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from sympy import (
    sqrt, root, Symbol, sqrtdenest, Integral, cos, Rational, I, Integer)
from sympy.simplify.sqrtdenest import (
    _subsets as subsets, _sqrt_numeric_denest)
from sympy.testing.pytest import slow

r2, r3, r5, r6, r7, r10, r15, r29 = [sqrt(x) for x in [2, 3, 5, 6, 7, 10,
                                          15, 29]]


def test_sqrtdenest():
    d = {sqrt(5 + 2 * r6): r2 + r3,
        sqrt(5. + 2 * r6): sqrt(5. + 2 * r6),
        sqrt(5. + 4*sqrt(5 + 2 * r6)): sqrt(5.0 + 4*r2 + 4*r3),
        sqrt(r2): sqrt(r2),
        sqrt(5 + r7): sqrt(5 + r7),
        sqrt(3 + sqrt(5 + 2*r7)):
         3*r2*(5 + 2*r7)**Rational(1, 4)/(2*sqrt(6 + 3*r7)) +
         r2*sqrt(6 + 3*r7)/(2*(5 + 2*r7)**Rational(1, 4)),
        sqrt(3 + 2*r3): 3**Rational(3, 4)*(r6/2 + 3*r2/2)/3}
    for i in d:
        assert sqrtdenest(i) == d[i], i


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_sqrtdenest_rec():
    assert sqrtdenest(sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 33)) == \
        -r2 + r3 + 2*r7
    assert sqrtdenest(sqrt(-28*r7 - 14*r5 + 4*sqrt(35) + 82)) == \
        -7 + r5 + 2*r7
    assert sqrtdenest(sqrt(6*r2/11 + 2*sqrt(22)/11 + 6*sqrt(11)/11 + 2)) == \
        sqrt(11)*(r2 + 3 + sqrt(11))/11
    assert sqrtdenest(sqrt(468*r3 + 3024*r2 + 2912*r6 + 19735)) == \
        9*r3 + 26 + 56*r6
    z = sqrt(-490*r3 - 98*sqrt(115) - 98*sqrt(345) - 2107)
    assert sqrtdenest(z) == sqrt(-1)*(7*r5 + 7*r15 + 7*sqrt(23))
    z = sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 34)
    assert sqrtdenest(z) == z
    assert sqrtdenest(sqrt(-8*r2 - 2*r5 + 18)) == -r10 + 1 + r2 + r5
    assert sqrtdenest(sqrt(8*r2 + 2*r5 - 18)) == \
        sqrt(-1)*(-r10 + 1 + r2 + r5)
    assert sqrtdenest(sqrt(8*r2/3 + 14*r5/3 + Rational(154, 9))) == \
        -r10/3 + r2 + r5 + 3
    assert sqrtdenest(sqrt(sqrt(2*r6 + 5) + sqrt(2*r7 + 8))) == \
        sqrt(1 + r2 + r3 + r7)
    assert sqrtdenest(sqrt(4*r15 + 8*r5 + 12*r3 + 24)) == 1 + r3 + r5 + r15

    w = 1 + r2 + r3 + r5 + r7
    assert sqrtdenest(sqrt((w**2).expand())) == w
    z = sqrt((w**2).expand() + 1)
    assert sqrtdenest(z) == z

    z = sqrt(2*r10 + 6*r2 + 4*r5 + 12 + 10*r15 + 30*r3)
    assert sqrtdenest(z) == z


def test_issue_6241():
    z = sqrt( -320 + 32*sqrt(5) + 64*r15)
    assert sqrtdenest(z) == z


def test_sqrtdenest3():
    z = sqrt(13 - 2*r10 + 2*r2*sqrt(-2*r10 + 11))
    assert sqrtdenest(z) == -1 + r2 + r10
    assert sqrtdenest(z, max_iter=1) == -1 + sqrt(2) + sqrt(10)
    z = sqrt(sqrt(r2 + 2) + 2)
    assert sqrtdenest(z) == z
    assert sqrtdenest(sqrt(-2*r10 + 4*r2*sqrt(-2*r10 + 11) + 20)) == \
        sqrt(-2*r10 - 4*r2 + 8*r5 + 20)
    assert sqrtdenest(sqrt((112 + 70*r2) + (46 + 34*r2)*r5)) == \
        r10 + 5 + 4*r2 + 3*r5
    z = sqrt(5 + sqrt(2*r6 + 5)*sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16))
    r = sqrt(-2*r29 + 11)
    assert sqrtdenest(z) == sqrt(r2*r + r3*r + r10 + r15 + 5)


