0]) - 3 * pio2_1t;
                return 3;
            } else {
                z = x + 3 * pio2_1;
                y[0] = z + 3 * pio2_1t;
                y[1] = (z - y[0]) + 3 * pio2_1t;
                return -3;
            }
        } else {
            if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
                goto medium;
            if (!sign) {
                z = x - 4 * pio2_1;
                y[0] = z - 4 * pio2_1t;
                y[1] = (z - y[0]) - 4 * pio2_1t;
                return 4;
            } else {
                z = x + 4 * pio2_1;
                y[0] = z + 4 * pio2_1t;
                y[1] = (z - y[0]) + 4 * pio2_1t;
                return -4;
            }
        }
    }
    if (ix < 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
    medium:
        /* rint(x/(pi/2)), Assume round-to-nearest. */
        fn = (double_t)x * invpio2 + toint - toint;
        n = (int32_t)fn;
        r = x - fn * pio2_1;
        w = fn * pio2_1t; /* 1st round, good to 85 bits */
        y[0] = r - w;
        u.f = y[0];
        ey = u.i >> 52 & 0x7ff;
        ex = ix >> 20;
        if (ex - ey > 16) { /* 2nd round, good to 118 bits */
            t = r;
            w = fn * pio2_2;
            r = t - w;
            w = fn * pio2_2t - ((t - r) - w);
            y[0] = r - w;
            u.f = y[0];
            ey = u.i >> 52 & 0x7ff;
            if (ex - ey > 49) { /* 3rd round, good to 151 bits, covers all cases */
                t = r;
                w = fn * pio2_3;
                r = t - w;
                w = fn * pio2_3t - ((t - r) - w);
                y[0] = r - w;
            }
        }
        y[1] = (r - y[0]) - w;
        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) */
    u.f = x;
    u.i &= (uint64_t)-1 >> 12;
    u.i |= (uint64_t)(0x3ff + 23) << 52;
    z = u.f;
    for (i = 0; i < 2; i++) {
        tx[i] = (double)(int32_t)z;
        z = (z - tx[i]) * 0x1p24;
    }
    tx[i] = z;
    /* skip zero terms, first term is non-zero */
    while (tx[i] == 0.0)
        i--;
    n = __rem_pio2_large(tx, ty, (int)(ix >> 20) - (0x3ff + 23), i + 1, 1);
    if (sign) {
        y[0] = -ty[0];
        y[1] = -ty[1];
        return -n;
    }
    y[0] = ty[0];
    y[1] = ty[1];
    return n;
}