def c_atanh(x, y):
    if not isfinite(x) or not isfinite(y):
        return atanh_special_values[special_type(x)][special_type(y)]

    # Reduce to case where x >= 0., using atanh(z) = -atanh(-z).
    if x < 0.:
        return c_neg(*c_atanh(*c_neg(x, y)))

    ay = fabs(y)
    if x > 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 y
        h = math.hypot(x/2., y/2.)   # safe from overflow
        real = x/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
        imag = -copysign(math.pi/2., -y)
    elif x == 1. and ay < CM_SQRT_DBL_MIN:
        # C99 standard says:  atanh(1+/-0.) should be inf +/- 0i
        if ay == 0.:
            raise ValueError("math domain error")
            #real = INF
            #imag = y
        else:
            real = -math.log(math.sqrt(ay)/math.sqrt(math.hypot(ay, 2.)))
            imag = copysign(math.atan2(2., -ay) / 2, y)
    else:
        real = log1p(4.*x/((1-x)*(1-x) + ay*ay))/4.
        imag = -math.atan2(-2.*y, (1-x)*(1+x) - ay*ay) / 2.
    return (real, imag)

def c_exp(x, y):
    if not isfinite(x) or not isfinite(y):
        if isinf(x) and isfinite(y) and y != 0.:
            if x > 0:
                real = copysign(INF, math.cos(y))
                imag = copysign(INF, math.sin(y))
            else:
                real = copysign(0., math.cos(y))
                imag = copysign(0., math.sin(y))
            r = (real, imag)
        else:
            r = exp_special_values[special_type(x)][special_type(y)]

        # need to raise ValueError if y is +/- infinity and x is not
        # a NaN and not -infinity
        if isinf(y) and (isfinite(x) or (isinf(x) and x > 0)):
            raise ValueError("math domain error")
        return r

    if x > CM_LOG_LARGE_DOUBLE:
        l = math.exp(x-1.)
        real = l * math.cos(y) * math.e
        imag = l * math.sin(y) * math.e
    else:
        l = math.exp(x)
        real = l * math.cos(y)
        imag = l * math.sin(y)
    if isinf(real) or isinf(imag):
        raise OverflowError("math range error")
    return real, imag

def c_rect(r, phi):
    if not isfinite(r) or not 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 isinf(r) and isfinite(phi) and phi != 0.:
            if r > 0:
                real = copysign(INF, math.cos(phi))
                imag = copysign(INF, math.sin(phi))
            else:
                real = -copysign(INF, math.cos(phi))
                imag = -copysign(INF, math.sin(phi))
            z = (real, imag)
        else:
            z = rect_special_values[special_type(r)][special_type(phi)]

        # need to raise ValueError if r is a nonzero number and phi
        # is infinite
        if r != 0. and not isnan(r) and isinf(phi):
            raise ValueError("math domain error")
        return z

    real = r * math.cos(phi)
    imag = r * math.sin(phi)
    return real, imag

def c_sinh(x, y):
    # special treatment for sinh(+/-inf + iy) if y is finite and nonzero
    if not isfinite(x) or not isfinite(y):
        if isinf(x) and isfinite(y) and y != 0.:
            if x > 0:
                real = copysign(INF, math.cos(y))
                imag = copysign(INF, math.sin(y))
            else:
                real = -copysign(INF, math.cos(y))
                imag = copysign(INF, math.sin(y))
            r = (real, imag)
        else:
            r = sinh_special_values[special_type(x)][special_type(y)]

        # need to raise ValueError if y is +/- infinity and x is not
        # a NaN
        if isinf(y) and not isnan(x):
            raise ValueError("math domain error")
        return r

    if fabs(x) > CM_LOG_LARGE_DOUBLE:
        x_minus_one = x - copysign(1., x)
        real = math.cos(y) * math.sinh(x_minus_one) * math.e
        imag = math.sin(y) * math.cosh(x_minus_one) * math.e
    else:
        real = math.cos(y) * math.sinh(x)
        imag = math.sin(y) * math.cosh(x)
    if isinf(real) or isinf(imag):
        raise OverflowError("math range error")
    return real, imag

def test_nan_and_special_values():
    from pypy.rlib.rfloat import isnan, isinf, isfinite, copysign
    inf = 1e300 * 1e300
    assert isinf(inf)
    nan = inf/inf
    assert isnan(nan)

