from sympy import (
    Abs, adjoint, arg, atan2, conjugate, cos, DiracDelta, E, exp, expand,
    Expr, Function, Heaviside, I, im, log, nan, oo, pi, Rational, re, S,
    sign, sin, sqrt, Symbol, symbols, transpose, zoo, exp_polar, Piecewise,
    Interval, comp, Integral)
from sympy.utilities.pytest import XFAIL, raises


def N_equals(a, b):
    """Check whether two complex numbers are numerically close"""
    return comp(a.n(), b.n(), 1.e-6)


def test_re():
    x, y = symbols('x,y')
    a, b = symbols('a,b', real=True)

    r = Symbol('r', real=True)
    i = Symbol('i', imaginary=True)

    assert re(nan) == nan

    assert re(oo) == oo
    assert re(-oo) == -oo

    assert re(0) == 0

    assert re(1) == 1
    assert re(-1) == -1

    assert re(E) == E
    assert re(-E) == -E

    assert re(x) == re(x)
    assert re(x*I) == -im(x)
    assert re(r*I) == 0
    assert re(r) == r
    assert re(i*I) == I * i
    assert re(i) == 0

    assert re(x + y) == re(x + y)
    assert re(x + r) == re(x) + r

    assert re(re(x)) == re(x)

    assert re(2 + I) == 2
    assert re(x + I) == re(x)

    assert re(x + y*I) == re(x) - im(y)
    assert re(x + r*I) == re(x)

    assert re(log(2*I)) == log(2)

    assert re((2 + I)**2).expand(complex=True) == 3

    assert re(conjugate(x)) == re(x)
    assert conjugate(re(x)) == re(x)

    assert re(x).as_real_imag() == (re(x), 0)

    assert re(i*r*x).diff(r) == re(i*x)
    assert re(i*r*x).diff(i) == I*r*im(x)

    assert re(
        sqrt(a + b*I)) == (a**2 + b**2)**Rational(1, 4)*cos(atan2(b, a)/2)
    assert re(a * (2 + b*I)) == 2*a

    assert re((1 + sqrt(a + b*I))/2) == \
        (a**2 + b**2)**Rational(1, 4)*cos(atan2(b, a)/2)/2 + Rational(1, 2)

    assert re(x).rewrite(im) == x - im(x)
    assert (x + re(y)).rewrite(re, im) == x + y - im(y)

    a = Symbol('a', algebraic=True)
    t = Symbol('t', transcendental=True)
    x = Symbol('x')
    assert re(a).is_algebraic
    assert re(x).is_algebraic is None
    assert re(t).is_algebraic is False


def test_im():
    x, y = symbols('x,y')
    a, b = symbols('a,b', real=True)

    r = Symbol('r', real=True)
    i = Symbol('i', imaginary=True)

    assert im(nan) == nan

    assert im(oo*I) == oo
    assert im(-oo*I) == -oo

    assert im(0) == 0

    assert im(1) == 0
    assert im(-1) == 0

    assert im(E*I) == E
    assert im(-E*I) == -E

    assert im(x) == im(x)
    assert im(x*I) == re(x)
    assert im(r*I) == r
    assert im(r) == 0
    assert im(i*I) == 0
    assert im(i) == -I * i

    assert im(x + y) == im(x + y)
    assert im(x + r) == im(x)
    assert im(x + r*I) == im(x) + r

    assert im(im(x)*I) == im(x)

    assert im(2 + I) == 1
    assert im(x + I) == im(x) + 1

    assert im(x + y*I) == im(x) + re(y)
    assert im(x + r*I) == im(x) + r

    assert im(log(2*I)) == pi/2

    assert im((2 + I)**2).expand(complex=True) == 4

    assert im(conjugate(x)) == -im(x)
    assert conjugate(im(x)) == im(x)

    assert im(x).as_real_imag() == (im(x), 0)

    assert im(i*r*x).diff(r) == im(i*x)
    assert im(i*r*x).diff(i) == -I * re(r*x)

