/* Complex math module */

/* much code borrowed from mathmodule.c */

#include "Python.h"

/* we need DBL_MAX, DBL_MIN, DBL_EPSILON, DBL_MANT_DIG and FLT_RADIX from

   float.h.  We assume that FLT_RADIX is either 2 or 16. */

#include <float.h>

#if (FLT_RADIX != 2 && FLT_RADIX != 16)

#error "Modules/cmathmodule.c expects FLT_RADIX to be 2 or 16"

#endif

#ifndef M_LN2

#define M_LN2 (0.6931471805599453094) /* natural log of 2 */

#endif

#ifndef M_LN10

#define M_LN10 (2.302585092994045684) /* natural log of 10 */

#endif

/*

   CM_LARGE_DOUBLE is used to avoid spurious overflow in the sqrt, log,

   inverse trig and inverse hyperbolic trig functions.  Its log is used in the

   evaluation of exp, cos, cosh, sin, sinh, tan, and tanh to avoid unecessary

   overflow.

 */

#define CM_LARGE_DOUBLE (DBL_MAX/4.)

#define CM_SQRT_LARGE_DOUBLE (sqrt(CM_LARGE_DOUBLE))

#define CM_LOG_LARGE_DOUBLE (log(CM_LARGE_DOUBLE))

#define CM_SQRT_DBL_MIN (sqrt(DBL_MIN))

/* 

   CM_SCALE_UP is an odd integer chosen such that multiplication by

   2**CM_SCALE_UP is sufficient to turn a subnormal into a normal.

   CM_SCALE_DOWN is (-(CM_SCALE_UP+1)/2).  These scalings are used to compute

   square roots accurately when the real and imaginary parts of the argument

   are subnormal.

*/

#if FLT_RADIX==2

#define CM_SCALE_UP (2*(DBL_MANT_DIG/2) + 1)

#elif FLT_RADIX==16

#define CM_SCALE_UP (4*DBL_MANT_DIG+1)

#endif

#define CM_SCALE_DOWN (-(CM_SCALE_UP+1)/2)

/* forward declarations */

static Py_complex c_asinh(Py_complex);

static Py_complex c_atanh(Py_complex);

static Py_complex c_cosh(Py_complex);

static Py_complex c_sinh(Py_complex);

static Py_complex c_sqrt(Py_complex);

static Py_complex c_tanh(Py_complex);

static PyObject * math_error(void);

/* Code to deal with special values (infinities, NaNs, etc.). */

/* special_type takes a double and returns an integer code indicating

   the type of the double as follows:

*/

enum special_types {

	ST_NINF,	/* 0, negative infinity */

	ST_NEG,		/* 1, negative finite number (nonzero) */

	ST_NZERO,	/* 2, -0. */

	ST_PZERO,	/* 3, +0. */

	ST_POS,		/* 4, positive finite number (nonzero) */

	ST_PINF,	/* 5, positive infinity */

	ST_NAN,		/* 6, Not a Number */

};

static enum special_types

special_type(double d)

{

	if (Py_IS_FINITE(d)) {

		if (d != 0) {

			if (copysign(1., d) == 1.)

				return ST_POS;

			else

				return ST_NEG;

		}

		else {

			if (copysign(1., d) == 1.)

				return ST_PZERO;

			else

				return ST_NZERO;

		}

	}

	if (Py_IS_NAN(d))

		return ST_NAN;

	if (copysign(1., d) == 1.)

		return ST_PINF;

	else

		return ST_NINF;

}

#define SPECIAL_VALUE(z, table)						\

	if (!Py_IS_FINITE((z).real) || !Py_IS_FINITE((z).imag)) {	\

		errno = 0;                                              \

		return table[special_type((z).real)]	                \

			    [special_type((z).imag)];			\

	}

#define P Py_MATH_PI

#define P14 0.25*Py_MATH_PI

#define P12 0.5*Py_MATH_PI

#define P34 0.75*Py_MATH_PI

#define INF Py_HUGE_VAL

#define N Py_NAN

#define U -9.5426319407711027e33 /* unlikely value, used as placeholder */

/* First, the C functions that do the real work.  Each of the c_*

   functions computes and returns the C99 Annex G recommended result

   and also sets errno as follows: errno = 0 if no floating-point

   exception is associated with the result; errno = EDOM if C99 Annex

   G recommends raising divide-by-zero or invalid for this result; and

   errno = ERANGE where the overflow floating-point signal should be

   raised.

