Library import 16 (#2433)

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1 files changed, 1381 insertions, 0 deletions

diff --git a/contrib/tools/python3/Modules/cmathmodule.c b/contrib/tools/python3/Modules/cmathmodule.c
new file mode 100644
index 0000000000..25491e6558
--- /dev/null
+++ b/contrib/tools/python3/Modules/cmathmodule.c
@@ -0,0 +1,1381 @@
+/* Complex math module */
+
+/* much code borrowed from mathmodule.c */
+
+#ifndef Py_BUILD_CORE_BUILTIN
+#  define Py_BUILD_CORE_MODULE 1
+#endif
+
+#include "Python.h"
+#include "pycore_pymath.h"        // _PY_SHORT_FLOAT_REPR
+/* 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>
+
+/* For _Py_log1p with workarounds for buggy handling of zeros. */
+#include "_math.h"
+
+#include "clinic/cmathmodule.c.h"
+/*[clinic input]
+module cmath
+[clinic start generated code]*/
+/*[clinic end generated code: output=da39a3ee5e6b4b0d input=308d6839f4a46333]*/
+
+/*[python input]
+class Py_complex_protected_converter(Py_complex_converter):
+    def modify(self):
+        return 'errno = 0;'
+
+
+class Py_complex_protected_return_converter(CReturnConverter):
+    type = "Py_complex"
+
+    def render(self, function, data):
+        self.declare(data)
+        data.return_conversion.append("""
+if (errno == EDOM) {
+    PyErr_SetString(PyExc_ValueError, "math domain error");
+    goto exit;
+}
+else if (errno == ERANGE) {
+    PyErr_SetString(PyExc_OverflowError, "math range error");
+    goto exit;
+}
+else {
+    return_value = PyComplex_FromCComplex(_return_value);
+}
+""".strip())
+[python start generated code]*/
+/*[python end generated code: output=da39a3ee5e6b4b0d input=8b27adb674c08321]*/
+
+#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 unnecessary
+   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 cmath_asinh_impl(PyObject *, Py_complex);
+static Py_complex cmath_atanh_impl(PyObject *, Py_complex);
+static Py_complex cmath_cosh_impl(PyObject *, Py_complex);
+static Py_complex cmath_sinh_impl(PyObject *, Py_complex);
+static Py_complex cmath_sqrt_impl(PyObject *, Py_complex);
+static Py_complex cmath_tanh_impl(PyObject *, 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];
+
+/*[clinic input]
+cmath.acos -> Py_complex_protected
+
+    z: Py_complex_protected
+    /
+
+Return the arc cosine of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_acos_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=40bd42853fd460ae input=bd6cbd78ae851927]*/
+{
+    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 = cmath_sqrt_impl(module, s1);
+        s2.real = 1.+z.real;
+        s2.imag = z.imag;
+        s2 = cmath_sqrt_impl(module, s2);
+        r.real = 2.*atan2(s1.real, s2.real);
+        r.imag = asinh(s2.real*s1.imag - s2.imag*s1.real);
+    }
+    errno = 0;
+    return r;
+}
+
+
+static Py_complex acosh_special_values[7][7];
+
+/*[clinic input]
+cmath.acosh = cmath.acos
+
+Return the inverse hyperbolic cosine of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_acosh_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=3e2454d4fcf404ca input=3f61bee7d703e53c]*/
+{
+    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 = cmath_sqrt_impl(module, s1);
+        s2.real = z.real + 1.;
+        s2.imag = z.imag;
+        s2 = cmath_sqrt_impl(module, s2);
+        r.real = asinh(s1.real*s2.real + s1.imag*s2.imag);
+        r.imag = 2.*atan2(s1.imag, s2.real);
+    }
+    errno = 0;
+    return r;
+}
+
+/*[clinic input]
+cmath.asin = cmath.acos
+
+Return the arc sine of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_asin_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=3b264cd1b16bf4e1 input=be0bf0cfdd5239c5]*/
+{
+    /* asin(z) = -i asinh(iz) */
+    Py_complex s, r;
+    s.real = -z.imag;
+    s.imag = z.real;
+    s = cmath_asinh_impl(module, s);