    assert im(
        sqrt(a + b*I)) == (a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)
    assert im(a * (2 + b*I)) == a*b

    assert im((1 + sqrt(a + b*I))/2) == \
        (a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)/2

    assert im(x).rewrite(re) == x - re(x)
    assert (x + im(y)).rewrite(im, re) == x + y - re(y)

    a = Symbol('a', algebraic=True)
    t = Symbol('t', transcendental=True)
    x = Symbol('x')
    assert re(a).is_algebraic
    assert re(x).is_algebraic is None
    assert re(t).is_algebraic is False


def test_sign():
    assert sign(1.2) == 1
    assert sign(-1.2) == -1
    assert sign(3*I) == I
    assert sign(-3*I) == -I
    assert sign(0) == 0
    assert sign(nan) == nan
    assert sign(2 + 2*I).doit() == sqrt(2)*(2 + 2*I)/4
    assert sign(2 + 3*I).simplify() == sign(2 + 3*I)
    assert sign(2 + 2*I).simplify() == sign(1 + I)
    assert sign(im(sqrt(1 - sqrt(3)))) == 1
    assert sign(sqrt(1 - sqrt(3))) == I

    x = Symbol('x')
    assert sign(x).is_finite is True
    assert sign(x).is_complex is True
    assert sign(x).is_imaginary is None
    assert sign(x).is_integer is None
    assert sign(x).is_real is None
    assert sign(x).is_zero is None
    assert sign(x).doit() == sign(x)
    assert sign(1.2*x) == sign(x)
    assert sign(2*x) == sign(x)
    assert sign(I*x) == I*sign(x)
    assert sign(-2*I*x) == -I*sign(x)
    assert sign(conjugate(x)) == conjugate(sign(x))

    p = Symbol('p', positive=True)
    n = Symbol('n', negative=True)
    m = Symbol('m', negative=True)
    assert sign(2*p*x) == sign(x)
    assert sign(n*x) == -sign(x)
    assert sign(n*m*x) == sign(x)

    x = Symbol('x', imaginary=True)
    assert sign(x).is_imaginary is True
    assert sign(x).is_integer is False
    assert sign(x).is_real is False
    assert sign(x).is_zero is False
    assert sign(x).diff(x) == 2*DiracDelta(-I*x)
    assert sign(x).doit() == x / Abs(x)
    assert conjugate(sign(x)) == -sign(x)

    x = Symbol('x', real=True)
    assert sign(x).is_imaginary is False
    assert sign(x).is_integer is True
    assert sign(x).is_real is True
    assert sign(x).is_zero is None
    assert sign(x).diff(x) == 2*DiracDelta(x)
    assert sign(x).doit() == sign(x)
    assert conjugate(sign(x)) == sign(x)

    x = Symbol('x', nonzero=True)
    assert sign(x).is_imaginary is False
    assert sign(x).is_integer is True
    assert sign(x).is_real is True
    assert sign(x).is_zero is False
    assert sign(x).doit() == x / Abs(x)
    assert sign(Abs(x)) == 1
    assert Abs(sign(x)) == 1

    x = Symbol('x', positive=True)
    assert sign(x).is_imaginary is False
    assert sign(x).is_integer is True
    assert sign(x).is_real is True
    assert sign(x).is_zero is False
    assert sign(x).doit() == x / Abs(x)
    assert sign(Abs(x)) == 1
    assert Abs(sign(x)) == 1

    x = 0
    assert sign(x).is_imaginary is False
    assert sign(x).is_integer is True
    assert sign(x).is_real is True
    assert sign(x).is_zero is True
    assert sign(x).doit() == 0
    assert sign(Abs(x)) == 0
    assert Abs(sign(x)) == 0

    nz = Symbol('nz', nonzero=True, integer=True)
    assert sign(nz).is_imaginary is False
    assert sign(nz).is_integer is True
    assert sign(nz).is_real is True
    assert sign(nz).is_zero is False
    assert sign(nz)**2 == 1
    assert (sign(nz)**3).args == (sign(nz), 3)

