/*      $NetBSD: catrigl.c,v 1.3 2022/04/19 20:32:16 rillig Exp $       */
/*-
 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG>
 * 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, 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 AND CONTRIBUTORS ``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 OR CONTRIBUTORS 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.
 */

/*
 * The algorithm is very close to that in "Implementing the complex arcsine
 * and arccosine functions using exception handling" by T. E. Hull, Thomas F.
 * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
 * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
 * http://dl.acm.org/citation.cfm?id=275324.
 *
 * The code for catrig.c contains complete comments.
 */
#include <sys/cdefs.h>
__RCSID("$NetBSD: catrigl.c,v 1.3 2022/04/19 20:32:16 rillig Exp $");

#include "namespace.h"
#ifdef __weak_alias
__weak_alias(casinl, _casinl)
#endif
#ifdef __weak_alias
__weak_alias(catanl, _catanl)
#endif


#include <sys/param.h>
#include <complex.h>
#include <float.h>
#include <math.h>
#ifdef notyet // missing log1pl __HAVE_LONG_DOUBLE

#include "math_private.h"

#undef isinf
#define isinf(x)        (fabsl(x) == INFINITY)
#undef isnan
#define isnan(x)        ((x) != (x))
#define raise_inexact() do { volatile float junk __unused = /*LINTED*/1 + tiny; } while (0)
#undef signbit
#define signbit(x)      (__builtin_signbitl(x)) 

#if __HAVE_LONG_DOUBLE + 0 == 128
// Ok
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
// XXX: Byte order
#define EXT_EXPBITS     15
struct ieee_ext {
        uint64_t ext_frac;
        uint16_t ext_exp:EXT_EXPBITS;
        uint16_t ext_sign:1;
        uint16_t ext_pad;
};
#define extu_exp        extu_ext.ext_exp
#define extu_sign       extu_ext.ext_sign
#define extu_frac       extu_ext.ext_frac
union ieee_ext_u {
        long double extu_ld;
        struct ieee_ext extu_ext;
};
#else
        #error "unsupported long double format"
#endif

#define GET_LDBL_EXPSIGN(r, s) \
    do { \
            union ieee_ext_u u; \
            u.extu_ld = s; \
            r = u.extu_sign; \
            r >>= EXT_EXPBITS - 1; \
    } while (0)
#define SET_LDBL_EXPSIGN(s, r) \
    do { \
            union ieee_ext_u u; \
            u.extu_ld = s; \
            u.extu_exp &= __BITS(0, EXT_EXPBITS - 1); \
            u.extu_exp |= (r) << (EXT_EXPBITS - 1); \
            s = u.extu_ld; \
    } while (0)

static const long double
A_crossover =           10,
B_crossover =           0.6417,
FOUR_SQRT_MIN =         0x1p-8189L,
QUARTER_SQRT_MAX =      0x1p8189L,
RECIP_EPSILON =         1/LDBL_EPSILON,
SQRT_MIN =              0x1p-8191L;

static const long double
m_e =           2.71828182845904523536028747135266250e0L,       /* 0x15bf0a8b1457695355fb8ac404e7a.0p-111 */
m_ln2 =         6.93147180559945309417232121458176568e-1L,      /* 0x162e42fefa39ef35793c7673007e6.0p-113 */
pio2_hi =      1.5707963267948966192313216916397514L, /* pi/2 */
SQRT_3_EPSILON = 2.40370335797945490975336727199878124e-17L,    /*  0x1bb67ae8584caa73b25742d7078b8.0p-168 */
SQRT_6_EPSILON = 3.39934988877629587239082586223300391e-17L;    /*  0x13988e1409212e7d0321914321a55.0p-167 */

static const volatile double
pio2_lo =               6.1232339957367659e-17; /*  0x11a62633145c07.0p-106 */
static const volatile float
tiny =                  0x1p-100;

static long double complex clog_for_large_values(long double complex z);

inline static long double
f(long double a, long double b, long double hypot_a_b)
{
        if (b < 0)
                return ((hypot_a_b - b) / 2);
        if (b == 0)
                return (a / 2);
        return (a * a / (hypot_a_b + b) / 2);
}

inline static void
do_hard_work(long double x, long double y, long double *rx, int *B_is_usable, long double *B, long double *sqrt_A2my2, long double *new_y)
{
        long double R, S, A;
        long double Am1, Amy;

        R = hypotl(x, y+1);
        S = hypotl(x, y-1);

