def test_sqrtdenest2():
    assert sqrtdenest(sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))) == \
            r5 + sqrt(11 - 2*r29)
    e = sqrt(-r5 + sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16))
    assert sqrtdenest(e) == root(-2*r29 + 11, 4)
    r = sqrt(1 + r7)
    assert sqrtdenest(sqrt(1 + r)) == sqrt(1 + r)
    e = sqrt(((1 + sqrt(1 + 2*sqrt(3 + r2 + r5)))**2).expand())
    assert sqrtdenest(e) == 1 + sqrt(1 + 2*sqrt(r2 + r5 + 3))
    assert sqrtdenest(sqrt(5*r3 + 6*r2)) == \
        sqrt(2)*root(3, 4) + root(3, 4)**3

    assert sqrtdenest(sqrt(((1 + r5 + sqrt(1 + r3))**2).expand())) == \
         1 + r5 + sqrt(1 + r3)

    assert sqrtdenest(sqrt(((1 + r5 + r7 + sqrt(1 + r3))**2).expand())) == \
        1 + sqrt(1 + r3) + r5 + r7

    e = sqrt(((1 + cos(2) + cos(3) + sqrt(1 + r3))**2).expand())
    assert sqrtdenest(e) == cos(3) + cos(2) + 1 + sqrt(1 + r3)

    e = sqrt(-2*r10 + 2*r2*sqrt(-2*r10 + 11) + 14)
    assert sqrtdenest(e) == sqrt(-2*r10 - 2*r2 + 4*r5 + 14)

    # check that the result is not more complicated than the input
    z= sqrt(-2*r29 + cos(2) + 2*sqrt(-10*r29 + 55) + 16)
    assert sqrtdenest(z) == z

    assert sqrtdenest(sqrt(r6 + sqrt(15))) == sqrt(r6 + sqrt(15))

    z = sqrt(15 - 2*sqrt(31) + 2*sqrt(55 - 10*r29))
    assert sqrtdenest(z) == z

def test_issue_from_PR1599():
    n1, n2, n3, n4 = symbols('n1 n2 n3 n4', negative=True)
    assert (powsimp(sqrt(n1)*sqrt(n2)*sqrt(n3)) ==
        -I*sqrt(-n1)*sqrt(-n2)*sqrt(-n3))
    assert (powsimp(root(n1, 3)*root(n2, 3)*root(n3, 3)*root(n4, 3)) ==
        -(-1)**(S(1)/3)*
        (-n1)**(S(1)/3)*(-n2)**(S(1)/3)*(-n3)**(S(1)/3)*(-n4)**(S(1)/3))

def norm(A_expr,axis=None,ord=2):

    assert axis is None or axis == 0 or axis == 1
    if axis is None:
        if not A_expr.is_Matrix:
            return sympy.Abs(A_expr)
        assert A_expr.cols == 1 or A_expr.rows == 1
        if A_expr.cols == 1 and A_expr.rows == 1:
            return sympy.Abs(A_expr)
        if A_expr.cols == 1:
            return sympy.root( sympy.Add( *[ a_expr**ord for a_expr in A_expr[:,0] ] ), ord )
        if A_expr.rows == 1:
            return sympy.root( sympy.Add( *[ a_expr**ord for a_expr in A_expr[0,:] ] ), ord )
    if axis == 0:
        A_norm_expr = sympy.Matrix.zeros(1,A_expr.cols)
        for c in range(A_expr.cols):
            A_norm_expr[0,c] = sympy.root( sympy.Add( *[ a_expr**ord for a_expr in A_expr[:,c] ] ), ord )
        return A_norm_expr
    if axis == 1:
        A_norm_expr = sympy.Matrix.zeros(A_expr.rows,1)
        for r in range(A_expr.rows):
            A_norm_expr[r,0] = sympy.root( sympy.Add( *[ a_expr**ord for a_expr in A_expr[r,:] ] ), ord )
        return A_norm_expr
    assert False
    return None

def test_issue_3109():
    from sympy import root, Rational
    I = S.ImaginaryUnit
    assert sqrt(33**(9*I/10)) == -33**(9*I/20)
    assert root((6*I)**(2*I), 3).as_base_exp()[1] == Rational(1, 3) # != 2*I/3
    assert root((6*I)**(I/3), 3).as_base_exp()[1] == I/9
    assert sqrt(exp(3*I)) == exp(3*I/2)
    assert sqrt(-sqrt(3)*(1 + 2*I)) == sqrt(sqrt(3))*sqrt(-1 - 2*I)

def test_factor_terms():
    A = Symbol('A', commutative=False)
    assert factor_terms(9*(x + x*y + 1) + (3*x + 3)**(2 + 2*x)) == \
        9*x*y + 9*x + _keep_coeff(S(3), x + 1)**_keep_coeff(S(2), x + 1) + 9
    assert factor_terms(9*(x + x*y + 1) + (3)**(2 + 2*x)) == \
        _keep_coeff(S(9), 3**(2*x) + x*y + x + 1)
    assert factor_terms(3**(2 + 2*x) + a*3**(2 + 2*x)) == \
        9*3**(2*x)*(a + 1)
    assert factor_terms(x + x*A) == \
        x*(1 + A)
    assert factor_terms(sin(x + x*A)) == \
        sin(x*(1 + A))
    assert factor_terms((3*x + 3)**((2 + 2*x)/3)) == \
        _keep_coeff(S(3), x + 1)**_keep_coeff(S(2)/3, x + 1)
    assert factor_terms(x + (x*y + x)**(3*x + 3)) == \
        x + (x*(y + 1))**_keep_coeff(S(3), x + 1)
    assert factor_terms(a*(x + x*y) + b*(x*2 + y*x*2)) == \
        x*(a + 2*b)*(y + 1)
    i = Integral(x, (x, 0, oo))
    assert factor_terms(i) == i