    assert sign(Symbol('x', nonnegative=True)).is_nonnegative
    assert sign(Symbol('x', nonnegative=True)).is_nonpositive is None
    assert sign(Symbol('x', nonpositive=True)).is_nonnegative is None
    assert sign(Symbol('x', nonpositive=True)).is_nonpositive
    assert sign(Symbol('x', real=True)).is_nonnegative is None
    assert sign(Symbol('x', real=True)).is_nonpositive is None
    assert sign(Symbol('x', real=True, zero=False)).is_nonpositive is None

    x, y = Symbol('x', real=True), Symbol('y')
    assert sign(x).rewrite(Piecewise) == \
        Piecewise((1, x > 0), (-1, x < 0), (0, True))
    assert sign(y).rewrite(Piecewise) == sign(y)
    assert sign(x).rewrite(Heaviside) == 2*Heaviside(x)-1
    assert sign(y).rewrite(Heaviside) == sign(y)

    # evaluate what can be evaluated
    assert sign(exp_polar(I*pi)*pi) is S.NegativeOne

    eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
    # if there is a fast way to know when and when you cannot prove an
    # expression like this is zero then the equality to zero is ok
    assert sign(eq).func is sign or sign(eq) == 0
    # but sometimes it's hard to do this so it's better not to load
    # abs down with tests that will be very slow
    q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
    p = expand(q**3)**Rational(1, 3)
    d = p - q
    assert sign(d).func is sign or sign(d) == 0


def test_as_real_imag():
    n = pi**1000
    # the special code for working out the real
    # and complex parts of a power with Integer exponent
    # should not run if there is no imaginary part, hence
    # this should not hang
    assert n.as_real_imag() == (n, 0)

    # issue 6261
    x = Symbol('x')
    assert sqrt(x).as_real_imag() == \
        ((re(x)**2 + im(x)**2)**(S(1)/4)*cos(atan2(im(x), re(x))/2),
     (re(x)**2 + im(x)**2)**(S(1)/4)*sin(atan2(im(x), re(x))/2))

    # issue 3853
    a, b = symbols('a,b', real=True)
    assert ((1 + sqrt(a + b*I))/2).as_real_imag() == \
           (
               (a**2 + b**2)**Rational(
                   1, 4)*cos(atan2(b, a)/2)/2 + Rational(1, 2),
               (a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)/2)

    assert sqrt(a**2).as_real_imag() == (sqrt(a**2), 0)
    i = symbols('i', imaginary=True)
    assert sqrt(i**2).as_real_imag() == (0, abs(i))


@XFAIL
def test_sign_issue_3068():
    n = pi**1000
    i = int(n)
    assert (n - i).round() == 1  # doesn't hang
    assert sign(n - i) == 1
    # perhaps it's not possible to get the sign right when
    # only 1 digit is being requested for this situtation;
    # 2 digits works
    assert (n - x).n(1, subs={x: i}) > 0
    assert (n - x).n(2, subs={x: i}) > 0


def test_Abs():
    raises(TypeError, lambda: Abs(Interval(2, 3)))  # issue 8717

    x, y = symbols('x,y')
    assert sign(sign(x)) == sign(x)
    assert sign(x*y).func is sign
    assert Abs(0) == 0
    assert Abs(1) == 1
    assert Abs(-1) == 1
    assert Abs(I) == 1
    assert Abs(-I) == 1
    assert Abs(nan) == nan
    assert Abs(I * pi) == pi
    assert Abs(-I * pi) == pi
    assert Abs(I * x) == Abs(x)
    assert Abs(-I * x) == Abs(x)
    assert Abs(-2*x) == 2*Abs(x)
    assert Abs(-2.0*x) == 2.0*Abs(x)
    assert Abs(2*pi*x*y) == 2*pi*Abs(x*y)
    assert Abs(conjugate(x)) == Abs(x)
    assert conjugate(Abs(x)) == Abs(x)
    assert Abs(x).expand(complex=True) == sqrt(re(x)**2 + im(x)**2)

