xref: /src/lib/libm/complex/catrig.c

/*	$NetBSD: catrig.c,v 1.3 2022/04/19 20:32:16 rillig Exp $	*/
/*-
 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen (at) 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.
 */

#include <sys/cdefs.h>
#if 0
__FBSDID("$FreeBSD: head/lib/msun/src/catrig.c 275819 2014-12-16 09:21:56Z ed $");
#endif
__RCSID("$NetBSD: catrig.c,v 1.3 2022/04/19 20:32:16 rillig Exp $");

#include "namespace.h"
#ifdef __weak_alias
__weak_alias(casin, _casin)
#endif
#ifdef __weak_alias
__weak_alias(catan, _catan)
#endif

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

#include "math.h"
#include "math_private.h"



#undef isinf
#define isinf(x)	(fabs(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_signbit(x))

/* We need that DBL_EPSILON^2/128 is larger than FOUR_SQRT_MIN. */
static const double
A_crossover =		10, /* Hull et al suggest 1.5, but 10 works better */
B_crossover =		0.6417,			/* suggested by Hull et al */
m_e =			2.7182818284590452e0,	/*  0x15bf0a8b145769.0p-51 */
m_ln2 =			6.9314718055994531e-1,	/*  0x162e42fefa39ef.0p-53 */
pio2_hi =		1.5707963267948966e0,	/*  0x1921fb54442d18.0p-52 */
RECIP_EPSILON =		1 / DBL_EPSILON,
SQRT_3_EPSILON =	2.5809568279517849e-8,	/*  0x1bb67ae8584caa.0p-78 */
SQRT_6_EPSILON =	3.6500241499888571e-8,	/*  0x13988e1409212e.0p-77 */
#if DBL_MAX_EXP == 1024	/* IEEE */
FOUR_SQRT_MIN =		0x1p-509,		/* >= 4 * sqrt(DBL_MIN) */
QUARTER_SQRT_MAX =	0x1p509,		/* <= sqrt(DBL_MAX) / 4 */
SQRT_MIN =		0x1p-511;		/* >= sqrt(DBL_MIN) */
#elif DBL_MAX_EXP == 127 /* VAX */
FOUR_SQRT_MIN =		0x1p-62,		/* >= 4 * sqrt(DBL_MIN) */
QUARTER_SQRT_MAX =	0x1p62,			/* <= sqrt(DBL_MAX) / 4 */
SQRT_MIN =		0x1p-64;		/* >= sqrt(DBL_MIN) */
#else
	#error "unsupported floating point format"
#endif


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

static double complex clog_for_large_values(double complex z);

/*
 * Testing indicates that all these functions are accurate up to 4 ULP.
 * The functions casin(h) and cacos(h) are about 2.5 times slower than asinh.
 * The functions catan(h) are a little under 2 times slower than atanh.
 *
 * The code for casinh, casin, cacos, and cacosh comes first.  The code is
 * rather complicated, and the four functions are highly interdependent.
 *
 * The code for catanh and catan comes at the end.  It is much simpler than
 * the other functions, and the code for these can be disconnected from the
 * rest of the code.
 */

/*
 *			================================
 *			| casinh, casin, cacos, cacosh |
 *			================================
 */

/*
 * 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.
 *
 * Throughout we use the convention z = x + I*y.
 *
 * casinh(z) = sign(x)*log(A+sqrt(A*A-1)) + I*asin(B)
 * where
 * A = (|z+I| + |z-I|) / 2
 * B = (|z+I| - |z-I|) / 2 = y/A
 *
 * These formulas become numerically unstable:
 *   (a) for Re(casinh(z)) when z is close to the line segment [-I, I] (that
 *       is, Re(casinh(z)) is close to 0);
 *   (b) for Im(casinh(z)) when z is close to either of the intervals
 *       [I, I*infinity) or (-I*infinity, -I] (that is, |Im(casinh(z))| is
 *       close to PI/2).
 *
 * These numerical problems are overcome by defining
 * f(a, b) = (hypot(a, b) - b) / 2 = a*a / (hypot(a, b) + b) / 2
 * Then if A < A_crossover, we use
 *   log(A + sqrt(A*A-1)) = log1p((A-1) + sqrt((A-1)*(A+1)))
 *   A-1 = f(x, 1+y) + f(x, 1-y)
 * and if B > B_crossover, we use
 *   asin(B) = atan2(y, sqrt(A*A - y*y)) = atan2(y, sqrt((A+y)*(A-y)))
 *   A-y = f(x, y+1) + f(x, y-1)
 * where without loss of generality we have assumed that x and y are
 * non-negative.
 *
 * Much of the difficulty comes because the intermediate computations may
 * produce overflows or underflows.  This is dealt with in the paper by Hull
 * et al by using exception handling.  We do this by detecting when
 * computations risk underflow or overflow.  The hardest part is handling the
 * underflows when computing f(a, b).
 *
 * Note that the function f(a, b) does not appear explicitly in the paper by
 * Hull et al, but the idea may be found on pages 308 and 309.  Introducing the
 * function f(a, b) allows us to concentrate many of the clever tricks in this
 * paper into one function.
 */