    for value, checker in [
            (inf,   lambda x: isinf(x) and x > 0.0),
            (-inf,  lambda x: isinf(x) and x < 0.0),
            (nan,   isnan),
            (42.0,  isfinite),
            (0.0,   lambda x: not x and copysign(1., x) == 1.),
            (-0.0,  lambda x: not x and copysign(1., x) == -1.),
            ]:
        def f():
            return value
        f1 = compile(f, [])
        res = f1()
        assert checker(res)

        l = [value]
        def g(x):
            return l[x]
        g2 = compile(g, [int])
        res = g2(0)
        assert checker(res)

        l2 = [(-value, -value), (value, value)]
        def h(x):
            return l2[x][1]
        h3 = compile(h, [int])
        res = h3(1)
        assert checker(res)

def str__Complex(space, w_complex):
    if w_complex.realval == 0 and copysign(1., w_complex.realval) == 1.:
        return space.wrap(str_format(w_complex.imagval) + 'j')
    sign = (copysign(1., w_complex.imagval) == 1. or
            isnan(w_complex.imagval)) and '+' or ''
    return space.wrap('(' + str_format(w_complex.realval)
                      + sign + str_format(w_complex.imagval) + 'j)')

def c_cosh(x, y):
    if not isfinite(x) or not isfinite(y):
        if isinf(x) and isfinite(y) and y != 0.:
            if x > 0:
                real = copysign(INF, math.cos(y))
                imag = copysign(INF, math.sin(y))
            else:
                real = copysign(INF, math.cos(y))
                imag = -copysign(INF, math.sin(y))
            r = (real, imag)
        else:
            r = cosh_special_values[special_type(x)][special_type(y)]

        # need to raise ValueError if y is +/- infinity and x is not
        # a NaN
        if isinf(y) and not isnan(x):
            raise ValueError("math domain error")
        return r

    if fabs(x) > CM_LOG_LARGE_DOUBLE:
        # deal correctly with cases where cosh(x) overflows but
        # cosh(z) does not.
        x_minus_one = x - copysign(1., x)
        real = math.cos(y) * math.cosh(x_minus_one) * math.e
        imag = math.sin(y) * math.sinh(x_minus_one) * math.e
    else:
        real = math.cos(y) * math.cosh(x)
        imag = math.sin(y) * math.sinh(x)
    if isinf(real) or isinf(imag):
        raise OverflowError("math range error")
    return real, imag

 def __eq__(self, other):
     if (type(self) is SomeFloat and type(other) is SomeFloat and
         self.is_constant() and other.is_constant()):
         from pypy.rlib.rfloat import isnan, copysign
         # NaN unpleasantness.
         if isnan(self.const) and isnan(other.const):
             return True
         # 0.0 vs -0.0 unpleasantness.
         if not self.const and not other.const:
             return copysign(1., self.const) == copysign(1., other.const)
         #
     return super(SomeFloat, self).__eq__(other)

def c_sqrt(x, y):
    # 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.

    if not isfinite(x) or not isfinite(y):
        return sqrt_special_values[special_type(x)][special_type(y)]

    if x == 0. and y == 0.:
        return (0., y)

    ax = fabs(x)
    ay = fabs(y)

    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)
        ay1= math.ldexp(ay, CM_SCALE_UP)
        s = math.ldexp(math.sqrt(ax + math.hypot(ax, ay1)),
                       CM_SCALE_DOWN)
    else:
        ax /= 8.
        s = 2.*math.sqrt(ax + math.hypot(ax, ay/8.))

    d = ay/(2.*s)

    if x >= 0.:
        return (s, copysign(d, y))
    else:
        return (d, copysign(s, y))

def rAssertAlmostEqual(a, b, rel_err = 2e-15, abs_err = 5e-323, msg=''):
    """Fail if the two floating-point numbers are not almost equal.

    Determine whether floating-point values a and b are equal to within
    a (small) rounding error.  The default values for rel_err and
    abs_err are chosen to be suitable for platforms where a float is
    represented by an IEEE 754 double.  They allow an error of between
    9 and 19 ulps.
    """

    # special values testing
    if isnan(a):
        if isnan(b):
            return
        raise AssertionError(msg + '%r should be nan' % (b,))

    if isinf(a):
        if a == b:
            return
        raise AssertionError(msg + 'finite result where infinity expected: '
                                   'expected %r, got %r' % (a, b))

    # if both a and b are zero, check whether they have the same sign
    # (in theory there are examples where it would be legitimate for a
    # and b to have opposite signs; in practice these hardly ever
    # occur).
    if not a and not b:
        # only check it if we are running on top of CPython >= 2.6
        if sys.version_info >= (2, 6) and copysign(1., a) != copysign(1., b):
            raise AssertionError(msg + 'zero has wrong sign: expected %r, '
                                       'got %r' % (a, b))

