--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/symbian-qemu-0.9.1-12/python-2.6.1/Modules/cmathmodule.c Fri Jul 31 15:01:17 2009 +0100
@@ -0,0 +1,1209 @@
+/* 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;
+
+ SPECIAL_VALUE(z, log_special_values);
+
+ ax = fabs(z.real);
+ ay = fabs(z.imag);
+
+ if (ax > CM_LARGE_DOUBLE || ay > CM_LARGE_DOUBLE) {
+ r.real = log(hypot(ax/2., ay/2.)) + M_LN2;
+ } else if (ax < DBL_MIN && ay < DBL_MIN) {
+ if (ax > 0. || ay > 0.) {
+ /* catch cases where hypot(ax, ay) is subnormal */
+ r.real = log(hypot(ldexp(ax, DBL_MANT_DIG),
+ ldexp(ay, DBL_MANT_DIG))) - DBL_MANT_DIG*M_LN2;
+ }
+ else {
+ /* log(+/-0. +/- 0i) */
+ r.real = -INF;
+ r.imag = atan2(z.imag, z.real);
+ errno = EDOM;
+ return r;
+ }
+ } else {
+ h = hypot(ax, ay);
+ if (0.71 <= h && h <= 1.73) {
+ am = ax > ay ? ax : ay; /* max(ax, ay) */
+ an = ax > ay ? ay : ax; /* min(ax, ay) */
+ r.real = log1p((am-1)*(am+1)+an*an)/2.;
+ } else {
+ r.real = log(h);
+ }
+ }
+ r.imag = atan2(z.imag, z.real);
+ errno = 0;
+ return r;
+}
+
+
+static Py_complex
+c_log10(Py_complex z)
+{
+ Py_complex r;
+ int errno_save;
+
+ r = c_log(z);
+ errno_save = errno; /* just in case the divisions affect errno */
+ r.real = r.real / M_LN10;
+ r.imag = r.imag / M_LN10;
+ errno = errno_save;
+ return r;
+}
+
+PyDoc_STRVAR(c_log10_doc,
+"log10(x)\n"
+"\n"
+"Return the base-10 logarithm of x.");
+
+
+static Py_complex
+c_sin(Py_complex z)
+{
+ /* sin(z) = -i sin(iz) */
+ Py_complex s, r;
+ s.real = -z.imag;
+ s.imag = z.real;
+ s = c_sinh(s);
+ r.real = s.imag;
+ r.imag = -s.real;
+ return r;
+}
+
+PyDoc_STRVAR(c_sin_doc,
+"sin(x)\n"
+"\n"
+"Return the sine of x.");
+
+
+/* sinh(infinity + i*y) needs to be dealt with specially */
+static Py_complex sinh_special_values[7][7];
+
+static Py_complex
+c_sinh(Py_complex z)
+{
+ Py_complex r;
+ double x_minus_one;
+
+ /* special treatment for sinh(+/-inf + iy) if y is finite and
+ nonzero */
+ 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 = sinh_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) {
+ x_minus_one = z.real - copysign(1., z.real);
+ r.real = cos(z.imag) * sinh(x_minus_one) * Py_MATH_E;
+ r.imag = sin(z.imag) * cosh(x_minus_one) * Py_MATH_E;
+ } else {
+ r.real = cos(z.imag) * sinh(z.real);
+ r.imag = sin(z.imag) * cosh(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_sinh_doc,
+"sinh(x)\n"
+"\n"
+"Return the hyperbolic sine of x.");
+
+
+static Py_complex sqrt_special_values[7][7];
+
+static Py_complex
+c_sqrt(Py_complex z)
+{
+ /*
+ Method: use symmetries to reduce to the case when x = z.real and y
+ = z.imag are nonnegative. Then the real part of the result is
+ given by
+
+ s = sqrt((x + hypot(x, y))/2)
+
+ and the imaginary part is
+
+ d = (y/2)/s
+
+ If either x or y is very large then there's a risk of overflow in
+ computation of the expression x + hypot(x, y). We can avoid this
+ by rewriting the formula for s as:
+
+ s = 2*sqrt(x/8 + hypot(x/8, y/8))
+
+ This costs us two extra multiplications/divisions, but avoids the
+ overhead of checking for x and y large.
