def test_posify(): assert sstr( posify(x + Symbol('p', positive=True) + Symbol('n', negative=True))) == '(n + p + _x, {_x: x})' eq, rep = posify(1 / x) assert log(eq).expand().subs(rep) == -log(x) assert sstr(posify([x, 1 + x])) == '([_x, _x + 1], {_x: x})' p = symbols('p', positive=True) n = symbols('n', negative=True) orig = [x, n, p] modified, reps = posify(orig) assert sstr(modified) == '[_x, n, p]' assert [w.subs(reps) for w in modified] == orig assert sstr(Integral(posify(1/x + y)[0], (y, 1, 3)).expand()) == \ 'Integral(1/_x, (y, 1, 3)) + Integral(_y, (y, 1, 3))' assert sstr(Sum(posify(1/x**n)[0], (n, 1, 3)).expand()) == \ 'Sum(_x**(-n), (n, 1, 3))'
def test_unrad1(): pytest.raises(NotImplementedError, lambda: unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3)) pytest.raises(NotImplementedError, lambda: unrad(sqrt(x) + cbrt(x + 1) + 2*sqrt(y))) s = symbols('s', cls=Dummy) # checkers to deal with possibility of answer coming # back with a sign change (cf issue sympy/sympy#5203) def check(rv, ans): assert bool(rv[1]) == bool(ans[1]) if ans[1]: return s_check(rv, ans) e = rv[0].expand() a = ans[0].expand() return e in [a, -a] and rv[1] == ans[1] def s_check(rv, ans): # get the dummy rv = list(rv) d = rv[0].atoms(Dummy) reps = list(zip(d, [s]*len(d))) # replace s with this dummy rv = (rv[0].subs(reps).expand(), [rv[1][0].subs(reps), rv[1][1].subs(reps)]) ans = (ans[0].subs(reps).expand(), [ans[1][0].subs(reps), ans[1][1].subs(reps)]) return str(rv[0]) in [str(ans[0]), str(-ans[0])] and \ str(rv[1]) == str(ans[1]) assert check(unrad(sqrt(x)), (x, [])) assert check(unrad(sqrt(x) + 1), (x - 1, [])) assert check(unrad(sqrt(x) + root(x, 3) + 2), (s**3 + s**2 + 2, [s, s**6 - x])) assert check(unrad(sqrt(x)*root(x, 3) + 2), (x**5 - 64, [])) assert check(unrad(sqrt(x) + cbrt(x + 1)), (x**3 - (x + 1)**2, [])) assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(2*x)), (-2*sqrt(2)*x - 2*x + 1, [])) assert check(unrad(sqrt(x) + sqrt(x + 1) + 2), (16*x - 9, [])) assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - x)), (5*x**2 - 4*x, [])) assert check(unrad(a*sqrt(x) + b*sqrt(x) + c*sqrt(y) + d*sqrt(y)), ((a*sqrt(x) + b*sqrt(x))**2 - (c*sqrt(y) + d*sqrt(y))**2, [])) assert check(unrad(sqrt(x) + sqrt(1 - x)), (2*x - 1, [])) assert check(unrad(sqrt(x) + sqrt(1 - x) - 3), (x**2 - x + 16, [])) assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)), (5*x**2 - 2*x + 1, [])) assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3) in [ (25*x**4 + 376*x**3 + 1256*x**2 - 2272*x + 784, []), (25*x**8 - 476*x**6 + 2534*x**4 - 1468*x**2 + 169, [])] assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - sqrt(1 - 2*x)) == \ (41*x**4 + 40*x**3 + 232*x**2 - 160*x + 16, []) # orig root at 0.487 assert check(unrad(sqrt(x) + sqrt(x + 1)), (Integer(1), [])) eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) assert check(unrad(eq), (16*x**2 - 9*x, [])) assert {s[x] for s in solve(eq, check=False)} == {0, Rational(9, 16)} assert solve(eq) == [] # but this one really does have those solutions assert ({s[x] for s in solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x)))} == {0, Rational(9, 16)}) assert