+    r.real = s.imag;
+    r.imag = -s.real;
+    return r;
+}
+
+
+static Py_complex asinh_special_values[7][7];
+
+/*[clinic input]
+cmath.asinh = cmath.acos
+
+Return the inverse hyperbolic sine of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_asinh_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=733d8107841a7599 input=5c09448fcfc89a79]*/
+{
+    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 = cmath_sqrt_impl(module, s1);
+        s2.real = 1.-z.imag;
+        s2.imag = z.real;
+        s2 = cmath_sqrt_impl(module, 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;
+}
+
+
+/*[clinic input]
+cmath.atan = cmath.acos
+
+Return the arc tangent of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_atan_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=b6bfc497058acba4 input=3b21ff7d5eac632a]*/
+{
+    /* atan(z) = -i atanh(iz) */
+    Py_complex s, r;
+    s.real = -z.imag;
+    s.imag = z.real;
+    s = cmath_atanh_impl(module, 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);
+}
+
+
+static Py_complex atanh_special_values[7][7];
+
+/*[clinic input]
+cmath.atanh = cmath.acos
+
+Return the inverse hyperbolic tangent of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_atanh_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=e83355f93a989c9e input=2b3fdb82fb34487b]*/
+{
+    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 _Py_c_neg(cmath_atanh_impl(module, _Py_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 = m_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;
+}
+
+
+/*[clinic input]
+cmath.cos = cmath.acos
+
+Return the cosine of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_cos_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=fd64918d5b3186db input=6022e39b77127ac7]*/
+{
+    /* cos(z) = cosh(iz) */
+    Py_complex r;
+    r.real = -z.imag;
+    r.imag = z.real;
+    r = cmath_cosh_impl(module, r);
+    return r;
+}
+
+
+/* cosh(infinity + i*y) needs to be dealt with specially */
+static Py_complex cosh_special_values[7][7];
+
+/*[clinic input]
+cmath.cosh = cmath.acos
+
+Return the hyperbolic cosine of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_cosh_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=2e969047da601bdb input=d6b66339e9cc332b]*/
+{
+    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;
+}
+
+
+/* exp(infinity + i*y) and exp(-infinity + i*y) need special treatment for
+   finite y */
+static Py_complex exp_special_values[7][7];
+
+/*[clinic input]
+cmath.exp = cmath.acos
+
+Return the exponential value e**z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_exp_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=edcec61fb9dfda6c input=8b9e6cf8a92174c3]*/
+{
+    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;
+}
+
+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 = m_log1p((am-1)*(am+1)+an*an)/2.;
+        } else {
+            r.real = log(h);
+        }
+    }
+    r.imag = atan2(z.imag, z.real);
+    errno = 0;
+    return r;
+}
+
+
+/*[clinic input]
+cmath.log10 = cmath.acos
+
+Return the base-10 logarithm of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_log10_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=2922779a7c38cbe1 input=cff5644f73c1519c]*/
+{
+    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;
+}
+
+
+/*[clinic input]
+cmath.sin = cmath.acos
+
+Return the sine of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_sin_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=980370d2ff0bb5aa input=2d3519842a8b4b85]*/
+{
+    /* sin(z) = -i sin(iz) */
+    Py_complex s, r;
+    s.real = -z.imag;
+    s.imag = z.real;
+    s = cmath_sinh_impl(module, s);
+    r.real = s.imag;
+    r.imag = -s.real;
+    return r;
+}
+
+
+/* sinh(infinity + i*y) needs to be dealt with specially */
