MLPY/Lib/site-packages/sympy/simplify/tests/test_powsimp.py

from sympy.core.function import Function
from sympy.core.mul import Mul
from sympy.core.numbers import (E, I, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import (root, sqrt)
from sympy.functions.elementary.trigonometric import sin
from sympy.functions.special.gamma_functions import gamma
from sympy.functions.special.hyper import hyper
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.simplify.powsimp import (powdenest, powsimp)
from sympy.simplify.simplify import (signsimp, simplify)
from sympy.core.symbol import Str

from sympy.abc import x, y, z, a, b


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**(a*Rational(2, 3))
    # eq != (x**a)**(2/3) (try x = -1 and a = 3 to see)
    assert powsimp(eq).exp == eq.exp == a*Rational(2, 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)'

    # issue 9183
    assert powsimp(-0.1**x) == -0.1**x

    # issue 10095
    assert powsimp((1/(2*E))**oo) == (exp(-1)/2)**oo

    # PR 13131
    eq = sin(2*x)**2*sin(2.0*x)**2
    assert powsimp(eq) == eq

    # issue 14615
    assert powsimp(x**2*y**3*(x*y**2)**Rational(3, 2)
        ) == x*y*(x*y**2)**Rational(5, 2)


def test_powsimp_negated_base():
    assert powsimp((-x + y)/sqrt(x - y)) == -sqrt(x - y)
    assert powsimp((-x + y)*(-z + y)/sqrt(x - y)/sqrt(z - y)) == sqrt(x - y)*sqrt(z - y)
    p = symbols('p', positive=True)
    reps = {p: 2, a: S.Half}
    assert powsimp((-p)**a/p**a).subs(reps) == ((-1)**a).subs(reps)
    assert powsimp((-p)**a*p**a).subs(reps) == ((-p**2)**a).subs(reps)
    n = symbols('n', negative=True)
    reps = {p: -2, a: S.Half}
    assert powsimp((-n)**a/n**a).subs(reps) == (-1)**(-a).subs(a, S.Half)
    assert powsimp((-n)**a*n**a).subs(reps) == ((-n**2)**a).subs(reps)
    # if x is 0 then the lhs is 0**a*oo**a which is not (-1)**a
    eq = (-x)**a/x**a
    assert powsimp(eq) == eq


def test_powsimp_nc():
    x, y, z = symbols('x,y,z')
    A, B, C = symbols('A B C', commutative=False)

    assert powsimp(A**x*A**y, combine='all') == A**(x + y)
    assert powsimp(A**x*A**y, combine='base') == A**x*A**y
    assert powsimp(A**x*A**y, combine='exp') == A**(x + y)

    assert powsimp(A**x*B**x, combine='all') == A**x*B**x
    assert powsimp(A**x*B**x, combine='base') == A**x*B**x
    assert powsimp(A**x*B**x, combine='exp') == A**x*B**x

    assert powsimp(B**x*A**x, combine='all') == B**x*A**x
    assert powsimp(B**x*A**x, combine='base') == B**x*A**x
    assert powsimp(B**x*A**x, combine='exp') == B**x*A**x

    assert powsimp(A**x*A**y*A**z, combine='all') == A**(x + y + z)
    assert powsimp(A**x*A**y*A**z, combine='base') == A**x*A**y*A**z
    assert powsimp(A**x*A**y*A**z, combine='exp') == A**(x + y + z)

    assert powsimp(A**x*B**x*C**x, combine='all') == A**x*B**x*C**x
    assert powsimp(A**x*B**x*C**x, combine='base') == A**x*B**x*C**x
    assert powsimp(A**x*B**x*C**x, combine='exp') == A**x*B**x*C**x

    assert powsimp(B**x*A**x*C**x, combine='all') == B**x*A**x*C**x
    assert powsimp(B**x*A**x*C**x, combine='base') == B**x*A**x*C**x
    assert powsimp(B**x*A**x*C**x, combine='exp') == B**x*A**x*C**x


def test_issue_6440():
    assert powsimp(16*2**a*8**b) == 2**(a + 3*b + 4)


def test_powdenest():
    x, y = symbols('x,y')
    p, q = symbols('p q', positive=True)
    i, j = symbols('i,j', integer=True)