*/

static Py_complex acos_special_values[7][7];

static Py_complex

c_acos(Py_complex z)

{

	Py_complex s1, s2, r;

	SPECIAL_VALUE(z, acos_special_values);

	if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) {

		/* avoid unnecessary overflow for large arguments */

		r.real = atan2(fabs(z.imag), z.real);

		/* split into cases to make sure that the branch cut has the

		   correct continuity on systems with unsigned zeros */

		if (z.real < 0.) {

			r.imag = -copysign(log(hypot(z.real/2., z.imag/2.)) +

					   M_LN2*2., z.imag);

		} else {

			r.imag = copysign(log(hypot(z.real/2., z.imag/2.)) +

					  M_LN2*2., -z.imag);

		}

	} else {

		s1.real = 1.-z.real;

		s1.imag = -z.imag;

		s1 = c_sqrt(s1);

		s2.real = 1.+z.real;

		s2.imag = z.imag;

		s2 = c_sqrt(s2);

		r.real = 2.*atan2(s1.real, s2.real);

		r.imag = asinh(s2.real*s1.imag - s2.imag*s1.real);

	}

	errno = 0;

	return r;

}

PyDoc_STRVAR(c_acos_doc,

"acos(x)\n"

"\n"

"Return the arc cosine of x.");

static Py_complex acosh_special_values[7][7];

static Py_complex

c_acosh(Py_complex z)

{

	Py_complex s1, s2, r;

	SPECIAL_VALUE(z, acosh_special_values);

	if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) {

		/* avoid unnecessary overflow for large arguments */

		r.real = log(hypot(z.real/2., z.imag/2.)) + M_LN2*2.;

		r.imag = atan2(z.imag, z.real);

	} else {

		s1.real = z.real - 1.;

		s1.imag = z.imag;

		s1 = c_sqrt(s1);

		s2.real = z.real + 1.;

		s2.imag = z.imag;

		s2 = c_sqrt(s2);

		r.real = asinh(s1.real*s2.real + s1.imag*s2.imag);

		r.imag = 2.*atan2(s1.imag, s2.real);

	}

	errno = 0;

	return r;

}

PyDoc_STRVAR(c_acosh_doc,

"acosh(x)\n"

"\n"

"Return the hyperbolic arccosine of x.");

static Py_complex

c_asin(Py_complex z)

{

	/* asin(z) = -i asinh(iz) */

	Py_complex s, r;

	s.real = -z.imag;

	s.imag = z.real;

	s = c_asinh(s);

	r.real = s.imag;

	r.imag = -s.real;

	return r;

}

PyDoc_STRVAR(c_asin_doc,

"asin(x)\n"

"\n"

"Return the arc sine of x.");

static Py_complex asinh_special_values[7][7];

static Py_complex

c_asinh(Py_complex z)

{

	Py_complex s1, s2, r;

	SPECIAL_VALUE(z, asinh_special_values);

	if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) {

		if (z.imag >= 0.) {

			r.real = copysign(log(hypot(z.real/2., z.imag/2.)) +

					  M_LN2*2., z.real);

		} else {

			r.real = -copysign(log(hypot(z.real/2., z.imag/2.)) +

					   M_LN2*2., -z.real);

		}

		r.imag = atan2(z.imag, fabs(z.real));

	} else {

		s1.real = 1.+z.imag;

		s1.imag = -z.real;

		s1 = c_sqrt(s1);

		s2.real = 1.-z.imag;

		s2.imag = z.real;

		s2 = c_sqrt(s2);

		r.real = asinh(s1.real*s2.imag-s2.real*s1.imag);

		r.imag = atan2(z.imag, s1.real*s2.real-s1.imag*s2.imag);

	}

	errno = 0;

	return r;

}

PyDoc_STRVAR(c_asinh_doc,

"asinh(x)\n"

"\n"

"Return the hyperbolic arc sine of x.");

static Py_complex

c_atan(Py_complex z)

{

	/* atan(z) = -i atanh(iz) */

	Py_complex s, r;

	s.real = -z.imag;

	s.imag = z.real;

	s = c_atanh(s);

	r.real = s.imag;

	r.imag = -s.real;

	return r;

}

/* Windows screws up atan2 for inf and nan, and alpha Tru64 5.1 doesn't follow

   C99 for atan2(0., 0.). */

static double

c_atan2(Py_complex z)

{

	if (Py_IS_NAN(z.real) || Py_IS_NAN(z.imag))

		return Py_NAN;

	if (Py_IS_INFINITY(z.imag)) {

		if (Py_IS_INFINITY(z.real)) {

			if (copysign(1., z.real) == 1.)

