codon/stdlib/cmath.codon

# Copyright (C) 2022-2023 Exaloop Inc. <https://exaloop.io>

import math

e = math.e
pi = math.pi
tau = math.tau
inf = math.inf
nan = math.nan
infj = complex(0.0, inf)
nanj = complex(0.0, nan)

# internal constants
_FLT_RADIX = 2
_M_LN2 = 0.6931471805599453094  # ln(2)
_M_LN10 = 2.302585092994045684  # ln(10)

@pure
@llvm
def _max_float() -> float:
    ret double 0x7FEFFFFFFFFFFFFF

@pure
@llvm
def _min_float() -> float:
    ret double 0x10000000000000

_DBL_MAX = _max_float()
_DBL_MIN = _min_float()
_DBL_MANT_DIG = 53

_CM_LARGE_DOUBLE = _DBL_MAX/4.
_CM_SQRT_LARGE_DOUBLE = math.sqrt(_CM_LARGE_DOUBLE)
_CM_LOG_LARGE_DOUBLE = math.log(_CM_LARGE_DOUBLE)
_CM_SQRT_DBL_MIN = math.sqrt(_DBL_MIN)
_CM_SCALE_UP = (2*(_DBL_MANT_DIG // 2) + 1)
_CM_SCALE_DOWN = (-(_CM_SCALE_UP+1)//2)

# 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

def _special_type(d: float):
    if math.isfinite(d):
        if d != 0:
            if math.copysign(1., d) == 1.:
                return _ST_POS
            else:
                return _ST_NEG
        else:
            if math.copysign(1., d) == 1.:
                return _ST_PZERO
            else:
                return _ST_NZERO
    if math.isnan(d):
        return _ST_NAN
    if math.copysign(1., d) == 1.:
        return _ST_PINF
    else:
        return _ST_NINF

def _acos_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    v = (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))
    return v

def _acosh_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    def C(a,b): return complex(a, b)
    v = (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))
    return v

def _asinh_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    v = (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))
    return v

def _atanh_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    v = (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))
    return v

def _cosh_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    v = (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))
    return v

def _exp_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    v = (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))
    return v

def _log_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    v = (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))
    return v

def _sinh_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    v = (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))
    return v

def _sqrt_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    v = (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))
    return v

def _tanh_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    v = (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))
    return v

def _rect_special():
    P = pi
    P14 = 0.25*pi
    P12 = 0.5*pi
    P34 = 0.75*pi
    INF = inf  # Py_HUGE_VAL
    N = nan
    U = -9.5426319407711027e33  # unlikely value, used as placeholder
    def C(a,b): return complex(a, b)
    v = (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 v

def _is_special(z):
    return (not math.isfinite(z.real)) or (not math.isfinite(z.imag))

def _special_get(z, table):
    t1 = _special_type(z.real)
    t2 = _special_type(z.imag)
    return table[7*t1 + t2]

def _sqrt_impl(z):
    if _is_special(z):
        return _special_get(z, _sqrt_special())

    r_real = 0.
    r_imag = 0.
    if z.real == 0. and z.imag == 0.:
        r_real = 0.
        r_imag = z.imag
        return complex(r_real, r_imag)

    ax = math.fabs(z.real)
    ay = math.fabs(z.imag)
    s = 0.
    if ax < _DBL_MIN and ay < _DBL_MIN and (ax > 0. or ay > 0.):
        # here we catch cases where hypot(ax, ay) is subnormal
        ax = math.ldexp(ax, _CM_SCALE_UP)
        s = math.ldexp(math.sqrt(ax + math.hypot(ax, math.ldexp(ay, _CM_SCALE_UP))), _CM_SCALE_DOWN)
    else:
        ax /= 8.
        s = 2.*math.sqrt(ax + math.hypot(ax, ay/8.))
    d = ay/(2.*s)

    if z.real >= 0.:
        r_real = s
        r_imag = math.copysign(d, z.imag)
    else:
        r_real = d
        r_imag = math.copysign(s, z.imag)
    # errno = 0
    return complex(r_real, r_imag)

def _acos_impl(z):
    if _is_special(z):
        return _special_get(z, _acos_special())