    assert powdenest(x) == x
    assert powdenest(x + 2*(x**(a*Rational(2, 3)))**(3*x)) == (x + 2*(x**(a*Rational(2, 3)))**(3*x))
    assert powdenest((exp(a*Rational(2, 3)))**(3*x))  # -X-> (exp(a/3))**(6*x)
    assert powdenest((x**(a*Rational(2, 3)))**(3*x)) == ((x**(a*Rational(2, 3)))**(3*x))
    assert powdenest(exp(3*x*log(2))) == 2**(3*x)
    assert powdenest(sqrt(p**2)) == p
    eq = p**(2*i)*q**(4*i)
    assert powdenest(eq) == (p*q**2)**(2*i)
    # -X-> (x**x)**i*(x**x)**j == x**(x*(i + j))
    assert powdenest((x**x)**(i + j))
    assert powdenest(exp(3*y*log(x))) == x**(3*y)
    assert powdenest(exp(y*(log(a) + log(b)))) == (a*b)**y
    assert powdenest(exp(3*(log(a) + log(b)))) == a**3*b**3
    assert powdenest(((x**(2*i))**(3*y))**x) == ((x**(2*i))**(3*y))**x
    assert powdenest(((x**(2*i))**(3*y))**x, force=True) == x**(6*i*x*y)
    assert powdenest(((x**(a*Rational(2, 3)))**(3*y/i))**x) == \
        (((x**(a*Rational(2, 3)))**(3*y/i))**x)
    assert powdenest((x**(2*i)*y**(4*i))**z, force=True) == (x*y**2)**(2*i*z)
    assert powdenest((p**(2*i)*q**(4*i))**j) == (p*q**2)**(2*i*j)
    e = ((p**(2*a))**(3*y))**x
    assert powdenest(e) == e
    e = ((x**2*y**4)**a)**(x*y)
    assert powdenest(e) == e
    e = (((x**2*y**4)**a)**(x*y))**3
    assert powdenest(e) == ((x**2*y**4)**a)**(3*x*y)
    assert powdenest((((x**2*y**4)**a)**(x*y)), force=True) == \
        (x*y**2)**(2*a*x*y)
    assert powdenest((((x**2*y**4)**a)**(x*y))**3, force=True) == \
        (x*y**2)**(6*a*x*y)
    assert powdenest((x**2*y**6)**i) != (x*y**3)**(2*i)
    x, y = symbols('x,y', positive=True)
    assert powdenest((x**2*y**6)**i) == (x*y**3)**(2*i)

    assert powdenest((x**(i*Rational(2, 3))*y**(i/2))**(2*i)) == (x**Rational(4, 3)*y)**(i**2)
    assert powdenest(sqrt(x**(2*i)*y**(6*i))) == (x*y**3)**i

    assert powdenest(4**x) == 2**(2*x)
    assert powdenest((4**x)**y) == 2**(2*x*y)
    assert powdenest(4**x*y) == 2**(2*x)*y


def test_powdenest_polar():
    x, y, z = symbols('x y z', polar=True)
    a, b, c = symbols('a b c')
    assert powdenest((x*y*z)**a) == x**a*y**a*z**a
    assert powdenest((x**a*y**b)**c) == x**(a*c)*y**(b*c)
    assert powdenest(((x**a)**b*y**c)**c) == x**(a*b*c)*y**(c**2)


def test_issue_5805():
    arg = ((gamma(x)*hyper((), (), x))*pi)**2
    assert powdenest(arg) == (pi*gamma(x)*hyper((), (), x))**2
    assert arg.is_positive is None


def test_issue_9324_powsimp_on_matrix_symbol():
    M = MatrixSymbol('M', 10, 10)
    expr = powsimp(M, deep=True)
    assert expr == M
    assert expr.args[0] == Str('M')


def test_issue_6367():
    z = -5*sqrt(2)/(2*sqrt(2*sqrt(29) + 29)) + sqrt(-sqrt(29)/29 + S.Half)
    assert Mul(*[powsimp(a) for a in Mul.make_args(z.normal())]) == 0
    assert powsimp(z.normal()) == 0
    assert simplify(z) == 0
    assert powsimp(sqrt(2 + sqrt(3))*sqrt(2 - sqrt(3)) + 1) == 2
    assert powsimp(z) != 0


def test_powsimp_polar():
    from sympy.functions.elementary.complexes import polar_lift
    from sympy.functions.elementary.exponential import exp_polar
    x, y, z = symbols('x y z')
    p, q, r = symbols('p q r', polar=True)

    assert (polar_lift(-1))**(2*x) == exp_polar(2*pi*I*x)
    assert powsimp(p**x * q**x) == (p*q)**x
    assert p**x * (1/p)**x == 1
    assert (1/p)**x == p**(-x)