/*
 * Function f(a, b, hypot_a_b) = (hypot(a, b) - b) / 2.
 * Pass hypot(a, b) as the third argument.
 */
static inline double
f(double a, double b, 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);
}

/*
 * All the hard work is contained in this function.
 * x and y are assumed positive or zero, and less than RECIP_EPSILON.
 * Upon return:
 * rx = Re(casinh(z)) = -Im(cacos(y + I*x)).
 * B_is_usable is set to 1 if the value of B is usable.
 * If B_is_usable is set to 0, sqrt_A2my2 = sqrt(A*A - y*y), and new_y = y.
 * If returning sqrt_A2my2 has potential to result in an underflow, it is
 * rescaled, and new_y is similarly rescaled.
 */
static inline void
do_hard_work(double x, double y, double *rx, int *B_is_usable, double *B,
    double *sqrt_A2my2, double *new_y)
{
	double R, S, A; /* A, B, R, and S are as in Hull et al. */
	double Am1, Amy; /* A-1, A-y. */

	R = hypot(x, y + 1);		/* |z+I| */
	S = hypot(x, y - 1);		/* |z-I| */

	/* A = (|z+I| + |z-I|) / 2 */
	A = (R + S) / 2;
	/*
	 * Mathematically A >= 1.  There is a small chance that this will not
	 * be so because of rounding errors.  So we will make certain it is
	 * so.
	 */
	if (A < 1)
		A = 1;

	if (A < A_crossover) {
		/*
		 * Am1 = fp + fm, where fp = f(x, 1+y), and fm = f(x, 1-y).
		 * rx = log1p(Am1 + sqrt(Am1*(A+1)))
		 */
		if (y == 1 && x < DBL_EPSILON * DBL_EPSILON / 128) {
			/*
			 * fp is of order x^2, and fm = x/2.
			 * A = 1 (inexactly).
			 */
			*rx = sqrt(x);
		} else if (x >= DBL_EPSILON * fabs(y - 1)) {
			/*
			 * Underflow will not occur because
			 * x >= DBL_EPSILON^2/128 >= FOUR_SQRT_MIN
			 */
			Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
			*rx = log1p(Am1 + sqrt(Am1 * (A + 1)));
		} else if (y < 1) {
			/*
			 * fp = x*x/(1+y)/4, fm = x*x/(1-y)/4, and
			 * A = 1 (inexactly).
			 */
			*rx = x / sqrt((1 - y) * (1 + y));
		} else {		/* if (y > 1) */
			/*
			 * A-1 = y-1 (inexactly).
			 */
			*rx = log1p((y - 1) + sqrt((y - 1) * (y + 1)));
		}
	} else {
		*rx = log(A + sqrt(A * A - 1));
	}

	*new_y = y;

	if (y < FOUR_SQRT_MIN) {
		/*
		 * Avoid a possible underflow caused by y/A.  For casinh this
		 * would be legitimate, but will be picked up by invoking atan2
		 * later on.  For cacos this would not be legitimate.
		 */
		*B_is_usable = 0;
		*sqrt_A2my2 = A * (2 / DBL_EPSILON);
		*new_y = y * (2 / DBL_EPSILON);
		return;
	}