check(unrad(sqrt(x) + root(x + 1, 3) + 2*sqrt(y), y), (2*sqrt(x)*cbrt(x + 1) + x - 4*y + (x + 1)**Rational(2, 3), [])) assert check(unrad(sqrt(x/(1 - x)) + cbrt(x + 1)), (x**5 - x**4 - x**3 + 2*x**2 + x - 1, [])) assert check(unrad(sqrt(x/(1 - x)) + 2*sqrt(y), y), (4*x*y + x - 4*y, [])) assert check(unrad(sqrt(x)*sqrt(1 - x) + 2, x), (x**2 - x + 4, [])) # http://tutorial.math.lamar.edu/ # Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a assert solve(Eq(x, sqrt(x + 6))) == [{x: 3}] assert solve(Eq(x + sqrt(x - 4), 4)) == [{x: 4}] assert solve(Eq(1, x + sqrt(2*x - 3))) == [] assert {s[x] for s in solve(Eq(sqrt(5*x + 6) - 2, x))} == {-1, 2} assert {s[x] for s in solve(Eq(sqrt(2*x - 1) - sqrt(x - 4), 2))} == {5, 13} assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [{x: -6}] # http://www.purplemath.com/modules/solverad.htm assert solve(cbrt(2*x - 5) - 3) == [{x: 16}] assert {s[x] for s in solve(x + 1 - root(x**4 + 4*x**3 - x, 4))} == {-Rational(1, 2), -Rational(1, 3)} assert {s[x] for s in solve(sqrt(2*x**2 - 7) - (3 - x))} == {-8, 2} assert solve(sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [{x: 0}] assert solve(sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)) == [{x: 5}] assert solve(sqrt(x)*sqrt(x - 7) - 12) == [{x: 16}] assert solve(sqrt(x - 3) + sqrt(x) - 3) == [{x: 4}] assert solve(sqrt(9*x**2 + 4) - (3*x + 2)) == [{x: 0}] assert solve(sqrt(x) - 2 - 5) == [{x: 49}] assert solve(sqrt(x - 3) - sqrt(x) - 3) == [] assert solve(sqrt(x - 1) - x + 7) == [{x: 10}] assert solve(sqrt(x - 2) - 5) == [{x: 27}] assert solve(sqrt(17*x - sqrt(x**2 - 5)) - 7) == [{x: 3}] assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == [] # don't posify the expression in unrad and do use _mexpand z = sqrt(2*x + 1)/sqrt(x) - sqrt(2 + 1/x) p = posify(z)[0] assert solve(p) == [] assert solve(z) == [] assert solve(z + 6*I) == [{x: -Rational(1, 11)}] assert solve(p + 6*I) == [] # issue sympy/sympy#8622 assert unrad((root(x + 1, 5) - root(x, 3))) == ( x**5 - x**3 - 3*x**2 - 3*x - 1, []) # issue sympy/sympy#8679 assert check(unrad(x + root(x, 3) + root(x, 3)**2 + sqrt(y), x), (s**3 + s**2 + s + sqrt(y), [s, s**3 - x])) # for coverage assert check(unrad(sqrt(x) + root(x, 3) + y), (s**3 + s**2 + y, [s, s**6 - x])) assert solve(sqrt(x) + root(x, 3) - 2) == [{x: 1}] pytest.raises(NotImplementedError, lambda: solve(sqrt(x) + root(x, 3) + root(x + 1, 5) - 2)) # fails through a different code path pytest.raises(NotImplementedError, lambda: solve(-sqrt(2) + cosh(x)/x)) # unrad some e = root(x + 1, 3) + root(x, 3) assert unrad(e) == (2*x + 1, []) eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5) assert check(unrad(eq), (15625*x**4 + 173000*x**3 + 355600*x**2 - 817920*x + 331776, [])) assert check(unrad(root(x, 4) + root(x, 4)**3 - 1), (s**3 + s - 1, [s, s**4 - x])) assert check(unrad(root(x, 2) + root(x, 2)**3 - 1), (x**3 + 2*x**2 + x - 1, [])) assert unrad(x**0.5) is None assert check(unrad(t + root(x + y, 5) + root(x + y, 5)**3), (s**3 + s + t, [s, s**5 - x - y])) assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, y), (s**3 + s + x, [s, s**5 - x - y])) assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, x), (s**5 + s**3 + s - y, [s, s**5 - x - y])) assert check(unrad(root(x - 1, 3) + root(x + 1, 5) + root(2, 5)), (s**5 + 5*root(2, 5)*s**4 + s**3 + 10*2**Rational(2, 5)*s**3 + 10*2**Rational(3, 5)*s**2 + 5*2**Rational(4, 5)*s + 4, [s, s**3 - x + 1])) pytest.raises(NotImplementedError, lambda: unrad((root(x, 2) + root(x, 3) + root(x, 4)).subs({x: x**5 - x + 1}))) # the simplify flag should be reset to False for unrad results; # if it's not then this next test will take a long time assert solve(root(x, 3) + root(x, 5) - 2) == [{x: 1}] eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5) assert check(unrad(eq), ((5*x - 4)*(3125*x**3 + 37100*x**2 + 100800*x - 82944), [])) ans = [{x: Rational(4, 5)}, {x: Rational(-1484, 375) + 172564/(140625*cbrt(114*sqrt(12657)/78125 + Rational(12459439, 52734375))) + 4*cbrt(114*sqrt(12657)/78125 + Rational(12459439, 52734375))}] assert solve(eq) == ans # duplicate radical handling assert check(unrad(sqrt(x + root(x + 1, 3)) - root(x + 1, 3) - 2), (s**3 - s**2 - 3*s - 5, [s, s**3 - x - 1])) # cov post-processing e = root(x**2 + 1, 3) - root(x**2 - 1, 5) - 2 assert check(unrad(e), (s**5 - 10*s**4 + 39*s**3 - 80*s**2 + 80*s - 30, [s, s**3 - x**2 - 1])) e = sqrt(x + root(x + 1, 2)) - root(x + 1, 3) - 2 assert check(unrad(e), (s**6 - 2*s**5 - 7*s**4 - 3*s**3 + 26*s**2 + 40*s + 25, [s, s**3 - x - 1])) assert check(unrad(e, _reverse=True), (s**6 - 14*s**5 + 73*s**4 - 187*s**3 + 276*s**2 - 228*s + 89, [s, s**2 - x - sqrt(x + 1)])) # this one needs r0, r1 reversal to work assert check(unrad(sqrt(x + sqrt(root(x, 3) - 1)) - root(x, 6) - 2), (s**12 - 2*s**8 - 8*s**7 - 8*s**6 + s**4 + 8*s**3 + 23*s**2 + 32*s + 17, [s, s**6 - x])) # is this needed? # assert unrad(root(cosh(x), 3)/x*root(x + 1, 5) - 1) == ( # x**15 - x**3*cosh(x)**5 - 3*x**2*cosh(x)**5 - 3*x*cosh(x)**5 - cosh(x)**5, []) pytest.raises(NotImplementedError, lambda: unrad(sqrt(cosh(x)/x) + root(x + 1, 3)*sqrt(x) - 1)) assert unrad((x+y)**(2*y/3) + cbrt(x+y) + 1) is None assert check(unrad((x+y)**(2*y/3) + cbrt(x+y) + 1, x), (s**(2*y) + s + 1, [s, s**3 - x - y])) # This tests two things: that if full unrad is attempted and fails # the solution should still be found; also it tests that the use of # composite assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3 assert len(solve(-512*y**3 + 1344*cbrt(x + 2)*y**2 - 1176*(x + 2)**Rational(2, 3)*y - 169*x + 686, y, _unrad=False)) == 3 # watch out for when the cov doesn't involve the symbol of interest eq = -x + (7*y/8 - cbrt(27*x/2 + 27*sqrt(x**2)/2)/3)**3 - 1 assert solve(eq, y) == [ {y: RootOf(-2304*x + 1029*y**3 - 1764*cbrt(4)*y**2*cbrt(x + sqrt(x**2)) + 2016*cbrt(2)*y*(x + sqrt(x**2))**Rational(2, 3) - 768*sqrt(x**2) - 1536, y, 0, evaluate=False)}, {y: RootOf(-2304*x + 1029*y**3 - 1764*cbrt(4)*y**2*cbrt(x + sqrt(x**2)) + 2016*cbrt(2)*y*(x + sqrt(x**2))**Rational(2, 3) - 768*sqrt(x**2) - 1536, y, 1, evaluate=False)}, {y: RootOf(-2304*x + 1029*y**3 - 1764*cbrt(4)*y**2*cbrt(x + sqrt(x**2)) + 2016*cbrt(2)*y*(x + sqrt(x**2))**Rational(2, 3) - 768*sqrt(x**2) - 1536, y, 2, evaluate=False)}] eq = root(x + 1, 3) - (root(x, 3) + root(x, 5)) assert check(unrad(eq), (3*s**13 + 3*s**11 + s**9 - 1, [s, s**15 - x])) assert check(unrad(eq - 2), (3*s**13 + 3*s**11 + 6*s**10 + s**9 + 12*s**8 + 6*s**6 + 12*s**5 + 12*s**3 + 7, [s, s**15 - x])) assert check(unrad(root(x, 3) - root(x + 1, 4)/2 + root(x + 2, 3)), (4096*s**13 + 960*s**12 + 48*s**11 - s**10 - 1728*s**4, [s, s**4 - x - 1])) # orig expr has two real roots: -1, -.389 assert check(unrad(root(x, 3) + root(x + 1, 4) - root(x + 2, 3)/2), (343*s**13 + 2904*s**12 + 1344*s**11 + 512*s**10 - 1323*s**9 - 3024*s**8 - 1728*s**7 + 1701*s**5 + 216*s**4 - 729*s, [s, s**4 - x - 1])) # orig expr has one real root: -0.048 assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3)), (729*s**13 - 216*s**12 + 1728*s**11 - 512*s**10 + 1701*s**9 - 3024*s**8 + 1344*s**7 + 1323*s**5 - 2904*s**4 + 343*s, [s, s**4 - x - 1])) # orig expr has 2 real roots: -0.91, -0.15 # orig expr has 1 real root: 19.53 assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3) - 2), (729*s**13 + 1242*s**12 + 18496*s**10 + 129701*s**9 + 388602*s**8 + 453312*s**7 - 612864*s**6 - 3337173*s**5 - 6332418*s**4 - 7134912*s**3 - 5064768*s**2 - 2111913*s - 398034, [s, s**4 - x - 1])) ans = solve(sqrt(x) + sqrt(x + 1) - sqrt(1 - x) - sqrt(2 + x)) assert len(ans) == 1 and NS(ans[0][x])[:4] == '0.73' # the fence optimization problem # https://github.com/sympy/sympy/issues/4793#issuecomment-36994519 eq = F - (2*x + 2*y + sqrt(x**2 + y**2)) ans = 2*F/7 - sqrt(2)*F/14 X = solve(eq, x, check=False) for xi in reversed(X): # reverse since currently, ans is the 2nd one Y = solve((x*y).subs(xi).diff(y), y, simplify=False, check=False) if any((a[y] - ans).expand().is_zero for a in Y): break else: assert None # no answer was found assert (solve(sqrt(x + 1) + root(x, 3) - 2) == [{x: (-11/(9*cbrt(Rational(47, 54) + sqrt(93)/6)) + Rational(1, 3) + cbrt(Rational(47, 54) + sqrt(93)/6))**3}]) assert (solve(sqrt(sqrt(x + 1)) + cbrt(x) - 2) == [{x: (-sqrt(-2*cbrt(Rational(-1, 16) + sqrt(6913)/16) + 6/cbrt(Rational(-1, 16) + sqrt(6913)/16) + Rational(17, 2) + 121/(4*sqrt(-6/cbrt(Rational(-1, 16) + sqrt(6913)/16) + 2*cbrt(Rational(-1, 16) + sqrt(6913)/16) + Rational(17, 4))))/2 + sqrt(-6/cbrt(Rational(-1, 16) + sqrt(6913)/16) + 2*cbrt(Rational(-1, 16) + sqrt(6913)/16) + Rational(17, 4))/2 + Rational(9, 4))**3}]) assert (solve(sqrt(x) + root(sqrt(x) + 1, 3) - 2) == [{x: (-cbrt(Rational(81, 2) + 3*sqrt(741)/2)/3 + (Rational(81, 2) + 3*sqrt(741)/2)**Rational(-1, 3) + 2)**2}]) eq = (-x + (Rational(1, 2) - sqrt(3)*I/2)*cbrt(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3 + x*(3*x**2 - 34) + 90)**2/4 - Rational(39304, 27)) - 45) + 34/(3*(Rational(1, 2) - sqrt(3)*I/2)*cbrt(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3 + x*(3*x**2 - 34) + 90)**2/4 - Rational(39304, 27)) - 45))) assert check(unrad(eq), (s**6 - sqrt(3)*s**6*I + 102*cbrt(12)*s**4 + 102*2**Rational(2, 3)*3**Rational(5, 6)*s**4*I + 1620*s**3 - 1620*sqrt(3)*s**3*I - 13872*cbrt(18)*s**2 + 471648 - 471648*sqrt(3)*I, [s, s**3 - 306*x - sqrt(3)*sqrt(31212*x**2 - 165240*x + 61484) + 810])) assert solve(eq, x, check=False) != [] # not other code errors
