import decimal
from sympy import (Rational, Symbol, Float, I, sqrt, oo, nan, pi, E, Integer,
                   S, factorial, Catalan, EulerGamma, GoldenRatio, cos, exp,
                   Number, zoo, log, Mul, Pow, Tuple, latex, Gt, Lt, Ge, Le,
                   AlgebraicNumber, simplify, sin)
from sympy.core.compatibility import long, u
from sympy.core.power import integer_nthroot
from sympy.core.logic import fuzzy_not
from sympy.core.numbers import (igcd, ilcm, igcdex, seterr, _intcache,
    mpf_norm, comp, mod_inverse)
from mpmath import mpf
from sympy.utilities.pytest import XFAIL, raises
import mpmath

t = Symbol('t', real=False)

def same_and_same_prec(a, b):
    # stricter matching for Floats
    return a == b and a._prec == b._prec


def test_integers_cache():
    python_int = 2**65 + 3175259

    while python_int in _intcache or hash(python_int) in _intcache:
        python_int += 1

    sympy_int = Integer(python_int)

    assert python_int in _intcache
    assert hash(python_int) not in _intcache

    sympy_int_int = Integer(sympy_int)

    assert python_int in _intcache
    assert hash(python_int) not in _intcache

    sympy_hash_int = Integer(hash(python_int))

    assert python_int in _intcache
    assert hash(python_int) in _intcache


def test_seterr():
    seterr(divide=True)
    raises(ValueError, lambda: S.Zero/S.Zero)
    seterr(divide=False)
    assert S.Zero / S.Zero == S.NaN


def test_mod():
    x = Rational(1, 2)
    y = Rational(3, 4)
    z = Rational(5, 18043)

    assert x % x == 0
    assert x % y == 1/S(2)
    assert x % z == 3/S(36086)
    assert y % x == 1/S(4)
    assert y % y == 0
    assert y % z == 9/S(72172)
    assert z % x == 5/S(18043)
    assert z % y == 5/S(18043)
    assert z % z == 0

    a = Float(2.6)

    assert (a % .2) == 0
    assert (a % 2).round(15) == 0.6
    assert (a % 0.5).round(15) == 0.1

    p = Symbol('p', infinite=True)

    assert zoo % 0 == nan
    assert oo % oo == nan
    assert zoo % oo == nan
    assert 5 % oo == nan
    assert p % 5 == nan

    # In these two tests, if the precision of m does
    # not match the precision of the ans, then it is
    # likely that the change made now gives an answer
    # with degraded accuracy.
    r = Rational(500, 41)
    f = Float('.36', 3)
    m = r % f
    ans = Float(r % Rational(f), 3)
    assert m == ans and m._prec == ans._prec
    f = Float('8.36', 3)
    m = f % r
    ans = Float(Rational(f) % r, 3)
    assert m == ans and m._prec == ans._prec

    s = S.Zero

    assert s % float(1) == S.Zero

    # No rounding required since these numbers can be represented
    # exactly.
    assert Rational(3, 4) % Float(1.1) == 0.75
    assert Float(1.5) % Rational(5, 4) == 0.25
    assert Rational(5, 4).__rmod__(Float('1.5')) == 0.25
    assert Float('1.5').__rmod__(Float('2.75')) == Float('1.25')
    assert 2.75 % Float('1.5') == Float('1.25')

    a = Integer(7)
    b = Integer(4)

    assert type(a % b) == Integer
    assert a % b == Integer(3)
    assert Integer(1) % Rational(2, 3) == Rational(1, 3)
    assert Rational(7, 5) % Integer(1) == Rational(2, 5)
    assert Integer(2) % 1.5 == 0.5

    assert Integer(3).__rmod__(Integer(10)) == Integer(1)
    assert Integer(10) % 4 == Integer(2)
    assert 15 % Integer(4) == Integer(3)


def test_divmod():
    assert divmod(S(12), S(8)) == Tuple(1, 4)
    assert divmod(-S(12), S(8)) == Tuple(-2, 4)
    assert divmod(S(0), S(1)) == Tuple(0, 0)
    raises(ZeroDivisionError, lambda: divmod(S(0), S(0)))
    raises(ZeroDivisionError, lambda: divmod(S(1), S(0)))
    assert divmod(S(12), 8) == Tuple(1, 4)
    assert divmod(12, S(8)) == Tuple(1, 4)

    assert divmod(S("2"), S("3/2")) == Tuple(S("1"), S("1/2"))
    assert divmod(S("3/2"), S("2")) == Tuple(S("0"), S("3/2"))
    assert divmod(S("2"), S("3.5")) == Tuple(S("0"), S("2"))
    assert divmod(S("3.5"), S("2")) == Tuple(S("1"), S("1.5"))
    assert divmod(S("2"), S("1/3")) == Tuple(S("6"), S("0"))
    assert divmod(S("1/3"), S("2")) == Tuple(S("0"), S("1/3"))
    assert divmod(S("2"), S("0.1")) == Tuple(S("20"), S("0"))
    assert divmod(S("0.1"), S("2")) == Tuple(S("0"), S("0.1"))
    assert divmod(S("2"), 2) == Tuple(S("1"), S("0"))
    assert divmod(2, S("2")) == Tuple(S("1"), S("0"))
    assert divmod(S("2"), 1.5) == Tuple(S("1"), S("0.5"))
    assert divmod(1.5, S("2")) == Tuple(S("0"), S("1.5"))
    assert divmod(0.3, S("2")) == Tuple(S("0"), S("0.3"))
    assert divmod(S("3/2"), S("3.5")) == Tuple(S("0"), S("3/2"))
    assert divmod(S("3.5"), S("3/2")) == Tuple(S("2"), S("0.5"))
    assert divmod(S("3/2"), S("1/3")) == Tuple(S("4"), Float("1/6"))
    assert divmod(S("1/3"), S("3/2")) == Tuple(S("0"), S("1/3"))
    assert divmod(S("3/2"), S("0.1")) == Tuple(S("15"), S("0"))
    assert divmod(S("0.1"), S("3/2")) == Tuple(S("0"), S("0.1"))
    assert divmod(S("3/2"), 2) == Tuple(S("0"), S("3/2"))
    assert divmod(2, S("3/2")) == Tuple(S("1"), S("0.5"))
    assert divmod(S("3/2"), 1.5) == Tuple(S("1"), S("0"))
    assert divmod(1.5, S("3/2")) == Tuple(S("1"), S("0"))
    assert divmod(S("3/2"), 0.3) == Tuple(S("5"), S("0"))
    assert divmod(0.3, S("3/2")) == Tuple(S("0"), S("0.3"))
    assert divmod(S("1/3"), S("3.5")) == Tuple(S("0"), S("1/3"))
    assert divmod(S("3.5"), S("0.1")) == Tuple(S("35"), S("0"))
    assert divmod(S("0.1"), S("3.5")) == Tuple(S("0"), S("0.1"))
    assert divmod(S("3.5"), 2) == Tuple(S("1"), S("1.5"))
    assert divmod(2, S("3.5")) == Tuple(S("0"), S("2"))
    assert divmod(S("3.5"), 1.5) == Tuple(S("2"), S("0.5"))
    assert divmod(1.5, S("3.5")) == Tuple(S("0"), S("1.5"))
    assert divmod(0.3, S("3.5")) == Tuple(S("0"), S("0.3"))
    assert divmod(S("0.1"), S("1/3")) == Tuple(S("0"), S("0.1"))
    assert divmod(S("1/3"), 2) == Tuple(S("0"), S("1/3"))
    assert divmod(2, S("1/3")) == Tuple(S("6"), S("0"))
    assert divmod(S("1/3"), 1.5) == Tuple(S("0"), S("1/3"))
    assert divmod(0.3, S("1/3")) == Tuple(S("0"), S("0.3"))
    assert divmod(S("0.1"), 2) == Tuple(S("0"), S("0.1"))
    assert divmod(2, S("0.1")) == Tuple(S("20"), S("0"))
    assert divmod(S("0.1"), 1.5) == Tuple(S("0"), S("0.1"))
    assert divmod(1.5, S("0.1")) == Tuple(S("15"), S("0"))
    assert divmod(S("0.1"), 0.3) == Tuple(S("0"), S("0.1"))