+
+ If both x and y are subnormal then hypot(x, y) may also be
+ subnormal, so will lack full precision. We solve this by rescaling
+ x and y by a sufficiently large power of 2 to ensure that x and y
+ are normal.
+ */
+
+
+ Py_complex r;
+ double s,d;
+ double ax, ay;
+
+ SPECIAL_VALUE(z, sqrt_special_values);
+
+ if (z.real == 0. && z.imag == 0.) {
+ r.real = 0.;
+ r.imag = z.imag;
+ return r;
+ }
+
+ ax = fabs(z.real);
+ ay = fabs(z.imag);
+
+ if (ax < DBL_MIN && ay < DBL_MIN && (ax > 0. || ay > 0.)) {
+ /* here we catch cases where hypot(ax, ay) is subnormal */
+ ax = ldexp(ax, CM_SCALE_UP);
+ s = ldexp(sqrt(ax + hypot(ax, ldexp(ay, CM_SCALE_UP))),
+ CM_SCALE_DOWN);
+ } else {
+ ax /= 8.;
+ s = 2.*sqrt(ax + hypot(ax, ay/8.));
+ }
+ d = ay/(2.*s);
+
+ if (z.real >= 0.) {
+ r.real = s;
+ r.imag = copysign(d, z.imag);
+ } else {
+ r.real = d;
+ r.imag = copysign(s, z.imag);
+ }
+ errno = 0;
+ return r;
+}
+
+PyDoc_STRVAR(c_sqrt_doc,
+"sqrt(x)\n"
+"\n"
+"Return the square root of x.");
+
+
+static Py_complex
+c_tan(Py_complex z)
+{
+ /* tan(z) = -i tanh(iz) */
+ Py_complex s, r;
+ s.real = -z.imag;
+ s.imag = z.real;
+ s = c_tanh(s);
+ r.real = s.imag;
+ r.imag = -s.real;
+ return r;
+}
+
+PyDoc_STRVAR(c_tan_doc,
+"tan(x)\n"
+"\n"
+"Return the tangent of x.");
+
+
+/* tanh(infinity + i*y) needs to be dealt with specially */
+static Py_complex tanh_special_values[7][7];
+
+static Py_complex
+c_tanh(Py_complex z)
+{
+ /* Formula:
+
+ tanh(x+iy) = (tanh(x)(1+tan(y)^2) + i tan(y)(1-tanh(x))^2) /
+ (1+tan(y)^2 tanh(x)^2)
+
+ To avoid excessive roundoff error, 1-tanh(x)^2 is better computed
+ as 1/cosh(x)^2. When abs(x) is large, we approximate 1-tanh(x)^2
+ by 4 exp(-2*x) instead, to avoid possible overflow in the
+ computation of cosh(x).