    a = Symbol('a', positive=True)
    assert Abs(2*pi*x*a) == 2*pi*a*Abs(x)
    assert Abs(2*pi*I*x*a) == 2*pi*a*Abs(x)

    x = Symbol('x', real=True)
    n = Symbol('n', integer=True)
    assert Abs((-1)**n) == 1
    assert x**(2*n) == Abs(x)**(2*n)
    assert Abs(x).diff(x) == sign(x)
    assert abs(x) == Abs(x)  # Python built-in
    assert Abs(x)**3 == x**2*Abs(x)
    assert Abs(x)**4 == x**4
    assert (
        Abs(x)**(3*n)).args == (Abs(x), 3*n)  # leave symbolic odd unchanged
    assert (1/Abs(x)).args == (Abs(x), -1)
    assert 1/Abs(x)**3 == 1/(x**2*Abs(x))
    assert Abs(x)**-3 == Abs(x)/(x**4)
    assert Abs(x**3) == x**2*Abs(x)

    x = Symbol('x', imaginary=True)
    assert Abs(x).diff(x) == -sign(x)

    eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
    # if there is a fast way to know when you can and when you cannot prove an
    # expression like this is zero then the equality to zero is ok
    assert abs(eq).func is Abs or abs(eq) == 0
    # but sometimes it's hard to do this so it's better not to load
    # abs down with tests that will be very slow
    q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
    p = expand(q**3)**Rational(1, 3)
    d = p - q
    assert abs(d).func is Abs or abs(d) == 0

    assert Abs(4*exp(pi*I/4)) == 4
    assert Abs(3**(2 + I)) == 9
    assert Abs((-3)**(1 - I)) == 3*exp(pi)

    assert Abs(oo) is oo
    assert Abs(-oo) is oo
    assert Abs(oo + I) is oo
    assert Abs(oo + I*oo) is oo

    a = Symbol('a', algebraic=True)
    t = Symbol('t', transcendental=True)
    x = Symbol('x')
    assert re(a).is_algebraic
    assert re(x).is_algebraic is None
    assert re(t).is_algebraic is False


def test_Abs_rewrite():
    x = Symbol('x', real=True)
    a = Abs(x).rewrite(Heaviside).expand()
    assert a == x*Heaviside(x) - x*Heaviside(-x)
    for i in [-2, -1, 0, 1, 2]:
        assert a.subs(x, i) == abs(i)
    y = Symbol('y')
    assert Abs(y).rewrite(Heaviside) == Abs(y)

    x, y = Symbol('x', real=True), Symbol('y')
    assert Abs(x).rewrite(Piecewise) == Piecewise((x, x >= 0), (-x, True))
    assert Abs(y).rewrite(Piecewise) == Abs(y)
    assert Abs(y).rewrite(sign) == y/sign(y)


def test_Abs_real():
    # test some properties of abs that only apply
    # to real numbers
    x = Symbol('x', complex=True)
    assert sqrt(x**2) != Abs(x)
    assert Abs(x**2) != x**2

    x = Symbol('x', real=True)
    assert sqrt(x**2) == Abs(x)
    assert Abs(x**2) == x**2

    # if the symbol is zero, the following will still apply
    nn = Symbol('nn', nonnegative=True, real=True)
    np = Symbol('np', nonpositive=True, real=True)
    assert Abs(nn) == nn
    assert Abs(np) == -np


def test_Abs_properties():
    x = Symbol('x')
    assert Abs(x).is_real is True
    assert Abs(x).is_rational is None
    assert Abs(x).is_positive is None
    assert Abs(x).is_nonnegative is True

    z = Symbol('z', complex=True, zero=False)
    assert Abs(z).is_real is True
    assert Abs(z).is_rational is None
    assert Abs(z).is_positive is True
    assert Abs(z).is_zero is False