+static Py_complex sinh_special_values[7][7];
+
+/*[clinic input]
+cmath.sinh = cmath.acos
+
+Return the hyperbolic sine of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_sinh_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=38b0a6cce26f3536 input=d2d3fc8c1ddfd2dd]*/
+{
+    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;
+}
+
+
+static Py_complex sqrt_special_values[7][7];
+
+/*[clinic input]
+cmath.sqrt = cmath.acos
+
+Return the square root of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_sqrt_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=b6507b3029c339fc input=7088b166fc9a58c7]*/
+{
+    /*
+       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) {
+        /* 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;
+}
+
+
+/*[clinic input]
+cmath.tan = cmath.acos
+
+Return the tangent of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_tan_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=7c5f13158a72eb13 input=fc167e528767888e]*/
+{
+    /* tan(z) = -i tanh(iz) */
+    Py_complex s, r;
+    s.real = -z.imag;
+    s.imag = z.real;
+    s = cmath_tanh_impl(module, s);
+    r.real = s.imag;
+    r.imag = -s.real;
+    return r;
+}
+
+
+/* tanh(infinity + i*y) needs to be dealt with specially */
+static Py_complex tanh_special_values[7][7];
+
+/*[clinic input]
+cmath.tanh = cmath.acos
+
+Return the hyperbolic tangent of z.
+[clinic start generated code]*/
+
+static Py_complex
+cmath_tanh_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=36d547ef7aca116c input=22f67f9dc6d29685]*/
+{
+    /* 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;
+}
+
+
+/*[clinic input]
+cmath.log
+
+    z as x: Py_complex
+    base as y_obj: object = NULL
+    /
+
+log(z[, base]) -> the logarithm of z to the given base.
+
+If the base is not specified, returns the natural logarithm (base e) of z.
+[clinic start generated code]*/
+
+static PyObject *
+cmath_log_impl(PyObject *module, Py_complex x, PyObject *y_obj)
+/*[clinic end generated code: output=4effdb7d258e0d94 input=e1f81d4fcfd26497]*/
+{
+    Py_complex y;
+
+    errno = 0;
+    x = c_log(x);
+    if (y_obj != NULL) {
+        y = PyComplex_AsCComplex(y_obj);
+        if (PyErr_Occurred()) {
+            return NULL;
+        }
+        y = c_log(y);
+        x = _Py_c_quot(x, y);
+    }
+    if (errno != 0)
+        return math_error();
+    return PyComplex_FromCComplex(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;
+}
+
+
+/*[clinic input]
+cmath.phase
+
+    z: Py_complex
+    /
+
+Return argument, also known as the phase angle, of a complex.
+[clinic start generated code]*/
+
+static PyObject *
+cmath_phase_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=50725086a7bfd253 input=5cf75228ba94b69d]*/
+{
+    double phi;
+
+    errno = 0;
+    phi = c_atan2(z); /* should not cause any exception */
+    if (errno != 0)
+        return math_error();
+    else
+        return PyFloat_FromDouble(phi);
+}
+
+/*[clinic input]
+cmath.polar
+
+    z: Py_complex
+    /
+
+Convert a complex from rectangular coordinates to polar coordinates.
+
+r is the distance from 0 and phi the phase angle.
+[clinic start generated code]*/
+
+static PyObject *
+cmath_polar_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=d0a8147c41dbb654 input=26c353574fd1a861]*/
+{
+    double r, phi;
+
+    errno = 0;
+    phi = c_atan2(z); /* should not cause any exception */
+    r = _Py_c_abs(z); /* sets errno to ERANGE on overflow */
+    if (errno != 0)
+        return math_error();
+    else
+        return Py_BuildValue("dd", r, phi);
+}
+
+/*
+  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];
+
+/*[clinic input]
+cmath.rect
+
+    r: double
+    phi: double
+    /
+
+Convert from polar coordinates to rectangular coordinates.