				/* atan2(+-inf, +inf) == +-pi/4 */

				return copysign(0.25*Py_MATH_PI, z.imag);

			else

				/* atan2(+-inf, -inf) == +-pi*3/4 */

				return copysign(0.75*Py_MATH_PI, z.imag);

		}

		/* atan2(+-inf, x) == +-pi/2 for finite x */

		return copysign(0.5*Py_MATH_PI, z.imag);

	}

	if (Py_IS_INFINITY(z.real) || z.imag == 0.) {

		if (copysign(1., z.real) == 1.)

			/* atan2(+-y, +inf) = atan2(+-0, +x) = +-0. */

			return copysign(0., z.imag);

		else

			/* atan2(+-y, -inf) = atan2(+-0., -x) = +-pi. */

			return copysign(Py_MATH_PI, z.imag);

	}

	return atan2(z.imag, z.real);

}

PyDoc_STRVAR(c_atan_doc,

"atan(x)\n"

"\n"

"Return the arc tangent of x.");

static Py_complex atanh_special_values[7][7];

static Py_complex

c_atanh(Py_complex z)

{

	Py_complex r;

	double ay, h;

	SPECIAL_VALUE(z, atanh_special_values);

	/* Reduce to case where z.real >= 0., using atanh(z) = -atanh(-z). */

	if (z.real < 0.) {

		return c_neg(c_atanh(c_neg(z)));

	}

	ay = fabs(z.imag);

	if (z.real > CM_SQRT_LARGE_DOUBLE || ay > CM_SQRT_LARGE_DOUBLE) {

		/*

		   if abs(z) is large then we use the approximation

		   atanh(z) ~ 1/z +/- i*pi/2 (+/- depending on the sign

		   of z.imag)

		*/

		h = hypot(z.real/2., z.imag/2.);  /* safe from overflow */

		r.real = z.real/4./h/h;

		/* the two negations in the next line cancel each other out

		   except when working with unsigned zeros: they're there to

		   ensure that the branch cut has the correct continuity on

		   systems that don't support signed zeros */

		r.imag = -copysign(Py_MATH_PI/2., -z.imag);

		errno = 0;

	} else if (z.real == 1. && ay < CM_SQRT_DBL_MIN) {

		/* C99 standard says:  atanh(1+/-0.) should be inf +/- 0i */

		if (ay == 0.) {

			r.real = INF;

			r.imag = z.imag;

			errno = EDOM;

		} else {

			r.real = -log(sqrt(ay)/sqrt(hypot(ay, 2.)));

			r.imag = copysign(atan2(2., -ay)/2, z.imag);

			errno = 0;

		}

	} else {

		r.real = log1p(4.*z.real/((1-z.real)*(1-z.real) + ay*ay))/4.;

		r.imag = -atan2(-2.*z.imag, (1-z.real)*(1+z.real) - ay*ay)/2.;

		errno = 0;

	}

	return r;

}

PyDoc_STRVAR(c_atanh_doc,

"atanh(x)\n"

"\n"

"Return the hyperbolic arc tangent of x.");

static Py_complex

c_cos(Py_complex z)

{

	/* cos(z) = cosh(iz) */

	Py_complex r;

	r.real = -z.imag;

	r.imag = z.real;

	r = c_cosh(r);

	return r;

}

PyDoc_STRVAR(c_cos_doc,

"cos(x)\n"

"n"

"Return the cosine of x.");

/* cosh(infinity + i*y) needs to be dealt with specially */

static Py_complex cosh_special_values[7][7];

static Py_complex

c_cosh(Py_complex z)

{

	Py_complex r;

	double x_minus_one;

	/* special treatment for cosh(+/-inf + iy) if y is not a NaN */

	if (!Py_IS_FINITE(z.real) || !Py_IS_FINITE(z.imag)) {

		if (Py_IS_INFINITY(z.real) && Py_IS_FINITE(z.imag) &&

		    (z.imag != 0.)) {

			if (z.real > 0) {

				r.real = copysign(INF, cos(z.imag));

				r.imag = copysign(INF, sin(z.imag));

			}

			else {

				r.real = copysign(INF, cos(z.imag));

				r.imag = -copysign(INF, sin(z.imag));