+
+ */
+
+ Py_complex r;
+ double tx, ty, cx, txty, denom;
+
+ /* special treatment for tanh(+/-inf + iy) if y is finite and
+ nonzero */
+ 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 = 1.0;
+ r.imag = copysign(0.,
+ 2.*sin(z.imag)*cos(z.imag));
+ }
+ else {
+ r.real = -1.0;
+ r.imag = copysign(0.,
+ 2.*sin(z.imag)*cos(z.imag));
+ }
+ }
+ else {
+ r = tanh_special_values[special_type(z.real)]
+ [special_type(z.imag)];
+ }
+ /* need to set errno = EDOM if z.imag is +/-infinity and
+ z.real is finite */
+ if (Py_IS_INFINITY(z.imag) && Py_IS_FINITE(z.real))
+ errno = EDOM;
+ else
+ errno = 0;
+ return r;
+ }
+
+ /* danger of overflow in 2.*z.imag !*/
+ if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) {
+ r.real = copysign(1., z.real);
+ r.imag = 4.*sin(z.imag)*cos(z.imag)*exp(-2.*fabs(z.real));
+ } else {
+ tx = tanh(z.real);
+ ty = tan(z.imag);
+ cx = 1./cosh(z.real);
+ txty = tx*ty;
+ denom = 1. + txty*txty;
+ r.real = tx*(1.+ty*ty)/denom;
+ r.imag = ((ty/denom)*cx)*cx;
+ }
+ errno = 0;
+ return r;
+}
+
+PyDoc_STRVAR(c_tanh_doc,
+"tanh(x)\n"
+"\n"
+"Return the hyperbolic tangent of x.");
+
+
+static PyObject *
+cmath_log(PyObject *self, PyObject *args)
+{
+ Py_complex x;
+ Py_complex y;
+
+ if (!PyArg_ParseTuple(args, "D|D", &x, &y))
+ return NULL;
+
+ errno = 0;
+ PyFPE_START_PROTECT("complex function", return 0)
+ x = c_log(x);
+ if (PyTuple_GET_SIZE(args) == 2) {
+ y = c_log(y);
+ x = c_quot(x, y);
+ }
+ PyFPE_END_PROTECT(x)
+ if (errno != 0)
+ return math_error();
+ return PyComplex_FromCComplex(x);
+}
+
+PyDoc_STRVAR(cmath_log_doc,
+"log(x[, base]) -> the logarithm of x to the given base.\n\
+If the base not specified, returns the natural logarithm (base e) of x.");
+
+
+/* And now the glue to make them available from Python: */
+
+static PyObject *
+math_error(void)
+{
+ if (errno == EDOM)
+ PyErr_SetString(PyExc_ValueError, "math domain error");
+ else if (errno == ERANGE)
+ PyErr_SetString(PyExc_OverflowError, "math range error");
+ else /* Unexpected math error */
+ PyErr_SetFromErrno(PyExc_ValueError);
+ return NULL;
+}
+
+static PyObject *
+math_1(PyObject *args, Py_complex (*func)(Py_complex))
+{
+ Py_complex x,r ;
+ if (!PyArg_ParseTuple(args, "D", &x))
+ return NULL;
+ errno = 0;
+ PyFPE_START_PROTECT("complex function", return 0);
+ r = (*func)(x);
+ PyFPE_END_PROTECT(r);
+ if (errno == EDOM) {
+ PyErr_SetString(PyExc_ValueError, "math domain error");
+ return NULL;
+ }
+ else if (errno == ERANGE) {
+ PyErr_SetString(PyExc_OverflowError, "math range error");
+ return NULL;
+ }
+ else {
+ return PyComplex_FromCComplex(r);
+ }
+}
+
+#define FUNC1(stubname, func) \
+ static PyObject * stubname(PyObject *self, PyObject *args) { \
+ return math_1(args, func); \
+ }
+
+FUNC1(cmath_acos, c_acos)
+FUNC1(cmath_acosh, c_acosh)
+FUNC1(cmath_asin, c_asin)