    # check radical extraction
    eq = sqrt(2) + sqrt(10)
    assert factor_terms(eq) == eq
    assert factor_terms(eq, radical=True) == sqrt(2)*(1 + sqrt(5))
    eq = root(-6, 3) + root(6, 3)
    assert factor_terms(eq, radical=True) == 6**(S(1)/3)*(1 + (-1)**(S(1)/3))

    eq = [x + x*y]
    ans = [x*(y + 1)]
    for c in [list, tuple, set]:
        assert factor_terms(c(eq)) == c(ans)
    assert factor_terms(Tuple(x + x*y)) == Tuple(x*(y + 1))
    assert factor_terms(Interval(0, 1)) == Interval(0, 1)
    e = 1/sqrt(a/2 + 1)
    assert factor_terms(e, clear=False) == 1/sqrt(a/2 + 1)
    assert factor_terms(e, clear=True) == sqrt(2)/sqrt(a + 2)

    eq = x/(x + 1/x) + 1/(x**2 + 1)
    assert factor_terms(eq, fraction=False) == eq
    assert factor_terms(eq, fraction=True) == 1

    assert factor_terms((1/(x**3 + x**2) + 2/x**2)*y) == \
        y*(2 + 1/(x + 1))/x**2

    # if not True, then processesing for this in factor_terms is not necessary
    assert gcd_terms(-x - y) == -x - y
    assert factor_terms(-x - y) == Mul(-1, x + y, evaluate=False)

    # if not True, then "special" processesing in factor_terms is not necessary
    assert gcd_terms(exp(Mul(-1, x + 1))) == exp(-x - 1)
    e = exp(-x - 2) + x
    assert factor_terms(e) == exp(Mul(-1, x + 2, evaluate=False)) + x
    assert factor_terms(e, sign=False) == e
    assert factor_terms(exp(-4*x - 2) - x) == -x + exp(Mul(-2, 2*x + 1, evaluate=False))

def test_trig_split():
    assert trig_split(cos(x), cos(y)) == (1, 1, 1, x, y, True)
    assert trig_split(2*cos(x), -2*cos(y)) == (2, 1, -1, x, y, True)
    assert trig_split(cos(x)*sin(y), cos(y)*sin(y)) == \
        (sin(y), 1, 1, x, y, True)

    assert trig_split(cos(x), -sqrt(3)*sin(x), two=True) == \
        (2, 1, -1, x, pi/6, False)
    assert trig_split(cos(x), sin(x), two=True) == \
        (sqrt(2), 1, 1, x, pi/4, False)
    assert trig_split(cos(x), -sin(x), two=True) == \
        (sqrt(2), 1, -1, x, pi/4, False)
    assert trig_split(sqrt(2)*cos(x), -sqrt(6)*sin(x), two=True) == \
        (2*sqrt(2), 1, -1, x, pi/6, False)
    assert trig_split(-sqrt(6)*cos(x), -sqrt(2)*sin(x), two=True) == \
        (-2*sqrt(2), 1, 1, x, pi/3, False)
    assert trig_split(cos(x)/sqrt(6), sin(x)/sqrt(2), two=True) == \
        (sqrt(6)/3, 1, 1, x, pi/6, False)
    assert trig_split(-sqrt(6)*cos(x)*sin(y),
            -sqrt(2)*sin(x)*sin(y), two=True) == \
        (-2*sqrt(2)*sin(y), 1, 1, x, pi/3, False)

    assert trig_split(cos(x), sin(x)) is None
    assert trig_split(cos(x), sin(z)) is None
    assert trig_split(2*cos(x), -sin(x)) is None
    assert trig_split(cos(x), -sqrt(3)*sin(x)) is None
    assert trig_split(cos(x)*cos(y), sin(x)*sin(z)) is None
    assert trig_split(cos(x)*cos(y), sin(x)*sin(y)) is None
    assert trig_split(-sqrt(6)*cos(x), sqrt(2)*sin(x)*sin(y), two=True) is \
        None

    assert trig_split(sqrt(3)*sqrt(x), cos(3), two=True) is None
    assert trig_split(sqrt(3)*root(x, 3), sin(3)*cos(2), two=True) is None
    assert trig_split(cos(5)*cos(6), cos(7)*sin(5), two=True) is None

def test_sqrtdenest():
    d = {sqrt(5 + 2 * sqrt(6)): sqrt(2) + sqrt(3),
        sqrt(sqrt(2)): sqrt(sqrt(2)),
        sqrt(5+sqrt(7)): sqrt(5+sqrt(7)),
        sqrt(3+sqrt(5+2*sqrt(7))):
            sqrt(6+3*sqrt(7))/(sqrt(2)*(5+2*sqrt(7))**Rational(1,4)) +
            3*(5+2*sqrt(7))**Rational(1,4)/(sqrt(2)*sqrt(6+3*sqrt(7))),
        sqrt(3+2*sqrt(3)): 3**Rational(1,4)/sqrt(2)+3/(sqrt(2)*3**Rational(1,4))}
    for i in d:
        assert sqrtdenest(i) == d[i] or denester([i])[0] == d[i]