    assert str(divmod(S("2"), 0.3)) == '(6, 0.2)'
    assert str(divmod(S("3.5"), S("1/3"))) == '(10, 0.166666666666667)'
    assert str(divmod(S("3.5"), 0.3)) == '(11, 0.2)'
    assert str(divmod(S("1/3"), S("0.1"))) == '(3, 0.0333333333333333)'
    assert str(divmod(1.5, S("1/3"))) == '(4, 0.166666666666667)'
    assert str(divmod(S("1/3"), 0.3)) == '(1, 0.0333333333333333)'
    assert str(divmod(0.3, S("0.1"))) == '(2, 0.1)'

    assert divmod(-3, S(2)) == (-2, 1)
    assert divmod(S(-3), S(2)) == (-2, 1)
    assert divmod(S(-3), 2) == (-2, 1)


def test_igcd():
    assert igcd(0, 0) == 0
    assert igcd(0, 1) == 1
    assert igcd(1, 0) == 1
    assert igcd(0, 7) == 7
    assert igcd(7, 0) == 7
    assert igcd(7, 1) == 1
    assert igcd(1, 7) == 1
    assert igcd(-1, 0) == 1
    assert igcd(0, -1) == 1
    assert igcd(-1, -1) == 1
    assert igcd(-1, 7) == 1
    assert igcd(7, -1) == 1
    assert igcd(8, 2) == 2
    assert igcd(4, 8) == 4
    assert igcd(8, 16) == 8
    assert igcd(7, -3) == 1
    assert igcd(-7, 3) == 1
    assert igcd(-7, -3) == 1
    assert igcd(*[10, 20, 30]) == 10
    raises(ValueError, lambda: igcd(45.1, 30))
    raises(ValueError, lambda: igcd(45, 30.1))


def test_ilcm():
    assert ilcm(0, 0) == 0
    assert ilcm(1, 0) == 0
    assert ilcm(0, 1) == 0
    assert ilcm(1, 1) == 1
    assert ilcm(2, 1) == 2
    assert ilcm(8, 2) == 8
    assert ilcm(8, 6) == 24
    assert ilcm(8, 7) == 56
    assert ilcm(*[10, 20, 30]) == 60
    raises(ValueError, lambda: ilcm(8.1, 7))
    raises(ValueError, lambda: ilcm(8, 7.1))


def test_igcdex():
    assert igcdex(2, 3) == (-1, 1, 1)
    assert igcdex(10, 12) == (-1, 1, 2)
    assert igcdex(100, 2004) == (-20, 1, 4)


def _strictly_equal(a, b):
    return (a.p, a.q, type(a.p), type(a.q)) == \
           (b.p, b.q, type(b.p), type(b.q))


def _test_rational_new(cls):
    """
    Tests that are common between Integer and Rational.
    """
    assert cls(0) is S.Zero
    assert cls(1) is S.One
    assert cls(-1) is S.NegativeOne
    # These look odd, but are similar to int():
    assert cls('1') is S.One
    assert cls(u('-1')) is S.NegativeOne

    i = Integer(10)
    assert _strictly_equal(i, cls('10'))
    assert _strictly_equal(i, cls(u('10')))
    assert _strictly_equal(i, cls(long(10)))
    assert _strictly_equal(i, cls(i))

    raises(TypeError, lambda: cls(Symbol('x')))


def test_Integer_new():
    """
    Test for Integer constructor
    """
    _test_rational_new(Integer)

    assert _strictly_equal(Integer(0.9), S.Zero)
    assert _strictly_equal(Integer(10.5), Integer(10))
    raises(ValueError, lambda: Integer("10.5"))
    assert Integer(Rational('1.' + '9'*20)) == 1


def test_Rational_new():
    """"
    Test for Rational constructor
    """
    _test_rational_new(Rational)

    n1 = Rational(1, 2)
    assert n1 == Rational(Integer(1), 2)
    assert n1 == Rational(Integer(1), Integer(2))
    assert n1 == Rational(1, Integer(2))
    assert n1 == Rational(Rational(1, 2))
    assert 1 == Rational(n1, n1)
    assert Rational(3, 2) == Rational(Rational(1, 2), Rational(1, 3))
    assert Rational(3, 1) == Rational(1, Rational(1, 3))
    n3_4 = Rational(3, 4)
    assert Rational('3/4') == n3_4
    assert -Rational('-3/4') == n3_4
    assert Rational('.76').limit_denominator(4) == n3_4
    assert Rational(19, 25).limit_denominator(4) == n3_4
    assert Rational('19/25').limit_denominator(4) == n3_4
    assert Rational(1.0, 3) == Rational(1, 3)
    assert Rational(1, 3.0) == Rational(1, 3)
    assert Rational(Float(0.5)) == Rational(1, 2)
    assert Rational('1e2/1e-2') == Rational(10000)
    assert Rational(-1, 0) == S.NegativeInfinity
    assert Rational(1, 0) == S.Infinity
    raises(TypeError, lambda: Rational('3**3'))
    raises(TypeError, lambda: Rational('1/2 + 2/3'))

    # handle fractions.Fraction instances
    try:
        import fractions
        assert Rational(fractions.Fraction(1, 2)) == Rational(1, 2)
    except ImportError:
        pass


def test_Number_new():
    """"
    Test for Number constructor
    """
    # Expected behavior on numbers and strings
    assert Number(1) is S.One
    assert Number(2).__class__ is Integer
    assert Number(-622).__class__ is Integer
    assert Number(5, 3).__class__ is Rational
    assert Number(5.3).__class__ is Float
    assert Number('1') is S.One
    assert Number('2').__class__ is Integer
    assert Number('-622').__class__ is Integer
    assert Number('5/3').__class__ is Rational
    assert Number('5.3').__class__ is Float
    raises(ValueError, lambda: Number('cos'))
    raises(TypeError, lambda: Number(cos))
    a = Rational(3, 5)
    assert Number(a) is a  # Check idempotence on Numbers


def test_Rational_cmp():
    n1 = Rational(1, 4)
    n2 = Rational(1, 3)
    n3 = Rational(2, 4)
    n4 = Rational(2, -4)
    n5 = Rational(0)
    n6 = Rational(1)
    n7 = Rational(3)
    n8 = Rational(-3)

    assert n8 < n5
    assert n5 < n6
    assert n6 < n7
    assert n8 < n7
    assert n7 > n8
    assert (n1 + 1)**n2 < 2
    assert ((n1 + n6)/n7) < 1

    assert n4 < n3
    assert n2 < n3
    assert n1 < n2
    assert n3 > n1
    assert not n3 < n1
    assert not (Rational(-1) > 0)
    assert Rational(-1) < 0

    raises(TypeError, lambda: n1 < S.NaN)
    raises(TypeError, lambda: n1 <= S.NaN)
    raises(TypeError, lambda: n1 > S.NaN)
    raises(TypeError, lambda: n1 >= S.NaN)


def test_Float():
    def eq(a, b):
        t = Float("1.0E-15")
        return (-t < a - b < t)

    a = Float(2) ** Float(3)
    assert eq(a.evalf(), Float(8))
    assert eq((pi ** -1).evalf(), Float("0.31830988618379067"))
    a = Float(2) ** Float(4)
    assert eq(a.evalf(), Float(16))
    assert (S(.3) == S(.5)) is False
    x_str = Float((0, '13333333333333', -52, 53))
    x2_str = Float((0, '26666666666666', -53, 53))
    x_hex = Float((0, long(0x13333333333333), -52, 53))
    x_dec = Float((0, 5404319552844595, -52, 53))
    x2_hex = Float((0, long(0x13333333333333)*2, -53, 53))
    assert x_str == x_hex == x_dec == x2_hex == Float(1.2)
    # x2_str and 1.2 are superficially the same
    assert str(x2_str) == str(Float(1.2))
    # but are different at the mpf level
    assert Float(1.2)._mpf_ == (0, long(5404319552844595), -52, 53)
    assert x2_str._mpf_ == (0, long(10808639105689190), -53, 53)

    assert Float((0, long(0), -123, -1)) == Float('nan')
    assert Float((0, long(0), -456, -2)) == Float('inf') == Float('+inf')
    assert Float((1, long(0), -789, -3)) == Float('-inf')

    raises(ValueError, lambda: Float((0, 7, 1, 3), ''))

    assert Float('+inf').is_finite is False
    assert Float('+inf').is_negative is False
    assert Float('+inf').is_positive is True
    assert Float('+inf').is_infinite is True
    assert Float('+inf').is_zero is False

    assert Float('-inf').is_finite is False
    assert Float('-inf').is_negative is True
    assert Float('-inf').is_positive is False
    assert Float('-inf').is_infinite is True
    assert Float('-inf').is_zero is False

    assert Float('0.0').is_finite is True
    assert Float('0.0').is_negative is False
    assert Float('0.0').is_positive is False
    assert Float('0.0').is_infinite is False
    assert Float('0.0').is_zero is True