        A = (R + S) / 2;
        if (A < 1)
                A = 1;

        if (A < A_crossover) {
                if (y == 1 && x < LDBL_EPSILON*LDBL_EPSILON/128) {
                        *rx = sqrtl(x);
                } else if (x >= LDBL_EPSILON * fabsl(y-1)) {
                        Am1 = f(x, 1+y, R) + f(x, 1-y, S);
                        *rx = log1pl(Am1 + sqrtl(Am1*(A+1)));
                } else if (y < 1) {
                        *rx = x/sqrtl((1-y)*(1+y));
                } else {
                        *rx = log1pl((y-1) + sqrtl((y-1)*(y+1)));
                }
        } else
                *rx = logl(A + sqrtl(A*A-1));

        *new_y = y;

        if (y < FOUR_SQRT_MIN) {
                *B_is_usable = 0;
                *sqrt_A2my2 = A * (2 / LDBL_EPSILON);
                *new_y= y * (2 / LDBL_EPSILON);
                return;
        }

        *B = y/A;
        *B_is_usable = 1;

        if (*B > B_crossover) {
                *B_is_usable = 0;
                if (y == 1 && x < LDBL_EPSILON/128) {
                        *sqrt_A2my2 = sqrtl(x)*sqrtl((A+y)/2);
                } else if (x >= LDBL_EPSILON * fabsl(y-1)) {
                        Amy = f(x, y+1, R) + f(x, y-1, S);
                        *sqrt_A2my2 = sqrtl(Amy*(A+y));
                } else if (y > 1) {
                        *sqrt_A2my2 = x * (4/LDBL_EPSILON/LDBL_EPSILON) * y /
                            sqrtl((y+1)*(y-1));
                        *new_y = y * (4/LDBL_EPSILON/LDBL_EPSILON);
                } else {
                        *sqrt_A2my2 = sqrtl((1-y)*(1+y));
                }
        }
}

long double complex
casinhl(long double complex z)
{
        long double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
        int B_is_usable;
        long double complex w;

        x = creall(z);
        y = cimagl(z);
        ax = fabsl(x);
        ay = fabsl(y);

        if (isnan(x) || isnan(y)) {
                if (isinf(x))
                        return (CMPLXL(x, y+y));
                if (isinf(y))
                        return (CMPLXL(y, x+x));
                if (y == 0) return (CMPLXL(x+x, y));
                return (CMPLXL(x+0.0L+(y+0), x+0.0L+(y+0)));
        }

        if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
                if (signbit(x) == 0)
                        w = clog_for_large_values(z) + m_ln2;
                else
                        w = clog_for_large_values(-z) + m_ln2;
                return (CMPLXL(copysignl(creall(w), x), copysignl(cimagl(w), y)));
        }

        if (x == 0 && y == 0)
                return (z);

        raise_inexact();

        if (ax < SQRT_6_EPSILON/4 && ay < SQRT_6_EPSILON/4)
                return (z);

        do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
        if (B_is_usable)
                ry = asinl(B);
        else
                ry = atan2l(new_y, sqrt_A2my2);
        return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
}

long double complex
casinl(long double complex z)
{
        long double complex w = casinhl(CMPLXL(cimagl(z), creall(z)));
        return (CMPLXL(cimagl(w), creall(w)));
}

long double complex
cacosl(long double complex z)
{
        long double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
        int sx, sy;
        int B_is_usable;
        long double complex w;

        x = creall(z);
        y = cimagl(z);
        sx = signbit(x);
        sy = signbit(y);
        ax = fabsl(x);
        ay = fabsl(y);

        if (isnan(x) || isnan(y)) {
                if (isinf(x))
                        return (CMPLXL(y+y, -INFINITY));
                if (isinf(y))
                        return (CMPLXL(x+x, -y));
                if (x == 0) return (CMPLXL(pio2_hi + pio2_lo, y+y));
                return (CMPLXL(x+0.0L+(y+0), x+0.0L+(y+0)));
        }

        if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
                w = clog_for_large_values(z);
                rx = fabsl(cimagl(w));
                ry = creall(w) + m_ln2;
                if (sy == 0)
                        ry = -ry;
                return (CMPLXL(rx, ry));
        }

        if (x == 1 && y == 0)
                return (CMPLXL(0, -y));

        raise_inexact();

        if (ax < SQRT_6_EPSILON/4 && ay < SQRT_6_EPSILON/4)
                return (CMPLXL(pio2_hi - (x - pio2_lo), -y));

        do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
        if (B_is_usable) {
                if (sx==0)
                        rx = acosl(B);
                else
                        rx = acosl(-B);
        } else {
                if (sx==0)
                        rx = atan2l(sqrt_A2mx2, new_x);
                else
                        rx = atan2l(sqrt_A2mx2, -new_x);
        }
        if (sy==0)
                ry = -ry;
        return (CMPLXL(rx, ry));
}

