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kronos-venv/lib/python3.12/site-packages/sympy/simplify/tests/test_sqrtdenest.py

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+from sympy.core.mul import Mul
+from sympy.core.numbers import (I, Integer, Rational)
+from sympy.core.symbol import Symbol
+from sympy.functions.elementary.miscellaneous import (root, sqrt)
+from sympy.functions.elementary.trigonometric import cos
+from sympy.integrals.integrals import Integral
+from sympy.simplify.sqrtdenest import sqrtdenest
+from sympy.simplify.sqrtdenest import (
+    _subsets as subsets, _sqrt_numeric_denest)
+
+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)
+
+    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) == r7*(1 + r6 + r7)/(Mul(7, (sqrt(-2*r29 + 11) + r5),
+                                                    evaluate=False))
+
+
+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