    # rationality properties
    assert Float(1).is_rational is None
    assert Float(1).is_irrational is None
    assert sqrt(2).n(15).is_rational is None
    assert sqrt(2).n(15).is_irrational is None

    # do not automatically evalf
    def teq(a):
        assert (a.evalf() == a) is False
        assert (a.evalf() != a) is True
        assert (a == a.evalf()) is False
        assert (a != a.evalf()) is True

    teq(pi)
    teq(2*pi)
    teq(cos(0.1, evaluate=False))

    # long integer
    i = 12345678901234567890
    assert same_and_same_prec(Float(12, ''), Float('12', ''))
    assert same_and_same_prec(Float(Integer(i), ''), Float(i, ''))
    assert same_and_same_prec(Float(i, ''), Float(str(i), 20))
    assert same_and_same_prec(Float(str(i)), Float(i, ''))
    assert same_and_same_prec(Float(i), Float(i, ''))

    # inexact floats (repeating binary = denom not multiple of 2)
    # cannot have precision greater than 15
    assert Float(.125, 22) == .125
    assert Float(2.0, 22) == 2
    assert float(Float('.12500000000000001', '')) == .125
    raises(ValueError, lambda: Float(.12500000000000001, ''))

    # allow spaces
    Float('123 456.123 456') == Float('123456.123456')
    Integer('123 456') == Integer('123456')
    Rational('123 456.123 456') == Rational('123456.123456')
    assert Float(' .3e2') == Float('0.3e2')

    # allow auto precision detection
    assert Float('.1', '') == Float(.1, 1)
    assert Float('.125', '') == Float(.125, 3)
    assert Float('.100', '') == Float(.1, 3)
    assert Float('2.0', '') == Float('2', 2)

    raises(ValueError, lambda: Float("12.3d-4", ""))
    raises(ValueError, lambda: Float(12.3, ""))
    raises(ValueError, lambda: Float('.'))
    raises(ValueError, lambda: Float('-.'))

    zero = Float('0.0')
    assert Float('-0') == zero
    assert Float('.0') == zero
    assert Float('-.0') == zero
    assert Float('-0.0') == zero
    assert Float(0.0) == zero
    assert Float(0) == zero
    assert Float(0, '') == Float('0', '')
    assert Float(1) == Float(1.0)
    assert Float(S.Zero) == zero
    assert Float(S.One) == Float(1.0)

    assert Float(decimal.Decimal('0.1'), 3) == Float('.1', 3)
    assert Float(decimal.Decimal('nan')) == S.NaN
    assert Float(decimal.Decimal('Infinity')) == S.Infinity
    assert Float(decimal.Decimal('-Infinity')) == S.NegativeInfinity

    assert '{0:.3f}'.format(Float(4.236622)) == '4.237'
    assert '{0:.35f}'.format(Float(pi.n(40), 40)) == '3.14159265358979323846264338327950288'

    assert Float(oo) == Float('+inf')
    assert Float(-oo) == Float('-inf')


def test_Float_default_to_highprec_from_str():
    s = str(pi.evalf(128))
    assert same_and_same_prec(Float(s), Float(s, ''))


def test_Float_eval():
    a = Float(3.2)
    assert (a**2).is_Float


def test_Float_issue_2107():
    a = Float(0.1, 10)
    b = Float("0.1", 10)

    assert a - a == 0
    assert a + (-a) == 0
    assert S.Zero + a - a == 0
    assert S.Zero + a + (-a) == 0

    assert b - b == 0
    assert b + (-b) == 0
    assert S.Zero + b - b == 0
    assert S.Zero + b + (-b) == 0


def test_Infinity():
    assert oo != 1
    assert 1*oo == oo
    assert 1 != oo
    assert oo != -oo
    assert oo != Symbol("x")**3
    assert oo + 1 == oo
    assert 2 + oo == oo
    assert 3*oo + 2 == oo
    assert S.Half**oo == 0
    assert S.Half**(-oo) == oo
    assert -oo*3 == -oo
    assert oo + oo == oo
    assert -oo + oo*(-5) == -oo
    assert 1/oo == 0
    assert 1/(-oo) == 0
    assert 8/oo == 0
    assert oo % 2 == nan
    assert 2 % oo == nan
    assert oo/oo == nan
    assert oo/-oo == nan
    assert -oo/oo == nan
    assert -oo/-oo == nan
    assert oo - oo == nan
    assert oo - -oo == oo
    assert -oo - oo == -oo
    assert -oo - -oo == nan
    assert oo + -oo == nan
    assert -oo + oo == nan
    assert oo + oo == oo
    assert -oo + oo == nan
    assert oo + -oo == nan
    assert -oo + -oo == -oo
    assert oo*oo == oo
    assert -oo*oo == -oo
    assert oo*-oo == -oo
    assert -oo*-oo == oo
    assert oo/0 == oo
    assert -oo/0 == -oo
    assert 0/oo == 0
    assert 0/-oo == 0
    assert oo*0 == nan
    assert -oo*0 == nan
    assert 0*oo == nan
    assert 0*-oo == nan
    assert oo + 0 == oo
    assert -oo + 0 == -oo
    assert 0 + oo == oo
    assert 0 + -oo == -oo
    assert oo - 0 == oo
    assert -oo - 0 == -oo
    assert 0 - oo == -oo
    assert 0 - -oo == oo
    assert oo/2 == oo
    assert -oo/2 == -oo
    assert oo/-2 == -oo
    assert -oo/-2 == oo
    assert oo*2 == oo
    assert -oo*2 == -oo
    assert oo*-2 == -oo
    assert 2/oo == 0
    assert 2/-oo == 0
    assert -2/oo == 0
    assert -2/-oo == 0
    assert 2*oo == oo
    assert 2*-oo == -oo
    assert -2*oo == -oo
    assert -2*-oo == oo
    assert 2 + oo == oo
    assert 2 - oo == -oo
    assert -2 + oo == oo
    assert -2 - oo == -oo
    assert 2 + -oo == -oo
    assert 2 - -oo == oo
    assert -2 + -oo == -oo
    assert -2 - -oo == oo
    assert S(2) + oo == oo
    assert S(2) - oo == -oo
    assert oo/I == -oo*I
    assert -oo/I == oo*I
    assert oo*float(1) == Float('inf') and (oo*float(1)).is_Float
    assert -oo*float(1) == Float('-inf') and (-oo*float(1)).is_Float
    assert oo/float(1) == Float('inf') and (oo/float(1)).is_Float
    assert -oo/float(1) == Float('-inf') and (-oo/float(1)).is_Float
    assert oo*float(-1) == Float('-inf') and (oo*float(-1)).is_Float
    assert -oo*float(-1) == Float('inf') and (-oo*float(-1)).is_Float
    assert oo/float(-1) == Float('-inf') and (oo/float(-1)).is_Float
    assert -oo/float(-1) == Float('inf') and (-oo/float(-1)).is_Float
    assert oo + float(1) == Float('inf') and (oo + float(1)).is_Float
    assert -oo + float(1) == Float('-inf') and (-oo + float(1)).is_Float
    assert oo - float(1) == Float('inf') and (oo - float(1)).is_Float
    assert -oo - float(1) == Float('-inf') and (-oo - float(1)).is_Float
    assert float(1)*oo == Float('inf') and (float(1)*oo).is_Float
    assert float(1)*-oo == Float('-inf') and (float(1)*-oo).is_Float
    assert float(1)/oo == 0
    assert float(1)/-oo == 0
    assert float(-1)*oo == Float('-inf') and (float(-1)*oo).is_Float
    assert float(-1)*-oo == Float('inf') and (float(-1)*-oo).is_Float
    assert float(-1)/oo == 0
    assert float(-1)/-oo == 0
    assert float(1) + oo == Float('inf')
    assert float(1) + -oo == Float('-inf')
    assert float(1) - oo == Float('-inf')
    assert float(1) - -oo == Float('inf')

