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