@slow
def test_sqrtdenest3_slow():
    # Slow because of the equals, not the sqrtdenest
    # Using == does not work as 7*(sqrt(-2*r29 + 11) + r5) is expanded
    # automatically
    n = sqrt(2*r6/7 + 2*r7/7 + 2*sqrt(42)/7 + 2)
    d = sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))
    assert sqrtdenest(n/d).equals(
        r7*(1 + r6 + r7)/(7*(sqrt(-2*r29 + 11) + r5)))

def test_sqrtdenest4():
    # see Denest_en.pdf in https://github.com/sympy/sympy/issues/3192
    z = sqrt(8 - r2*sqrt(5 - r5) - sqrt(3)*(1 + r5))
    z1 = sqrtdenest(z)
    c = sqrt(-r5 + 5)
    z1 = ((-r15*c - r3*c + c + r5*c - r6 - r2 + r10 + sqrt(30))/4).expand()
    assert sqrtdenest(z) == z1

    z = sqrt(2*r2*sqrt(r2 + 2) + 5*r2 + 4*sqrt(r2 + 2) + 8)
    assert sqrtdenest(z) == r2 + sqrt(r2 + 2) + 2

    w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3)
    z = sqrt((w**2).expand())
    assert sqrtdenest(z) == w.expand()


def test_sqrt_symbolic_denest():
    x = Symbol('x')
    z = sqrt(((1 + sqrt(sqrt(2 + x) + 3))**2).expand())
    assert sqrtdenest(z) == sqrt((1 + sqrt(sqrt(2 + x) + 3))**2)
    z = sqrt(((1 + sqrt(sqrt(2 + cos(1)) + 3))**2).expand())
    assert sqrtdenest(z) == 1 + sqrt(sqrt(2 + cos(1)) + 3)
    z = ((1 + cos(2))**4 + 1).expand()
    assert sqrtdenest(z) == z
    z = sqrt(((1 + sqrt(sqrt(2 + cos(3*x)) + 3))**2 + 1).expand())
    assert sqrtdenest(z) == z
    c = cos(3)
    c2 = c**2
    assert sqrtdenest(sqrt(2*sqrt(1 + r3)*c + c2 + 1 + r3*c2)) == \
        -1 - sqrt(1 + r3)*c
    ra = sqrt(1 + r3)
    z = sqrt(20*ra*sqrt(3 + 3*r3) + 12*r3*ra*sqrt(3 + 3*r3) + 64*r3 + 112)
    assert sqrtdenest(z) == z


def test_issue_5857():
    from sympy.abc import x, y
    z = sqrt(1/(4*r3 + 7) + 1)
    ans = (r2 + r6)/(r3 + 2)
    assert sqrtdenest(z) == ans
    assert sqrtdenest(1 + z) == 1 + ans
    assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == \
        Integral(1 + ans, (x, 1, 2))
    assert sqrtdenest(x + sqrt(y)) == x + sqrt(y)
    ans = (r2 + r6)/(r3 + 2)
    assert sqrtdenest(z) == ans
    assert sqrtdenest(1 + z) == 1 + ans
    assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == \
        Integral(1 + ans, (x, 1, 2))
    assert sqrtdenest(x + sqrt(y)) == x + sqrt(y)


def test_subsets():
    assert subsets(1) == [[1]]
    assert subsets(4) == [
        [1, 0, 0, 0], [0, 1, 0, 0], [1, 1, 0, 0], [0, 0, 1, 0], [1, 0, 1, 0],
        [0, 1, 1, 0], [1, 1, 1, 0], [0, 0, 0, 1], [1, 0, 0, 1], [0, 1, 0, 1],
        [1, 1, 0, 1], [0, 0, 1, 1], [1, 0, 1, 1], [0, 1, 1, 1], [1, 1, 1, 1]]


def test_issue_5653():
    assert sqrtdenest(
        sqrt(2 + sqrt(2 + sqrt(2)))) == sqrt(2 + sqrt(2 + sqrt(2)))

def test_issue_12420():
    assert sqrtdenest((3 - sqrt(2)*sqrt(4 + 3*I) + 3*I)/2) == I
    e = 3 - sqrt(2)*sqrt(4 + I) + 3*I
    assert sqrtdenest(e) == e

def test_sqrt_ratcomb():
    assert sqrtdenest(sqrt(1 + r3) + sqrt(3 + 3*r3) - sqrt(10 + 6*r3)) == 0

def test_issue_18041():
    e = -sqrt(-2 + 2*sqrt(3)*I)
    assert sqrtdenest(e) == -1 - sqrt(3)*I

def test_issue_19914():
    a = Integer(-8)
    b = Integer(-1)
    r = Integer(63)
    d2 = a*a - b*b*r

    assert _sqrt_numeric_denest(a, b, r, d2) == \
        sqrt(14)*I/2 + 3*sqrt(2)*I/2
    assert sqrtdenest(sqrt(-8-sqrt(63))) == sqrt(14)*I/2 + 3*sqrt(2)*I/2

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