    assert Float('nan') == nan
    assert nan*1.0 == nan
    assert -1.0*nan == nan
    assert nan*oo == nan
    assert nan*-oo == nan
    assert nan/oo == nan
    assert nan/-oo == nan
    assert nan + oo == nan
    assert nan + -oo == nan
    assert nan - oo == nan
    assert nan - -oo == nan
    assert -oo * S.Zero == nan

    assert oo*nan == nan
    assert -oo*nan == nan
    assert oo/nan == nan
    assert -oo/nan == nan
    assert oo + nan == nan
    assert -oo + nan == nan
    assert oo - nan == nan
    assert -oo - nan == nan
    assert S.Zero * oo == nan
    assert oo.is_Rational is False
    assert isinstance(oo, Rational) is False

    assert S.One/oo == 0
    assert -S.One/oo == 0
    assert S.One/-oo == 0
    assert -S.One/-oo == 0
    assert S.One*oo == oo
    assert -S.One*oo == -oo
    assert S.One*-oo == -oo
    assert -S.One*-oo == oo
    assert S.One/nan == nan
    assert S.One - -oo == oo
    assert S.One + nan == nan
    assert S.One - nan == nan
    assert nan - S.One == nan
    assert nan/S.One == nan
    assert -oo - S.One == -oo


def test_Infinity_2():
    x = Symbol('x')
    assert oo*x != oo
    assert oo*(pi - 1) == oo
    assert oo*(1 - pi) == -oo

    assert (-oo)*x != -oo
    assert (-oo)*(pi - 1) == -oo
    assert (-oo)*(1 - pi) == oo

    assert (-1)**S.NaN is S.NaN
    assert oo - Float('inf') is S.NaN
    assert oo + Float('-inf') is S.NaN
    assert oo*0 is S.NaN
    assert oo/Float('inf') is S.NaN
    assert oo/Float('-inf') is S.NaN
    assert oo**S.NaN is S.NaN
    assert -oo + Float('inf') is S.NaN
    assert -oo - Float('-inf') is S.NaN
    assert -oo*S.NaN is S.NaN
    assert -oo*0 is S.NaN
    assert -oo/Float('inf') is S.NaN
    assert -oo/Float('-inf') is S.NaN
    assert -oo/S.NaN is S.NaN
    assert abs(-oo) == oo
    assert all((-oo)**i is S.NaN for i in (oo, -oo, S.NaN))
    assert (-oo)**3 == -oo
    assert (-oo)**2 == oo
    assert abs(S.ComplexInfinity) == oo


def test_Mul_Infinity_Zero():
    assert 0*Float('inf') == nan
    assert 0*Float('-inf') == nan
    assert 0*Float('inf') == nan
    assert 0*Float('-inf') == nan
    assert Float('inf')*0 == nan
    assert Float('-inf')*0 == nan
    assert Float('inf')*0 == nan
    assert Float('-inf')*0 == nan
    assert Float(0)*Float('inf') == nan
    assert Float(0)*Float('-inf') == nan
    assert Float(0)*Float('inf') == nan
    assert Float(0)*Float('-inf') == nan
    assert Float('inf')*Float(0) == nan
    assert Float('-inf')*Float(0) == nan
    assert Float('inf')*Float(0) == nan
    assert Float('-inf')*Float(0) == nan


def test_Div_By_Zero():
    assert 1/S(0) == zoo
    assert 1/Float(0) == Float('inf')
    assert 0/S(0) == nan
    assert 0/Float(0) == nan
    assert S(0)/0 == nan
    assert Float(0)/0 == nan
    assert -1/S(0) == zoo
    assert -1/Float(0) == Float('-inf')


def test_Infinity_inequations():
    assert oo > pi
    assert not (oo < pi)
    assert exp(-3) < oo

    assert Float('+inf') > pi
    assert not (Float('+inf') < pi)
    assert exp(-3) < Float('+inf')

    raises(TypeError, lambda: oo < I)
    raises(TypeError, lambda: oo <= I)
    raises(TypeError, lambda: oo > I)
    raises(TypeError, lambda: oo >= I)
    raises(TypeError, lambda: -oo < I)
    raises(TypeError, lambda: -oo <= I)
    raises(TypeError, lambda: -oo > I)
    raises(TypeError, lambda: -oo >= I)

    raises(TypeError, lambda: I < oo)
    raises(TypeError, lambda: I <= oo)
    raises(TypeError, lambda: I > oo)
    raises(TypeError, lambda: I >= oo)
    raises(TypeError, lambda: I < -oo)
    raises(TypeError, lambda: I <= -oo)
    raises(TypeError, lambda: I > -oo)
    raises(TypeError, lambda: I >= -oo)

    assert oo > -oo and oo >= -oo
    assert (oo < -oo) == False and (oo <= -oo) == False
    assert -oo < oo and -oo <= oo
    assert (-oo > oo) == False and (-oo >= oo) == False

    assert (oo < oo) == False  # issue 7775
    assert (oo > oo) == False
    assert (-oo > -oo) == False and (-oo < -oo) == False
    assert oo >= oo and oo <= oo and -oo >= -oo and -oo <= -oo
    assert (-oo < -Float('inf')) ==  False
    assert (oo > Float('inf')) == False
    assert -oo >= -Float('inf')
    assert oo <= Float('inf')

    x = Symbol('x')
    b = Symbol('b', finite=True, real=True)
    assert (x < oo) == Lt(x, oo)  # issue 7775
    assert b < oo and b > -oo and b <= oo and b >= -oo
    assert oo > b and oo >= b and (oo < b) == False and (oo <= b) == False
    assert (-oo > b) == False and (-oo >= b) == False and -oo < b and -oo <= b
    assert (oo < x) == Lt(oo, x) and (oo > x) == Gt(oo, x)
    assert (oo <= x) == Le(oo, x) and (oo >= x) == Ge(oo, x)
    assert (-oo < x) == Lt(-oo, x) and (-oo > x) == Gt(-oo, x)
    assert (-oo <= x) == Le(-oo, x) and (-oo >= x) == Ge(-oo, x)


def test_NaN():
    assert nan == nan
    assert nan != 1
    assert 1*nan == nan
    assert 1 != nan
    assert nan == -nan
    assert oo != Symbol("x")**3
    assert nan + 1 == nan
    assert 2 + nan == nan
    assert 3*nan + 2 == nan
    assert -nan*3 == nan
    assert nan + nan == nan
    assert -nan + nan*(-5) == nan
    assert 1/nan == nan
    assert 1/(-nan) == nan
    assert 8/nan == nan
    raises(TypeError, lambda: nan > 0)
    raises(TypeError, lambda: nan < 0)
    raises(TypeError, lambda: nan >= 0)
    raises(TypeError, lambda: nan <= 0)
    raises(TypeError, lambda: 0 < nan)
    raises(TypeError, lambda: 0 > nan)
    raises(TypeError, lambda: 0 <= nan)
    raises(TypeError, lambda: 0 >= nan)
    assert S.One + nan == nan
    assert S.One - nan == nan
    assert S.One*nan == nan
    assert S.One/nan == nan
    assert nan - S.One == nan
    assert nan*S.One == nan
    assert nan + S.One == nan
    assert nan/S.One == nan
    assert nan**0 == 1  # as per IEEE 754
    assert 1**nan == nan # IEEE 754 is not the best choice for symbolic work
    # test Pow._eval_power's handling of NaN
    assert Pow(nan, 0, evaluate=False)**2 == 1