+FUNC1(cmath_asinh, c_asinh)
+FUNC1(cmath_atan, c_atan)
+FUNC1(cmath_atanh, c_atanh)
+FUNC1(cmath_cos, c_cos)
+FUNC1(cmath_cosh, c_cosh)
+FUNC1(cmath_exp, c_exp)
+FUNC1(cmath_log10, c_log10)
+FUNC1(cmath_sin, c_sin)
+FUNC1(cmath_sinh, c_sinh)
+FUNC1(cmath_sqrt, c_sqrt)
+FUNC1(cmath_tan, c_tan)
+FUNC1(cmath_tanh, c_tanh)
+
+static PyObject *
+cmath_phase(PyObject *self, PyObject *args)
+{
+ Py_complex z;
+ double phi;
+ if (!PyArg_ParseTuple(args, "D:phase", &z))
+ return NULL;
+ errno = 0;
+ PyFPE_START_PROTECT("arg function", return 0)
+ phi = c_atan2(z);
+ PyFPE_END_PROTECT(phi)
+ if (errno != 0)
+ return math_error();
+ else
+ return PyFloat_FromDouble(phi);
+}
+
+PyDoc_STRVAR(cmath_phase_doc,
+"phase(z) -> float\n\n\
+Return argument, also known as the phase angle, of a complex.");
+
+static PyObject *
+cmath_polar(PyObject *self, PyObject *args)
+{
+ Py_complex z;
+ double r, phi;
+ if (!PyArg_ParseTuple(args, "D:polar", &z))
+ return NULL;
+ PyFPE_START_PROTECT("polar function", return 0)
+ phi = c_atan2(z); /* should not cause any exception */
+ r = c_abs(z); /* sets errno to ERANGE on overflow; otherwise 0 */
+ PyFPE_END_PROTECT(r)
+ if (errno != 0)
+ return math_error();
+ else
+ return Py_BuildValue("dd", r, phi);
+}
+
+PyDoc_STRVAR(cmath_polar_doc,
+"polar(z) -> r: float, phi: float\n\n\
+Convert a complex from rectangular coordinates to polar coordinates. r is\n\
+the distance from 0 and phi the phase angle.");
+
+/*
+ rect() isn't covered by the C99 standard, but it's not too hard to
+ figure out 'spirit of C99' rules for special value handing:
+
+ rect(x, t) should behave like exp(log(x) + it) for positive-signed x
+ rect(x, t) should behave like -exp(log(-x) + it) for negative-signed x
+ rect(nan, t) should behave like exp(nan + it), except that rect(nan, 0)
+ gives nan +- i0 with the sign of the imaginary part unspecified.
+
+*/
+
+static Py_complex rect_special_values[7][7];
+
+static PyObject *
+cmath_rect(PyObject *self, PyObject *args)
+{
+ Py_complex z;
+ double r, phi;
+ if (!PyArg_ParseTuple(args, "dd:rect", &r, &phi))
+ return NULL;
+ errno = 0;
+ PyFPE_START_PROTECT("rect function", return 0)
+
+ /* deal with special values */
+ if (!Py_IS_FINITE(r) || !Py_IS_FINITE(phi)) {
+ /* if r is +/-infinity and phi is finite but nonzero then
+ result is (+-INF +-INF i), but we need to compute cos(phi)
+ and sin(phi) to figure out the signs. */
+ if (Py_IS_INFINITY(r) && (Py_IS_FINITE(phi)
+ && (phi != 0.))) {
+ if (r > 0) {
+ z.real = copysign(INF, cos(phi));
+ z.imag = copysign(INF, sin(phi));
+ }
+ else {
+ z.real = -copysign(INF, cos(phi));
+ z.imag = -copysign(INF, sin(phi));
+ }
+ }
+ else {
+ z = rect_special_values[special_type(r)]
+ [special_type(phi)];
+ }
+ /* need to set errno = EDOM if r is a nonzero number and phi
+ is infinite */