	/* B = (|z+I| - |z-I|) / 2 = y/A */
	*B = y / A;
	*B_is_usable = 1;

	if (*B > B_crossover) {
		*B_is_usable = 0;
		/*
		 * Amy = fp + fm, where fp = f(x, y+1), and fm = f(x, y-1).
		 * sqrt_A2my2 = sqrt(Amy*(A+y))
		 */
		if (y == 1 && x < DBL_EPSILON / 128) {
			/*
			 * fp is of order x^2, and fm = x/2.
			 * A = 1 (inexactly).
			 */
			*sqrt_A2my2 = sqrt(x) * sqrt((A + y) / 2);
		} else if (x >= DBL_EPSILON * fabs(y - 1)) {
			/*
			 * Underflow will not occur because
			 * x >= DBL_EPSILON/128 >= FOUR_SQRT_MIN
			 * and
			 * x >= DBL_EPSILON^2 >= FOUR_SQRT_MIN
			 */
			Amy = f(x, y + 1, R) + f(x, y - 1, S);
			*sqrt_A2my2 = sqrt(Amy * (A + y));
		} else if (y > 1) {
			/*
			 * fp = x*x/(y+1)/4, fm = x*x/(y-1)/4, and
			 * A = y (inexactly).
			 *
			 * y < RECIP_EPSILON.  So the following
			 * scaling should avoid any underflow problems.
			 */
			*sqrt_A2my2 = x * (4 / DBL_EPSILON / DBL_EPSILON) * y /
			    sqrt((y + 1) * (y - 1));
			*new_y = y * (4 / DBL_EPSILON / DBL_EPSILON);
		} else {		/* if (y < 1) */
			/*
			 * fm = 1-y >= DBL_EPSILON, fp is of order x^2, and
			 * A = 1 (inexactly).
			 */
			*sqrt_A2my2 = sqrt((1 - y) * (1 + y));
		}
	}
}

/*
 * casinh(z) = z + O(z^3)   as z -> 0
 *
 * casinh(z) = sign(x)*clog(sign(x)*z) + O(1/z^2)   as z -> infinity
 * The above formula works for the imaginary part as well, because
 * Im(casinh(z)) = sign(x)*atan2(sign(x)*y, fabs(x)) + O(y/z^3)
 *    as z -> infinity, uniformly in y
 */
double complex
casinh(double complex z)
{
	double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
	int B_is_usable;
	double complex w;

	x = creal(z);
	y = cimag(z);
	ax = fabs(x);
	ay = fabs(y);

	if (isnan(x) || isnan(y)) {
		/* casinh(+-Inf + I*NaN) = +-Inf + I*NaN */
		if (isinf(x))
			return (CMPLX(x, y + y));
		/* casinh(NaN + I*+-Inf) = opt(+-)Inf + I*NaN */
		if (isinf(y))
			return (CMPLX(y, x + x));
		/* casinh(NaN + I*0) = NaN + I*0 */
		if (y == 0)
			return (CMPLX(x + x, y));
		/*
		 * All other cases involving NaN return NaN + I*NaN.
		 * C99 leaves it optional whether to raise invalid if one of
		 * the arguments is not NaN, so we opt not to raise it.
		 */
		return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
	}

	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
		/* clog...() will raise inexact unless x or y is infinite. */
		if (signbit(x) == 0)
			w = clog_for_large_values(z) + m_ln2;
		else
			w = clog_for_large_values(-z) + m_ln2;
		return (CMPLX(copysign(creal(w), x), copysign(cimag(w), y)));
	}

	/* Avoid spuriously raising inexact for z = 0. */
	if (x == 0 && y == 0)
		return (z);

	/* All remaining cases are inexact. */
	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 = asin(B);
	else
		ry = atan2(new_y, sqrt_A2my2);
	return (CMPLX(copysign(rx, x), copysign(ry, y)));
}

/*
 * casin(z) = reverse(casinh(reverse(z)))
 * where reverse(x + I*y) = y + I*x = I*conj(z).
 */
double complex
casin(double complex z)
{
	double complex w = casinh(CMPLX(cimag(z), creal(z)));

	return (CMPLX(cimag(w), creal(w)));
}

/*
 * cacos(z) = PI/2 - casin(z)
 * but do the computation carefully so cacos(z) is accurate when z is
 * close to 1.
 *
 * cacos(z) = PI/2 - z + O(z^3)   as z -> 0
 *
 * cacos(z) = -sign(y)*I*clog(z) + O(1/z^2)   as z -> infinity
 * The above formula works for the real part as well, because
 * Re(cacos(z)) = atan2(fabs(y), x) + O(y/z^3)
 *    as z -> infinity, uniformly in y
 */
double complex
cacos(double complex z)
{
	double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
	int sx, sy;
	int B_is_usable;
	double complex w;

	x = creal(z);
	y = cimag(z);
	sx = signbit(x);
	sy = signbit(y);
	ax = fabs(x);
	ay = fabs(y);