    p = Symbol('p', positive=True)
    assert Abs(p).is_real is True
    assert Abs(p).is_rational is None
    assert Abs(p).is_positive is True
    assert Abs(p).is_zero is False

    q = Symbol('q', rational=True)
    assert Abs(q).is_rational is True
    assert Abs(q).is_integer is None
    assert Abs(q).is_positive is None
    assert Abs(q).is_nonnegative is True

    i = Symbol('i', integer=True)
    assert Abs(i).is_integer is True
    assert Abs(i).is_positive is None
    assert Abs(i).is_nonnegative is True

    e = Symbol('n', even=True)
    ne = Symbol('ne', real=True, even=False)
    assert Abs(e).is_even
    assert Abs(ne).is_even is False
    assert Abs(i).is_even is None

    o = Symbol('n', odd=True)
    no = Symbol('no', real=True, odd=False)
    assert Abs(o).is_odd
    assert Abs(no).is_odd is False
    assert Abs(i).is_odd is None


def test_abs():
    # this tests that abs calls Abs; don't rename to
    # test_Abs since that test is already above
    a = Symbol('a', positive=True)
    assert abs(I*(1 + a)**2) == (1 + a)**2


def test_arg():
    assert arg(0) == nan
    assert arg(1) == 0
    assert arg(-1) == pi
    assert arg(I) == pi/2
    assert arg(-I) == -pi/2
    assert arg(1 + I) == pi/4
    assert arg(-1 + I) == 3*pi/4
    assert arg(1 - I) == -pi/4
    f = Function('f')
    assert not arg(f(0) + I*f(1)).atoms(re)

    p = Symbol('p', positive=True)
    assert arg(p) == 0

    n = Symbol('n', negative=True)
    assert arg(n) == pi

    x = Symbol('x')
    assert conjugate(arg(x)) == arg(x)

    e = p + I*p**2
    assert arg(e) == arg(1 + p*I)
    # make sure sign doesn't swap
    e = -2*p + 4*I*p**2
    assert arg(e) == arg(-1 + 2*p*I)
    # make sure sign isn't lost
    x = symbols('x', real=True)  # could be zero
    e = x + I*x
    assert arg(e) == arg(x*(1 + I))
    assert arg(e/p) == arg(x*(1 + I))
    e = p*cos(p) + I*log(p)*exp(p)
    assert arg(e).args[0] == e
    # keep it simple -- let the user do more advanced cancellation
    e = (p + 1) + I*(p**2 - 1)
    assert arg(e).args[0] == e

    f = Function('f')
    e = 2*x*(f(0) - 1) - 2*x*f(0)
    assert arg(e) == arg(-2*x)
    assert arg(f(0)).func == arg and arg(f(0)).args == (f(0),)


def test_arg_rewrite():
    assert arg(1 + I) == atan2(1, 1)

    x = Symbol('x', real=True)
    y = Symbol('y', real=True)
    assert arg(x + I*y).rewrite(atan2) == atan2(y, x)


def test_adjoint():
    a = Symbol('a', antihermitian=True)
    b = Symbol('b', hermitian=True)
    assert adjoint(a) == -a
    assert adjoint(I*a) == I*a
    assert adjoint(b) == b
    assert adjoint(I*b) == -I*b
    assert adjoint(a*b) == -b*a
    assert adjoint(I*a*b) == I*b*a

    x, y = symbols('x y')
    assert adjoint(adjoint(x)) == x
    assert adjoint(x + y) == adjoint(x) + adjoint(y)
    assert adjoint(x - y) == adjoint(x) - adjoint(y)
    assert adjoint(x * y) == adjoint(x) * adjoint(y)
    assert adjoint(x / y) == adjoint(x) / adjoint(y)
    assert adjoint(-x) == -adjoint(x)