+ if (r != 0. && !Py_IS_NAN(r) && Py_IS_INFINITY(phi))
+ errno = EDOM;
+ else
+ errno = 0;
+ }
+ else {
+ z.real = r * cos(phi);
+ z.imag = r * sin(phi);
+ errno = 0;
+ }
+
+ PyFPE_END_PROTECT(z)
+ if (errno != 0)
+ return math_error();
+ else
+ return PyComplex_FromCComplex(z);
+}
+
+PyDoc_STRVAR(cmath_rect_doc,
+"rect(r, phi) -> z: complex\n\n\
+Convert from polar coordinates to rectangular coordinates.");
+
+static PyObject *
+cmath_isnan(PyObject *self, PyObject *args)
+{
+ Py_complex z;
+ if (!PyArg_ParseTuple(args, "D:isnan", &z))
+ return NULL;
+ return PyBool_FromLong(Py_IS_NAN(z.real) || Py_IS_NAN(z.imag));
+}
+
+PyDoc_STRVAR(cmath_isnan_doc,
+"isnan(z) -> bool\n\
+Checks if the real or imaginary part of z not a number (NaN)");
+
+static PyObject *
+cmath_isinf(PyObject *self, PyObject *args)
+{
+ Py_complex z;
+ if (!PyArg_ParseTuple(args, "D:isnan", &z))
+ return NULL;
+ return PyBool_FromLong(Py_IS_INFINITY(z.real) ||
+ Py_IS_INFINITY(z.imag));
+}
+
+PyDoc_STRVAR(cmath_isinf_doc,
+"isinf(z) -> bool\n\
+Checks if the real or imaginary part of z is infinite.");
+
+
+PyDoc_STRVAR(module_doc,
+"This module is always available. It provides access to mathematical\n"
+"functions for complex numbers.");
+
+static PyMethodDef cmath_methods[] = {
+ {"acos", cmath_acos, METH_VARARGS, c_acos_doc},
+ {"acosh", cmath_acosh, METH_VARARGS, c_acosh_doc},
+ {"asin", cmath_asin, METH_VARARGS, c_asin_doc},
+ {"asinh", cmath_asinh, METH_VARARGS, c_asinh_doc},
+ {"atan", cmath_atan, METH_VARARGS, c_atan_doc},
+ {"atanh", cmath_atanh, METH_VARARGS, c_atanh_doc},
+ {"cos", cmath_cos, METH_VARARGS, c_cos_doc},
+ {"cosh", cmath_cosh, METH_VARARGS, c_cosh_doc},
+ {"exp", cmath_exp, METH_VARARGS, c_exp_doc},
+ {"isinf", cmath_isinf, METH_VARARGS, cmath_isinf_doc},
+ {"isnan", cmath_isnan, METH_VARARGS, cmath_isnan_doc},
+ {"log", cmath_log, METH_VARARGS, cmath_log_doc},
+ {"log10", cmath_log10, METH_VARARGS, c_log10_doc},
+ {"phase", cmath_phase, METH_VARARGS, cmath_phase_doc},
+ {"polar", cmath_polar, METH_VARARGS, cmath_polar_doc},
+ {"rect", cmath_rect, METH_VARARGS, cmath_rect_doc},
+ {"sin", cmath_sin, METH_VARARGS, c_sin_doc},
+ {"sinh", cmath_sinh, METH_VARARGS, c_sinh_doc},
+ {"sqrt", cmath_sqrt, METH_VARARGS, c_sqrt_doc},
+ {"tan", cmath_tan, METH_VARARGS, c_tan_doc},
+ {"tanh", cmath_tanh, METH_VARARGS, c_tanh_doc},
+ {NULL, NULL} /* sentinel */
+};
+
+PyMODINIT_FUNC
+initcmath(void)
+{
+ PyObject *m;
+
+ m = Py_InitModule3("cmath", cmath_methods, module_doc);
+ if (m == NULL)
+ return;
+
+ PyModule_AddObject(m, "pi",
+ PyFloat_FromDouble(Py_MATH_PI));
+ PyModule_AddObject(m, "e", PyFloat_FromDouble(Py_MATH_E));
+
+ /* initialize special value tables */
+
+#define INIT_SPECIAL_VALUES(NAME, BODY) { Py_complex* p = (Py_complex*)NAME; BODY }
