/*-
 * Copyright (c) 2026 Steven G. Kargl
 * 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 unmodified, 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.
 */

/**
 * Compute the inverse sqrt of x, i.e., rsqrt(x) = 1 / sqrt(x).
 *
 * First, filter out special cases:
 *
 *   1. rsqrt(+-0) = +-inf, and raise FE_DIVBYZERO exception.
 *   2. rsqrt(nan) = NaN.
 *   3. rsqrt(+inf) returns +0.
 *   2. rsqrt(x<0) = NaN, and raises FE_INVALID.
 *
 * If x is a subnormal, scale x into the normal range by x*0x1pN; while
 * recording the exponent of the scale factor N.  Split the possibly
 * scaled x into f*2^n with f in [0.5,1).  Set m=n or m=n-N (subnormal).
 * If n is odd, then set f = f/2 and increase n to n+1.  Thus, f is
 * in [0.25,1) with n even.
 *
 * An initial estimate of y = rqrt[f](x) is 1 / sqrt[f](x).  Exhaustive
 * testing of rsqrtf() gave a max ULP of 1.49; while testing 500M x in
 * [0,1000] gave a max ULP of 1.24 for rsqrt().  The value of y is then
 * used with one iteration of Goldschmidt's algorithm:
 *
 *      z = x * y
 *      h = y / 2
 *      r = 0.5 - h * z
 *      y = h * r + h
 *
 * A factor of 2 appears missing in the above, but it is included in the
 * exponent m.
 */
#include <fenv.h>
#include <float.h>
#include "math.h"
#include "math_private.h"

#pragma STDC FENV_ACCESS ON

#ifdef _CC
#undef _CC
#endif
#define _CC (0x1p27 + 1)

double
rsqrt(double x)
{
        volatile static const double vzero = 0;
        static const double half = 0.5;
        int hx, m, rnd;
        uint32_t lx, ux;
        double h, ph, pl, rh, rl, y, zh, zl;

        EXTRACT_WORDS(hx, lx, x);
        ux = (uint32_t)hx & 0x7fffffff;

        /* x = +-0.  Raise exception. */
        if ((ux | lx) == 0)
            return (1 / x);

        /* x is NaN. */
        if (ux > 0x7ff00000)
            return (x + x);

        /* x is +-inf. */
        if (ux == 0x7ff00000)
            return (hx & 0x80000000 ? vzero / vzero : 0.);

        /* x < 0.  Raise exception. */
        if (hx < 0)
            return (vzero / vzero);

        /*
         * If x is subnormal, then scale it into the normal range.
         * Split x into significand and exponent, x = f * 2^m, with
         * f in [0.5,1) and m a biased exponent.
         */
        m = 0;
        if (hx < 0x00100000) {          /* Subnormal */
            x *= 0x1p54;
            GET_HIGH_WORD(hx, x);
            m = -54;
        }
        m += (hx >> 20) - 1022;
        hx = (hx & 0x000fffff) | 0x3fe00000;
        SET_HIGH_WORD(x, hx);

        /* m is odd.  Put x into [0.25,0.5) and increase m. */
        if (m & 1) {
            x /= 2;
            m += 1;
        }
        m = -(m >> 1);                  /* Prepare for 2^(-m/2). */

        y = 1 / sqrt(x);                /* ~52-bit estimate. */

        h = y / 2;

        /*
         * For values of x with a representation of 0x1.ffffffffffffepN
         * with N an odd integer, the computed rsqrt() is not correctly
         * rounded in round-to-nearest without toggling the rounding mode
         * to FE_TOWARDZERO.  Note, FE_DOWNWARD also works.  However,
         * messing with the rounding mode is expensive, so only do it
         * when necessary. Example, x = 3.9999999999999991
         * gives y --> hx = 0x3ff00000, lx = 0x00000001
         */
        EXTRACT_WORDS(hx, lx, y);
        if ((hx & 0x000fffff) == 0 && lx == 1) {
            rnd = fegetround();
            fesetround(FE_TOWARDZERO);
            _MUL(x, y, zh, zl);
            _XMUL(zh, zl, h, 0, ph, pl);
            fesetround(rnd);
        } else {
            _MUL(x, y, zh, zl);
            _XMUL(zh, zl, h, 0, ph, pl);
        }

        _XADD(-ph, -pl, half, 0, rh, rl);
        y = rh * h + h;

        ux = (m + 1024) << 20;
        INSERT_WORDS(x, ux, 0);
        return (y *= x);
}

#if LDBL_MANT_DIG == 53
__weak_reference(rsqrt, rsqrtl);
#endif