    x, y = symbols('x y', commutative=False)
    assert adjoint(adjoint(x)) == x
    assert adjoint(x + y) == adjoint(x) + adjoint(y)
    assert adjoint(x - y) == adjoint(x) - adjoint(y)
    assert adjoint(x * y) == adjoint(y) * adjoint(x)
    assert adjoint(x / y) == 1 / adjoint(y) * adjoint(x)
    assert adjoint(-x) == -adjoint(x)


def test_conjugate():
    a = Symbol('a', real=True)
    b = Symbol('b', imaginary=True)
    assert conjugate(a) == a
    assert conjugate(I*a) == -I*a
    assert conjugate(b) == -b
    assert conjugate(I*b) == I*b
    assert conjugate(a*b) == -a*b
    assert conjugate(I*a*b) == I*a*b

    x, y = symbols('x y')
    assert conjugate(conjugate(x)) == x
    assert conjugate(x + y) == conjugate(x) + conjugate(y)
    assert conjugate(x - y) == conjugate(x) - conjugate(y)
    assert conjugate(x * y) == conjugate(x) * conjugate(y)
    assert conjugate(x / y) == conjugate(x) / conjugate(y)
    assert conjugate(-x) == -conjugate(x)

    a = Symbol('a', algebraic=True)
    t = Symbol('t', transcendental=True)
    assert re(a).is_algebraic
    assert re(x).is_algebraic is None
    assert re(t).is_algebraic is False


def test_conjugate_transpose():
    x = Symbol('x')
    assert conjugate(transpose(x)) == adjoint(x)
    assert transpose(conjugate(x)) == adjoint(x)
    assert adjoint(transpose(x)) == conjugate(x)
    assert transpose(adjoint(x)) == conjugate(x)
    assert adjoint(conjugate(x)) == transpose(x)
    assert conjugate(adjoint(x)) == transpose(x)

    class Symmetric(Expr):
        def _eval_adjoint(self):
            return None

        def _eval_conjugate(self):
            return None

        def _eval_transpose(self):
            return self
    x = Symmetric()
    assert conjugate(x) == adjoint(x)
    assert transpose(x) == x


def test_transpose():
    a = Symbol('a', complex=True)
    assert transpose(a) == a
    assert transpose(I*a) == I*a

    x, y = symbols('x y')
    assert transpose(transpose(x)) == x
    assert transpose(x + y) == transpose(x) + transpose(y)
    assert transpose(x - y) == transpose(x) - transpose(y)
    assert transpose(x * y) == transpose(x) * transpose(y)
    assert transpose(x / y) == transpose(x) / transpose(y)
    assert transpose(-x) == -transpose(x)

    x, y = symbols('x y', commutative=False)
    assert transpose(transpose(x)) == x
    assert transpose(x + y) == transpose(x) + transpose(y)
    assert transpose(x - y) == transpose(x) - transpose(y)
    assert transpose(x * y) == transpose(y) * transpose(x)
    assert transpose(x / y) == 1 / transpose(y) * transpose(x)
    assert transpose(-x) == -transpose(x)


def test_polarify():
    from sympy import polar_lift, polarify
    x = Symbol('x')
    z = Symbol('z', polar=True)
    f = Function('f')
    ES = {}

    assert polarify(-1) == (polar_lift(-1), ES)
    assert polarify(1 + I) == (polar_lift(1 + I), ES)

    assert polarify(exp(x), subs=False) == exp(x)
    assert polarify(1 + x, subs=False) == 1 + x
    assert polarify(f(I) + x, subs=False) == f(polar_lift(I)) + x

    assert polarify(x, lift=True) == polar_lift(x)
    assert polarify(z, lift=True) == z
    assert polarify(f(x), lift=True) == f(polar_lift(x))
    assert polarify(1 + x, lift=True) == polar_lift(1 + x)
    assert polarify(1 + f(x), lift=True) == polar_lift(1 + f(polar_lift(x)))

    newex, subs = polarify(f(x) + z)
    assert newex.subs(subs) == f(x) + z

    mu = Symbol("mu")
    sigma = Symbol("sigma", positive=True)