+[clinic start generated code]*/
+
+static PyObject *
+cmath_rect_impl(PyObject *module, double r, double phi)
+/*[clinic end generated code: output=385a0690925df2d5 input=24c5646d147efd69]*/
+{
+    Py_complex z;
+    errno = 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 if (phi == 0.0) {
+        /* Workaround for buggy results with phi=-0.0 on OS X 10.8.  See
+           bugs.python.org/issue18513. */
+        z.real = r;
+        z.imag = r * phi;
+        errno = 0;
+    }
+    else {
+        z.real = r * cos(phi);
+        z.imag = r * sin(phi);
+        errno = 0;
+    }
+
+    if (errno != 0)
+        return math_error();
+    else
+        return PyComplex_FromCComplex(z);
+}
+
+/*[clinic input]
+cmath.isfinite = cmath.polar
+
+Return True if both the real and imaginary parts of z are finite, else False.
+[clinic start generated code]*/
+
+static PyObject *
+cmath_isfinite_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=ac76611e2c774a36 input=848e7ee701895815]*/
+{
+    return PyBool_FromLong(Py_IS_FINITE(z.real) && Py_IS_FINITE(z.imag));
+}
+
+/*[clinic input]
+cmath.isnan = cmath.polar
+
+Checks if the real or imaginary part of z not a number (NaN).
+[clinic start generated code]*/
+
+static PyObject *
+cmath_isnan_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=e7abf6e0b28beab7 input=71799f5d284c9baf]*/
+{
+    return PyBool_FromLong(Py_IS_NAN(z.real) || Py_IS_NAN(z.imag));
+}
+
+/*[clinic input]
+cmath.isinf = cmath.polar
+
+Checks if the real or imaginary part of z is infinite.
+[clinic start generated code]*/
+
+static PyObject *
+cmath_isinf_impl(PyObject *module, Py_complex z)
+/*[clinic end generated code: output=502a75a79c773469 input=363df155c7181329]*/
+{
+    return PyBool_FromLong(Py_IS_INFINITY(z.real) ||
+                           Py_IS_INFINITY(z.imag));
+}
+
+/*[clinic input]
+cmath.isclose -> bool
+
+    a: Py_complex
+    b: Py_complex
+    *
+    rel_tol: double = 1e-09
+        maximum difference for being considered "close", relative to the
+        magnitude of the input values
+    abs_tol: double = 0.0
+        maximum difference for being considered "close", regardless of the
+        magnitude of the input values
+
+Determine whether two complex numbers are close in value.
+
+Return True if a is close in value to b, and False otherwise.
+
+For the values to be considered close, the difference between them must be
+smaller than at least one of the tolerances.
+
+-inf, inf and NaN behave similarly to the IEEE 754 Standard. That is, NaN is
+not close to anything, even itself. inf and -inf are only close to themselves.
+[clinic start generated code]*/
+
+static int
+cmath_isclose_impl(PyObject *module, Py_complex a, Py_complex b,
+                   double rel_tol, double abs_tol)
+/*[clinic end generated code: output=8a2486cc6e0014d1 input=df9636d7de1d4ac3]*/
+{
+    double diff;
+
+    /* sanity check on the inputs */
+    if (rel_tol < 0.0 || abs_tol < 0.0 ) {
+        PyErr_SetString(PyExc_ValueError,
+                        "tolerances must be non-negative");
+        return -1;
+    }
+
+    if ( (a.real == b.real) && (a.imag == b.imag) ) {
+        /* short circuit exact equality -- needed to catch two infinities of
+           the same sign. And perhaps speeds things up a bit sometimes.
+        */
+        return 1;
+    }
+
+    /* This catches the case of two infinities of opposite sign, or
+       one infinity and one finite number. Two infinities of opposite
+       sign would otherwise have an infinite relative tolerance.
+       Two infinities of the same sign are caught by the equality check
+       above.