    # if a-b overflows, or b is infinite, return False.  Again, in
    # theory there are examples where a is within a few ulps of the
    # max representable float, and then b could legitimately be
    # infinite.  In practice these examples are rare.
    try:
        absolute_error = abs(b-a)
    except OverflowError:
        pass
    else:
        # test passes if either the absolute error or the relative
        # error is sufficiently small.  The defaults amount to an
        # error of between 9 ulps and 19 ulps on an IEEE-754 compliant
        # machine.
        if absolute_error <= max(abs_err, rel_err * abs(a)):
            return
    raise AssertionError(msg + '%r and %r are not sufficiently close' % (a, b))

def float_hex__Float(space, w_float):
    value = w_float.floatval
    if not isfinite(value):
        return str__Float(space, w_float)
    if value == 0.0:
        if copysign(1., value) == -1.:
            return space.wrap("-0x0.0p+0")
        else:
            return space.wrap("0x0.0p+0")
    mant, exp = math.frexp(value)
    shift = 1 - max(rfloat.DBL_MIN_EXP - exp, 0)
    mant = math.ldexp(mant, shift)
    mant = abs(mant)
    exp -= shift
    result = ['\0'] * ((TOHEX_NBITS - 1) // 4 + 2)
    result[0] = _char_from_hex(int(mant))
    mant -= int(mant)
    result[1] = "."
    for i in range((TOHEX_NBITS - 1) // 4):
        mant *= 16.0
        result[i + 2] = _char_from_hex(int(mant))
        mant -= int(mant)
    if exp < 0:
        sign = "-"
    else:
        sign = "+"
    exp = abs(exp)
    s = ''.join(result)
    if value < 0.0:
        return space.wrap("-0x%sp%s%d" % (s, sign, exp))
    else:
        return space.wrap("0x%sp%s%d" % (s, sign, exp))

def test_special_values():
    from pypy.module.cmath.special_value import sqrt_special_values
    assert len(sqrt_special_values) == 7
    assert len(sqrt_special_values[4]) == 7
    assert isinstance(sqrt_special_values[5][1], tuple)
    assert sqrt_special_values[5][1][0] == 1e200 * 1e200
    assert sqrt_special_values[5][1][1] == -0.
    assert copysign(1., sqrt_special_values[5][1][1]) == -1.

 def div(self, v1, v2):
     # XXX this won't work after translation, probably requires ovfcheck
     try:
         return v1 / v2
     except ZeroDivisionError:
         if v1 == v2 == 0.0:
             return rfloat.NAN
         return rfloat.copysign(rfloat.INFINITY, v1 * v2)

def c_asinh(x, y):
    if not isfinite(x) or not isfinite(y):
        return asinh_special_values[special_type(x)][special_type(y)]

    if fabs(x) > CM_LARGE_DOUBLE or fabs(y) > CM_LARGE_DOUBLE:
        if y >= 0.:
            real = copysign(math.log(math.hypot(x/2., y/2.)) +
                            M_LN2*2., x)
        else:
            real = -copysign(math.log(math.hypot(x/2., y/2.)) +
                             M_LN2*2., -x)
        imag = math.atan2(y, fabs(x))
    else:
        s1x, s1y = c_sqrt(1.+y, -x)
        s2x, s2y = c_sqrt(1.-y, x)
        real = asinh(s1x*s2y - s2x*s1y)
        imag = math.atan2(y, s1x*s2x - s1y*s2y)
    return (real, imag)

 def pow(self, v1, v2):
     try:
         return math.pow(v1, v2)
     except ValueError:
         return rfloat.NAN
     except OverflowError:
         if math.modf(v2)[0] == 0 and math.modf(v2 / 2)[0] != 0:
             # Odd integer powers result in the same sign as the base
             return rfloat.copysign(rfloat.INFINITY, v1)
         return rfloat.INFINITY

def c_acos(x, y):
    if not isfinite(x) or not isfinite(y):
        return acos_special_values[special_type(x)][special_type(y)]

    if fabs(x) > CM_LARGE_DOUBLE or fabs(y) > CM_LARGE_DOUBLE:
        # avoid unnecessary overflow for large arguments
        real = math.atan2(fabs(y), x)
        # split into cases to make sure that the branch cut has the
        # correct continuity on systems with unsigned zeros
        if x < 0.:
            imag = -copysign(math.log(math.hypot(x/2., y/2.)) +
                             M_LN2*2., y)
        else:
            imag = copysign(math.log(math.hypot(x/2., y/2.)) +
                            M_LN2*2., -y)
    else:
        s1x, s1y = c_sqrt(1.-x, -y)
        s2x, s2y = c_sqrt(1.+x, y)
        real = 2.*math.atan2(s1x, s2x)
        imag = asinh(s2x*s1y - s2y*s1x)
    return (real, imag)

def c_tanh(x, y):
    # 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).