def test_special_numbers():
    assert isinstance(S.NaN, Number) is True
    assert isinstance(S.Infinity, Number) is True
    assert isinstance(S.NegativeInfinity, Number) is True

    assert S.NaN.is_number is True
    assert S.Infinity.is_number is True
    assert S.NegativeInfinity.is_number is True
    assert S.ComplexInfinity.is_number is True

    assert isinstance(S.NaN, Rational) is False
    assert isinstance(S.Infinity, Rational) is False
    assert isinstance(S.NegativeInfinity, Rational) is False

    assert S.NaN.is_rational is not True
    assert S.Infinity.is_rational is not True
    assert S.NegativeInfinity.is_rational is not True


def test_powers():
    assert integer_nthroot(1, 2) == (1, True)
    assert integer_nthroot(1, 5) == (1, True)
    assert integer_nthroot(2, 1) == (2, True)
    assert integer_nthroot(2, 2) == (1, False)
    assert integer_nthroot(2, 5) == (1, False)
    assert integer_nthroot(4, 2) == (2, True)
    assert integer_nthroot(123**25, 25) == (123, True)
    assert integer_nthroot(123**25 + 1, 25) == (123, False)
    assert integer_nthroot(123**25 - 1, 25) == (122, False)
    assert integer_nthroot(1, 1) == (1, True)
    assert integer_nthroot(0, 1) == (0, True)
    assert integer_nthroot(0, 3) == (0, True)
    assert integer_nthroot(10000, 1) == (10000, True)
    assert integer_nthroot(4, 2) == (2, True)
    assert integer_nthroot(16, 2) == (4, True)
    assert integer_nthroot(26, 2) == (5, False)
    assert integer_nthroot(1234567**7, 7) == (1234567, True)
    assert integer_nthroot(1234567**7 + 1, 7) == (1234567, False)
    assert integer_nthroot(1234567**7 - 1, 7) == (1234566, False)
    b = 25**1000
    assert integer_nthroot(b, 1000) == (25, True)
    assert integer_nthroot(b + 1, 1000) == (25, False)
    assert integer_nthroot(b - 1, 1000) == (24, False)
    c = 10**400
    c2 = c**2
    assert integer_nthroot(c2, 2) == (c, True)
    assert integer_nthroot(c2 + 1, 2) == (c, False)
    assert integer_nthroot(c2 - 1, 2) == (c - 1, False)
    assert integer_nthroot(2, 10**10) == (1, False)

    p, r = integer_nthroot(int(factorial(10000)), 100)
    assert p % (10**10) == 5322420655
    assert not r

    # Test that this is fast
    assert integer_nthroot(2, 10**10) == (1, False)


def test_integer_nthroot_overflow():
    assert integer_nthroot(10**(50*50), 50) == (10**50, True)
    assert integer_nthroot(10**100000, 10000) == (10**10, True)


def test_powers_Integer():
    """Test Integer._eval_power"""
    # check infinity
    assert S(1) ** S.Infinity == S.NaN
    assert S(-1)** S.Infinity == S.NaN
    assert S(2) ** S.Infinity == S.Infinity
    assert S(-2)** S.Infinity == S.Infinity + S.Infinity * S.ImaginaryUnit
    assert S(0) ** S.Infinity == 0

    # check Nan
    assert S(1) ** S.NaN == S.NaN
    assert S(-1) ** S.NaN == S.NaN

    # check for exact roots
    assert S(-1) ** Rational(6, 5) == - (-1)**(S(1)/5)
    assert sqrt(S(4)) == 2
    assert sqrt(S(-4)) == I * 2
    assert S(16) ** Rational(1, 4) == 2
    assert S(-16) ** Rational(1, 4) == 2 * (-1)**Rational(1, 4)
    assert S(9) ** Rational(3, 2) == 27
    assert S(-9) ** Rational(3, 2) == -27*I
    assert S(27) ** Rational(2, 3) == 9
    assert S(-27) ** Rational(2, 3) == 9 * (S(-1) ** Rational(2, 3))
    assert (-2) ** Rational(-2, 1) == Rational(1, 4)

    # not exact roots
    assert sqrt(-3) == I*sqrt(3)
    assert (3) ** (S(3)/2) == 3 * sqrt(3)
    assert (-3) ** (S(3)/2) == - 3 * sqrt(-3)
    assert (-3) ** (S(5)/2) == 9 * I * sqrt(3)
    assert (-3) ** (S(7)/2) == - I * 27 * sqrt(3)
    assert (2) ** (S(3)/2) == 2 * sqrt(2)
    assert (2) ** (S(-3)/2) == sqrt(2) / 4
    assert (81) ** (S(2)/3) == 9 * (S(3) ** (S(2)/3))
    assert (-81) ** (S(2)/3) == 9 * (S(-3) ** (S(2)/3))
    assert (-3) ** Rational(-7, 3) == \
        -(-1)**Rational(2, 3)*3**Rational(2, 3)/27
    assert (-3) ** Rational(-2, 3) == \
        -(-1)**Rational(1, 3)*3**Rational(1, 3)/3

    # join roots
    assert sqrt(6) + sqrt(24) == 3*sqrt(6)
    assert sqrt(2) * sqrt(3) == sqrt(6)

    # separate symbols & constansts
    x = Symbol("x")
    assert sqrt(49 * x) == 7 * sqrt(x)
    assert sqrt((3 - sqrt(pi)) ** 2) == 3 - sqrt(pi)

    # check that it is fast for big numbers
    assert (2**64 + 1) ** Rational(4, 3)
    assert (2**64 + 1) ** Rational(17, 25)

    # negative rational power and negative base
    assert (-3) ** Rational(-7, 3) == \
        -(-1)**Rational(2, 3)*3**Rational(2, 3)/27
    assert (-3) ** Rational(-2, 3) == \
        -(-1)**Rational(1, 3)*3**Rational(1, 3)/3

    assert S(1234).factors() == {617: 1, 2: 1}
    assert Rational(2*3, 3*5*7).factors() == {2: 1, 5: -1, 7: -1}

    # test that eval_power factors numbers bigger than
    # the current limit in factor_trial_division (2**15)
    from sympy import nextprime
    n = nextprime(2**15)
    assert sqrt(n**2) == n
    assert sqrt(n**3) == n*sqrt(n)
    assert sqrt(4*n) == 2*sqrt(n)

    # check that factors of base with powers sharing gcd with power are removed
    assert (2**4*3)**Rational(1, 6) == 2**Rational(2, 3)*3**Rational(1, 6)
    assert (2**4*3)**Rational(5, 6) == 8*2**Rational(1, 3)*3**Rational(5, 6)