    r_real = 0.
    r_imag = 0.
    if math.fabs(z.real) > _CM_LARGE_DOUBLE or math.fabs(z.imag) > _CM_LARGE_DOUBLE:
        # avoid unnecessary overflow for large arguments
        r_real = math.atan2(math.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 = -math.copysign(math.log(math.hypot(z.real/2., z.imag/2.)) + _M_LN2*2, z.imag)
        else:
            r_imag = math.copysign(math.log(math.hypot(z.real/2., z.imag/2.)) + _M_LN2*2, -z.imag)
    else:
        s1 = _sqrt_impl(complex(1. - z.real, -z.imag))
        s2 = _sqrt_impl(complex(1. + z.real, z.imag))
        r_real = 2.*math.atan2(s1.real, s2.real)
        r_imag = math.asinh(s2.real*s1.imag - s2.imag*s1.real)
    return complex(r_real, r_imag)

def _acosh_impl(z):
    if _is_special(z):
        return _special_get(z, _acosh_special())

    r_real = 0.
    r_imag = 0.
    if math.fabs(z.real) > _CM_LARGE_DOUBLE or math.fabs(z.imag) > _CM_LARGE_DOUBLE:
        # avoid unnecessary overflow for large arguments
        r_real = math.log(math.hypot(z.real/2., z.imag/2.)) + _M_LN2*2.
        r_imag = math.atan2(z.imag, z.real)
    else:
        s1 = _sqrt_impl(complex(z.real - 1., z.imag))
        s2 = _sqrt_impl(complex(z.real + 1., z.imag))
        r_real = math.asinh(s1.real*s2.real + s1.imag*s2.imag)
        r_imag = 2.*math.atan2(s1.imag, s2.real)
    return complex(r_real, r_imag)

def _asinh_impl(z):
    if _is_special(z):
        return _special_get(z, _asinh_special())

    r_real = 0.
    r_imag = 0.
    if math.fabs(z.real) > _CM_LARGE_DOUBLE or math.fabs(z.imag) > _CM_LARGE_DOUBLE:
        if z.imag >= 0.:
            r_real = math.copysign(math.log(math.hypot(z.real/2., z.imag/2.)) + _M_LN2*2, z.real)
        else:
            r_real = -math.copysign(math.log(math.hypot(z.real/2., z.imag/2.)) + _M_LN2*2, -z.real)
        r_imag = math.atan2(z.imag, math.fabs(z.real))
    else:
        s1 = _sqrt_impl(complex(1. + z.imag, -z.real))
        s2 = _sqrt_impl(complex(1. - z.imag, z.real))
        r_real = math.asinh(s1.real*s2.imag - s2.real*s1.imag)
        r_imag = math.atan2(z.imag, s1.real*s2.real - s1.imag*s2.imag)
    return complex(r_real, r_imag)

def _asin_impl(z):
    s = _asinh_impl(complex(-z.imag, z.real))
    r_real = s.imag
    r_imag = -s.real
    return complex(r_real, r_imag)

def _atanh_impl(z):
    if _is_special(z):
        return _special_get(z, _atanh_special())

    # Reduce to case where z.real >= 0., using atanh(z) = -atanh(-z).
    if z.real < 0.:
        return -_atanh_impl(-z)

    r_real = 0.
    r_imag = 0.
    ay = math.fabs(z.imag)
    if z.real > _CM_SQRT_LARGE_DOUBLE or 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 = math.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 = -math.copysign(pi/2., -z.imag)
        # errno = 0
    elif z.real == 1. and 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 = -math.log(math.sqrt(ay)/math.sqrt(math.hypot(ay, 2.)))
            r_imag = math.copysign(math.atan2(2., -ay)/2, z.imag)
            # errno = 0
    else:
        r_real = math.log1p(4.*z.real/((1-z.real)*(1-z.real) + ay*ay))/4.
        r_imag = -math.atan2(-2.*z.imag, (1-z.real)*(1+z.real) - ay*ay)/2.
        # errno = 0
    return complex(r_real, r_imag)

def _atan_impl(z):
    s = _atanh_impl(complex(-z.imag, z.real))
    r_real = s.imag
    r_imag = -s.real
    return complex(r_real, r_imag)

def _cosh_impl(z):
    r_real = 0.
    r_imag = 0.
    # special treatment for cosh(+/-inf + iy) if y is not a NaN
    if (not math.isfinite(z.real)) or (not math.isfinite(z.imag)):
        if math.isinf(z.real) and math.isfinite(z.imag) and z.imag != 0.:
            if z.real > 0:
                r_real = math.copysign(inf, math.cos(z.imag))
                r_imag = math.copysign(inf, math.sin(z.imag))
            else:
                r_real = math.copysign(inf, math.cos(z.imag))
                r_imag = -math.copysign(inf, math.sin(z.imag))
        else:
            r = _special_get(z, _cosh_special())
            r_real = r.real
            r_imag = r.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 complex(r_real, r_imag)

