path: root/Modules/cmathmodule.c

/* Complex math module */

/* much code borrowed from mathmodule.c */

#include "Python.h"
#include "_math.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>

#include "clinic/cmathmodule.c.h"
/*[clinic input]
output preset file
module cmath
[clinic start generated code]*/
/*[clinic end generated code: output=da39a3ee5e6b4b0d input=ef7e0fdd8a143c03]*/

/*[python input]
class Py_complex_protected_converter(Py_complex_converter):
    def modify(self):
        return 'errno = 0; PyFPE_START_PROTECT("complex function", goto exit);'


class Py_complex_protected_return_converter(CReturnConverter):
    type = "Py_complex"

    def render(self, function, data):
        self.declare(data)
        data.return_conversion.append("""
PyFPE_END_PROTECT(_return_value);
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=231019039a6fbb9a]*/

#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(PyModuleDef *, Py_complex);
static Py_complex cmath_atanh_impl(PyModuleDef *, Py_complex);
static Py_complex cmath_cosh_impl(PyModuleDef *, Py_complex);
static Py_complex cmath_sinh_impl(PyModuleDef *, Py_complex);
static Py_complex cmath_sqrt_impl(PyModuleDef *, Py_complex);
static Py_complex cmath_tanh_impl(PyModuleDef *, 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=7c1dd21ff818db6b 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 = m_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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=c23c776429def981 input=bc016412080bb3e9]*/
{
    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 = m_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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=42d2346d46690826 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=0c6664823c7b1b35 input=5a21fa0242928c9b]*/
{
    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 = m_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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=b7d44f02c6a5c3b5 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=279e0b9fefc8da7c input=df19cdc9f9d431c9]*/
{
    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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=9d1cdc1b5e761667 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=f3b5d3282b3024d3 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=6f8825eb2bcad9ba 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=c7c426ca0e782341 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=e7f5e2b253825ac7 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=d71fff8298043a95 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=b6bda283d0c5a7b4 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 && (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;
}


/*[clinic input]
cmath.tan = cmath.acos

Return the tangent of z.
[clinic start generated code]*/

static Py_complex
cmath_tan_impl(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=df374bacf36d99b4 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=f578773d27a18e96 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

    x: Py_complex
    y_obj: object = NULL
    /

The logarithm of z to the given base.

If the base not specified, returns the natural logarithm (base e) of z.
[clinic start generated code]*/

static PyObject *
cmath_log_impl(PyModuleDef *module, Py_complex x, PyObject *y_obj)
/*[clinic end generated code: output=35e2a1e5229b5a46 input=ee0e823a7c6e68ea]*/
{
    Py_complex y;

    errno = 0;
    PyFPE_START_PROTECT("complex function", return 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);
    }
    PyFPE_END_PROTECT(x)
    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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=e09eaf373cb624c3 input=5cf75228ba94b69d]*/
{
    double phi;

    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);
}

/*[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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=07d41b16c877875a input=26c353574fd1a861]*/
{
    double r, phi;

    PyFPE_START_PROTECT("polar function", return 0)
    phi = c_atan2(z); /* should not cause any exception */
    r = _Py_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);
}

/*
  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(PyModuleDef *module, double r, double phi)
/*[clinic end generated code: output=d97a8749bd63e9d5 input=24c5646d147efd69]*/
{
    Py_complex z;
    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 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;
    }

    PyFPE_END_PROTECT(z)
    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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=8f6682fa93de45d6 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=b85fe8c2047718ee 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(PyModuleDef *module, Py_complex z)
/*[clinic end generated code: output=8ca9c6109e468bf4 input=363df155c7181329]*/
{
    return PyBool_FromLong(Py_IS_INFINITY(z.real) ||
                           Py_IS_INFINITY(z.imag));
}


PyDoc_STRVAR(module_doc,
"This module is always available. It provides access to mathematical\n"
"functions for complex 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_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 struct PyModuleDef cmathmodule = {
    PyModuleDef_HEAD_INIT,
    "cmath",
    module_doc,
    -1,
    cmath_methods,
    NULL,
    NULL,
    NULL,
    NULL
};

PyMODINIT_FUNC
PyInit_cmath(void)
{
    PyObject *m;

    m = PyModule_Create(&cmathmodule);
    if (m == NULL)
        return NULL;

    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)
    })
    return m;
}