	if (isnan(x) || isnan(y)) {
		/* cacos(+-Inf + I*NaN) = NaN + I*opt(-)Inf */
		if (isinf(x))
			return (CMPLX(y + y, -INFINITY));
		/* cacos(NaN + I*+-Inf) = NaN + I*-+Inf */
		if (isinf(y))
			return (CMPLX(x + x, -y));
		/* cacos(0 + I*NaN) = PI/2 + I*NaN with inexact */
		if (x == 0)
			return (CMPLX(pio2_hi + pio2_lo, y + y));
		/*
		 * All other cases involving NaN return NaN + I*NaN.
		 * C99 leaves it optional whether to raise invalid if one of
		 * the arguments is not NaN, so we opt not to raise it.
		 */
		return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
	}

	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
		/* clog...() will raise inexact unless x or y is infinite. */
		w = clog_for_large_values(z);
		rx = fabs(cimag(w));
		ry = creal(w) + m_ln2;
		if (sy == 0)
			ry = -ry;
		return (CMPLX(rx, ry));
	}

	/* Avoid spuriously raising inexact for z = 1. */
	if (x == 1 && y == 0)
		return (CMPLX(0, -y));

	/* All remaining cases are inexact. */
	raise_inexact();

	if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
		return (CMPLX(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 = acos(B);
		else
			rx = acos(-B);
	} else {
		if (sx == 0)
			rx = atan2(sqrt_A2mx2, new_x);
		else
			rx = atan2(sqrt_A2mx2, -new_x);
	}
	if (sy == 0)
		ry = -ry;
	return (CMPLX(rx, ry));
}

/*
 * cacosh(z) = I*cacos(z) or -I*cacos(z)
 * where the sign is chosen so Re(cacosh(z)) >= 0.
 */
double complex
cacosh(double complex z)
{
	double complex w;
	double rx, ry;

	w = cacos(z);
	rx = creal(w);
	ry = cimag(w);
	/* cacosh(NaN + I*NaN) = NaN + I*NaN */
	if (isnan(rx) && isnan(ry))
		return (CMPLX(ry, rx));
	/* cacosh(NaN + I*+-Inf) = +Inf + I*NaN */
	/* cacosh(+-Inf + I*NaN) = +Inf + I*NaN */
	if (isnan(rx))
		return (CMPLX(fabs(ry), rx));
	/* cacosh(0 + I*NaN) = NaN + I*NaN */
	if (isnan(ry))
		return (CMPLX(ry, ry));
	return (CMPLX(fabs(ry), copysign(rx, cimag(z))));
}

/*
 * Optimized version of clog() for |z| finite and larger than ~RECIP_EPSILON.
 */
static double complex
clog_for_large_values(double complex z)
{
	double x, y;
	double ax, ay, t;

	x = creal(z);
	y = cimag(z);
	ax = fabs(x);
	ay = fabs(y);
	if (ax < ay) {
		t = ax;
		ax = ay;
		ay = t;
	}

	/*
	 * Avoid overflow in hypot() when x and y are both very large.
	 * Divide x and y by E, and then add 1 to the logarithm.  This depends
	 * on E being larger than sqrt(2).
	 * Dividing by E causes an insignificant loss of accuracy; however
	 * this method is still poor since it is uneccessarily slow.
	 */
	if (ax > DBL_MAX / 2)
		return (CMPLX(log(hypot(x / m_e, y / m_e)) + 1, atan2(y, x)));

	/*
	 * Avoid overflow when x or y is large.  Avoid underflow when x or
	 * y is small.
	 */
	if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
		return (CMPLX(log(hypot(x, y)), atan2(y, x)));

	return (CMPLX(log(ax * ax + ay * ay) / 2, atan2(y, x)));
}

/*
 *				=================
 *				| catanh, catan |
 *				=================
 */

/*
 * sum_squares(x,y) = x*x + y*y (or just x*x if y*y would underflow).
 * Assumes x*x and y*y will not overflow.
 * Assumes x and y are finite.
 * Assumes y is non-negative.
 * Assumes fabs(x) >= DBL_EPSILON.
 */
static inline double
sum_squares(double x, double y)
{

	/* Avoid underflow when y is small. */
	if (y < SQRT_MIN)
		return (x * x);

