/*      $OpenBSD: s_csqrtl.c,v 1.4 2016/09/12 19:47:02 guenther Exp $   */

/*
 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
 *
 * Permission to use, copy, modify, and distribute this software for any
 * purpose with or without fee is hereby granted, provided that the above
 * copyright notice and this permission notice appear in all copies.
 *
 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
 */

/*                                                      csqrtl()
 *
 *      Complex square root
 *
 *
 *
 * SYNOPSIS:
 *
 * long double complex csqrtl();
 * long double complex z, w;
 *
 * w = csqrtl( z );
 *
 *
 *
 * DESCRIPTION:
 *
 *
 * If z = x + iy,  r = |z|, then
 *
 *                       1/2
 * Re w  =  [ (r + x)/2 ]   ,
 *
 *                       1/2
 * Im w  =  [ (r - x)/2 ]   .
 *
 * Cancellation error in r-x or r+x is avoided by using the
 * identity  2 Re w Im w  =  y.
 *
 * Note that -w is also a square root of z.  The root chosen
 * is always in the right half plane and Im w has the same sign as y.
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      -10,+10     500000      1.1e-19     3.0e-20
 *
 */

#include <complex.h>
#include <math.h>

long double complex
csqrtl(long double complex z)
{
        long double complex w;
        long double x, y, r, t, scale;

        x = creall(z);
        y = cimagl(z);

        if (y == 0.0L) {
                if (x < 0.0L) {
                        w = 0.0L + copysign(sqrtl(-x), y) * I;
                        return (w);
                }
                else {
                        w = sqrtl(x) + 0.0L * I;
                        return (w);
                }
        }

        if (x == 0.0L) {
                r = fabsl(y);
                r = sqrtl(0.5L * r);
                if (y > 0.0L)
                        w = r + r * I;
                else
                        w = r - r * I;
                return (w);
        }

        /* Rescale to avoid internal overflow or underflow.  */
        if ((fabsl(x) > 4.0L) || (fabsl(y) > 4.0L)) {
                x *= 0.25L;
                y *= 0.25L;
                scale = 2.0L;
        }
        else {
#if 1
                x *= 7.3786976294838206464e19;  /* 2^66 */
                y *= 7.3786976294838206464e19;
                scale = 1.16415321826934814453125e-10;  /* 2^-33 */
#else
                x *= 4.0L;
                y *= 4.0L;
                scale = 0.5L;
#endif
        }
        w = x + y * I;
        r = cabsl(w);
        if (x > 0) {
                t = sqrtl(0.5L * r + 0.5L * x);
                r = scale * fabsl((0.5L * y) / t);
                t *= scale;
        }
        else {
                r = sqrtl(0.5L * r - 0.5L * x);
                t = scale * fabsl((0.5L * y) / r);
                r *= scale;
        }
        if (y < 0)
                w = t - r * I;
        else
                w = t + r * I;
        return (w);
}
DEF_STD(csqrtl);