    # this test caused a pattern recognition failure in sqrtdenest
    # nest = sqrt(2) + sqrt(5) - sqrt(7)
    nest = symbols('nest')
    x0, x1, x2, x3, x4, x5, x6 = symbols('x:7')
    l = sqrt(2) + sqrt(5)
    r = sqrt(7) + nest
    s = (l**2 - r**2).expand() + nest**2 # == nest**2
    ok = solve(nest**4 - s**2, nest)[1] # this will change if results order changes
    assert abs((l - r).subs(nest, ok).n()) < 1e-12
    x0 = sqrt(3)
    x2 = root(45*I*x0 - 28, 3)
    x3 = 19/x2
    x4 = x2 + x3
    x5 = -x4 - 14
    x6 = sqrt(-x5)
    ans = -x0*x6/3 + x0*sqrt(-x4 + 28 - 6*sqrt(210)*x6/x5)/3
    assert expand_mul(radsimp(ok) - ans) == 0
    # issue 2554
    eq = sqrt(sqrt(sqrt(2) + 2) + 2)
    assert sqrtdenest(eq) == eq

def test_latex_functions():
    assert latex(exp(x)) == "e^{x}"
    assert latex(exp(1)+exp(2)) == "e + e^{2}"

    f = Function('f')
    assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}'

    beta = Function('beta')

    assert latex(beta(x)) == r"\beta{\left (x \right )}"
    assert latex(sin(x)) == r"\sin{\left (x \right )}"
    assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
    assert latex(sin(2*x**2), fold_func_brackets=True) == \
    r"\sin {2 x^{2}}"
    assert latex(sin(x**2), fold_func_brackets=True) == \
    r"\sin {x^{2}}"

    assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
    assert latex(asin(x)**2,inv_trig_style="full") == \
        r"\arcsin^{2}{\left (x \right )}"
    assert latex(asin(x)**2,inv_trig_style="power") == \
        r"\sin^{-1}{\left (x \right )}^{2}"
    assert latex(asin(x**2),inv_trig_style="power",fold_func_brackets=True) == \
        r"\sin^{-1} {x^{2}}"

    assert latex(factorial(k)) == r"k!"
    assert latex(factorial(-k)) == r"\left(- k\right)!"

    assert latex(factorial2(k)) == r"k!!"
    assert latex(factorial2(-k)) == r"\left(- k\right)!!"

    assert latex(binomial(2,k)) == r"{\binom{2}{k}}"

    assert latex(FallingFactorial(3,k)) == r"{\left(3\right)}_{\left(k\right)}"
    assert latex(RisingFactorial(3,k)) == r"{\left(3\right)}^{\left(k\right)}"

    assert latex(floor(x)) == r"\lfloor{x}\rfloor"
    assert latex(ceiling(x)) == r"\lceil{x}\rceil"
    assert latex(Abs(x)) == r"\lvert{x}\rvert"
    assert latex(re(x)) == r"\Re{x}"
    assert latex(re(x+y)) == r"\Re {\left (x + y \right )}"
    assert latex(im(x)) == r"\Im{x}"
    assert latex(conjugate(x)) == r"\overline{x}"
    assert latex(gamma(x)) == r"\Gamma\left(x\right)"
    assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
    assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
    assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'

    assert latex(cot(x)) == r'\cot{\left (x \right )}'
    assert latex(coth(x)) == r'\coth{\left (x \right )}'
    assert latex(re(x)) == r'\Re{x}'
    assert latex(im(x)) == r'\Im{x}'
    assert latex(root(x,y)) == r'x^{\frac{1}{y}}'
    assert latex(arg(x)) == r'\arg{\left (x \right )}'
    assert latex(zeta(x)) == r'\zeta{\left (x \right )}'

def test_roots_cyclotomic():
    assert roots_cyclotomic(cyclotomic_poly(1, x, polys=True)) == [1]
    assert roots_cyclotomic(cyclotomic_poly(2, x, polys=True)) == [-1]
    assert roots_cyclotomic(cyclotomic_poly(
        3, x, polys=True)) == [-S(1)/2 - I*sqrt(3)/2, -S(1)/2 + I*sqrt(3)/2]
    assert roots_cyclotomic(cyclotomic_poly(4, x, polys=True)) == [-I, I]
    assert roots_cyclotomic(cyclotomic_poly(
        6, x, polys=True)) == [S(1)/2 - I*sqrt(3)/2, S(1)/2 + I*sqrt(3)/2]

    assert roots_cyclotomic(cyclotomic_poly(7, x, polys=True)) == [
        -cos(pi/7) - I*sin(pi/7),
        -cos(pi/7) + I*sin(pi/7),
        -cos(3*pi/7) - I*sin(3*pi/7),
        -cos(3*pi/7) + I*sin(3*pi/7),
        cos(2*pi/7) - I*sin(2*pi/7),
        cos(2*pi/7) + I*sin(2*pi/7),
    ]

    assert roots_cyclotomic(cyclotomic_poly(8, x, polys=True)) == [
        -sqrt(2)/2 - I*sqrt(2)/2,
        -sqrt(2)/2 + I*sqrt(2)/2,
        sqrt(2)/2 - I*sqrt(2)/2,
        sqrt(2)/2 + I*sqrt(2)/2,
    ]

    assert roots_cyclotomic(cyclotomic_poly(12, x, polys=True)) == [
        -sqrt(3)/2 - I/2,
        -sqrt(3)/2 + I/2,
        sqrt(3)/2 - I/2,
        sqrt(3)/2 + I/2,
    ]

    assert roots_cyclotomic(
        cyclotomic_poly(1, x, polys=True), factor=True) == [1]
    assert roots_cyclotomic(
        cyclotomic_poly(2, x, polys=True), factor=True) == [-1]

    assert roots_cyclotomic(cyclotomic_poly(3, x, polys=True), factor=True) == \
        [-root(-1, 3), -1 + root(-1, 3)]
    assert roots_cyclotomic(cyclotomic_poly(4, x, polys=True), factor=True) == \
        [-I, I]
    assert roots_cyclotomic(cyclotomic_poly(5, x, polys=True), factor=True) == \
        [-root(-1, 5), -root(-1, 5)**3, root(-1, 5)**2, -1 - root(-1, 5)**2 + root(-1, 5) + root(-1, 5)**3]

    assert roots_cyclotomic(cyclotomic_poly(6, x, polys=True), factor=True) == \
        [1 - root(-1, 3), root(-1, 3)]