def test_unrad1(): pytest.raises(NotImplementedError, lambda: unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3)) pytest.raises(NotImplementedError, lambda: unrad(sqrt(x) + cbrt(x + 1) + 2 * sqrt(y))) s = symbols('s', cls=Dummy) # checkers to deal with possibility of answer coming # back with a sign change (cf issue sympy/sympy#5203) def check(rv, ans): assert bool(rv[1]) == bool(ans[1]) if ans[1]: return s_check(rv, ans) e = rv[0].expand() a = ans[0].expand() return e in [a, -a] and rv[1] == ans[1] def s_check(rv, ans): # get the dummy rv = list(rv) d = rv[0].atoms(Dummy) reps = list(zip(d, [s] * len(d))) # replace s with this dummy rv = (rv[0].subs(reps).expand(), [rv[1][0].subs(reps), rv[1][1].subs(reps)]) ans = (ans[0].subs(reps).expand(), [ans[1][0].subs(reps), ans[1][1].subs(reps)]) return str(rv[0]) in [str(ans[0]), str(-ans[0])] and \ str(rv[1]) == str(ans[1]) assert check(unrad(sqrt(x)), (x, [])) assert check(unrad(sqrt(x) + 1), (x - 1, [])) assert check(unrad(sqrt(x) + root(x, 3) + 2), (s**3 + s**2 + 2, [s, s**6 - x])) assert check(unrad(sqrt(x) * root(x, 3) + 2), (x**5 - 64, [])) assert check(unrad(sqrt(x) + cbrt(x + 1)), (x**3 - (x + 1)**2, [])) assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(2 * x)), (-2 * sqrt(2) * x - 2 * x + 1, [])) assert check(unrad(sqrt(x) + sqrt(x + 1) + 2), (16 * x - 9, [])) assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - x)), (5 * x**2 - 4 * x, [])) assert check(unrad(a * sqrt(x) + b * sqrt(x) + c * sqrt(y) + d * sqrt(y)), ((a * sqrt(x) + b * sqrt(x))**2 - (c * sqrt(y) + d * sqrt(y))**2, [])) assert check(unrad(sqrt(x) + sqrt(1 - x)), (2 * x - 1, [])) assert check(unrad(sqrt(x) + sqrt(1 - x) - 3), (x**2 - x + 16, [])) assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)), (5 * x**2 - 2 * x + 1, [])) assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3) in [ (25 * x**4 + 376 * x**3 + 1256 * x**2 - 2272 * x + 784, []), (25 * x**8 - 476 * x**6 + 2534 * x**4 - 1468 * x**2 + 169, []) ] assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - sqrt(1 - 2*x)) == \ (41*x**4 + 40*x**3 + 232*x**2 - 160*x + 16, []) # orig root at 0.487 assert check(unrad(sqrt(x) + sqrt(x + 1)), (Integer(1), [])) eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) assert check(unrad(eq), (16 * x**2 - 9 * x, [])) assert {s[x] for s in solve(eq, check=False)} == {0, Rational(9, 16)} assert solve(eq) == [] # but this one really does have those solutions assert ({s[x] for s in solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x))) } == {0, Rational(9, 16)}) assert check(unrad(sqrt(x) + root(x + 1, 3) + 2 * sqrt(y), y), (2 * sqrt(x) * cbrt(x + 1) + x - 4 * y + (x + 1)**Rational(2, 3), [])) assert check(unrad(sqrt(x / (1 - x)) + cbrt(x + 1)), (x**5 - x**4 - x**3 + 2 * x**2 + x - 1, [])) assert check(unrad(sqrt(x / (1 - x)) + 2 * sqrt(y), y), (4 * x * y + x - 4 * y, [])) assert check(unrad(sqrt(x) * sqrt(1 - x) + 2, x), (x**2 - x + 4, [])) # http://tutorial.math.lamar.edu/ # Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a assert solve(Eq(x, sqrt(x + 6))) == [{x: 3}] assert solve(Eq(x + sqrt(x - 4), 4)) == [{x: 4}] assert solve(Eq(1, x + sqrt(2 * x - 3))) == [] assert {s[x] for s in solve(Eq(sqrt(5 * x + 6) - 2, x))} == {-1, 2} assert {s[x] for s in