			}

		}

		else {

			r = cosh_special_values[special_type(z.real)]

				               [special_type(z.imag)];

		}

		/* need to set errno = EDOM if y is +/- infinity and x is not

		   a NaN */

		if (Py_IS_INFINITY(z.imag) && !Py_IS_NAN(z.real))

			errno = EDOM;

		else

			errno = 0;

		return r;

	}

	if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) {

		/* deal correctly with cases where cosh(z.real) overflows but

		   cosh(z) does not. */

		x_minus_one = z.real - copysign(1., z.real);

		r.real = cos(z.imag) * cosh(x_minus_one) * Py_MATH_E;

		r.imag = sin(z.imag) * sinh(x_minus_one) * Py_MATH_E;

	} else {

		r.real = cos(z.imag) * cosh(z.real);

		r.imag = sin(z.imag) * sinh(z.real);

	}

	/* detect overflow, and set errno accordingly */

	if (Py_IS_INFINITY(r.real) || Py_IS_INFINITY(r.imag))

		errno = ERANGE;

	else

		errno = 0;

	return r;

}

PyDoc_STRVAR(c_cosh_doc,

"cosh(x)\n"

"n"

"Return the hyperbolic cosine of x.");

/* exp(infinity + i*y) and exp(-infinity + i*y) need special treatment for

   finite y */

static Py_complex exp_special_values[7][7];

static Py_complex

c_exp(Py_complex z)

{

	Py_complex r;

	double l;

	if (!Py_IS_FINITE(z.real) || !Py_IS_FINITE(z.imag)) {

		if (Py_IS_INFINITY(z.real) && Py_IS_FINITE(z.imag)

		    && (z.imag != 0.)) {

			if (z.real > 0) {

				r.real = copysign(INF, cos(z.imag));

				r.imag = copysign(INF, sin(z.imag));

			}

			else {

				r.real = copysign(0., cos(z.imag));

				r.imag = copysign(0., sin(z.imag));

			}

		}

		else {

			r = exp_special_values[special_type(z.real)]

				              [special_type(z.imag)];

		}

		/* need to set errno = EDOM if y is +/- infinity and x is not

		   a NaN and not -infinity */

		if (Py_IS_INFINITY(z.imag) &&

		    (Py_IS_FINITE(z.real) ||

		     (Py_IS_INFINITY(z.real) && z.real > 0)))

			errno = EDOM;

		else

			errno = 0;

		return r;

	}

	if (z.real > CM_LOG_LARGE_DOUBLE) {

		l = exp(z.real-1.);

		r.real = l*cos(z.imag)*Py_MATH_E;

		r.imag = l*sin(z.imag)*Py_MATH_E;

	} else {

		l = exp(z.real);

		r.real = l*cos(z.imag);

		r.imag = l*sin(z.imag);

	}

	/* detect overflow, and set errno accordingly */

	if (Py_IS_INFINITY(r.real) || Py_IS_INFINITY(r.imag))

		errno = ERANGE;

	else

		errno = 0;

	return r;

}

PyDoc_STRVAR(c_exp_doc,

"exp(x)\n"

"\n"

"Return the exponential value e**x.");

static Py_complex log_special_values[7][7];

static Py_complex

c_log(Py_complex z)

{

	/*

	   The usual formula for the real part is log(hypot(z.real, z.imag)).

	   There are four situations where this formula is potentially

	   problematic:

	   (1) the absolute value of z is subnormal.  Then hypot is subnormal,

	   so has fewer than the usual number of bits of accuracy, hence may

	   have large relative error.  This then gives a large absolute error

	   in the log.  This can be solved by rescaling z by a suitable power

	   of 2.

	   (2) the absolute value of z is greater than DBL_MAX (e.g. when both

	   z.real and z.imag are within a factor of 1/sqrt(2) of DBL_MAX)

	   Again, rescaling solves this.

	   (3) the absolute value of z is close to 1.  In this case it's

	   difficult to achieve good accuracy, at least in part because a

	   change of 1ulp in the real or imaginary part of z can result in a

	   change of billions of ulps in the correctly rounded answer.

	   (4) z = 0.  The simplest thing to do here is to call the

	   floating-point log with an argument of 0, and let its behaviour

	   (returning -infinity, signaling a floating-point exception, setting

	   errno, or whatever) determine that of c_log.  So the usual formula

	   is fine here.

	 */

	Py_complex r;

	double ax, ay, am, an, h;