def test_sqrtdenest2():
    assert sqrtdenest(sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))) == \
            r5 + sqrt(11 - 2*r29)
    e = sqrt(-r5 + sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16))
    assert sqrtdenest(e) == root(-2*r29 + 11, 4)
    r = sqrt(1 + r7)
    assert sqrtdenest(sqrt(1 + r)) == sqrt(1 + r)
    e = sqrt(((1 + sqrt(1 + 2*sqrt(3 + r2 + r5)))**2).expand())
    assert sqrtdenest(e) == 1 + sqrt(1 + 2*sqrt(r2 + r5 + 3))
    assert sqrtdenest(sqrt(5*r3 + 6*r2)) == \
        root(3, 4)**3*(r6 + 3)/3

    assert sqrtdenest(sqrt(((1 + r5 + sqrt(1 + r3))**2).expand())) == \
         1 + r5 + sqrt(1 + r3)

    assert sqrtdenest(sqrt(((1 + r5 + r7 + sqrt(1 + r3))**2).expand())) == \
        1 + sqrt(1 + r3) + r5 + r7

    e = sqrt(((1 + cos(2) + cos(3) + sqrt(1 + r3))**2).expand())
    assert sqrtdenest(e) == cos(3) + cos(2) + 1 + sqrt(1 + r3)

    e = sqrt(-2*r10 + 2*r2*sqrt(-2*r10 + 11) + 14)
    assert sqrtdenest(e) == sqrt(-2*r10 - 2*r2 + 4*r5 + 14)

    # check that the result is not more complicated than the input
    z= sqrt(-2*r29 + cos(2) + 2*sqrt(-10*r29 + 55) + 16)
    assert sqrtdenest(z) == z

    assert sqrtdenest(sqrt(r6 + sqrt(15))) == sqrt(r6 + sqrt(15))

    # no assertion error when the 'r's are not the same in _denester
    z = sqrt(15 - 2*sqrt(31) + 2*sqrt(55 - 10*r29))
    assert sqrtdenest(z) == z

    # currently cannot denest this; check one does not get a wrong answer
    z = sqrt(8 - r2*sqrt(5 - r5) - 3*(1 + r5))
    assert (sqrtdenest(z) - z).evalf() < 1.0e-100

def test_powsimp():
    x, y, z, n = symbols('x,y,z,n')
    f = Function('f')
    assert powsimp( 4**x * 2**(-x) * 2**(-x) ) == 1
    assert powsimp( (-4)**x * (-2)**(-x) * 2**(-x) ) == 1

    assert powsimp(
        f(4**x * 2**(-x) * 2**(-x)) ) == f(4**x * 2**(-x) * 2**(-x))
    assert powsimp( f(4**x * 2**(-x) * 2**(-x)), deep=True ) == f(1)
    assert exp(x)*exp(y) == exp(x)*exp(y)
    assert powsimp(exp(x)*exp(y)) == exp(x + y)
    assert powsimp(exp(x)*exp(y)*2**x*2**y) == (2*E)**(x + y)
    assert powsimp(exp(x)*exp(y)*2**x*2**y, combine='exp') == \
        exp(x + y)*2**(x + y)
    assert powsimp(exp(x)*exp(y)*exp(2)*sin(x) + sin(y) + 2**x*2**y) == \
        exp(2 + x + y)*sin(x) + sin(y) + 2**(x + y)
    assert powsimp(sin(exp(x)*exp(y))) == sin(exp(x)*exp(y))
    assert powsimp(sin(exp(x)*exp(y)), deep=True) == sin(exp(x + y))
    assert powsimp(x**2*x**y) == x**(2 + y)
    # This should remain factored, because 'exp' with deep=True is supposed
    # to act like old automatic exponent combining.
    assert powsimp((1 + E*exp(E))*exp(-E), combine='exp', deep=True) == \
        (1 + exp(1 + E))*exp(-E)
    assert powsimp((1 + E*exp(E))*exp(-E), deep=True) == \
        (1 + exp(1 + E))*exp(-E)
    assert powsimp((1 + E*exp(E))*exp(-E)) == (1 + exp(1 + E))*exp(-E)
    assert powsimp((1 + E*exp(E))*exp(-E), combine='exp') == \
        (1 + exp(1 + E))*exp(-E)
    assert powsimp((1 + E*exp(E))*exp(-E), combine='base') == \
        (1 + E*exp(E))*exp(-E)
    x, y = symbols('x,y', nonnegative=True)
    n = Symbol('n', real=True)
    assert powsimp(y**n * (y/x)**(-n)) == x**n
    assert powsimp(x**(x**(x*y)*y**(x*y))*y**(x**(x*y)*y**(x*y)), deep=True) \
        == (x*y)**(x*y)**(x*y)
    assert powsimp(2**(2**(2*x)*x), deep=False) == 2**(2**(2*x)*x)
    assert powsimp(2**(2**(2*x)*x), deep=True) == 2**(x*4**x)
    assert powsimp(
        exp(-x + exp(-x)*exp(-x*log(x))), deep=False, combine='exp') == \
        exp(-x + exp(-x)*exp(-x*log(x)))
    assert powsimp(
        exp(-x + exp(-x)*exp(-x*log(x))), deep=False, combine='exp') == \
        exp(-x + exp(-x)*exp(-x*log(x)))
    assert powsimp((x + y)/(3*z), deep=False, combine='exp') == (x + y)/(3*z)
    assert powsimp((x/3 + y/3)/z, deep=True, combine='exp') == (x/3 + y/3)/z
    assert powsimp(exp(x)/(1 + exp(x)*exp(y)), deep=True) == \
        exp(x)/(1 + exp(x + y))
    assert powsimp(x*y**(z**x*z**y), deep=True) == x*y**(z**(x + y))
    assert powsimp((z**x*z**y)**x, deep=True) == (z**(x + y))**x
    assert powsimp(x*(z**x*z**y)**x, deep=True) == x*(z**(x + y))**x
    p = symbols('p', positive=True)
    assert powsimp((1/x)**log(2)/x) == (1/x)**(1 + log(2))
    assert powsimp((1/p)**log(2)/p) == p**(-1 - log(2))

