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_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_sympyissue_6249():
A = MatrixSymbol('A', 3, 3)
B = MatrixSymbol('B', 3, 3)
p = A * B - B * A
assert cancel(p) == p
assert combsimp(p) == p
assert factor(p) == p
assert separatevars(p) == p
assert sqrtdenest(p) == p
M = MatrixSymbol('M', 2, 1)
assert simplify(M[0] / 2) == M[0] / 2
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*root(5 + 2*r7, 4)/(2*sqrt(6 + 3*r7)) +
r2*sqrt(6 + 3*r7)/(2*root(5 + 2*r7, 4)),
sqrt(3 + 2*r3): 3**Rational(3, 4)*(r6/2 + 3*r2/2)/3}
for i in d:
assert sqrtdenest(i) == d[i]
def test_sympyissue_5857():
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_sympyissue_5857():
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_issue_5857():
from diofant.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_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)
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)))
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)
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)
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)))
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)
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 * root(5 + 2 * r7, 4) / (2 * sqrt(6 + 3 * r7)) +
r2 * sqrt(6 + 3 * r7) / (2 * root(5 + 2 * r7, 4)),
sqrt(3 + 2 * r3):
3**Rational(3, 4) * (r6 / 2 + 3 * r2 / 2) / 3
}
for i in d:
assert sqrtdenest(i) == d[i]
def test_sqrt_symbolic_denest():
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_sqrt_symbolic_denest():
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_sympyissue_5653():
assert sqrtdenest(
sqrt(2 + sqrt(2 + sqrt(2)))) == sqrt(2 + sqrt(2 + sqrt(2)))
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_sympyissue_5653():
assert sqrtdenest(sqrt(2 + sqrt(2 + sqrt(2)))) == sqrt(2 +
sqrt(2 + sqrt(2)))
def test_sympyissue_6241():
z = sqrt(-320 + 32 * sqrt(5) + 64 * r15)
assert sqrtdenest(z) == z
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(27, 4)
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_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(27, 4)
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_sympyissue_6241():
z = sqrt( -320 + 32*sqrt(5) + 64*r15)
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