    if math.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 - math.copysign(1., z.real)
        r_real = math.cos(z.imag) * math.cosh(x_minus_one) * e
        r_imag = math.sin(z.imag) * math.sinh(x_minus_one) * e
    else:
        r_real = math.cos(z.imag) * math.cosh(z.real)
        r_imag = math.sin(z.imag) * math.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 complex(r_real, r_imag)

def _cos_impl(z):
    r = _cosh_impl(complex(-z.imag, z.real))
    return r

def _exp_impl(z):
    r_real = 0.
    r_imag = 0.
    if (not math.isfinite(z.real)) or (not math.isfinite(z.imag)):
        if math.isinf(z.real) and math.isfinite(z.imag) and z.imag != 0.:
            if z.real > 0:
                r_real = math.copysign(inf, math.cos(z.imag))
                r_imag = math.copysign(inf, math.sin(z.imag))
            else:
                r_real = math.copysign(0., math.cos(z.imag))
                r_imag = math.copysign(0., math.sin(z.imag))
        else:
            r = _special_get(z, _exp_special())
            r_real = r.real
            r_imag = r.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 complex(r_real, r_imag)

    if z.real > _CM_LOG_LARGE_DOUBLE:
        l = math.exp(z.real - 1.)
        r_real = l*math.cos(z.imag)*e
        r_imag = l*math.sin(z.imag)*e
    else:
        l = math.exp(z.real)
        r_real = l*math.cos(z.imag)
        r_imag = l*math.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 complex(r_real, r_imag)

def _c_log(z):
    if _is_special(z):
        return _special_get(z, _log_special())

    ax = math.fabs(z.real)
    ay = math.fabs(z.imag)

    r_real = 0.
    r_imag = 0.
    if ax > _CM_LARGE_DOUBLE or ay > _CM_LARGE_DOUBLE:
        r_real = math.log(math.hypot(ax/2., ay/2.)) + _M_LN2
    elif ax < _DBL_MIN and ay < _DBL_MIN:
        if ax > 0. or ay > 0.:
            # catch cases where hypot(ax, ay) is subnormal
            r_real = math.log(math.hypot(math.ldexp(ax, _DBL_MANT_DIG), math.ldexp(ay, _DBL_MANT_DIG))) - _DBL_MANT_DIG*_M_LN2
        else:
            # log(+/-0. +/- 0i)
            r_real = -inf
            r_imag = math.atan2(z.imag, z.real)
            # errno = EDOM
            return complex(r_real, r_imag)
    else:
        h = math.hypot(ax, ay)
        if 0.71 <= h <= 1.73:
            am = max(ax, ay)
            an = min(ax, ay)
            r_real = math.log1p((am-1)*(am+1) + an*an)/2.
        else:
            r_real = math.log(h)
    r_imag = math.atan2(z.imag, z.real)
    # errno = 0
    return complex(r_real, r_imag)

def _log10_impl(z):
    s = _c_log(z)
    return complex(s.real / _M_LN10, s.imag / _M_LN10)

def _sinh_impl(z):
    r_real = 0.
    r_imag = 0.
    if (not math.isfinite(z.real)) or (not math.isfinite(z.imag)):
        if math.isinf(z.real) and math.isfinite(z.imag) and z.imag != 0.:
            if z.real > 0:
                r_real = math.copysign(inf, math.cos(z.imag))
                r_imag = math.copysign(inf, math.sin(z.imag))
            else:
                r_real = -math.copysign(inf, math.cos(z.imag))
                r_imag = math.copysign(inf, math.sin(z.imag))
        else:
            r = _special_get(z, _sinh_special())
            r_real = r.real
            r_imag = r.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 complex(r_real, r_imag)

    if math.fabs(z.real) > _CM_LOG_LARGE_DOUBLE:
        x_minus_one = z.real - math.copysign(1., z.real)
        r_real = math.cos(z.imag) * math.sinh(x_minus_one) * e
        r_imag = math.sin(z.imag) * math.cosh(x_minus_one) * e
    else:
        r_real = math.cos(z.imag) * math.sinh(z.real)
        r_imag = math.sin(z.imag) * math.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 complex(r_real, r_imag)

def _sin_impl(z):
    s = _sinh_impl(complex(-z.imag, z.real))
    r = complex(s.imag, -s.real)
    return r