    assert exp_polar(x)*exp_polar(y) == exp_polar(x)*exp_polar(y)
    assert powsimp(exp_polar(x)*exp_polar(y)) == exp_polar(x + y)
    assert powsimp(exp_polar(x)*exp_polar(y)*p**x*p**y) == \
        (p*exp_polar(1))**(x + y)
    assert powsimp(exp_polar(x)*exp_polar(y)*p**x*p**y, combine='exp') == \
        exp_polar(x + y)*p**(x + y)
    assert powsimp(
        exp_polar(x)*exp_polar(y)*exp_polar(2)*sin(x) + sin(y) + p**x*p**y) \
        == p**(x + y) + sin(x)*exp_polar(2 + x + y) + sin(y)
    assert powsimp(sin(exp_polar(x)*exp_polar(y))) == \
        sin(exp_polar(x)*exp_polar(y))
    assert powsimp(sin(exp_polar(x)*exp_polar(y)), deep=True) == \
        sin(exp_polar(x + y))


def test_issue_5728():
    b = x*sqrt(y)
    a = sqrt(b)
    c = sqrt(sqrt(x)*y)
    assert powsimp(a*b) == sqrt(b)**3
    assert powsimp(a*b**2*sqrt(y)) == sqrt(y)*a**5
    assert powsimp(a*x**2*c**3*y) == c**3*a**5
    assert powsimp(a*x*c**3*y**2) == c**7*a
    assert powsimp(x*c**3*y**2) == c**7
    assert powsimp(x*c**3*y) == x*y*c**3
    assert powsimp(sqrt(x)*c**3*y) == c**5
    assert powsimp(sqrt(x)*a**3*sqrt(y)) == sqrt(x)*sqrt(y)*a**3
    assert powsimp(Mul(sqrt(x)*c**3*sqrt(y), y, evaluate=False)) == \
        sqrt(x)*sqrt(y)**3*c**3
    assert powsimp(a**2*a*x**2*y) == a**7

    # symbolic powers work, too
    b = x**y*y
    a = b*sqrt(b)
    assert a.is_Mul is True
    assert powsimp(a) == sqrt(b)**3

    # as does exp
    a = x*exp(y*Rational(2, 3))
    assert powsimp(a*sqrt(a)) == sqrt(a)**3
    assert powsimp(a**2*sqrt(a)) == sqrt(a)**5
    assert powsimp(a**2*sqrt(sqrt(a))) == sqrt(sqrt(a))**9


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)**Rational(1, 3)*
        (-n1)**Rational(1, 3)*(-n2)**Rational(1, 3)*(-n3)**Rational(1, 3)*(-n4)**Rational(1, 3))


def test_issue_10195():
    a = Symbol('a', integer=True)
    l = Symbol('l', even=True, nonzero=True)
    n = Symbol('n', odd=True)
    e_x = (-1)**(n/2 - S.Half) - (-1)**(n*Rational(3, 2) - S.Half)
    assert powsimp((-1)**(l/2)) == I**l
    assert powsimp((-1)**(n/2)) == I**n
    assert powsimp((-1)**(n*Rational(3, 2))) == -I**n
    assert powsimp(e_x) == (-1)**(n/2 - S.Half) + (-1)**(n*Rational(3, 2) +
            S.Half)
    assert powsimp((-1)**(a*Rational(3, 2))) == (-I)**a

def test_issue_15709():
    assert powsimp(3**x*Rational(2, 3)) == 2*3**(x-1)
    assert powsimp(2*3**x/3) == 2*3**(x-1)


def test_issue_11981():
    x, y = symbols('x y', commutative=False)
    assert powsimp((x*y)**2 * (y*x)**2) == (x*y)**2 * (y*x)**2


def test_issue_17524():
    a = symbols("a", real=True)
    e = (-1 - a**2)*sqrt(1 + a**2)
    assert signsimp(powsimp(e)) == signsimp(e) == -(a**2 + 1)**(S(3)/2)


def test_issue_19627():
    # if you use force the user must verify
    assert powdenest(sqrt(sin(x)**2), force=True) == sin(x)
    assert powdenest((x**(S.Half/y))**(2*y), force=True) == x
    from sympy.core.function import expand_power_base
    e = 1 - a
    expr = (exp(z/e)*x**(b/e)*y**((1 - b)/e))**e
    assert powdenest(expand_power_base(expr, force=True), force=True
        ) == x**b*y**(1 - b)*exp(z)


def test_issue_22546():
    p1, p2 = symbols('p1, p2', positive=True)
    ref = powsimp(p1**z/p2**z)
    e = z + 1
    ans = ref.subs(z, e)
    assert ans.is_Pow
    assert powsimp(p1**e/p2**e) == ans
    i = symbols('i', integer=True)
    ref = powsimp(x**i/y**i)
    e = i + 1
    ans = ref.subs(i, e)
    assert ans.is_Pow
    assert powsimp(x**e/y**e) == ans