+#define C(REAL, IMAG) p->real = REAL; p->imag = IMAG; ++p;
+
+ INIT_SPECIAL_VALUES(acos_special_values, {
+ C(P34,INF) C(P,INF) C(P,INF) C(P,-INF) C(P,-INF) C(P34,-INF) C(N,INF)
+ C(P12,INF) C(U,U) C(U,U) C(U,U) C(U,U) C(P12,-INF) C(N,N)
+ C(P12,INF) C(U,U) C(P12,0.) C(P12,-0.) C(U,U) C(P12,-INF) C(P12,N)
+ C(P12,INF) C(U,U) C(P12,0.) C(P12,-0.) C(U,U) C(P12,-INF) C(P12,N)
+ C(P12,INF) C(U,U) C(U,U) C(U,U) C(U,U) C(P12,-INF) C(N,N)
+ C(P14,INF) C(0.,INF) C(0.,INF) C(0.,-INF) C(0.,-INF) C(P14,-INF) C(N,INF)
+ C(N,INF) C(N,N) C(N,N) C(N,N) C(N,N) C(N,-INF) C(N,N)
+ })
+
+ INIT_SPECIAL_VALUES(acosh_special_values, {
+ C(INF,-P34) C(INF,-P) C(INF,-P) C(INF,P) C(INF,P) C(INF,P34) C(INF,N)
+ C(INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,P12) C(N,N)
+ C(INF,-P12) C(U,U) C(0.,-P12) C(0.,P12) C(U,U) C(INF,P12) C(N,N)
+ C(INF,-P12) C(U,U) C(0.,-P12) C(0.,P12) C(U,U) C(INF,P12) C(N,N)
+ C(INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,P12) C(N,N)
+ C(INF,-P14) C(INF,-0.) C(INF,-0.) C(INF,0.) C(INF,0.) C(INF,P14) C(INF,N)
+ C(INF,N) C(N,N) C(N,N) C(N,N) C(N,N) C(INF,N) C(N,N)
+ })
+
+ INIT_SPECIAL_VALUES(asinh_special_values, {
+ C(-INF,-P14) C(-INF,-0.) C(-INF,-0.) C(-INF,0.) C(-INF,0.) C(-INF,P14) C(-INF,N)
+ C(-INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(-INF,P12) C(N,N)
+ C(-INF,-P12) C(U,U) C(-0.,-0.) C(-0.,0.) C(U,U) C(-INF,P12) C(N,N)
+ C(INF,-P12) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(INF,P12) C(N,N)
+ C(INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,P12) C(N,N)
+ C(INF,-P14) C(INF,-0.) C(INF,-0.) C(INF,0.) C(INF,0.) C(INF,P14) C(INF,N)
+ C(INF,N) C(N,N) C(N,-0.) C(N,0.) C(N,N) C(INF,N) C(N,N)
+ })
+
+ INIT_SPECIAL_VALUES(atanh_special_values, {
+ C(-0.,-P12) C(-0.,-P12) C(-0.,-P12) C(-0.,P12) C(-0.,P12) C(-0.,P12) C(-0.,N)
+ C(-0.,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(-0.,P12) C(N,N)
+ C(-0.,-P12) C(U,U) C(-0.,-0.) C(-0.,0.) C(U,U) C(-0.,P12) C(-0.,N)
+ C(0.,-P12) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(0.,P12) C(0.,N)
+ C(0.,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(0.,P12) C(N,N)
+ C(0.,-P12) C(0.,-P12) C(0.,-P12) C(0.,P12) C(0.,P12) C(0.,P12) C(0.,N)
+ C(0.,-P12) C(N,N) C(N,N) C(N,N) C(N,N) C(0.,P12) C(N,N)
+ })
+
+ INIT_SPECIAL_VALUES(cosh_special_values, {
+ C(INF,N) C(U,U) C(INF,0.) C(INF,-0.) C(U,U) C(INF,N) C(INF,N)
+ C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
+ C(N,0.) C(U,U) C(1.,0.) C(1.,-0.) C(U,U) C(N,0.) C(N,0.)
+ C(N,0.) C(U,U) C(1.,-0.) C(1.,0.) C(U,U) C(N,0.) C(N,0.)
+ C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
+ C(INF,N) C(U,U) C(INF,-0.) C(INF,0.) C(U,U) C(INF,N) C(INF,N)
+ C(N,N) C(N,N) C(N,0.) C(N,0.) C(N,N) C(N,N) C(N,N)
+ })
+
+ INIT_SPECIAL_VALUES(exp_special_values, {
+ C(0.,0.) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(0.,0.) C(0.,0.)