    # Make sure polarify(lift=True) doesn't try to lift the integration
    # variable
    assert polarify(
        Integral(sqrt(2)*x*exp(-(-mu + x)**2/(2*sigma**2))/(2*sqrt(pi)*sigma),
        (x, -oo, oo)), lift=True) == Integral(sqrt(2)*(sigma*exp_polar(0))**exp_polar(I*pi)*
        exp((sigma*exp_polar(0))**(2*exp_polar(I*pi))*exp_polar(I*pi)*polar_lift(-mu + x)**
        (2*exp_polar(0))/2)*exp_polar(0)*polar_lift(x)/(2*sqrt(pi)), (x, -oo, oo))


def test_unpolarify():
    from sympy import (exp_polar, polar_lift, exp, unpolarify,
                       principal_branch)
    from sympy import gamma, erf, sin, tanh, uppergamma, Eq, Ne
    from sympy.abc import x
    p = exp_polar(7*I) + 1
    u = exp(7*I) + 1

    assert unpolarify(1) == 1
    assert unpolarify(p) == u
    assert unpolarify(p**2) == u**2
    assert unpolarify(p**x) == p**x
    assert unpolarify(p*x) == u*x
    assert unpolarify(p + x) == u + x
    assert unpolarify(sqrt(sin(p))) == sqrt(sin(u))

    # Test reduction to principal branch 2*pi.
    t = principal_branch(x, 2*pi)
    assert unpolarify(t) == x
    assert unpolarify(sqrt(t)) == sqrt(t)

    # Test exponents_only.
    assert unpolarify(p**p, exponents_only=True) == p**u
    assert unpolarify(uppergamma(x, p**p)) == uppergamma(x, p**u)

    # Test functions.
    assert unpolarify(sin(p)) == sin(u)
    assert unpolarify(tanh(p)) == tanh(u)
    assert unpolarify(gamma(p)) == gamma(u)
    assert unpolarify(erf(p)) == erf(u)
    assert unpolarify(uppergamma(x, p)) == uppergamma(x, p)

    assert unpolarify(uppergamma(sin(p), sin(p + exp_polar(0)))) == \
        uppergamma(sin(u), sin(u + 1))
    assert unpolarify(uppergamma(polar_lift(0), 2*exp_polar(0))) == \
        uppergamma(0, 2)

    assert unpolarify(Eq(p, 0)) == Eq(u, 0)
    assert unpolarify(Ne(p, 0)) == Ne(u, 0)
    assert unpolarify(polar_lift(x) > 0) == (x > 0)

    # Test bools
    assert unpolarify(True) is True


def test_issue_4035():
    x = Symbol('x')
    assert Abs(x).expand(trig=True) == Abs(x)
    assert sign(x).expand(trig=True) == sign(x)
    assert arg(x).expand(trig=True) == arg(x)


def test_issue_3206():
    x = Symbol('x')
    assert Abs(Abs(x)) == Abs(x)


def test_issue_4754_derivative_conjugate():
    x = Symbol('x', real=True)
    y = Symbol('y', imaginary=True)
    f = Function('f')
    assert (f(x).conjugate()).diff(x) == (f(x).diff(x)).conjugate()
    assert (f(y).conjugate()).diff(y) == -(f(y).diff(y)).conjugate()


def test_derivatives_issue_4757():
    x = Symbol('x', real=True)
    y = Symbol('y', imaginary=True)
    f = Function('f')
    assert re(f(x)).diff(x) == re(f(x).diff(x))
    assert im(f(x)).diff(x) == im(f(x).diff(x))
    assert re(f(y)).diff(y) == -I*im(f(y).diff(y))
    assert im(f(y)).diff(y) == -I*re(f(y).diff(y))
    assert Abs(f(x)).diff(x).subs(f(x), 1 + I*x).doit() == x/sqrt(1 + x**2)
    assert arg(f(x)).diff(x).subs(f(x), 1 + I*x**2).doit() == 2*x/(1 + x**4)
    assert Abs(f(y)).diff(y).subs(f(y), 1 + y).doit() == -y/sqrt(1 - y**2)
    assert arg(f(y)).diff(y).subs(f(y), I + y**2).doit() == 2*y/(1 + y**4)