    # coefficient of exponent can only be simplified for positive bases
    assert powsimp(2**(2*x)) == 4**x
    assert powsimp((-1)**(2*x)) == (-1)**(2*x)
    i = symbols('i', integer=True)
    assert powsimp((-1)**(2*i)) == 1
    assert powsimp((-1)**(-x)) != (-1)**x  # could be 1/((-1)**x), but is not
    # force=True overrides assumptions
    assert powsimp((-1)**(2*x), force=True) == 1

    # rational exponents allow combining of negative terms
    w, n, m = symbols('w n m', negative=True)
    e = i/a  # not a rational exponent if `a` is unknown
    ex = w**e*n**e*m**e
    assert powsimp(ex) == m**(i/a)*n**(i/a)*w**(i/a)
    e = i/3
    ex = w**e*n**e*m**e
    assert powsimp(ex) == (-1)**i*(-m*n*w)**(i/3)
    e = (3 + i)/i
    ex = w**e*n**e*m**e
    assert powsimp(ex) == (-1)**(3*e)*(-m*n*w)**e

    eq = x**(2*a/3)
    # eq != (x**a)**(2/3) (try x = -1 and a = 3 to see)
    assert powsimp(eq).exp == eq.exp == 2*a/3
    # powdenest goes the other direction
    assert powsimp(2**(2*x)) == 4**x

    assert powsimp(exp(p/2)) == exp(p/2)

    # issue 6368
    eq = Mul(*[sqrt(Dummy(imaginary=True)) for i in range(3)])
    assert powsimp(eq) == eq and eq.is_Mul

    assert all(powsimp(e) == e for e in (sqrt(x**a), sqrt(x**2)))

    # issue 8836
    assert str( powsimp(exp(I*pi/3)*root(-1,3)) ) == '(-1)**(2/3)'

def test_slow_general_univariate():
    r = rootof(x**5 - x**2 + 1, 0)
    assert solve(sqrt(x) + 1/root(x, 3) > 1) == \
        Or(And(S(0) < x, x < r**6), And(r**6 < x, x < oo))

#.........这里部分代码省略.........
    assert radsimp(1/(r2*a + r2*b + r3 + r7)) == (
        (sqrt(2)*a*(a + b)**2 - 5*sqrt(2)*a + sqrt(42)*a + sqrt(2)*b*(a +
        b)**2 - 5*sqrt(2)*b + sqrt(42)*b - sqrt(7)*(a + b)**2 - sqrt(3)*(a +
        b)**2 - 2*sqrt(3) + 2*sqrt(7))/(2*a**4 + 8*a**3*b + 12*a**2*b**2 -
        20*a**2 + 8*a*b**3 - 40*a*b + 2*b**4 - 20*b**2 + 8))
    assert radsimp(1/(r2*a + r2*b + r2*c + r2*d)) == \
        sqrt(2)/(2*a + 2*b + 2*c + 2*d)
    assert radsimp(1/(1 + r2*a + r2*b + r2*c + r2*d)) == (
        (sqrt(2)*a + sqrt(2)*b + sqrt(2)*c + sqrt(2)*d - 1)/(2*a**2 + 4*a*b +
        4*a*c + 4*a*d + 2*b**2 + 4*b*c + 4*b*d + 2*c**2 + 4*c*d + 2*d**2 - 1))
    assert radsimp((y**2 - x)/(y - sqrt(x))) == \
        sqrt(x) + y
    assert radsimp(-(y**2 - x)/(y - sqrt(x))) == \
        -(sqrt(x) + y)
    assert radsimp(1/(1 - I + a*I)) == \
        (-I*a + 1 + I)/(a**2 - 2*a + 2)
    assert radsimp(1/((-x + y)*(x - sqrt(y)))) == \
        (-x - sqrt(y))/((x - y)*(x**2 - y))
    e = (3 + 3*sqrt(2))*x*(3*x - 3*sqrt(y))
    assert radsimp(e) == x*(3 + 3*sqrt(2))*(3*x - 3*sqrt(y))
    assert radsimp(1/e) == (
        (-9*x + 9*sqrt(2)*x - 9*sqrt(y) + 9*sqrt(2)*sqrt(y))/(9*x*(9*x**2 -
        9*y)))
    assert radsimp(1 + 1/(1 + sqrt(3))) == \
        Mul(S.Half, -1 + sqrt(3), evaluate=False) + 1
    A = symbols("A", commutative=False)
    assert radsimp(x**2 + sqrt(2)*x**2 - sqrt(2)*x*A) == \
        x**2 + sqrt(2)*x**2 - sqrt(2)*x*A
    assert radsimp(1/sqrt(5 + 2 * sqrt(6))) == -sqrt(2) + sqrt(3)
    assert radsimp(1/sqrt(5 + 2 * sqrt(6))**3) == -(-sqrt(3) + sqrt(2))**3

    # issue 6532
    assert fraction(radsimp(1/sqrt(x))) == (sqrt(x), x)
    assert fraction(radsimp(1/sqrt(2*x + 3))) == (sqrt(2*x + 3), 2*x + 3)
    assert fraction(radsimp(1/sqrt(2*(x + 3)))) == (sqrt(2*x + 6), 2*x + 6)

    # issue 5994
    e = S('-(2 + 2*sqrt(2) + 4*2**(1/4))/'
        '(1 + 2**(3/4) + 3*2**(1/4) + 3*sqrt(2))')
    assert radsimp(e).expand() == -2*2**(S(3)/4) - 2*2**(S(1)/4) + 2 + 2*sqrt(2)