    # check that bases sharing a gcd are exptracted
    assert 2**Rational(1, 3)*3**Rational(1, 4)*6**Rational(1, 5) == \
        2**Rational(8, 15)*3**Rational(9, 20)
    assert sqrt(8)*24**Rational(1, 3)*6**Rational(1, 5) == \
        4*2**Rational(7, 10)*3**Rational(8, 15)
    assert sqrt(8)*(-24)**Rational(1, 3)*(-6)**Rational(1, 5) == \
        4*(-3)**Rational(8, 15)*2**Rational(7, 10)
    assert 2**Rational(1, 3)*2**Rational(8, 9) == 2*2**Rational(2, 9)
    assert 2**Rational(2, 3)*6**Rational(1, 3) == 2*3**Rational(1, 3)
    assert 2**Rational(2, 3)*6**Rational(8, 9) == \
        2*2**Rational(5, 9)*3**Rational(8, 9)
    assert (-2)**Rational(2, S(3))*(-4)**Rational(1, S(3)) == -2*2**Rational(1, 3)
    assert 3*Pow(3, 2, evaluate=False) == 3**3
    assert 3*Pow(3, -1/S(3), evaluate=False) == 3**(2/S(3))
    assert (-2)**(1/S(3))*(-3)**(1/S(4))*(-5)**(5/S(6)) == \
        -(-1)**Rational(5, 12)*2**Rational(1, 3)*3**Rational(1, 4) * \
        5**Rational(5, 6)

    assert Integer(-2)**Symbol('', even=True) == \
        Integer(2)**Symbol('', even=True)
    assert (-1)**Float(.5) == 1.0*I


def test_powers_Rational():
    """Test Rational._eval_power"""
    # check infinity
    assert Rational(1, 2) ** S.Infinity == 0
    assert Rational(3, 2) ** S.Infinity == S.Infinity
    assert Rational(-1, 2) ** S.Infinity == 0
    assert Rational(-3, 2) ** S.Infinity == \
        S.Infinity + S.Infinity * S.ImaginaryUnit

    # check Nan
    assert Rational(3, 4) ** S.NaN == S.NaN
    assert Rational(-2, 3) ** S.NaN == S.NaN

    # exact roots on numerator
    assert sqrt(Rational(4, 3)) == 2 * sqrt(3) / 3
    assert Rational(4, 3) ** Rational(3, 2) == 8 * sqrt(3) / 9
    assert sqrt(Rational(-4, 3)) == I * 2 * sqrt(3) / 3
    assert Rational(-4, 3) ** Rational(3, 2) == - I * 8 * sqrt(3) / 9
    assert Rational(27, 2) ** Rational(1, 3) == 3 * (2 ** Rational(2, 3)) / 2
    assert Rational(5**3, 8**3) ** Rational(4, 3) == Rational(5**4, 8**4)

    # exact root on denominator
    assert sqrt(Rational(1, 4)) == Rational(1, 2)
    assert sqrt(Rational(1, -4)) == I * Rational(1, 2)
    assert sqrt(Rational(3, 4)) == sqrt(3) / 2
    assert sqrt(Rational(3, -4)) == I * sqrt(3) / 2
    assert Rational(5, 27) ** Rational(1, 3) == (5 ** Rational(1, 3)) / 3

    # not exact roots
    assert sqrt(Rational(1, 2)) == sqrt(2) / 2
    assert sqrt(Rational(-4, 7)) == I * sqrt(Rational(4, 7))
    assert Rational(-3, 2)**Rational(-7, 3) == \
        -4*(-1)**Rational(2, 3)*2**Rational(1, 3)*3**Rational(2, 3)/27
    assert Rational(-3, 2)**Rational(-2, 3) == \
        -(-1)**Rational(1, 3)*2**Rational(2, 3)*3**Rational(1, 3)/3

    # negative integer power and negative rational base
    assert Rational(-2, 3) ** Rational(-2, 1) == Rational(9, 4)

    a = Rational(1, 10)
    assert a**Float(a, 2) == Float(a, 2)**Float(a, 2)
    assert Rational(-2, 3)**Symbol('', even=True) == \
        Rational(2, 3)**Symbol('', even=True)


def test_powers_Float():
    assert str((S('-1/10')**S('3/10')).n()) == str(Float(-.1)**(.3))


def test_abs1():
    assert Rational(1, 6) != Rational(-1, 6)
    assert abs(Rational(1, 6)) == abs(Rational(-1, 6))


def test_accept_int():
    assert Float(4) == 4


def test_dont_accept_str():
    assert Float("0.2") != "0.2"
    assert not (Float("0.2") == "0.2")


def test_int():
    a = Rational(5)
    assert int(a) == 5
    a = Rational(9, 10)
    assert int(a) == int(-a) == 0
    assert 1/(-1)**Rational(2, 3) == -(-1)**Rational(1, 3)
    assert int(pi) == 3
    assert int(E) == 2
    assert int(GoldenRatio) == 1
    # issue 10368
    a = S(32442016954)/78058255275
    assert type(int(a)) is type(int(-a)) is int


def test_long():
    a = Rational(5)
    assert long(a) == 5
    a = Rational(9, 10)
    assert long(a) == long(-a) == 0
    a = Integer(2**100)
    assert long(a) == a
    assert long(pi) == 3
    assert long(E) == 2
    assert long(GoldenRatio) == 1

def test_real_bug():
    x = Symbol("x")
    assert str(2.0*x*x) in ["(2.0*x)*x", "2.0*x**2", "2.00000000000000*x**2"]
    assert str(2.1*x*x) != "(2.0*x)*x"


def test_bug_sqrt():
    assert ((sqrt(Rational(2)) + 1)*(sqrt(Rational(2)) - 1)).expand() == 1


def test_pi_Pi():
    "Test that pi (instance) is imported, but Pi (class) is not"
    from sympy import pi
    with raises(ImportError):
        from sympy import Pi


def test_no_len():
    # there should be no len for numbers
    raises(TypeError, lambda: len(Rational(2)))
    raises(TypeError, lambda: len(Rational(2, 3)))
    raises(TypeError, lambda: len(Integer(2)))


def test_issue_3321():
    assert sqrt(Rational(1, 5)) == sqrt(Rational(1, 5))
    assert 5 * sqrt(Rational(1, 5)) == sqrt(5)


def test_issue_3692():
    assert ((-1)**Rational(1, 6)).expand(complex=True) == I/2 + sqrt(3)/2
    assert ((-5)**Rational(1, 6)).expand(complex=True) == \
        5**Rational(1, 6)*I/2 + 5**Rational(1, 6)*sqrt(3)/2
    assert ((-64)**Rational(1, 6)).expand(complex=True) == I + sqrt(3)


def test_issue_3423():
    x = Symbol("x")
    assert sqrt(x - 1).as_base_exp() == (x - 1, S.Half)
    assert sqrt(x - 1) != I*sqrt(1 - x)


def test_issue_3449():
    x = Symbol("x")
    assert sqrt(x - 1).subs(x, 5) == 2


def test_Integer_factors():
    def F(i):
        return Integer(i).factors()

    assert F(1) == {}
    assert F(2) == {2: 1}
    assert F(3) == {3: 1}
    assert F(4) == {2: 2}
    assert F(5) == {5: 1}
    assert F(6) == {2: 1, 3: 1}
    assert F(7) == {7: 1}
    assert F(8) == {2: 3}
    assert F(9) == {3: 2}
    assert F(10) == {2: 1, 5: 1}
    assert F(11) == {11: 1}
    assert F(12) == {2: 2, 3: 1}
    assert F(13) == {13: 1}
    assert F(14) == {2: 1, 7: 1}
    assert F(15) == {3: 1, 5: 1}
    assert F(16) == {2: 4}
    assert F(17) == {17: 1}
    assert F(18) == {2: 1, 3: 2}
    assert F(19) == {19: 1}
    assert F(20) == {2: 2, 5: 1}
    assert F(21) == {3: 1, 7: 1}
    assert F(22) == {2: 1, 11: 1}
    assert F(23) == {23: 1}
    assert F(24) == {2: 3, 3: 1}
    assert F(25) == {5: 2}
    assert F(26) == {2: 1, 13: 1}
    assert F(27) == {3: 3}
    assert F(28) == {2: 2, 7: 1}
    assert F(29) == {29: 1}
    assert F(30) == {2: 1, 3: 1, 5: 1}
    assert F(31) == {31: 1}
    assert F(32) == {2: 5}
    assert F(33) == {3: 1, 11: 1}
    assert F(34) == {2: 1, 17: 1}
    assert F(35) == {5: 1, 7: 1}
    assert F(36) == {2: 2, 3: 2}
    assert F(37) == {37: 1}
    assert F(38) == {2: 1, 19: 1}
    assert F(39) == {3: 1, 13: 1}
    assert F(40) == {2: 3, 5: 1}
    assert F(41) == {41: 1}
    assert F(42) == {2: 1, 3: 1, 7: 1}
    assert F(43) == {43: 1}
    assert F(44) == {2: 2, 11: 1}
    assert F(45) == {3: 2, 5: 1}
    assert F(46) == {2: 1, 23: 1}
    assert F(47) == {47: 1}
    assert F(48) == {2: 4, 3: 1}
    assert F(49) == {7: 2}
    assert F(50) == {2: 1, 5: 2}
    assert F(51) == {3: 1, 17: 1}