def test_periodic_argument():
    from sympy import (periodic_argument, unbranched_argument, oo,
                       principal_branch, polar_lift, pi)
    x = Symbol('x')
    p = Symbol('p', positive=True)

    assert unbranched_argument(2 + I) == periodic_argument(2 + I, oo)
    assert unbranched_argument(1 + x) == periodic_argument(1 + x, oo)
    assert N_equals(unbranched_argument((1 + I)**2), pi/2)
    assert N_equals(unbranched_argument((1 - I)**2), -pi/2)
    assert N_equals(periodic_argument((1 + I)**2, 3*pi), pi/2)
    assert N_equals(periodic_argument((1 - I)**2, 3*pi), -pi/2)

    assert unbranched_argument(principal_branch(x, pi)) == \
        periodic_argument(x, pi)

    assert unbranched_argument(polar_lift(2 + I)) == unbranched_argument(2 + I)
    assert periodic_argument(polar_lift(2 + I), 2*pi) == \
        periodic_argument(2 + I, 2*pi)
    assert periodic_argument(polar_lift(2 + I), 3*pi) == \
        periodic_argument(2 + I, 3*pi)
    assert periodic_argument(polar_lift(2 + I), pi) == \
        periodic_argument(polar_lift(2 + I), pi)

    assert unbranched_argument(polar_lift(1 + I)) == pi/4
    assert periodic_argument(2*p, p) == periodic_argument(p, p)
    assert periodic_argument(pi*p, p) == periodic_argument(p, p)

    assert Abs(polar_lift(1 + I)) == Abs(1 + I)


@XFAIL
def test_principal_branch_fail():
    # TODO XXX why does abs(x)._eval_evalf() not fall back to global evalf?
    assert N_equals(principal_branch((1 + I)**2, pi/2), 0)


def test_principal_branch():
    from sympy import principal_branch, polar_lift, exp_polar
    p = Symbol('p', positive=True)
    x = Symbol('x')
    neg = Symbol('x', negative=True)

    assert principal_branch(polar_lift(x), p) == principal_branch(x, p)
    assert principal_branch(polar_lift(2 + I), p) == principal_branch(2 + I, p)
    assert principal_branch(2*x, p) == 2*principal_branch(x, p)
    assert principal_branch(1, pi) == exp_polar(0)
    assert principal_branch(-1, 2*pi) == exp_polar(I*pi)
    assert principal_branch(-1, pi) == exp_polar(0)
    assert principal_branch(exp_polar(3*pi*I)*x, 2*pi) == \
        principal_branch(exp_polar(I*pi)*x, 2*pi)
    assert principal_branch(neg*exp_polar(pi*I), 2*pi) == neg*exp_polar(-I*pi)

    assert N_equals(principal_branch((1 + I)**2, 2*pi), 2*I)
    assert N_equals(principal_branch((1 + I)**2, 3*pi), 2*I)
    assert N_equals(principal_branch((1 + I)**2, 1*pi), 2*I)

    # test argument sanitization
    assert principal_branch(x, I).func is principal_branch
    assert principal_branch(x, -4).func is principal_branch
    assert principal_branch(x, -oo).func is principal_branch
    assert principal_branch(x, zoo).func is principal_branch


@XFAIL
def test_issue_6167_6151():
    n = pi**1000
    i = int(n)
    assert sign(n - i) == 1
    assert abs(n - i) == n - i
    eps = pi**-1500
    big = pi**1000
    one = cos(x)**2 + sin(x)**2
    e = big*one - big + eps
    assert sign(simplify(e)) == 1
    for xi in (111, 11, 1, S(1)/10):
        assert sign(e.subs(x, xi)) == 1