long double complex
cacoshl(long double complex z)
{
        long double complex w;
        long double rx, ry;

        w = cacosl(z);
        rx = creall(w);
        ry = cimagl(w);
        if (isnan(rx) && isnan(ry))
                return (CMPLXL(ry, rx));
        if (isnan(rx))
                return (CMPLXL(fabsl(ry), rx));
        if (isnan(ry))
                return (CMPLXL(ry, ry));
        return (CMPLXL(fabsl(ry), copysignl(rx, cimagl(z))));
}

static long double complex
clog_for_large_values(long double complex z)
{
        long double x, y;
        long double ax, ay, t;

        x = creall(z);
        y = cimagl(z);
        ax = fabsl(x);
        ay = fabsl(y);
        if (ax < ay) {
                t = ax;
                ax = ay;
                ay = t;
        }

        if (ax > LDBL_MAX / 2)
                return (CMPLXL(logl(hypotl(x / m_e, y / m_e)) + 1, atan2l(y, x)));

        if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
                return (CMPLXL(logl(hypotl(x, y)), atan2l(y, x)));

        return (CMPLXL(logl(ax*ax + ay*ay) / 2, atan2l(y, x)));
}

inline static long double
sum_squares(long double x, long double y)
{
        if (y < SQRT_MIN)
                return (x*x);

        return (x*x + y*y);
}

inline static long double
real_part_reciprocal(long double x, long double y)
{
        long double scale;
        uint16_t hx, hy;
        int16_t ix, iy;

        GET_LDBL_EXPSIGN(hx, x);
        ix = hx & 0x7fff;
        GET_LDBL_EXPSIGN(hy, y);
        iy = hy & 0x7fff;
#define BIAS    (LDBL_MAX_EXP - 1)
#define CUTOFF  (LDBL_MANT_DIG / 2 + 1)
        if (ix - iy >= CUTOFF || isinf(x))
                return (1/x);
        if (iy - ix >= CUTOFF)
                return (x/y/y);
        if (ix <= BIAS + LDBL_MAX_EXP / 2 - CUTOFF)
                return (x/(x*x + y*y));
        scale = 1;
        SET_LDBL_EXPSIGN(scale, 0x7fff - ix);
        x *= scale;
        y *= scale;
        return (x/(x*x + y*y) * scale);
}

long double complex
catanhl(long double complex z)
{
        long double x, y, ax, ay, rx, ry;

        x = creall(z);
        y = cimagl(z);
        ax = fabsl(x);
        ay = fabsl(y);

        if (y == 0 && ax <= 1)
                return (CMPLXL(atanhl(x), y));  /* XXX need atanhl() */

        if (x == 0)
                return (CMPLXL(x, atanl(y)));

        if (isnan(x) || isnan(y)) {
                if (isinf(x))
                        return (CMPLXL(copysignl(0, x), y+y));
                if (isinf(y))
                        return (CMPLXL(copysignl(0, x), copysignl(pio2_hi + pio2_lo, y)));
                return (CMPLXL(x+0.0L+(y+0), x+0.0L+(y+0)));
        }

        if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
                return (CMPLXL(real_part_reciprocal(x, y), copysignl(pio2_hi + pio2_lo, y)));

        if (ax < SQRT_3_EPSILON/2 && ay < SQRT_3_EPSILON/2) {
                raise_inexact();
                return (z);
        }

        if (ax == 1 && ay < LDBL_EPSILON) {
#if 0
                if (ay > 2*LDBL_MIN)
                        rx = - logl(ay/2) / 2;
                else
#endif
                        rx = - (logl(ay) - m_ln2) / 2;
        } else
                rx = log1pl(4*ax / sum_squares(ax-1, ay)) / 4;

        if (ax == 1)
                ry = atan2l(2, -ay) / 2;
        else if (ay < LDBL_EPSILON)
                ry = atan2l(2*ay, (1-ax)*(1+ax)) / 2;
        else
                ry = atan2l(2*ay, (1-ax)*(1+ax) - ay*ay) / 2;

        return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
}

long double complex
catanl(long double complex z)
{
        long double complex w = catanhl(CMPLXL(cimagl(z), creall(z)));
        return (CMPLXL(cimagl(w), creall(w)));
}

#else
__strong_alias(_casinl, casin)
__strong_alias(_catanl, catan)
__strong_alias(cacoshl, cacosh)
__strong_alias(cacosl, cacos)
__strong_alias(casinhl, casinh)
__strong_alias(catanhl, catanh)
#endif