def test_Rational_factors():
    def F(p, q, visual=None):
        return Rational(p, q).factors(visual=visual)

    assert F(2, 3) == {2: 1, 3: -1}
    assert F(2, 9) == {2: 1, 3: -2}
    assert F(2, 15) == {2: 1, 3: -1, 5: -1}
    assert F(6, 10) == {3: 1, 5: -1}


def test_issue_4107():
    assert pi*(E + 10) + pi*(-E - 10) != 0
    assert pi*(E + 10**10) + pi*(-E - 10**10) != 0
    assert pi*(E + 10**20) + pi*(-E - 10**20) != 0
    assert pi*(E + 10**80) + pi*(-E - 10**80) != 0

    assert (pi*(E + 10) + pi*(-E - 10)).expand() == 0
    assert (pi*(E + 10**10) + pi*(-E - 10**10)).expand() == 0
    assert (pi*(E + 10**20) + pi*(-E - 10**20)).expand() == 0
    assert (pi*(E + 10**80) + pi*(-E - 10**80)).expand() == 0


def test_IntegerInteger():
    a = Integer(4)
    b = Integer(a)

    assert a == b


def test_Rational_gcd_lcm_cofactors():
    assert Integer(4).gcd(2) == Integer(2)
    assert Integer(4).lcm(2) == Integer(4)
    assert Integer(4).gcd(Integer(2)) == Integer(2)
    assert Integer(4).lcm(Integer(2)) == Integer(4)

    assert Integer(4).gcd(3) == Integer(1)
    assert Integer(4).lcm(3) == Integer(12)
    assert Integer(4).gcd(Integer(3)) == Integer(1)
    assert Integer(4).lcm(Integer(3)) == Integer(12)

    assert Rational(4, 3).gcd(2) == Rational(2, 3)
    assert Rational(4, 3).lcm(2) == Integer(4)
    assert Rational(4, 3).gcd(Integer(2)) == Rational(2, 3)
    assert Rational(4, 3).lcm(Integer(2)) == Integer(4)

    assert Integer(4).gcd(Rational(2, 9)) == Rational(2, 9)
    assert Integer(4).lcm(Rational(2, 9)) == Integer(4)

    assert Rational(4, 3).gcd(Rational(2, 9)) == Rational(2, 9)
    assert Rational(4, 3).lcm(Rational(2, 9)) == Rational(4, 3)
    assert Rational(4, 5).gcd(Rational(2, 9)) == Rational(2, 45)
    assert Rational(4, 5).lcm(Rational(2, 9)) == Integer(4)

    assert Integer(4).cofactors(2) == (Integer(2), Integer(2), Integer(1))
    assert Integer(4).cofactors(Integer(2)) == \
        (Integer(2), Integer(2), Integer(1))

    assert Integer(4).gcd(Float(2.0)) == S.One
    assert Integer(4).lcm(Float(2.0)) == Float(8.0)
    assert Integer(4).cofactors(Float(2.0)) == (S.One, Integer(4), Float(2.0))

    assert Rational(1, 2).gcd(Float(2.0)) == S.One
    assert Rational(1, 2).lcm(Float(2.0)) == Float(1.0)
    assert Rational(1, 2).cofactors(Float(2.0)) == \
        (S.One, Rational(1, 2), Float(2.0))


def test_Float_gcd_lcm_cofactors():
    assert Float(2.0).gcd(Integer(4)) == S.One
    assert Float(2.0).lcm(Integer(4)) == Float(8.0)
    assert Float(2.0).cofactors(Integer(4)) == (S.One, Float(2.0), Integer(4))

    assert Float(2.0).gcd(Rational(1, 2)) == S.One
    assert Float(2.0).lcm(Rational(1, 2)) == Float(1.0)
    assert Float(2.0).cofactors(Rational(1, 2)) == \
        (S.One, Float(2.0), Rational(1, 2))


def test_issue_4611():
    assert abs(pi._evalf(50) - 3.14159265358979) < 1e-10
    assert abs(E._evalf(50) - 2.71828182845905) < 1e-10
    assert abs(Catalan._evalf(50) - 0.915965594177219) < 1e-10
    assert abs(EulerGamma._evalf(50) - 0.577215664901533) < 1e-10
    assert abs(GoldenRatio._evalf(50) - 1.61803398874989) < 1e-10
    x = Symbol("x")
    assert (pi + x).evalf() == pi.evalf() + x
    assert (E + x).evalf() == E.evalf() + x
    assert (Catalan + x).evalf() == Catalan.evalf() + x
    assert (EulerGamma + x).evalf() == EulerGamma.evalf() + x
    assert (GoldenRatio + x).evalf() == GoldenRatio.evalf() + x


def test_conversion_to_mpmath():
    assert mpmath.mpmathify(Integer(1)) == mpmath.mpf(1)
    assert mpmath.mpmathify(Rational(1, 2)) == mpmath.mpf(0.5)
    assert mpmath.mpmathify(Float('1.23', 15)) == mpmath.mpf('1.23')


def test_relational():
    # real
    x = S(.1)
    assert (x != cos) is True
    assert (x == cos) is False

    # rational
    x = Rational(1, 3)
    assert (x != cos) is True
    assert (x == cos) is False

    # integer defers to rational so these tests are omitted

    # number symbol
    x = pi
    assert (x != cos) is True
    assert (x == cos) is False


def test_Integer_as_index():
    assert 'hello'[Integer(2):] == 'llo'


def test_Rational_int():
    assert int( Rational(7, 5)) == 1
    assert int( Rational(1, 2)) == 0
    assert int(-Rational(1, 2)) == 0
    assert int(-Rational(7, 5)) == -1


def test_zoo():
    b = Symbol('b', finite=True)
    nz = Symbol('nz', nonzero=True)
    p = Symbol('p', positive=True)
    n = Symbol('n', negative=True)
    im = Symbol('i', imaginary=True)
    c = Symbol('c', complex=True)
    pb = Symbol('pb', positive=True, finite=True)
    nb = Symbol('nb', negative=True, finite=True)
    imb = Symbol('ib', imaginary=True, finite=True)
    for i in [I, S.Infinity, S.NegativeInfinity, S.Zero, S.One, S.Pi, S.Half, S(3), log(3),
              b, nz, p, n, im, pb, nb, imb, c]:
        if i.is_finite and (i.is_real or i.is_imaginary):
            assert i + zoo is zoo
            assert i - zoo is zoo
            assert zoo + i is zoo
            assert zoo - i is zoo
        elif i.is_finite is not False:
            assert (i + zoo).is_Add
            assert (i - zoo).is_Add
            assert (zoo + i).is_Add
            assert (zoo - i).is_Add
        else:
            assert (i + zoo) is S.NaN
            assert (i - zoo) is S.NaN
            assert (zoo + i) is S.NaN
            assert (zoo - i) is S.NaN

        if fuzzy_not(i.is_zero) and (i.is_real or i.is_imaginary):
            assert i*zoo is zoo
            assert zoo*i is zoo
        elif i.is_zero:
            assert i*zoo is S.NaN
            assert zoo*i is S.NaN
        else:
            assert (i*zoo).is_Mul
            assert (zoo*i).is_Mul

        if fuzzy_not((1/i).is_zero) and (i.is_real or i.is_imaginary):
            assert zoo/i is zoo
        elif (1/i).is_zero:
            assert zoo/i is S.NaN
        elif i.is_zero:
            assert zoo/i is zoo
        else:
            assert (zoo/i).is_Mul

    assert (I*oo).is_Mul  # allow directed infinity
    assert zoo + zoo is S.NaN
    assert zoo * zoo is zoo
    assert zoo - zoo is S.NaN
    assert zoo/zoo is S.NaN
    assert zoo**zoo is S.NaN
    assert zoo**0 is S.One
    assert zoo**2 is zoo
    assert 1/zoo is S.Zero

    assert Mul.flatten([S(-1), oo, S(0)]) == ([S.NaN], [], None)


def test_issue_4122():
    x = Symbol('x', nonpositive=True)
    assert (oo + x).is_Add
    x = Symbol('x', finite=True)
    assert (oo + x).is_Add  # x could be imaginary
    x = Symbol('x', nonnegative=True)
    assert oo + x == oo
    x = Symbol('x', finite=True, real=True)
    assert oo + x == oo