def _tanh_impl(z):
    r_real = 0.
    r_imag = 0.
    # special treatment for tanh(+/-inf + iy) if y is finite and
    # nonzero
    if (not math.isfinite(z.real)) or (not math.isfinite(z.imag)):
        if math.isinf(z.real) and math.isfinite(z.imag) and z.imag != 0.:
            if z.real > 0:
                r_real = 1.0
                r_imag = math.copysign(0., 2.*math.sin(z.imag)*math.cos(z.imag))
            else:
                r_real = -1.0
                r_imag = math.copysign(0., 2.*math.sin(z.imag)*math.cos(z.imag))
        else:
            r = _special_get(z, _tanh_special())
            r_real = r.real
            r_imag = r.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 complex(r_real, r_imag)

    # danger of overflow in 2.*z.imag !
    if math.fabs(z.real) > _CM_LOG_LARGE_DOUBLE:
        r_real = math.copysign(1., z.real)
        r_imag = 4.*math.sin(z.imag)*math.cos(z.imag)*math.exp(-2.*math.fabs(z.real))
    else:
        tx = math.tanh(z.real)
        ty = math.tan(z.imag)
        cx = 1./math.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 complex(r_real, r_imag)

def _tan_impl(z):
    s = _tanh_impl(complex(-z.imag, z.real))
    r = complex(s.imag, -s.real)
    return r

def phase(x):
    z = complex(x)
    return z._phase()

def polar(x):
    z = complex(x)
    return complex(x)._polar()

def rect(r, phi):
    z_real = 0.
    z_imag = 0.
    if (not math.isfinite(r)) or (not math.isfinite(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 math.isinf(r) and (math.isfinite(phi) and phi != 0.):
            if r > 0:
                z_real = math.copysign(inf, math.cos(phi))
                z_imag = math.copysign(inf, math.sin(phi))
            else:
                z_real = -math.copysign(inf, math.cos(phi))
                z_imag = -math.copysign(inf, math.sin(phi))
        else:
            z = _special_get(complex(r, phi), _rect_special())
            z_real = z.real
            z_imag = z.imag
        '''
        /* 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;
        '''
    elif 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 * math.cos(phi)
        z_imag = r * math.sin(phi)
        # errno = 0
    return complex(z_real, z_imag)

def exp(x):
    z = complex(x)
    return _exp_impl(z)

def log(x, base = e):
    z = complex(x)
    y = complex(base)
    r = _c_log(z)
    if y == complex(e, 0.0):
        return r
    else:
        return r/_c_log(y)

def log10(x):
    z = complex(x)
    return _log10_impl(z)

def sqrt(x):
    z = complex(x)
    return _sqrt_impl(z)

def asin(x):
    z = complex(x)
    return _asin_impl(z)

def acos(x):
    z = complex(x)
    return _acos_impl(z)

def atan(x):
    z = complex(x)
    return _atan_impl(z)

def sin(x):
    z = complex(x)
    return _sin_impl(z)

def cos(x):
    z = complex(x)
    return _cos_impl(z)

def tan(x):
    z = complex(x)
    return _tan_impl(z)

def asinh(x):
    z = complex(x)
    return _asinh_impl(z)

def acosh(x):
    z = complex(x)
    return _acosh_impl(z)

def atanh(x):
    z = complex(x)
    return _atanh_impl(z)

def sinh(x):
    z = complex(x)
    return _sinh_impl(z)

def cosh(x):
    z = complex(x)
    return _cosh_impl(z)

def tanh(x):
    z = complex(x)
    return _tanh_impl(z)

def isfinite(x):
    z = complex(x)
    return math.isfinite(z.real) and math.isfinite(z.imag)

def isinf(x):
    z = complex(x)
    return math.isinf(z.real) or math.isinf(z.imag)

def isnan(x):
    z = complex(x)
    return math.isnan(z.real) or math.isnan(z.imag)

def isclose(a, b, rel_tol: float = 1e-09, abs_tol: float = 0.0):
    if rel_tol < 0. or abs_tol < 0.:
        raise ValueError("tolerances must be non-negative")

    x = complex(a)
    y = complex(b)

    if x.real == y.real and x.imag == y.imag:
        return True

    if (math.isinf(x.real) or math.isinf(x.imag) or
        math.isinf(y.real) or math.isinf(y.imag)):
        return False

    diff = abs(x - y)
    return (((diff <= rel_tol * abs(y)) or
             (diff <= rel_tol * abs(x))) or
             (diff <= abs_tol))