/*      $OpenBSD: s_casinl.c,v 1.5 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.
 */

/*                                                      casinl()
 *
 *      Complex circular arc sine
 *
 *
 *
 * SYNOPSIS:
 *
 * long double complex casinl();
 * long double complex z, w;
 *
 * w = casinl( z );
 *
 *
 *
 * DESCRIPTION:
 *
 * Inverse complex sine:
 *
 *                               2
 * w = -i clog( iz + csqrt( 1 - z ) ).
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    DEC       -10,+10     10100       2.1e-15     3.4e-16
 *    IEEE      -10,+10     30000       2.2e-14     2.7e-15
 * Larger relative error can be observed for z near zero.
 * Also tested by csin(casin(z)) = z.
 */

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

#if     LDBL_MANT_DIG == 64
static const long double MACHEPL= 5.42101086242752217003726400434970855712890625E-20L;
#elif   LDBL_MANT_DIG == 113
static const long double MACHEPL = 9.629649721936179265279889712924636592690508e-35L;
#endif

static const long double PIO2L = 1.570796326794896619231321691639751442098585L;

long double complex
casinl(long double complex z)
{
        long double complex w;
        long double x, y, b;
        static long double complex ca, ct, zz, z2;

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

#if 0
        if (y == 0.0L) {
                if (fabsl(x) > 1.0L) {
                        w = PIO2L + 0.0L * I;
                        /*mtherr( "casinl", DOMAIN );*/
                }
                else {
                        w = asinl(x) + 0.0L * I;
                }
                return (w);
        }
#endif

        /* Power series expansion */
        b = cabsl(z);
        if (b < 0.125L) {
                long double complex sum;
                long double n, cn;

                z2 = (x - y) * (x + y) + (2.0L * x * y) * I;
                cn = 1.0L;
                n = 1.0L;
                ca = x + y * I;
                sum = x + y * I;
                do {
                        ct = z2 * ca;
                        ca = ct;

                        cn *= n;
                        n += 1.0L;
                        cn /= n;
                        n += 1.0L;
                        b = cn/n;

                        ct *= b;
                        sum += ct;
                        b = cabsl(ct);
                }

                while (b > MACHEPL);
                w = sum;
                return w;
        }

        ca = x + y * I;
        ct = ca * I;    /* iz */
        /* sqrt(1 - z*z) */
        /* cmul(&ca, &ca, &zz) */
        /* x * x  -  y * y */
        zz = (x - y) * (x + y) + (2.0L * x * y) * I;
        zz = 1.0L - creall(zz) - cimagl(zz) * I;
        z2 = csqrtl(zz);

        zz = ct + z2;
        zz = clogl(zz);
        /* multiply by 1/i = -i */
        w = zz * (-1.0L * I);
        return (w);
}
DEF_STD(casinl);