+    */
+
+    if (Py_IS_INFINITY(a.real) || Py_IS_INFINITY(a.imag) ||
+        Py_IS_INFINITY(b.real) || Py_IS_INFINITY(b.imag)) {
+        return 0;
+    }
+
+    /* now do the regular computation
+       this is essentially the "weak" test from the Boost library
+    */
+
+    diff = _Py_c_abs(_Py_c_diff(a, b));
+
+    return (((diff <= rel_tol * _Py_c_abs(b)) ||
+             (diff <= rel_tol * _Py_c_abs(a))) ||
+            (diff <= abs_tol));
+}
+
+PyDoc_STRVAR(module_doc,
+"This module provides access to mathematical functions for complex\n"
+"numbers.");
+
+static PyMethodDef cmath_methods[] = {
+    CMATH_ACOS_METHODDEF
+    CMATH_ACOSH_METHODDEF
+    CMATH_ASIN_METHODDEF
+    CMATH_ASINH_METHODDEF
+    CMATH_ATAN_METHODDEF
+    CMATH_ATANH_METHODDEF
+    CMATH_COS_METHODDEF
+    CMATH_COSH_METHODDEF
+    CMATH_EXP_METHODDEF
+    CMATH_ISCLOSE_METHODDEF
+    CMATH_ISFINITE_METHODDEF
+    CMATH_ISINF_METHODDEF
+    CMATH_ISNAN_METHODDEF
+    CMATH_LOG_METHODDEF
+    CMATH_LOG10_METHODDEF
+    CMATH_PHASE_METHODDEF
+    CMATH_POLAR_METHODDEF
+    CMATH_RECT_METHODDEF
+    CMATH_SIN_METHODDEF
+    CMATH_SINH_METHODDEF
+    CMATH_SQRT_METHODDEF
+    CMATH_TAN_METHODDEF
+    CMATH_TANH_METHODDEF
+    {NULL, NULL}  /* sentinel */
+};
+
+static int
+cmath_exec(PyObject *mod)
+{
+    if (_PyModule_Add(mod, "pi", PyFloat_FromDouble(Py_MATH_PI)) < 0) {
+        return -1;
+    }
+    if (_PyModule_Add(mod, "e", PyFloat_FromDouble(Py_MATH_E)) < 0) {
+        return -1;
+    }
+    // 2pi
+    if (_PyModule_Add(mod, "tau", PyFloat_FromDouble(Py_MATH_TAU)) < 0) {
+        return -1;
+    }
+    if (_PyModule_Add(mod, "inf", PyFloat_FromDouble(Py_INFINITY)) < 0) {
+        return -1;
+    }
+
+    Py_complex infj = {0.0, Py_INFINITY};
+    if (_PyModule_Add(mod, "infj", PyComplex_FromCComplex(infj)) < 0) {
+        return -1;
+    }
+    if (_PyModule_Add(mod, "nan", PyFloat_FromDouble(fabs(Py_NAN))) < 0) {
+        return -1;
+    }
+    Py_complex nanj = {0.0, fabs(Py_NAN)};
+    if (_PyModule_Add(mod, "nanj", PyComplex_FromCComplex(nanj)) < 0) {
+        return -1;
+    }
+
+    /* 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)
+    })
+    return 0;
+}
+
+static PyModuleDef_Slot cmath_slots[] = {
+    {Py_mod_exec, cmath_exec},
+    {Py_mod_multiple_interpreters, Py_MOD_PER_INTERPRETER_GIL_SUPPORTED},
+    {0, NULL}
+};
+
+static struct PyModuleDef cmathmodule = {
+    PyModuleDef_HEAD_INIT,
+    .m_name = "cmath",
+    .m_doc = module_doc,
+    .m_size = 0,
+    .m_methods = cmath_methods,
+    .m_slots = cmath_slots
+};
+
+PyMODINIT_FUNC
+PyInit_cmath(void)
+{
+    return PyModuleDef_Init(&cmathmodule);
+}