    # issue 5986 (modifications to radimp didn't initially recognize this so
    # the test is included here)
    assert radsimp(1/(-sqrt(5)/2 - S(1)/2 + (-sqrt(5)/2 - S(1)/2)**2)) == 1

    # from issue 5934
    eq = (
        (-240*sqrt(2)*sqrt(sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) -
        360*sqrt(2)*sqrt(-8*sqrt(5) + 40)*sqrt(-sqrt(5) + 5) -
        120*sqrt(10)*sqrt(-8*sqrt(5) + 40)*sqrt(-sqrt(5) + 5) +
        120*sqrt(2)*sqrt(-sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) +
        120*sqrt(2)*sqrt(-8*sqrt(5) + 40)*sqrt(sqrt(5) + 5) +
        120*sqrt(10)*sqrt(-sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) +
        120*sqrt(10)*sqrt(-8*sqrt(5) + 40)*sqrt(sqrt(5) + 5))/(-36000 -
        7200*sqrt(5) + (12*sqrt(10)*sqrt(sqrt(5) + 5) +
        24*sqrt(10)*sqrt(-sqrt(5) + 5))**2))
    assert radsimp(eq) is S.NaN  # it's 0/0

    # work with normal form
    e = 1/sqrt(sqrt(7)/7 + 2*sqrt(2) + 3*sqrt(3) + 5*sqrt(5)) + 3
    assert radsimp(e) == (
        -sqrt(sqrt(7) + 14*sqrt(2) + 21*sqrt(3) +
        35*sqrt(5))*(-11654899*sqrt(35) - 1577436*sqrt(210) - 1278438*sqrt(15)
        - 1346996*sqrt(10) + 1635060*sqrt(6) + 5709765 + 7539830*sqrt(14) +
        8291415*sqrt(21))/1300423175 + 3)

    # obey power rules
    base = sqrt(3) - sqrt(2)
    assert radsimp(1/base**3) == (sqrt(3) + sqrt(2))**3
    assert radsimp(1/(-base)**3) == -(sqrt(2) + sqrt(3))**3
    assert radsimp(1/(-base)**x) == (-base)**(-x)
    assert radsimp(1/base**x) == (sqrt(2) + sqrt(3))**x
    assert radsimp(root(1/(-1 - sqrt(2)), -x)) == (-1)**(-1/x)*(1 + sqrt(2))**(1/x)

    # recurse
    e = cos(1/(1 + sqrt(2)))
    assert radsimp(e) == cos(-sqrt(2) + 1)
    assert radsimp(e/2) == cos(-sqrt(2) + 1)/2
    assert radsimp(1/e) == 1/cos(-sqrt(2) + 1)
    assert radsimp(2/e) == 2/cos(-sqrt(2) + 1)
    assert fraction(radsimp(e/sqrt(x))) == (sqrt(x)*cos(-sqrt(2)+1), x)

    # test that symbolic denominators are not processed
    r = 1 + sqrt(2)
    assert radsimp(x/r, symbolic=False) == -x*(-sqrt(2) + 1)
    assert radsimp(x/(y + r), symbolic=False) == x/(y + 1 + sqrt(2))
    assert radsimp(x/(y + r)/r, symbolic=False) == \
        -x*(-sqrt(2) + 1)/(y + 1 + sqrt(2))

    # issue 7408
    eq = sqrt(x)/sqrt(y)
    assert radsimp(eq) == umul(sqrt(x), sqrt(y), 1/y)
    assert radsimp(eq, symbolic=False) == eq

    # issue 7498
    assert radsimp(sqrt(x)/sqrt(y)**3) == umul(sqrt(x), sqrt(y**3), 1/y**3)

    # for coverage
    eq = sqrt(x)/y**2
    assert radsimp(eq) == eq

def test_latex_functions():
    assert latex(exp(x)) == "e^{x}"
    assert latex(exp(1) + exp(2)) == "e + e^{2}"

    f = Function("f")
    assert latex(f(x)) == "\\operatorname{f}{\\left (x \\right )}"

    beta = Function("beta")

    assert latex(beta(x)) == r"\beta{\left (x \right )}"
    assert latex(sin(x)) == r"\sin{\left (x \right )}"
    assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
    assert latex(sin(2 * x ** 2), fold_func_brackets=True) == r"\sin {2 x^{2}}"
    assert latex(sin(x ** 2), fold_func_brackets=True) == r"\sin {x^{2}}"

    assert latex(asin(x) ** 2) == r"\operatorname{asin}^{2}{\left (x \right )}"
    assert latex(asin(x) ** 2, inv_trig_style="full") == r"\arcsin^{2}{\left (x \right )}"
    assert latex(asin(x) ** 2, inv_trig_style="power") == r"\sin^{-1}{\left (x \right )}^{2}"
    assert latex(asin(x ** 2), inv_trig_style="power", fold_func_brackets=True) == r"\sin^{-1} {x^{2}}"

    assert latex(factorial(k)) == r"k!"
    assert latex(factorial(-k)) == r"\left(- k\right)!"

    assert latex(subfactorial(k)) == r"!k"
    assert latex(subfactorial(-k)) == r"!\left(- k\right)"

    assert latex(factorial2(k)) == r"k!!"
    assert latex(factorial2(-k)) == r"\left(- k\right)!!"