solve(Eq(sqrt(2 * x - 1) - sqrt(x - 4), 2))} == {5, 13} assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [{x: -6}] # http://www.purplemath.com/modules/solverad.htm assert solve(cbrt(2 * x - 5) - 3) == [{x: 16}] assert {s[x] for s in solve(x + 1 - root(x**4 + 4 * x**3 - x, 4)) } == {-Rational(1, 2), -Rational(1, 3)} assert {s[x] for s in solve(sqrt(2 * x**2 - 7) - (3 - x))} == {-8, 2} assert solve(sqrt(2 * x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [{x: 0}] assert solve(sqrt(x + 4) + sqrt(2 * x - 1) - 3 * sqrt(x - 1)) == [{x: 5}] assert solve(sqrt(x) * sqrt(x - 7) - 12) == [{x: 16}] assert solve(sqrt(x - 3) + sqrt(x) - 3) == [{x: 4}] assert solve(sqrt(9 * x**2 + 4) - (3 * x + 2)) == [{x: 0}] assert solve(sqrt(x) - 2 - 5) == [{x: 49}] assert solve(sqrt(x - 3) - sqrt(x) - 3) == [] assert solve(sqrt(x - 1) - x + 7) == [{x: 10}] assert solve(sqrt(x - 2) - 5) == [{x: 27}] assert solve(sqrt(17 * x - sqrt(x**2 - 5)) - 7) == [{x: 3}] assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == [] # don't posify the expression in unrad and do use _mexpand z = sqrt(2 * x + 1) / sqrt(x) - sqrt(2 + 1 / x) p = posify(z)[0] assert solve(p) == [] assert solve(z) == [] assert solve(z + 6 * I) == [{x: -Rational(1, 11)}] assert solve(p + 6 * I) == [] # issue sympy/sympy#8622 assert unrad( (root(x + 1, 5) - root(x, 3))) == (x**5 - x**3 - 3 * x**2 - 3 * x - 1, []) # issue sympy/sympy#8679 assert check(unrad(x + root(x, 3) + root(x, 3)**2 + sqrt(y), x), (s**3 + s**2 + s + sqrt(y), [s, s**3 - x])) # for coverage assert check(unrad(sqrt(x) + root(x, 3) + y), (s**3 + s**2 + y, [s, s**6 - x])) assert solve(sqrt(x) + root(x, 3) - 2) == [{x: 1}] pytest.raises(NotImplementedError, lambda: solve(sqrt(x) + root(x, 3) + root(x + 1, 5) - 2)) # fails through a different code path pytest.raises(NotImplementedError, lambda: solve(-sqrt(2) + cosh(x) / x)) # unrad some e = root(x + 1, 3) + root(x, 3) assert unrad(e) == (2 * x + 1, []) eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6 * sqrt(5) / 5) assert check(unrad(eq), (15625 * x**4 + 173000 * x**3 + 355600 * x**2 - 817920 * x + 331776, [])) assert check(unrad(root(x, 4) + root(x, 4)**3 - 1), (s**3 + s - 1, [s, s**4 - x])) assert check(unrad(root(x, 2) + root(x, 2)**3 - 1), (x**3 + 2 * x**2 + x - 1, [])) assert unrad(x**0.5) is None assert check(unrad(t + root(x + y, 5) + root(x + y, 5)**3), (s**3 + s + t, [s, s**5 - x - y])) assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, y), (s**3 + s + x, [s, s**5 - x - y])) assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, x), (s**5 + s**3 + s - y, [s, s**5 - x - y])) assert check( unrad(root(x - 1, 3) + root(x + 1, 5) + root(2, 5)), (s**5 + 5 * root(2, 5) * s**4 + s**3 + 10 * 2**Rational(2, 5) * s**3 + 10 * 2**Rational(3, 5) * s**2 + 5 * 2**Rational(4, 5) * s + 4, [s, s**3 - x + 1])) pytest.raises( NotImplementedError, lambda: unrad( (root(x, 2) + root(x, 3) + root(x, 4)).subs({x: x**5 - x + 1}))) # the simplify flag should be reset to False for unrad results; # if it's not then this next test will take a long time assert solve(root(x, 3) + root(x, 5) - 2) == [{x: 1}] eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6 * sqrt(5) / 5) assert check(unrad(eq), ((5 * x - 4) * (3125 * x**3 + 37100 * x**2 + 100800 * x - 82944), [])) ans = [{ x: Rational(4, 5) }, { x: Rational(-1484, 375) + 172564 / (140625 * cbrt(114 * sqrt(12657) / 78125 + Rational(12459439, 52734375))) + 4 * cbrt(114 * sqrt(12657) / 78125 + Rational(12459439, 52734375)) }] assert solve(eq) == ans # duplicate radical handling assert check(unrad(sqrt(x + root(x + 1, 3)) - root(x + 1, 3) - 2), (s**3 - s**2 - 3 * s - 5, [s, s**3 - x - 1])) # cov post-processing e = root(x**2 + 1, 3) - root(x**2 - 1, 5) - 2 assert check(unrad(e), (s**5 - 10 * s**4 + 39 * s**3 - 80 * s**2 + 80 * s - 30, [s, s**3 - x**2 - 1])) e = sqrt(x + root(x + 1, 2)) - root(x + 1, 3) - 2 assert check(unrad(e), (s**6 - 2 * s**5 - 7 * s**4 - 3 * s**3 + 26 * s**2 + 40 * s + 25, [s, s**3 - x - 1])) assert check(unrad(e, _reverse=True), (s**6 - 14 * s**5 + 73 * s**4 - 187 * s**3 + 276 * s**2 - 228 * s + 89, [s, s**2 - x - sqrt(x + 1)])) # this one needs r0, r1 reversal to work assert check(unrad(sqrt(x + sqrt(root(x, 3) - 1)) - root(x, 6) - 2), (s**12 - 2 * s**8 - 8 * s**7 - 8 * s**6 + s**4 + 8 * s**3 + 23 * s**2 + 32 * s + 17, [s, s**6 - x])) # is this needed? # assert unrad(root(cosh(x), 3)/x*root(x + 1, 5) - 1) == ( # x**15 - x**3*cosh(x)**5 - 3*x**2*cosh(x)**5 - 3*x*cosh(x)**5 - cosh(x)**5, []) pytest.raises( NotImplementedError, lambda: unrad(sqrt(cosh(x) / x) + root(x + 1, 3) * sqrt(x) - 1)) assert unrad((x + y)**(2 * y / 3) + cbrt(x + y) + 1) is None assert check(unrad((x + y)**(2 * y / 3) + cbrt(x + y) + 1, x), (s**(2 * y) + s + 1, [s, s**3 - x - y])) # This tests two things: that if full unrad is attempted and fails # the solution should still be found; also it tests that the use of # composite assert len(solve(sqrt(y) * x + x**3 - 1, x)) == 3 assert len( solve(-512 * y**3 + 1344 * cbrt(x + 2) * y**2 - 1176 * (x + 2)**Rational(2, 3) * y - 169 * x + 686, y, _unrad=False)) == 3 # watch out for when the cov doesn't involve the symbol of interest eq = -x + (7 * y / 8 - cbrt(27 * x / 2 + 27 * sqrt(x**2) / 2) / 3)**3 - 1 assert solve(eq, y) == [{ y: RootOf( -768 * x + 343 * y**3 - 588 * cbrt(4) * y**2 * cbrt(x + sqrt(x**2)) + 672 * cbrt(2) * y * cbrt(x + sqrt(x**2))**2 - 256 * sqrt(x**2) - 512, y, 0) }, { y: RootOf( -768 * x + 343 * y**3 - 588 * cbrt(4) * y**2 * cbrt(x + sqrt(x**2)) + 672 * cbrt(2) * y * cbrt(x + sqrt(x**2))**2 - 256 * sqrt(x**2) - 512, y, 1) }, { y: RootOf( -768 * x + 343 * y**3 - 588 * cbrt(4) * y**2 * cbrt(x + sqrt(x**2)) + 672 * cbrt(2) * y * cbrt(x + sqrt(x**2))**2 - 256 * sqrt(x**2) - 512, y, 2) }] eq = root(x + 1, 3) - (root(x, 3) + root(x, 5)) assert check(unrad(eq), (3 * s**13 + 3 * s**11 + s**9 - 1, [s, s**15 - x])) assert check(unrad(eq - 2), (3 * s**13 + 3 * s**11 + 6 * s**10 + s**9 + 12 * s**8 + 6 * s**6 + 12 * s**5 + 12 * s**3 + 7, [s, s**15 - x])) assert check( unrad(root(x, 3) - root(x + 1, 4) / 2 + root(x + 2, 3)), (4096 * s**13 + 960 * s**12 + 48 * s**11 - s**10 - 1728 * s**4, [s, s**4 - x - 1])) # orig expr has two real roots: -1, -.389 assert check( unrad(root(x, 3) + root(x + 1, 4) - root(x + 2, 3) / 2), (343 * s**13 + 2904 * s**12 + 1344 * s**11 + 512 * s**10 - 1323 * s**9 - 3024 * s**8 - 1728 * s**7 + 1701 * s**5 + 216 * s**4 - 729 * s, [s, s**4 - x - 1])) # orig expr has one real root: -0.048 assert check( unrad(root(x, 3) / 2 - root(x + 1, 4) + root(x + 2, 3)), (729 * s**13 - 216 * s**12 + 1728 * s**11 - 512 * s**10 + 1701 * s**9 - 3024 * s**8 + 1344 * s**7 + 1323 * s**5 - 2904 * s**4 + 343 * s, [s, s**4 - x - 1])) # orig expr has 2 real roots: -0.91, -0.15 # orig expr has 1 real root: 19.53 assert check(unrad(root(x, 3) / 2 - root(x + 1, 4) + root(x + 2, 3) - 2), (729 * s**13 + 1242 * s**12 + 18496 * s**10 + 129701 * s**9 + 388602 * s**8 + 453312 * s**7 - 612864 * s**6 - 3337173 * s**5 - 6332418 * s**4 - 7134912 * s**3 - 5064768 * s**2 - 2111913 * s - 398034, [s, s**4 - x - 1])) ans = solve(sqrt(x) + sqrt(x + 1) - sqrt(1 - x) - sqrt(2 + x)) assert len(ans) == 1 and NS(ans[0][x])[:4] == '0.73' # the fence optimization problem # https://github.com/sympy/sympy/issues/4793#issuecomment-36994519 eq = F - (2 * x + 2 * y + sqrt(x**2 + y**2)) ans = 2 * F / 7 - sqrt(2) * F / 14 X = solve(eq, x, check=False) for xi in reversed(X): # reverse since currently, ans is the 2nd one Y = solve((x * y).subs(xi).diff(y), y, simplify=False, check=False) if any((a[y] - ans).expand().is_zero for a in Y): break else: assert None # no answer was found assert (solve(sqrt(x + 1) + root(x, 3) - 2) == [{ x: (-11 / (9 * cbrt(Rational(47, 54) + sqrt(93) / 6)) + Rational(1, 3) + cbrt(Rational(47, 54) + sqrt(93) / 6))**3 }]) assert (solve(sqrt(sqrt(x + 1)) + cbrt(x) - 2) == [{ x: (-sqrt(-2 * cbrt(Rational(-1, 16) + sqrt(6913) / 16) + 6 / cbrt(Rational(-1, 16) + sqrt(6913) / 16) + Rational(17, 2) + 121 / (4 * sqrt(-6 / cbrt(Rational(-1, 16) + sqrt(6913) / 16) + 2 * cbrt(Rational(-1, 16) + sqrt(6913) / 16) + Rational(17, 4)))) / 2 + sqrt(-6 / cbrt(Rational(-1, 16) + sqrt(6913) / 16) + 2 * cbrt(Rational(-1, 16) + sqrt(6913) / 16) + Rational(17, 4)) / 2 + Rational(9, 4))**3 }]) assert (solve(sqrt(x) + root(sqrt(x) + 1, 3) - 2) == [{ x: (-cbrt(Rational(81, 2) + 3 * sqrt(741) / 2) / 3 + (Rational(81, 2) + 3 * sqrt(741) / 2)**Rational(-1, 3) + 2)**2 }]) eq = (-x + (Rational(1, 2) - sqrt(3) * I / 2) * cbrt(3 * x**3 / 2 - x * (3 * x**2 - 34) / 2 + sqrt( (-3 * x**3 + x * (3 * x**2 - 34) + 90)**2 / 4 - Rational(39304, 27)) - 45) + 34 / (3 * (Rational(1, 2) - sqrt(3) * I / 2) * cbrt(3 * x**3 / 2 - x * (3 * x**2 - 34) / 2 + sqrt( (-3 * x**3 + x * (3 * x**2 - 34) + 90)**2 / 4 - Rational(39304, 27)) - 45))) assert check(unrad(eq), (s**6 - sqrt(3) * s**6 * I + 102 * cbrt(12) * s**4 + 102 * 2**Rational(2, 3) * 3**Rational(5, 6) * s**4 * I + 1620 * s**3 - 1620 * sqrt(3) * s**3 * I - 13872 * cbrt(18) * s**2 + 471648 - 471648 * sqrt(3) * I, [ s, s**3 - 306 * x - sqrt(3) * sqrt(31212 * x**2 - 165240 * x + 61484) + 810 ])) assert solve(eq, x, check=False) != [] # not other code errors
def test_posify(): assert posify(A)[0].is_commutative is False for q in (A * B / A, (A * B / A)**2, (A * B)**2, A * B - B * A): p = posify(q) assert p[0].subs(p[1]) == q
def test_posify(): assert posify(A)[0].is_commutative is False for q in (A*B/A, (A*B/A)**2, (A*B)**2, A*B - B*A): p = posify(q) assert p[0].subs(p[1]) == q