def cos(x):
    """Evaluates the cos of an interval"""
    if isinstance(x, (int, float)):
        return interval(np.sin(x))
    elif isinstance(x, interval):
        if not (np.isfinite(x.start) and np.isfinite(x.end)):
            return interval(-1, 1, is_valid=x.is_valid)
        na, __ = divmod(x.start, np.pi / 2.0)
        nb, __ = divmod(x.end, np.pi / 2.0)
        start = min(np.cos(x.start), np.cos(x.end))
        end = max(np.cos(x.start), np.cos(x.end))
        if nb - na > 4:
            #differ more than 2*pi
            return interval(-1, 1, is_valid=x.is_valid)
        elif na == nb:
            #in the same quadarant
            return interval(start, end, is_valid=x.is_valid)
        else:
            if (na) // 4 != (nb) // 4:
                #cos has max
                end = 1
            if (na - 2) // 4 != (nb - 2) // 4:
                #cos has min
                start = -1
            return interval(start, end, is_valid=x.is_valid)
    else:
        raise NotImplementedError

def sin(x):
    """evaluates the sine of an interval"""
    if isinstance(x, (int, float)):
        return interval(np.sin(x))
    elif isinstance(x, interval):
        if not x.is_valid:
            return interval(-1, 1, is_valid=x.is_valid)
        na, __ = divmod(x.start, np.pi / 2.0)
        nb, __ = divmod(x.end, np.pi / 2.0)
        start = min(np.sin(x.start), np.sin(x.end))
        end = max(np.sin(x.start), np.sin(x.end))
        if nb - na > 4:
            return interval(-1, 1, is_valid=x.is_valid)
        elif na == nb:
            return interval(start, end, is_valid=x.is_valid)
        else:
            if (na - 1) // 4 != (nb - 1) // 4:
                #sin has max
                end = 1
            if (na - 3) // 4 != (nb - 3) // 4:
                #sin has min
                start = -1
            return interval(start, end)
    else:
        raise NotImplementedError

def test_tan():
    a = tan(interval(0, np.pi / 4))
    assert a.start == 0
    assert a.end == np.tan(np.pi / 4)

    a = tan(interval(np.pi / 4, 3 * np.pi / 4))
    #discontinuity
    assert a.is_valid is None

def exp(x):
    """evaluates the exponential of an interval"""
    if isinstance(x, (int, float)):
        return interval(np.exp(x), np.exp(x))
    elif isinstance(x, interval):
        return interval(np.exp(x.start), np.exp(x.end), is_valid=x.is_valid)
    else:
        raise NotImplementedError

def tanh(x):
    """Evaluates the hyperbolic tan of an interval"""
    if isinstance(x, (int, float)):
        return interval(np.tanh(x), np.tanh(x))
    elif isinstance(x, interval):
        return interval(np.tanh(x.start), np.tanh(x.end), is_valid=x.is_valid)
    else:
        raise NotImplementedError

def test_tan():
    a = tan(interval(0, np.pi / 4))
    assert a.start == 0
    # must match lib_interval definition of tan:
    assert a.end == np.sin(np.pi / 4)/np.cos(np.pi / 4)

    a = tan(interval(np.pi / 4, 3 * np.pi / 4))
    #discontinuity
    assert a.is_valid is None

def sinh(x):
    """Evaluates the hyperbolic sine of an interval"""
    np = import_module('numpy')
    if isinstance(x, (int, float)):
        return interval(np.sinh(x), np.sinh(x))
    elif isinstance(x, interval):
        return interval(np.sinh(x.start), np.sinh(x.end), is_valid=x.is_valid)
    else:
        raise NotImplementedError

def atan(x):
    """evaluates the tan inverse of an interval"""
    if isinstance(x, (int, float)):
        return interval(np.arctan(x))
    elif isinstance(x, interval):
        start = np.arctan(x.start)
        end = np.arctan(x.end)
        return interval(start, end, is_valid=x.is_valid)
    else:
        raise NotImplementedError

def asinh(x):
    """Evaluates the inverse hyperbolic sine of an interval"""
    if isinstance(x, (int, float)):
        return interval(np.arcsinh(x))
    elif isinstance(x, interval):
        start = np.arcsinh(x.start)
        end = np.arcsinh(x.end)
        return interval(start, end, is_valid=x.is_valid)
    else:
        return NotImplementedError