    assert latex(binomial(2, k)) == r"{\binom{2}{k}}"

    assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{\left(k\right)}"
    assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"

    assert latex(floor(x)) == r"\lfloor{x}\rfloor"
    assert latex(ceiling(x)) == r"\lceil{x}\rceil"
    assert latex(Min(x, 2, x ** 3)) == r"\min\left(2, x, x^{3}\right)"
    assert latex(Min(x, y) ** 2) == r"\min\left(x, y\right)^{2}"
    assert latex(Max(x, 2, x ** 3)) == r"\max\left(2, x, x^{3}\right)"
    assert latex(Max(x, y) ** 2) == r"\max\left(x, y\right)^{2}"
    assert latex(Abs(x)) == r"\lvert{x}\rvert"
    assert latex(re(x)) == r"\Re{x}"
    assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
    assert latex(im(x)) == r"\Im{x}"
    assert latex(conjugate(x)) == r"\overline{x}"
    assert latex(gamma(x)) == r"\Gamma\left(x\right)"
    assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
    assert latex(lowergamma(x, y)) == r"\gamma\left(x, y\right)"
    assert latex(uppergamma(x, y)) == r"\Gamma\left(x, y\right)"

    assert latex(cot(x)) == r"\cot{\left (x \right )}"
    assert latex(coth(x)) == r"\coth{\left (x \right )}"
    assert latex(re(x)) == r"\Re{x}"
    assert latex(im(x)) == r"\Im{x}"
    assert latex(root(x, y)) == r"x^{\frac{1}{y}}"
    assert latex(arg(x)) == r"\arg{\left (x \right )}"
    assert latex(zeta(x)) == r"\zeta\left(x\right)"

    assert latex(zeta(x)) == r"\zeta\left(x\right)"
    assert latex(zeta(x) ** 2) == r"\zeta^{2}\left(x\right)"
    assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
    assert latex(zeta(x, y) ** 2) == r"\zeta^{2}\left(x, y\right)"
    assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
    assert latex(dirichlet_eta(x) ** 2) == r"\eta^{2}\left(x\right)"
    assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
    assert latex(polylog(x, y) ** 2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
    assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
    assert latex(lerchphi(x, y, n) ** 2) == r"\Phi^{2}\left(x, y, n\right)"

    assert latex(Ei(x)) == r"\operatorname{Ei}{\left (x \right )}"
    assert latex(Ei(x) ** 2) == r"\operatorname{Ei}^{2}{\left (x \right )}"
    assert latex(expint(x, y) ** 2) == r"\operatorname{E}_{x}^{2}\left(y\right)"
    assert latex(Shi(x) ** 2) == r"\operatorname{Shi}^{2}{\left (x \right )}"
    assert latex(Si(x) ** 2) == r"\operatorname{Si}^{2}{\left (x \right )}"
    assert latex(Ci(x) ** 2) == r"\operatorname{Ci}^{2}{\left (x \right )}"
    assert latex(Chi(x) ** 2) == r"\operatorname{Chi}^{2}{\left (x \right )}"

    assert latex(jacobi(n, a, b, x)) == r"P_{n}^{\left(a,b\right)}\left(x\right)"
    assert latex(jacobi(n, a, b, x) ** 2) == r"\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}"
    assert latex(gegenbauer(n, a, x)) == r"C_{n}^{\left(a\right)}\left(x\right)"
    assert latex(gegenbauer(n, a, x) ** 2) == r"\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}"
    assert latex(chebyshevt(n, x)) == r"T_{n}\left(x\right)"
    assert latex(chebyshevt(n, x) ** 2) == r"\left(T_{n}\left(x\right)\right)^{2}"
    assert latex(chebyshevu(n, x)) == r"U_{n}\left(x\right)"
    assert latex(chebyshevu(n, x) ** 2) == r"\left(U_{n}\left(x\right)\right)^{2}"
    assert latex(legendre(n, x)) == r"P_{n}\left(x\right)"
    assert latex(legendre(n, x) ** 2) == r"\left(P_{n}\left(x\right)\right)^{2}"
    assert latex(assoc_legendre(n, a, x)) == r"P_{n}^{\left(a\right)}\left(x\right)"
    assert latex(assoc_legendre(n, a, x) ** 2) == r"\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}"
    assert latex(laguerre(n, x)) == r"L_{n}\left(x\right)"
    assert latex(laguerre(n, x) ** 2) == r"\left(L_{n}\left(x\right)\right)^{2}"
    assert latex(assoc_laguerre(n, a, x)) == r"L_{n}^{\left(a\right)}\left(x\right)"
    assert latex(assoc_laguerre(n, a, x) ** 2) == r"\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}"
    assert latex(hermite(n, x)) == r"H_{n}\left(x\right)"
    assert latex(hermite(n, x) ** 2) == r"\left(H_{n}\left(x\right)\right)^{2}"

    # Test latex printing of function names with "_"
    assert latex(polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
    assert latex(polar_lift(0) ** 3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"

def convert_func(func):
    if func.func_normal():
        if func.L_PAREN():  # function called with parenthesis
            arg = convert_func_arg(func.func_arg())
        else:
            arg = convert_func_arg(func.func_arg_noparens())

        name = func.func_normal().start.text[1:]

        # change arc<trig> -> a<trig>
        if name in [
                "arcsin", "arccos", "arctan", "arccsc", "arcsec", "arccot"
        ]:
            name = "a" + name[3:]
            expr = getattr(sympy.functions, name)(arg, evaluate=False)
        if name in ["arsinh", "arcosh", "artanh"]:
            name = "a" + name[2:]
            expr = getattr(sympy.functions, name)(arg, evaluate=False)

        if (name == "log" or name == "ln"):
            if func.subexpr():
                base = convert_expr(func.subexpr().expr())
            elif name == "log":
                base = 10
            elif name == "ln":
                base = sympy.E
            expr = sympy.log(arg, base, evaluate=False)

        func_pow = None
        should_pow = True
        if func.supexpr():
            if func.supexpr().expr():
                func_pow = convert_expr(func.supexpr().expr())
            else:
                func_pow = convert_atom(func.supexpr().atom())

        if name in [
                "sin", "cos", "tan", "csc", "sec", "cot", "sinh", "cosh",
                "tanh"
        ]:
            if func_pow == -1:
                name = "a" + name
                should_pow = False
            expr = getattr(sympy.functions, name)(arg, evaluate=False)

        if func_pow and should_pow:
            expr = sympy.Pow(expr, func_pow, evaluate=False)