    if not isfinite(x) or not isfinite(y):
        if isinf(x) and isfinite(y) and y != 0.:
            if x > 0:
                real = 1.0        # vv XXX why is the 2. there?
                imag = copysign(0., 2. * math.sin(y) * math.cos(y))
            else:
                real = -1.0
                imag = copysign(0., 2. * math.sin(y) * math.cos(y))
            r = (real, imag)
        else:
            r = tanh_special_values[special_type(x)][special_type(y)]

        # need to raise ValueError if y is +/-infinity and x is finite
        if isinf(y) and isfinite(x):
            raise ValueError("math domain error")
        return r

    if fabs(x) > CM_LOG_LARGE_DOUBLE:
        real = copysign(1., x)
        imag = 4. * math.sin(y) * math.cos(y) * math.exp(-2.*fabs(x))
    else:
        tx = math.tanh(x)
        ty = math.tan(y)
        cx = 1. / math.cosh(x)
        txty = tx * ty
        denom = 1. + txty * txty
        real = tx * (1. + ty*ty) / denom
        imag = ((ty / denom) * cx) * cx
    return real, imag

 def _make_key(self, obj):
     # see the tests 'test_zeros_not_mixed*' in ../test/test_compiler.py
     space = self.space
     w_type = space.type(obj)
     if space.is_w(w_type, space.w_float):
         val = space.float_w(obj)
         if val == 0.0 and rfloat.copysign(1., val) < 0:
             w_key = space.newtuple([obj, space.w_float, space.w_None])
         else:
             w_key = space.newtuple([obj, space.w_float])
     elif space.is_w(w_type, space.w_complex):
         w_real = space.getattr(obj, space.wrap("real"))
         w_imag = space.getattr(obj, space.wrap("imag"))
         real = space.float_w(w_real)
         imag = space.float_w(w_imag)
         real_negzero = (real == 0.0 and
                         rfloat.copysign(1., real) < 0)
         imag_negzero = (imag == 0.0 and
                         rfloat.copysign(1., imag) < 0)
         if real_negzero and imag_negzero:
             tup = [obj, space.w_complex, space.w_None, space.w_None,
                    space.w_None]
         elif imag_negzero:
             tup = [obj, space.w_complex, space.w_None, space.w_None]
         elif real_negzero:
             tup = [obj, space.w_complex, space.w_None]
         else:
             tup = [obj, space.w_complex]
         w_key = space.newtuple(tup)
     elif space.is_w(w_type, space.w_tuple):
         result_w = [obj, w_type]
         for w_item in space.fixedview(obj):
             result_w.append(self._make_key(w_item))
         w_key = space.newtuple(result_w[:])
     else:
         w_key = space.newtuple([obj, w_type])
     return w_key

def _sinpi(x):
    y = math.fmod(abs(x), 2.)
    n = int(rfloat.round_away(2. * y))
    if n == 0:
        r = math.sin(math.pi * y)
    elif n == 1:
        r = math.cos(math.pi * (y - .5))
    elif n == 2:
        r = math.sin(math.pi * (1. - y))
    elif n == 3:
        r = -math.cos(math.pi * (y - 1.5))
    elif n == 4:
        r = math.sin(math.pi * (y - 2.))
    else:
        raise AssertionError("should not reach")
    return rfloat.copysign(1., x) * r

 def __init__(self, value):
     self.value = value     # a concrete value
     # try to be smart about constant mutable or immutable values
     key = type(self.value), self.value  # to avoid confusing e.g. 0 and 0.0
     #
     # we also have to avoid confusing 0.0 and -0.0 (needed e.g. for
     # translating the cmath module)
     if key[0] is float and not self.value:
         from pypy.rlib.rfloat import copysign
         if copysign(1., self.value) == -1.:    # -0.0
             key = (float, "-0.0")
     #
     try:
         hash(key)
     except TypeError:
         key = id(self.value)
     self.key = key