def test_exp():
    a = exp(interval(-np.inf, 0))
    assert a.start == np.exp(-np.inf)
    assert a.end == np.exp(0)
    a = exp(interval(1, 2))
    assert a.start == np.exp(1)
    assert a.end == np.exp(2)
    a = exp(1)
    assert a.start == np.exp(1)
    assert a.end == np.exp(1)

def Abs(x):
    if isinstance(x, (int, float)):
        return interval(abs(x))
    elif isinstance(x, interval):
        if x.start < 0 and x.end > 0:
            return interval(0, max(abs(x.start), abs(x.end)), is_valid=x.is_valid)
        else:
            return interval(abs(x.start), abs(x.end))
    else:
        raise NotImplementedError

def test_log10():
    a = log10(interval(1, 2))
    assert a.start == 0
    assert a.end == np.log10(2)
    a = log10(interval(-1, 1))
    assert a.is_valid is None
    a = log10(interval(-3, -1))
    assert a.is_valid is False
    a = log10(-3)
    assert a.is_valid is False
    a = log10(2)
    assert a.start == np.log10(2)
    assert a.end == np.log10(2)

def test_hashable():
    '''
    test that interval objects are hashable.
    this is required in order to be able to put them into the cache, which
    appears to be necessary for plotting in py3k. For details, see:

    https://github.com/sympy/sympy/pull/2101
    https://github.com/sympy/sympy/issues/6533
    '''
    hash(interval(1, 1))
    hash(interval(1, 1, is_valid=True))
    hash(interval(-4, -0.5))
    hash(interval(-2, -0.5))
    hash(interval(0.25, 8.0))

def test_cosh():
    a = cosh(interval(1, 2))
    assert a.start == np.cosh(1)
    assert a.end == np.cosh(2)
    a = cosh(interval(-2, -1))
    assert a.start == np.cosh(-1)
    assert a.end == np.cosh(-2)

    a = cosh(interval(-2, 1))
    assert a.start == 1
    assert a.end == np.cosh(-2)

    a = cosh(1)
    assert a.start == np.cosh(1)
    assert a.end == np.cosh(1)

def test_acosh():
    a = acosh(interval(3, 5))
    assert a.start == np.arccosh(3)
    assert a.end == np.arccosh(5)

    a = acosh(interval(0, 3))
    assert a.is_valid is None
    a = acosh(interval(-3, 0.5))
    assert a.is_valid is False

    a = acosh(0.5)
    assert a.is_valid is False

    a = acosh(2)
    assert a.start == np.arccosh(2)
    assert a.end == np.arccosh(2)

def cosh(x):
    """Evaluates the hyperbolic cos of an interval"""
    if isinstance(x, (int, float)):
        return interval(np.cosh(x), np.cosh(x))
    elif isinstance(x, interval):
        #both signs
        if x.start < 0 and x.end > 0:
            end = max(np.cosh(x.start), np.cosh(x.end))
            return interval(1, end, is_valid=x.is_valid)
        else:
            #Monotonic
            start = np.cosh(x.start)
            end = np.cosh(x.end)
            return interval(start, end, is_valid=x.is_valid)
    else:
        raise NotImplementedError

def test_floor():
    a = floor(interval(0.2, 0.5))
    assert a.start == 0
    assert a.end == 0

    a = floor(interval(0.5, 1.5))
    assert a.start == 0
    assert a.end == 1
    assert a.is_valid is None

    a = floor(interval(-5, 5))
    assert a.is_valid is None

    a = floor(5.4)
    assert a.start == 5
    assert a.end == 5

def test_ceil():
    a = ceil(interval(0.2, 0.5))
    assert a.start == 1
    assert a.end == 1

    a = ceil(interval(0.5, 1.5))
    assert a.start == 1
    assert a.end == 2
    assert a.is_valid is None

    a = ceil(interval(-5, 5))
    assert a.is_valid is None

    a = ceil(5.4)
    assert a.start == 6
    assert a.end == 6

def test_atan():
    a = atan(interval(0, 1))
    assert a.start == np.arctan(0)
    assert a.end == np.arctan(1)
    a = atan(1)
    assert a.start == np.arctan(1)
    assert a.end == np.arctan(1)

def test_tanh():
    a = tanh(interval(-3, 3))
    assert a.start == np.tanh(-3)
    assert a.end == np.tanh(3)

    a = tanh(3)
    assert a.start == np.tanh(3)
    assert a.end == np.tanh(3)