+ C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
+ C(N,N) C(U,U) C(1.,-0.) C(1.,0.) C(U,U) C(N,N) C(N,N)
+ C(N,N) C(U,U) C(1.,-0.) C(1.,0.) C(U,U) C(N,N) C(N,N)
+ C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
+ C(INF,N) C(U,U) C(INF,-0.) C(INF,0.) C(U,U) C(INF,N) C(INF,N)
+ C(N,N) C(N,N) C(N,-0.) C(N,0.) C(N,N) C(N,N) C(N,N)
+ })
+
+ INIT_SPECIAL_VALUES(log_special_values, {
+ C(INF,-P34) C(INF,-P) C(INF,-P) C(INF,P) C(INF,P) C(INF,P34) C(INF,N)
+ C(INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,P12) C(N,N)
+ C(INF,-P12) C(U,U) C(-INF,-P) C(-INF,P) C(U,U) C(INF,P12) C(N,N)
+ C(INF,-P12) C(U,U) C(-INF,-0.) C(-INF,0.) C(U,U) C(INF,P12) C(N,N)
+ C(INF,-P12) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,P12) C(N,N)
+ C(INF,-P14) C(INF,-0.) C(INF,-0.) C(INF,0.) C(INF,0.) C(INF,P14) C(INF,N)
+ C(INF,N) C(N,N) C(N,N) C(N,N) C(N,N) C(INF,N) C(N,N)
+ })
+
+ INIT_SPECIAL_VALUES(sinh_special_values, {
+ C(INF,N) C(U,U) C(-INF,-0.) C(-INF,0.) C(U,U) C(INF,N) C(INF,N)
+ C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
+ C(0.,N) C(U,U) C(-0.,-0.) C(-0.,0.) C(U,U) C(0.,N) C(0.,N)
+ C(0.,N) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(0.,N) C(0.,N)
+ C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
+ C(INF,N) C(U,U) C(INF,-0.) C(INF,0.) C(U,U) C(INF,N) C(INF,N)
+ C(N,N) C(N,N) C(N,-0.) C(N,0.) C(N,N) C(N,N) C(N,N)
+ })
+
+ INIT_SPECIAL_VALUES(sqrt_special_values, {
+ C(INF,-INF) C(0.,-INF) C(0.,-INF) C(0.,INF) C(0.,INF) C(INF,INF) C(N,INF)
+ C(INF,-INF) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,INF) C(N,N)
+ C(INF,-INF) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(INF,INF) C(N,N)
+ C(INF,-INF) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(INF,INF) C(N,N)
+ C(INF,-INF) C(U,U) C(U,U) C(U,U) C(U,U) C(INF,INF) C(N,N)
+ C(INF,-INF) C(INF,-0.) C(INF,-0.) C(INF,0.) C(INF,0.) C(INF,INF) C(INF,N)
+ C(INF,-INF) C(N,N) C(N,N) C(N,N) C(N,N) C(INF,INF) C(N,N)
+ })
+
+ INIT_SPECIAL_VALUES(tanh_special_values, {
+ C(-1.,0.) C(U,U) C(-1.,-0.) C(-1.,0.) C(U,U) C(-1.,0.) C(-1.,0.)
+ C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
+ C(N,N) C(U,U) C(-0.,-0.) C(-0.,0.) C(U,U) C(N,N) C(N,N)
+ C(N,N) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(N,N) C(N,N)
+ C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
+ C(1.,0.) C(U,U) C(1.,-0.) C(1.,0.) C(U,U) C(1.,0.) C(1.,0.)
+ C(N,N) C(N,N) C(N,-0.) C(N,0.) C(N,N) C(N,N) C(N,N)
+ })
+
+ INIT_SPECIAL_VALUES(rect_special_values, {
+ C(INF,N) C(U,U) C(-INF,0.) C(-INF,-0.) C(U,U) C(INF,N) C(INF,N)
+ C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
+ C(0.,0.) C(U,U) C(-0.,0.) C(-0.,-0.) C(U,U) C(0.,0.) C(0.,0.)
+ C(0.,0.) C(U,U) C(0.,-0.) C(0.,0.) C(U,U) C(0.,0.) C(0.,0.)
+ C(N,N) C(U,U) C(U,U) C(U,U) C(U,U) C(N,N) C(N,N)
+ C(INF,N) C(U,U) C(INF,-0.) C(INF,0.) C(U,U) C(INF,N) C(INF,N)
+ C(N,N) C(N,N) C(N,0.) C(N,0.) C(N,N) C(N,N) C(N,N)
+ })
+}