    # similarily for negative infinity
    x = Symbol('x', nonnegative=True)
    assert (-oo + x).is_Add
    x = Symbol('x', finite=True)
    assert (-oo + x).is_Add
    x = Symbol('x', nonpositive=True)
    assert -oo + x == -oo
    x = Symbol('x', finite=True, real=True)
    assert -oo + x == -oo


def test_GoldenRatio_expand():
    assert GoldenRatio.expand(func=True) == S.Half + sqrt(5)/2


def test_as_content_primitive():
    assert S.Zero.as_content_primitive() == (1, 0)
    assert S.Half.as_content_primitive() == (S.Half, 1)
    assert (-S.Half).as_content_primitive() == (S.Half, -1)
    assert S(3).as_content_primitive() == (3, 1)
    assert S(3.1).as_content_primitive() == (1, 3.1)


def test_hashing_sympy_integers():
    # Test for issue 5072
    assert set([Integer(3)]) == set([int(3)])
    assert hash(Integer(4)) == hash(int(4))


def test_issue_4172():
    assert int((E**100).round()) == \
        26881171418161354484126255515800135873611119
    assert int((pi**100).round()) == \
        51878483143196131920862615246303013562686760680406
    assert int((Rational(1)/EulerGamma**100).round()) == \
        734833795660954410469466


@XFAIL
def test_mpmath_issues():
    from mpmath.libmp.libmpf import _normalize
    import mpmath.libmp as mlib
    rnd = mlib.round_nearest
    mpf = (0, long(0), -123, -1, 53, rnd)  # nan
    assert _normalize(mpf, 53) != (0, long(0), 0, 0)
    mpf = (0, long(0), -456, -2, 53, rnd)  # +inf
    assert _normalize(mpf, 53) != (0, long(0), 0, 0)
    mpf = (1, long(0), -789, -3, 53, rnd)  # -inf
    assert _normalize(mpf, 53) != (0, long(0), 0, 0)

    from mpmath.libmp.libmpf import fnan
    assert mlib.mpf_eq(fnan, fnan)


def test_Catalan_EulerGamma_prec():
    n = GoldenRatio
    f = Float(n.n(), 5)
    assert f._mpf_ == (0, long(212079), -17, 18)
    assert f._prec == 20
    assert n._as_mpf_val(20) == f._mpf_

    n = EulerGamma
    f = Float(n.n(), 5)
    assert f._mpf_ == (0, long(302627), -19, 19)
    assert f._prec == 20
    assert n._as_mpf_val(20) == f._mpf_


def test_Float_eq():
    assert Float(.12, 3) != Float(.12, 4)
    assert Float(.12, 3) == .12
    assert 0.12 == Float(.12, 3)
    assert Float('.12', 22) != .12


def test_int_NumberSymbols():
    assert [int(i) for i in [pi, EulerGamma, E, GoldenRatio, Catalan]] == \
        [3, 0, 2, 1, 0]


def test_issue_6640():
    from mpmath.libmp.libmpf import finf, fninf
    # fnan is not included because Float no longer returns fnan,
    # but otherwise, the same sort of test could apply
    assert Float(finf).is_zero is False
    assert Float(fninf).is_zero is False
    assert bool(Float(0)) is False


def test_issue_6349():
    assert Float('23.e3', '')._prec == 10
    assert Float('23e3', '')._prec == 20
    assert Float('23000', '')._prec == 20
    assert Float('-23000', '')._prec == 20

def test_mpf_norm():
    assert mpf_norm((1, 0, 1, 0), 10) == mpf('0')._mpf_
    assert Float._new((1, 0, 1, 0), 10)._mpf_ == mpf('0')._mpf_

def test_latex():
    assert latex(pi) == r"\pi"
    assert latex(E) == r"e"
    assert latex(GoldenRatio) == r"\phi"
    assert latex(EulerGamma) == r"\gamma"
    assert latex(oo) == r"\infty"
    assert latex(-oo) == r"-\infty"
    assert latex(zoo) == r"\tilde{\infty}"
    assert latex(nan) == r"\mathrm{NaN}"
    assert latex(I) == r"i"


def test_issue_7742():
    assert -oo % 1 == nan


def test_simplify_AlgebraicNumber():
    A = AlgebraicNumber
    e = 3**(S(1)/6)*(3 + (135 + 78*sqrt(3))**(S(2)/3))/(45 + 26*sqrt(3))**(S(1)/3)
    assert simplify(A(e)) == A(12)  # wester test_C20

    e = (41 + 29*sqrt(2))**(S(1)/5)
    assert simplify(A(e)) == A(1 + sqrt(2))  # wester test_C21

    e = (3 + 4*I)**(Rational(3, 2))
    assert simplify(A(e)) == A(2 + 11*I)  # issue 4401


def test_Float_idempotence():
    x = Float('1.23', '')
    y = Float(x)
    z = Float(x, 15)
    assert same_and_same_prec(y, x)
    assert not same_and_same_prec(z, x)
    x = Float(10**20)
    y = Float(x)
    z = Float(x, 15)
    assert same_and_same_prec(y, x)
    assert not same_and_same_prec(z, x)


def test_comp():
    # sqrt(2) = 1.414213 5623730950...
    a = sqrt(2).n(7)
    assert comp(a, 1.41421346) is False
    assert comp(a, 1.41421347)
    assert comp(a, 1.41421366)
    assert comp(a, 1.41421367) is False
    assert comp(sqrt(2).n(2), '1.4')
    assert comp(sqrt(2).n(2), Float(1.4, 2), '')
    raises(ValueError, lambda: comp(sqrt(2).n(2), 1.4, ''))
    assert comp(sqrt(2).n(2), Float(1.4, 3), '') is False


def test_issue_9491():
    assert oo**zoo == nan


def test_issue_10063():
    assert 2**Float(3) == Float(8)


def test_issue_10020():
    assert oo**I is S.NaN
    assert oo**(1 + I) is S.ComplexInfinity
    assert oo**(-1 + I) is S.Zero
    assert (-oo)**I is S.NaN
    assert (-oo)**(-1 + I) is S.Zero
    assert oo**t == Pow(oo, t, evaluate=False)
    assert (-oo)**t == Pow(-oo, t, evaluate=False)


def test_invert_numbers():
    assert S(2).invert(5) == 3
    assert S(2).invert(S(5)/2) == S.Half
    assert S(2).invert(5.) == 3
    assert S(2).invert(S(5)) == 3
    assert S(2.).invert(5) == 3
    assert S(sqrt(2)).invert(5) == 1/sqrt(2)
    assert S(sqrt(2)).invert(sqrt(3)) == 1/sqrt(2)


def test_mod_inverse():
    assert mod_inverse(3, 11) == 4
    assert mod_inverse(5, 11) == 9
    assert mod_inverse(21124921, 521512) == 7713
    assert mod_inverse(124215421, 5125) == 2981
    assert mod_inverse(214, 12515) == 1579
    assert mod_inverse(5823991, 3299) == 1442
    assert mod_inverse(123, 44) == 39
    assert mod_inverse(2, 5) == 3
    assert mod_inverse(-2, 5) == -3
    x = Symbol('x')
    assert S(2).invert(x) == S.Half
    raises(TypeError, lambda: mod_inverse(2, x))
    raises(ValueError, lambda: mod_inverse(2, S.Half))
    raises(ValueError, lambda: mod_inverse(2, cos(1)**2 + sin(1)**2))