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

/*
 * real_part_reciprocal(x, y) = Re(1/(x+I*y)) = x/(x*x + y*y).
 * Assumes x and y are not NaN, and one of x and y is larger than
 * RECIP_EPSILON.  We avoid unwarranted underflow.  It is important to not use
 * the code creal(1/z), because the imaginary part may produce an unwanted
 * underflow.
 * This is only called in a context where inexact is always raised before
 * the call, so no effort is made to avoid or force inexact.
 */
static inline double
real_part_reciprocal(double x, double y)
{
	double scale;
	uint32_t hx, hy;
	int32_t ix, iy;

	/*
	 * This code is inspired by the C99 document n1124.pdf, Section G.5.1,
	 * example 2.
	 */
	GET_HIGH_WORD(hx, x);
	ix = hx & 0x7ff00000;
	GET_HIGH_WORD(hy, y);
	iy = hy & 0x7ff00000;
#define	BIAS	(DBL_MAX_EXP - 1)
/* XXX more guard digits are useful iff there is extra precision. */
#define	CUTOFF	(DBL_MANT_DIG / 2 + 1)	/* just half or 1 guard digit */
	if (ix - iy >= CUTOFF << 20 || isinf(x))
		return (1 / x);		/* +-Inf -> +-0 is special */
	if (iy - ix >= CUTOFF << 20)
		return (x / y / y);	/* should avoid double div, but hard */
	if (ix <= (BIAS + DBL_MAX_EXP / 2 - CUTOFF) << 20)
		return (x / (x * x + y * y));
	scale = 1;
	SET_HIGH_WORD(scale, 0x7ff00000 - ix);	/* 2**(1-ilogb(x)) */
	x *= scale;
	y *= scale;
	return (x / (x * x + y * y) * scale);
}

/*
 * catanh(z) = log((1+z)/(1-z)) / 2
 *           = log1p(4*x / |z-1|^2) / 4
 *             + I * atan2(2*y, (1-x)*(1+x)-y*y) / 2
 *
 * catanh(z) = z + O(z^3)   as z -> 0
 *
 * catanh(z) = 1/z + sign(y)*I*PI/2 + O(1/z^3)   as z -> infinity
 * The above formula works for the real part as well, because
 * Re(catanh(z)) = x/|z|^2 + O(x/z^4)
 *    as z -> infinity, uniformly in x
 */
double complex
catanh(double complex z)
{
	double x, y, ax, ay, rx, ry;

	x = creal(z);
	y = cimag(z);
	ax = fabs(x);
	ay = fabs(y);

	/* This helps handle many cases. */
	if (y == 0 && ax <= 1)
		return (CMPLX(atanh(x), y));

	/* To ensure the same accuracy as atan(), and to filter out z = 0. */
	if (x == 0)
		return (CMPLX(x, atan(y)));

	if (isnan(x) || isnan(y)) {
		/* catanh(+-Inf + I*NaN) = +-0 + I*NaN */
		if (isinf(x))
			return (CMPLX(copysign(0, x), y + y));
		/* catanh(NaN + I*+-Inf) = sign(NaN)0 + I*+-PI/2 */
		if (isinf(y))
			return (CMPLX(copysign(0, x),
			    copysign(pio2_hi + pio2_lo, y)));
		/*
		 * All other cases involving NaN return NaN + I*NaN.
		 * C99 leaves it optional whether to raise invalid if one of
		 * the arguments is not NaN, so we opt not to raise it.
		 */
		return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
	}

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

	if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
		/*
		 * z = 0 was filtered out above.  All other cases must raise
		 * inexact, but this is the only only that needs to do it
		 * explicitly.
		 */
		raise_inexact();
		return (z);
	}

	if (ax == 1 && ay < DBL_EPSILON)
		rx = (m_ln2 - log(ay)) / 2;
	else
		rx = log1p(4 * ax / sum_squares(ax - 1, ay)) / 4;

	if (ax == 1)
		ry = atan2(2, -ay) / 2;
	else if (ay < DBL_EPSILON)
		ry = atan2(2 * ay, (1 - ax) * (1 + ax)) / 2;
	else
		ry = atan2(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;

	return (CMPLX(copysign(rx, x), copysign(ry, y)));
}

/*
 * catan(z) = reverse(catanh(reverse(z)))
 * where reverse(x + I*y) = y + I*x = I*conj(z).
 */
double complex
catan(double complex z)
{
	double complex w = catanh(CMPLX(cimag(z), creal(z)));

	return (CMPLX(cimag(w), creal(w)));
}