        return expr
    elif func.LETTER() or func.SYMBOL():
        if func.LETTER():
            fname = func.LETTER().getText()
        elif func.SYMBOL():
            fname = func.SYMBOL().getText()[1:]
        fname = str(fname)  # can't be unicode
        if func.subexpr():
            subscript = None
            if func.subexpr().expr():  # subscript is expr
                subscript = convert_expr(func.subexpr().expr())
            else:  # subscript is atom
                subscript = convert_atom(func.subexpr().atom())
            subscriptName = StrPrinter().doprint(subscript)
            fname += '_{' + subscriptName + '}'
        input_args = func.args()
        output_args = []
        while input_args.args():  # handle multiple arguments to function
            output_args.append(convert_expr(input_args.expr()))
            input_args = input_args.args()
        output_args.append(convert_expr(input_args.expr()))
        return sympy.Function(fname)(*output_args)
    elif func.FUNC_INT():
        return handle_integral(func)
    elif func.FUNC_SQRT():
        expr = convert_expr(func.base)
        if func.root:
            r = convert_expr(func.root)
            return sympy.root(expr, r)
        else:
            return sympy.sqrt(expr)
    elif func.FUNC_SUM():
        return handle_sum_or_prod(func, "summation")
    elif func.FUNC_PROD():
        return handle_sum_or_prod(func, "product")
    elif func.FUNC_LIM():
        return handle_limit(func)

def test_AlgebraicNumber():
    a = AlgebraicNumber(sqrt(2))
    sT(a, "AlgebraicNumber(Pow(Integer(2), Rational(1, 2)), [Integer(1), Integer(0)])")
    a = AlgebraicNumber(root(-2, 3))
    sT(a, "AlgebraicNumber(Pow(Integer(-2), Rational(1, 3)), [Integer(1), Integer(0)])")

def test_issue_3109_fail():
    from sympy import root, Rational
    I = S.ImaginaryUnit
    assert sqrt(exp(5*I)) == -exp(5*I/2)
    assert root(exp(5*I), 3).exp == Rational(1, 3)

def root4(x):
    return root(x, 4)

 ("x_a", Symbol('x_{a}')),
 ("x_{b}", Symbol('x_{b}')),
 ("h_\\theta", Symbol('h_{theta}')),
 ("h_{\\theta}", Symbol('h_{theta}')),
 ("h_{\\theta}(x_0, x_1)",
  Symbol('h_{theta}')(Symbol('x_{0}'), Symbol('x_{1}'))),
 ("x!", _factorial(x)),
 ("100!", _factorial(100)),
 ("\\theta!", _factorial(theta)),
 ("(x + 1)!", _factorial(_Add(x, 1))),
 ("(x!)!", _factorial(_factorial(x))),
 ("x!!!", _factorial(_factorial(_factorial(x)))),
 ("5!7!", _Mul(_factorial(5), _factorial(7))),
 ("\\sqrt{x}", sqrt(x)),
 ("\\sqrt{x + b}", sqrt(_Add(x, b))),
 ("\\sqrt[3]{\\sin x}", root(sin(x), 3)),
 ("\\sqrt[y]{\\sin x}", root(sin(x), y)),
 ("\\sqrt[\\theta]{\\sin x}", root(sin(x), theta)),
 ("x < y", StrictLessThan(x, y)),
 ("x \\leq y", LessThan(x, y)),
 ("x > y", StrictGreaterThan(x, y)),
 ("x \\geq y", GreaterThan(x, y)),
 ("\\mathit{x}", Symbol('x')),
 ("\\mathit{test}", Symbol('test')),
 ("\\mathit{TEST}", Symbol('TEST')),
 ("\\mathit{HELLO world}", Symbol('HELLO world')),
 ("\\sum_{k = 1}^{3} c", Sum(c, (k, 1, 3))),
 ("\\sum_{k = 1}^3 c", Sum(c, (k, 1, 3))),
 ("\\sum^{3}_{k = 1} c", Sum(c, (k, 1, 3))),
 ("\\sum^3_{k = 1} c", Sum(c, (k, 1, 3))),
 ("\\sum_{k = 1}^{10} k^2", Sum(k**2, (k, 1, 10))),

    def expr(self, it):
        name = self.name

        if name == "frac":
            it[0] = expr(self.upper) / expr(self.lower)
        elif name == "over":
            it.zip(lambda l,r: expr(l) / expr(r))
        elif name == "times":
            it.zip(lambda l,r: expr(l) * expr(r))

        elif name == "sin":
            it[0] = sympy.sin( expr(self.token) )
        elif name == "cos":
            it[0] = sympy.cos( expr(self.token) )
        elif name == "tan":
            it[0] = sympy.tan( expr(self.token) )
        elif name == "choose":
            it.zip(lambda l,r: sympy.binomial(expr(l), expr(r)))
        elif name == "sqrt":
            it[0] = sympy.root( expr(self.token), expr(self.n) )
        elif name == "pi":
            it[0] = sympy.pi
        elif name == "int":
            # TODO: This needs to work more like the parenthesis transform
            i = it.next()
            instart = True
            lower = None
            upper = None

            # TODO: Try to find spacing first, then fall back to mathrm
            # Pattern: [(int), (limits)?, (^|_){0-2}, (.+), (<spacing>)?, (mathrm of d or del), (.)
            while type(i[0]) != tex_commands["mathrm"] or str(i[0].token) != 'd': # TODO: Do better letter checking
                if instart:
                    t = i[0]
                    if type(t) == tex_commands["limits"]:
                        del i[0]
                    elif isinstance(t, TexSpecial) and t.data == "^":
                        upper = expr(t.arg)
                        del i[0]
                    elif isinstance(t, TexSpecial) and t.data == "_":
                        lower = expr(t.arg)
                        del i[0]
                    else: