def _solve_radical(f, symbol, solveset_solver):
""" Helper function to solve equations with radicals """
eq, cov = unrad(f)
if not cov:
result = solveset_solver(eq, symbol) - Union(*[solveset_solver(g, symbol) for g in denoms(f, [symbol])])
else:
y, yeq = cov
if not solveset_solver(y - I, y):
yreal = Dummy("yreal", real=True)
yeq = yeq.xreplace({y: yreal})
eq = eq.xreplace({y: yreal})
y = yreal
g_y_s = solveset_solver(yeq, symbol)
f_y_sols = solveset_solver(eq, y)
result = Union(*[imageset(Lambda(y, g_y), f_y_sols) for g_y in g_y_s])
return FiniteSet(*[s for s in result if checksol(f, symbol, s) is True])
def _solve_radical(f, symbol, solveset_solver):
""" Helper function to solve equations with radicals """
eq, cov = unrad(f)
if not cov:
result = solveset_solver(eq, symbol) - \
Union(*[solveset_solver(g, symbol) for g in denoms(f, [symbol])])
else:
y, yeq = cov
if not solveset_solver(y - I, y):
yreal = Dummy('yreal', real=True)
yeq = yeq.xreplace({y: yreal})
eq = eq.xreplace({y: yreal})
y = yreal
g_y_s = solveset_solver(yeq, symbol)
f_y_sols = solveset_solver(eq, y)
result = Union(*[imageset(Lambda(y, g_y), f_y_sols) for g_y in g_y_s])
return FiniteSet(*[s for s in result if checksol(f, symbol, s) is True])
def _solve_radical(f, symbol, solveset_solver):
""" Helper function to solve equations with radicals """
from sympy.solvers.solvers import unrad
try:
eq, cov, dens = unrad(f)
if cov == []:
result = solveset_solver(eq, symbol) - \
Union(*[solveset_solver(g, symbol) for g in dens])
else:
if len(cov) > 1:
raise NotImplementedError("Multivariate solver is "
"not implemented.")
else:
y = cov[0][0]
g_y_s = solveset_solver(cov[0][1], symbol)
f_y_sols = solveset_solver(eq, y)
result = Union(*[imageset(Lambda(y, g_y), f_y_sols)
for g_y in g_y_s])
return FiniteSet(*[s for s in result if checksol(f, symbol, s) is True])
except ValueError:
raise NotImplementedError
def _solve_radical(f, symbol, solveset_solver):
""" Helper function to solve equations with radicals """
from sympy.solvers.solvers import unrad
try:
eq, cov, dens = unrad(f)
if cov == []:
result = solveset_solver(eq, symbol) - \
Union(*[solveset_solver(g, symbol) for g in dens])
else:
if len(cov) > 1:
raise NotImplementedError("Multivariate solver is "
"not implemented.")
else:
y = cov[0][0]
g_y_s = solveset_solver(cov[0][1], symbol)
f_y_sols = solveset_solver(eq, y)
result = Union(
*[imageset(Lambda(y, g_y), f_y_sols) for g_y in g_y_s])
return FiniteSet(
*[s for s in result if checksol(f, symbol, s) is True])
except ValueError:
raise NotImplementedError
def test_unrad():
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue 2104)
def check(rv, ans):
rv, ans = list(rv), list(ans)
rv[0] = rv[0].expand()
ans[0] = ans[0].expand()
return rv[0] in [ans[0], -ans[0]] and rv[1:] == ans[1:]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = zip(d, [s]*len(d))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [(p[0].subs(reps), p[1].subs(reps))
for p in rv[1]],
[a.subs(reps) for a in rv[2]])
ans = (ans[0].subs(reps).expand(), [(p[0].subs(reps), p[1].subs(reps))
for p in ans[1]],
[a.subs(reps) for a in ans[2]])
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 s_check(unrad(sqrt(x) + x**Rational(1, 3) + 2),
(2 + s**2 + s**3, [(s, x - s**6)], []))
assert check(unrad(sqrt(x)*x**Rational(1, 3) + 2),
(x**5 - 64, [], []))
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3)),
(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)),
(-4*x + 5*x**2, [], []))
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),
(9*x + (x - 5)**2 - 9, [], []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(-5*x**2 + 2*x - 1, [], []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3),
(-25*x**4 - 376*x**3 - 1256*x**2 + 2272*x - 784, [], []))
assert check(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, [], []))
assert check(unrad(sqrt(x) + sqrt(x + 1)), (S(1), [], []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq),
(16*x**3 - 9*x**2, [], []))
assert set(solve(eq, check=False)) == set([S(0), S(9)/16])
assert solve(eq) == []
# but this one really does have those solutions
assert set(solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x)))) == \
set([S.Zero, S(9)/16])
'''NOTE
real_root changes the value of the result if the solution is
simplified; `a` in the text below is the root that is not 4/5:
>>> eq
sqrt(x) + sqrt(-x + 1) + sqrt(x + 1) - 6*sqrt(5)/5
>>> eq.subs(x, a).n()
-0.e-123 + 0.e-127*I
>>> real_root(eq.subs(x, a)).n()
-0.e-123 + 0.e-127*I
>>> (eq.subs(x,simplify(a))).n()
-0.e-126
>>> real_root(eq.subs(x, simplify(a))).n()
0.194825975605452 + 2.15093623885838*I
>>> sqrt(x).subs(x, real_root(a)).n()
0.809823827278194 - 0.e-25*I
>>> sqrt(x).subs(x, (a)).n()
0.809823827278194 - 0.e-25*I
>>> sqrt(x).subs(x, simplify(a)).n()
0.809823827278194 - 5.32999467690853e-25*I
>>> sqrt(x).subs(x, real_root(simplify(a))).n()
0.49864610868139 + 1.44572604257047*I
'''
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
ra = S('''-1484/375 - 4*(-1/2 + sqrt(3)*I/2)*(-12459439/52734375 +
114*sqrt(12657)/78125)**(1/3) - 172564/(140625*(-1/2 +
sqrt(3)*I/2)*(-12459439/52734375 + 114*sqrt(12657)/78125)**(1/3))''')
rb = S(4)/5
ans = solve(sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert all(abs(eq.subs(x, i).n()) < 1e-10 for i in (ra, rb)) and \
len(ans) == 2 and \
set([i.n(chop=True) for i in ans]) == \
set([i.n(chop=True) for i in (ra, rb)])
raises(ValueError, lambda:
unrad(-root(x,3)**2 + 2**pi*root(x,3) - x + 2**pi))
raises(ValueError, lambda:
unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3))
raises(ValueError, lambda:
unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2*sqrt(y)))
# same as last but consider only y
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2*sqrt(y), y),
(4*y - (sqrt(x) + (x + 1)**(S(1)/3))**2, [], []))
assert check(unrad(sqrt(x/(1 - x)) + (x + 1)**Rational(1, 3)),
(x**3/(-x + 1)**3 - (x + 1)**2, [], [(-x + 1)**3]))
# same as last but consider only y; no y-containing denominators now
assert s_check(unrad(sqrt(x/(1 - x)) + 2*sqrt(y), y),
(x/(-x + 1) - 4*y, [], []))
assert check(unrad(sqrt(x)*sqrt(1 - x) + 2, x),
(x*(-x + 1) - 4, [], []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [3]
assert solve(Eq(x + sqrt(x - 4), 4)) == [4]
assert solve(Eq(1, x + sqrt(2*x - 3))) == []
assert set(solve(Eq(sqrt(5*x + 6) - 2, x))) == set([-S(1), S(2)])
assert set(solve(Eq(sqrt(2*x - 1) - sqrt(x - 4), 2))) == set([S(5), S(13)])
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6]
# http://www.purplemath.com/modules/solverad.htm
assert solve((2*x - 5)**Rational(1, 3) - 3) == [16]
assert solve((x**3 - 3*x**2)**Rational(1, 3) + 1 - x) == []
assert set(solve(x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4))) == \
set([-S(1)/2, -S(1)/3])
assert set(solve(sqrt(2*x**2 - 7) - (3 - x))) == set([-S(8), S(2)])
assert solve(sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [0]
assert solve(sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)) == [5]
assert solve(sqrt(x)*sqrt(x - 7) - 12) == [16]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [4]
assert solve(sqrt(9*x**2 + 4) - (3*x + 2)) == [0]
assert solve(sqrt(x) - 2 - 5) == [49]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [10]
assert solve(sqrt(x - 2) - 5) == [27]
assert solve(sqrt(17*x - sqrt(x**2 - 5)) - 7) == [3]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
# don't posify the expression in unrad and 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) == [-S(1)/11]
assert solve(p + 6*I) == []
eq = sqrt(2 + I) + 2*I
assert unrad(eq - x, x, all=True) == (x**4 + 4*x**2 + 8*x + 37, [], [])
ans = (81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 -
31608*x**2 - 189648*x + 141358, [], [])
r = sqrt(sqrt(2)/3 + 7)
eq = sqrt(r) + r - x
assert unrad(eq, all=1)
r2 = sqrt(sqrt(2) + 21)/sqrt(3)
assert r != r2 and r.equals(r2)
assert unrad(eq - r + r2, all=True) == ans
def test_unrad():
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue 2104)
def check(rv, ans):
rv, ans = list(rv), list(ans)
rv[0] = rv[0].expand()
ans[0] = ans[0].expand()
return rv[0] in [ans[0], -ans[0]] and rv[1:] == ans[1:]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = zip(d, [s]*len(d))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [(p[0].subs(reps), p[1].subs(reps))
for p in rv[1]],
[a.subs(reps) for a in rv[2]])
ans = (ans[0].subs(reps).expand(), [(p[0].subs(reps), p[1].subs(reps))
for p in ans[1]],
[a.subs(reps) for a in ans[2]])
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 s_check(unrad(sqrt(x) + x**Rational(1,3) + 2),
(2 + s**2 + s**3, [(s, x - s**6)], []))
assert check(unrad(sqrt(x)*x**Rational(1, 3) + 2),
(x**5 - 64, [], []))
assert check(unrad(sqrt(x) + (x + 1)**Rational(1,3)),
(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)),
(-4*x + 5*x**2, [], []))
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),
(36*x + (2*x - 10)**2 - 36, [], []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(-5*x**2 + 2*x - 1, [], []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3),
(-25*x**4 - 376*x**3 - 1256*x**2 + 2272*x - 784, [], []))
assert check(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, [], []))
assert check(unrad(sqrt(x) + sqrt(x + 1)), (S(1), [], []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq),
(16*x**3 - 9*x**2, [], []))
assert solve(eq, check=False) == [0, S(9)/16]
assert solve(eq) == []
# but this one really does have those solutions
assert solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x))) == [0, S(9)/16]
'''real_root changes the value of the result if the solution is
simplified; `a` in the text below is the root that is not 4/5:
>>> eq
sqrt(x) + sqrt(-x + 1) + sqrt(x + 1) - 6*sqrt(5)/5
>>> eq.subs(x, a).n()
-0.e-123 + 0.e-127*I
>>> real_root(eq.subs(x, a)).n()
-0.e-123 + 0.e-127*I
>>> (eq.subs(x,simplify(a))).n()
-0.e-126
>>> real_root(eq.subs(x, simplify(a))).n()
0.194825975605452 + 2.15093623885838*I
>>> sqrt(x).subs(x, real_root(a)).n()
0.809823827278194 - 0.e-25*I
>>> sqrt(x).subs(x, (a)).n()
0.809823827278194 - 0.e-25*I
>>> sqrt(x).subs(x, simplify(a)).n()
0.809823827278194 - 5.32999467690853e-25*I
>>> sqrt(x).subs(x, real_root(simplify(a))).n()
0.49864610868139 + 1.44572604257047*I
'''
eq=(sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
ra = S('''-1484/375 - 4*(-1/2 + sqrt(3)*I/2)*(-12459439/52734375 +
114*sqrt(12657)/78125)**(1/3) - 172564/(140625*(-1/2 +
sqrt(3)*I/2)*(-12459439/52734375 + 114*sqrt(12657)/78125)**(1/3))''')
rb = S(4)/5
ans = solve(sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert all(abs(eq.subs(x, i).n()) < 1e-10 for i in (ra, rb)) and \
len(ans) == 2 and \
sorted([i.n(chop=True) for i in ans]) == \
sorted([i.n(chop=True) for i in (ra, rb)])
ans = solve(sqrt(x) + sqrt(x + 1) - \
sqrt(1 - x) - sqrt(2 + x))
assert len(ans) == 1 and NS(ans[0])[:4] == '0.73'
# the fence optimization problem
# http://code.google.com/p/sympy/issues/detail?id=1694#c159
F = Symbol('F')
eq = F - (2*x + 2*y + sqrt(x**2 + y**2))
X = solve(eq, x, hint='minimal')[0]
Y = solve((x*y).subs(x, X).diff(y), y, simplify=False, minimal=True)
ans = 2*F/7 - sqrt(2)*F/14
assert any((a - ans).expand().is_zero for a in Y)
raises(ValueError, 'unrad(sqrt(x) + sqrt(x+1) + sqrt(1-sqrt(x)) + 3)')
raises(ValueError, 'unrad(sqrt(x) + (x+1)**Rational(1,3) + 2*sqrt(y))')
# same as last but consider only y
assert check(unrad(sqrt(x) + (x + 1)**Rational(1,3) + 2*sqrt(y), y),
(4*y - (sqrt(x) + (x + 1)**(S(1)/3))**2, [], []))
assert check(unrad(sqrt(x/(1 - x)) + (x + 1)**Rational(1,3)),
(x**3/(-x + 1)**3 - (x + 1)**2, [], [(-x + 1)**3]))
# same as last but consider only y; no y-containing denominators now
assert s_check(unrad(sqrt(x/(1 - x)) + 2*sqrt(y), y),
(x/(-x + 1) - 4*y, [], []))
assert check(unrad(sqrt(x)*sqrt(1 - x) + 2, x),
(x*(-x + 1) - 4, [], []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [3]
assert solve(Eq(x + sqrt(x - 4), 4)) == [4]
assert solve(Eq(1, x + sqrt(2*x - 3))) == []
assert solve(Eq(sqrt(5*x + 6) - 2, x)) == [-1, 2]
assert solve(Eq(sqrt(2*x - 1) - sqrt(x - 4), 2)) == [5, 13]
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6]
# http://www.purplemath.com/modules/solverad.htm
assert solve((2*x - 5)**Rational(1, 3) - 3) == [16]
assert solve((x**3 - 3*x**2)**Rational(1, 3) + 1 - x) == []
assert solve(x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4)) == \
[-S(1)/2, -S(1)/3]
assert solve(sqrt(2*x**2 - 7) - (3 - x)) == [-8, 2]
assert solve(sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [0]
assert solve(sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)) == [5]
assert solve(sqrt(x)*sqrt(x - 7) - 12) == [16]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [4]
assert solve(sqrt(9*x**2 + 4) - (3*x + 2)) == [0]
assert solve(sqrt(x) - 2 - 5) == [49]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [10]
assert solve(sqrt(x - 2) - 5) == [27]
assert solve(sqrt(17*x - sqrt(x**2 - 5)) - 7) == [3]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
def test_unrad():
from sympy.abc import x, y, a, b, c, d
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue 2104)
def check(rv, ans):
rv, ans = list(rv), list(ans)
rv[0] = rv[0].expand()
ans[0] = ans[0].expand()
return rv[0] in [ans[0], -ans[0]] and rv[1:] == ans[1:]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = zip(d, [s]*len(d))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [(p[0].subs(reps), p[1].subs(reps))
for p in rv[1]],
[a.subs(reps) for a in rv[2]])
ans = (ans[0].subs(reps).expand(), [(p[0].subs(reps), p[1].subs(reps))
for p in ans[1]],
[a.subs(reps) for a in ans[2]])
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 s_check(unrad(sqrt(x) + x**Rational(1,3) + 2),
(2 + s**2 + s**3, [(s, x - s**6)], []))
assert check(unrad(sqrt(x)*x**Rational(1, 3) + 2),
(x**5 - 64, [], []))
assert check(unrad(sqrt(x) + (x + 1)**Rational(1,3)),
(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)),
(-4*x + 5*x**2, [], []))
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),
(36*x + (2*x - 10)**2 - 36, [], []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(-5*x**2 + 2*x - 1, [], []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3),
(-25*x**4 - 376*x**3 - 1256*x**2 + 2272*x - 784, [], []))
assert check(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, [], []))
assert check(unrad(sqrt(x) + sqrt(x + 1)), (S(1), [], []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq),
(16*x**3 - 9*x**2, [], []))
assert solve(eq, check=False) == [0, S(9)/16]
assert solve(eq) == []
# but this one really does have those solutions
assert solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x))) == [0, S(9)/16]
ans = solve(sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert len(ans) == 2 and S(4)/5 in ans
ans = solve(sqrt(x) + sqrt(x + 1) - \
sqrt(1 - x) - sqrt(2 + x))
assert len(ans) == 1 and NS(ans[0])[:4] == '0.73'
# the fence optimization problem
# http://code.google.com/p/sympy/issues/detail?id=1694#c159
x, y, a, F = symbols('x y a F')
eq = F - (2*x + 2*y + sqrt(x**2 + y**2))
X = solve(eq, x, hint='minimal')[0]
Y = solve((x*y).subs(x, X).diff(y), y, simplify=False, minimal=True)
ans = 2*F/7 - sqrt(2)*F/14
assert any((a - ans).expand().is_zero for a in Y)
raises(ValueError, 'unrad(sqrt(x) + sqrt(x+1) + sqrt(1-sqrt(x)) + 3)')
raises(ValueError, 'unrad(sqrt(x) + (x+1)**Rational(1,3) + 2*sqrt(y))')
# same as last but consider only y
assert check(unrad(sqrt(x) + (x+1)**Rational(1,3) + 2*sqrt(y), y),
(4*y - (sqrt(x) + (x + 1)**(S(1)/3))**2, [], []))
assert check(unrad(sqrt(x/(1 - x)) + (x+1)**Rational(1,3)),
(x**3/(-x + 1)**3 - (x + 1)**2, [], [(-x + 1)**3]))
# same as last but consider only y; no y-containing denominators now
assert s_check(unrad(sqrt(x/(1 - x)) + 2*sqrt(y), y),
(x/(-x + 1) - 4*y, [], []))
assert check(unrad(sqrt(x)*sqrt(1-x) + 2, x),
(x*(-x + 1) - 4, [], []))
# http://tutorial.math.lamar.edu/Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [3]
assert solve(Eq(x + sqrt(x - 4), 4)) == [4]
assert solve(Eq(1, x + sqrt(2*x - 3))) == []
assert solve(Eq(sqrt(5*x + 6) - 2, x)) == [-1, 2]
assert solve(Eq(sqrt(2*x - 1) - sqrt(x - 4), 2)) == [5, 13]
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6]
# http://www.purplemath.com/modules/solverad.htm
assert solve((2*x-5)**Rational(1,3)-3) == [16]
assert solve((x**3-3*x**2)**Rational(1,3)+1-x) == []
assert solve(x+1-(x**4+4*x**3-x)**Rational(1,4)) == [-S(1)/2, -S(1)/3]
assert solve(sqrt(2*x**2-7)-(3-x)) == [-8, 2]
assert solve(sqrt(2*x+9)-sqrt(x+1)-sqrt(x+4)) == [0]
assert solve(sqrt(x+4)+sqrt(2*x-1)-3*sqrt(x-1)) == [5]
assert solve(sqrt(x)*sqrt(x-7)-12) == [16]
assert solve(sqrt(x-3)+sqrt(x)-3) == [4]
assert solve(sqrt(9*x**2+4)-(3*x+2)) == [0]
assert solve(sqrt(x)-2-5) == [49]
assert solve(sqrt(x-3)-sqrt(x)-3) == []
assert solve(sqrt(x-1)-x+7) == [10]
assert solve(sqrt(x-2)-5) == [27]
def test_unrad():
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue 2104)
def check(rv, ans):
rv, ans = list(rv), list(ans)
rv[0] = rv[0].expand()
ans[0] = ans[0].expand()
return rv[0] in [ans[0], -ans[0]] and rv[1:] == ans[1:]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = zip(d, [s] * len(d))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [
(p[0].subs(reps), p[1].subs(reps)) for p in rv[1]
], [a.subs(reps) for a in rv[2]])
ans = (ans[0].subs(reps).expand(), [
(p[0].subs(reps), p[1].subs(reps)) for p in ans[1]
], [a.subs(reps) for a in ans[2]])
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 s_check(unrad(sqrt(x) + x**Rational(1, 3) + 2),
(2 + s**2 + s**3, [(s, x - s**6)], []))
assert check(unrad(sqrt(x) * x**Rational(1, 3) + 2), (x**5 - 64, [], []))
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3)),
(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)),
(-4 * x + 5 * x**2, [], []))
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),
(9 * x + (x - 5)**2 - 9, [], []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(-5 * x**2 + 2 * x - 1, [], []))
assert check(
unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3),
(-25 * x**4 - 376 * x**3 - 1256 * x**2 + 2272 * x - 784, [], []))
assert check(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, [], []))
assert check(unrad(sqrt(x) + sqrt(x + 1)), (S(1), [], []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq), (16 * x**3 - 9 * x**2, [], []))
assert set(solve(eq, check=False)) == set([S(0), S(9) / 16])
assert solve(eq) == []
# but this one really does have those solutions
assert set(solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x)))) == \
set([S.Zero, S(9)/16])
'''NOTE
real_root changes the value of the result if the solution is
simplified; `a` in the text below is the root that is not 4/5:
>>> eq
sqrt(x) + sqrt(-x + 1) + sqrt(x + 1) - 6*sqrt(5)/5
>>> eq.subs(x, a).n()
-0.e-123 + 0.e-127*I
>>> real_root(eq.subs(x, a)).n()
-0.e-123 + 0.e-127*I
>>> (eq.subs(x,simplify(a))).n()
-0.e-126
>>> real_root(eq.subs(x, simplify(a))).n()
0.194825975605452 + 2.15093623885838*I
>>> sqrt(x).subs(x, real_root(a)).n()
0.809823827278194 - 0.e-25*I
>>> sqrt(x).subs(x, (a)).n()
0.809823827278194 - 0.e-25*I
>>> sqrt(x).subs(x, simplify(a)).n()
0.809823827278194 - 5.32999467690853e-25*I
>>> sqrt(x).subs(x, real_root(simplify(a))).n()
0.49864610868139 + 1.44572604257047*I
'''
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6 * sqrt(5) / 5)
ra = S('''-1484/375 - 4*(-1/2 + sqrt(3)*I/2)*(-12459439/52734375 +
114*sqrt(12657)/78125)**(1/3) - 172564/(140625*(-1/2 +
sqrt(3)*I/2)*(-12459439/52734375 + 114*sqrt(12657)/78125)**(1/3))''')
rb = S(4) / 5
ans = solve(sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6 * sqrt(5) / 5)
assert all(abs(eq.subs(x, i).n()) < 1e-10 for i in (ra, rb)) and \
len(ans) == 2 and \
set([i.n(chop=True) for i in ans]) == \
set([i.n(chop=True) for i in (ra, rb)])
raises(ValueError,
lambda: unrad(-root(x, 3)**2 + 2**pi * root(x, 3) - x + 2**pi))
raises(ValueError,
lambda: unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3))
raises(ValueError,
lambda: unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2 * sqrt(y)))
# same as last but consider only y
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2 * sqrt(y), y),
(4 * y - (sqrt(x) + (x + 1)**(S(1) / 3))**2, [], []))
assert check(unrad(sqrt(x / (1 - x)) + (x + 1)**Rational(1, 3)),
(x**3 / (-x + 1)**3 - (x + 1)**2, [], [(-x + 1)**3]))
# same as last but consider only y; no y-containing denominators now
assert s_check(unrad(sqrt(x / (1 - x)) + 2 * sqrt(y), y),
(x / (-x + 1) - 4 * y, [], []))
assert check(unrad(sqrt(x) * sqrt(1 - x) + 2, x),
(x * (-x + 1) - 4, [], []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [3]
assert solve(Eq(x + sqrt(x - 4), 4)) == [4]
assert solve(Eq(1, x + sqrt(2 * x - 3))) == []
assert set(solve(Eq(sqrt(5 * x + 6) - 2, x))) == set([-S(1), S(2)])
assert set(solve(Eq(sqrt(2 * x - 1) - sqrt(x - 4),
2))) == set([S(5), S(13)])
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6]
# http://www.purplemath.com/modules/solverad.htm
assert solve((2 * x - 5)**Rational(1, 3) - 3) == [16]
assert solve((x**3 - 3 * x**2)**Rational(1, 3) + 1 - x) == []
assert set(solve(x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4))) == \
set([-S(1)/2, -S(1)/3])
assert set(solve(sqrt(2 * x**2 - 7) - (3 - x))) == set([-S(8), S(2)])
assert solve(sqrt(2 * x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [0]
assert solve(sqrt(x + 4) + sqrt(2 * x - 1) - 3 * sqrt(x - 1)) == [5]
assert solve(sqrt(x) * sqrt(x - 7) - 12) == [16]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [4]
assert solve(sqrt(9 * x**2 + 4) - (3 * x + 2)) == [0]
assert solve(sqrt(x) - 2 - 5) == [49]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [10]
assert solve(sqrt(x - 2) - 5) == [27]
assert solve(sqrt(17 * x - sqrt(x**2 - 5)) - 7) == [3]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
# don't posify the expression in unrad and 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) == [-S(1) / 11]
assert solve(p + 6 * I) == []
eq = sqrt(2 + I) + 2 * I
assert unrad(eq - x, x, all=True) == (x**4 + 4 * x**2 + 8 * x + 37, [], [])
ans = (81 * x**8 - 2268 * x**6 - 4536 * x**5 + 22644 * x**4 +
63216 * x**3 - 31608 * x**2 - 189648 * x + 141358, [], [])
r = sqrt(sqrt(2) / 3 + 7)
eq = sqrt(r) + r - x
assert unrad(eq, all=1)
r2 = sqrt(sqrt(2) + 21) / sqrt(3)
assert r != r2 and r.equals(r2)
assert unrad(eq - r + r2, all=True) == ans
def test_unrad():
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue 2104)
def check(rv, ans):
rv, ans = list(rv), list(ans)
rv[0] = rv[0].expand()
ans[0] = ans[0].expand()
return rv[0] in [ans[0], -ans[0]] and rv[1:] == ans[1:]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = zip(d, [s] * len(d))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [
(p[0].subs(reps), p[1].subs(reps)) for p in rv[1]
], [a.subs(reps) for a in rv[2]])
ans = (ans[0].subs(reps).expand(), [
(p[0].subs(reps), p[1].subs(reps)) for p in ans[1]
], [a.subs(reps) for a in ans[2]])
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 s_check(unrad(sqrt(x) + x**Rational(1, 3) + 2),
(2 + s**2 + s**3, [(s, x - s**6)], []))
assert check(unrad(sqrt(x) * x**Rational(1, 3) + 2), (x**5 - 64, [], []))
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3)),
(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)),
(-4 * x + 5 * x**2, [], []))
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),
(36 * x + (2 * x - 10)**2 - 36, [], []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(-5 * x**2 + 2 * x - 1, [], []))
assert check(
unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3),
(-25 * x**4 - 376 * x**3 - 1256 * x**2 + 2272 * x - 784, [], []))
assert check(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, [], []))
assert check(unrad(sqrt(x) + sqrt(x + 1)), (S(1), [], []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq), (16 * x**3 - 9 * x**2, [], []))
assert solve(eq, check=False) == [0, S(9) / 16]
assert solve(eq) == []
# but this one really does have those solutions
assert solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x))) == [0, S(9) / 16]
ans = solve(sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6 * sqrt(5) / 5)
assert len(ans) == 2 and S(4) / 5 in ans
ans = solve(sqrt(x) + sqrt(x + 1) - \
sqrt(1 - x) - sqrt(2 + x))
assert len(ans) == 1 and NS(ans[0])[:4] == '0.73'
# the fence optimization problem
# http://code.google.com/p/sympy/issues/detail?id=1694#c159
F = Symbol('F')
eq = F - (2 * x + 2 * y + sqrt(x**2 + y**2))
X = solve(eq, x, hint='minimal')[0]
Y = solve((x * y).subs(x, X).diff(y), y, simplify=False, minimal=True)
ans = 2 * F / 7 - sqrt(2) * F / 14
assert any((a - ans).expand().is_zero for a in Y)
raises(ValueError, 'unrad(sqrt(x) + sqrt(x+1) + sqrt(1-sqrt(x)) + 3)')
raises(ValueError, 'unrad(sqrt(x) + (x+1)**Rational(1,3) + 2*sqrt(y))')
# same as last but consider only y
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2 * sqrt(y), y),
(4 * y - (sqrt(x) + (x + 1)**(S(1) / 3))**2, [], []))
assert check(unrad(sqrt(x / (1 - x)) + (x + 1)**Rational(1, 3)),
(x**3 / (-x + 1)**3 - (x + 1)**2, [], [(-x + 1)**3]))
# same as last but consider only y; no y-containing denominators now
assert s_check(unrad(sqrt(x / (1 - x)) + 2 * sqrt(y), y),
(x / (-x + 1) - 4 * y, [], []))
assert check(unrad(sqrt(x) * sqrt(1 - x) + 2, x),
(x * (-x + 1) - 4, [], []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [3]
assert solve(Eq(x + sqrt(x - 4), 4)) == [4]
assert solve(Eq(1, x + sqrt(2 * x - 3))) == []
assert solve(Eq(sqrt(5 * x + 6) - 2, x)) == [-1, 2]
assert solve(Eq(sqrt(2 * x - 1) - sqrt(x - 4), 2)) == [5, 13]
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6]
# http://www.purplemath.com/modules/solverad.htm
assert solve((2 * x - 5)**Rational(1, 3) - 3) == [16]
assert solve((x**3 - 3 * x**2)**Rational(1, 3) + 1 - x) == [S(1) / 3]
assert solve(x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4)) == \
[-S(1)/2, -S(1)/3]
assert solve(sqrt(2 * x**2 - 7) - (3 - x)) == [-8, 2]
assert solve(sqrt(2 * x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [0]
assert solve(sqrt(x + 4) + sqrt(2 * x - 1) - 3 * sqrt(x - 1)) == [5]
assert solve(sqrt(x) * sqrt(x - 7) - 12) == [16]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [4]
assert solve(sqrt(9 * x**2 + 4) - (3 * x + 2)) == [0]
assert solve(sqrt(x) - 2 - 5) == [49]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [10]
assert solve(sqrt(x - 2) - 5) == [27]
assert solve(sqrt(17 * x - sqrt(x**2 - 5)) - 7) == [3]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
def test_unrad():
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue 2104)
def check(rv, ans):
rv, ans = list(rv), list(ans)
rv[0] = rv[0].expand()
ans[0] = ans[0].expand()
return rv[0] in [ans[0], -ans[0]] and rv[1:] == ans[1:]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = zip(d, [s] * len(d))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [
(p[0].subs(reps), p[1].subs(reps)) for p in rv[1]
], [a.subs(reps) for a in rv[2]])
ans = (ans[0].subs(reps).expand(), [
(p[0].subs(reps), p[1].subs(reps)) for p in ans[1]
], [a.subs(reps) for a in ans[2]])
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 s_check(unrad(sqrt(x) + x**Rational(1, 3) + 2),
(2 + s**2 + s**3, [(s, x - s**6)], []))
assert check(unrad(sqrt(x) * x**Rational(1, 3) + 2), (x**5 - 64, [], []))
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3)),
(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)),
(-4 * x + 5 * x**2, [], []))
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),
(36 * x + (2 * x - 10)**2 - 36, [], []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(-5 * x**2 + 2 * x - 1, [], []))
assert check(
unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3),
(-25 * x**4 - 376 * x**3 - 1256 * x**2 + 2272 * x - 784, [], []))
assert check(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, [], []))
assert check(unrad(sqrt(x) + sqrt(x + 1)), (S(1), [], []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq), (16 * x**3 - 9 * x**2, [], []))
assert solve(eq, check=False) == [0, S(9) / 16]
assert solve(eq) == []
# but this one really does have those solutions
assert solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x))) == [0, S(9) / 16]
'''real_root changes the value of the result if the solution is
simplified; `a` in the text below is the root that is not 4/5:
>>> eq
sqrt(x) + sqrt(-x + 1) + sqrt(x + 1) - 6*sqrt(5)/5
>>> eq.subs(x, a).n()
-0.e-123 + 0.e-127*I
>>> real_root(eq.subs(x, a)).n()
-0.e-123 + 0.e-127*I
>>> (eq.subs(x,simplify(a))).n()
-0.e-126
>>> real_root(eq.subs(x, simplify(a))).n()
0.194825975605452 + 2.15093623885838*I
>>> sqrt(x).subs(x, real_root(a)).n()
0.809823827278194 - 0.e-25*I
>>> sqrt(x).subs(x, (a)).n()
0.809823827278194 - 0.e-25*I
>>> sqrt(x).subs(x, simplify(a)).n()
0.809823827278194 - 5.32999467690853e-25*I
>>> sqrt(x).subs(x, real_root(simplify(a))).n()
0.49864610868139 + 1.44572604257047*I
'''
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6 * sqrt(5) / 5)
ra = S('''-1484/375 - 4*(-1/2 + sqrt(3)*I/2)*(-12459439/52734375 +
114*sqrt(12657)/78125)**(1/3) - 172564/(140625*(-1/2 +
sqrt(3)*I/2)*(-12459439/52734375 + 114*sqrt(12657)/78125)**(1/3))''')
rb = S(4) / 5
ans = solve(sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6 * sqrt(5) / 5)
assert all(abs(eq.subs(x, i).n()) < 1e-10 for i in (ra, rb)) and \
len(ans) == 2 and \
sorted([i.n(chop=True) for i in ans]) == \
sorted([i.n(chop=True) for i in (ra, rb)])
ans = solve(sqrt(x) + sqrt(x + 1) - \
sqrt(1 - x) - sqrt(2 + x))
assert len(ans) == 1 and NS(ans[0])[:4] == '0.73'
# the fence optimization problem
# http://code.google.com/p/sympy/issues/detail?id=1694#c159
F = Symbol('F')
eq = F - (2 * x + 2 * y + sqrt(x**2 + y**2))
X = solve(eq, x, hint='minimal')[0]
Y = solve((x * y).subs(x, X).diff(y), y, simplify=False, minimal=True)
ans = 2 * F / 7 - sqrt(2) * F / 14
assert any((a - ans).expand().is_zero for a in Y)
raises(ValueError,
lambda: unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3))
raises(ValueError,
lambda: unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2 * sqrt(y)))
# same as last but consider only y
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2 * sqrt(y), y),
(4 * y - (sqrt(x) + (x + 1)**(S(1) / 3))**2, [], []))
assert check(unrad(sqrt(x / (1 - x)) + (x + 1)**Rational(1, 3)),
(x**3 / (-x + 1)**3 - (x + 1)**2, [], [(-x + 1)**3]))
# same as last but consider only y; no y-containing denominators now
assert s_check(unrad(sqrt(x / (1 - x)) + 2 * sqrt(y), y),
(x / (-x + 1) - 4 * y, [], []))
assert check(unrad(sqrt(x) * sqrt(1 - x) + 2, x),
(x * (-x + 1) - 4, [], []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [3]
assert solve(Eq(x + sqrt(x - 4), 4)) == [4]
assert solve(Eq(1, x + sqrt(2 * x - 3))) == []
assert solve(Eq(sqrt(5 * x + 6) - 2, x)) == [-1, 2]
assert solve(Eq(sqrt(2 * x - 1) - sqrt(x - 4), 2)) == [5, 13]
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6]
# http://www.purplemath.com/modules/solverad.htm
assert solve((2 * x - 5)**Rational(1, 3) - 3) == [16]
assert solve((x**3 - 3 * x**2)**Rational(1, 3) + 1 - x) == []
assert solve(x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4)) == \
[-S(1)/2, -S(1)/3]
assert solve(sqrt(2 * x**2 - 7) - (3 - x)) == [-8, 2]
assert solve(sqrt(2 * x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [0]
assert solve(sqrt(x + 4) + sqrt(2 * x - 1) - 3 * sqrt(x - 1)) == [5]
assert solve(sqrt(x) * sqrt(x - 7) - 12) == [16]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [4]
assert solve(sqrt(9 * x**2 + 4) - (3 * x + 2)) == [0]
assert solve(sqrt(x) - 2 - 5) == [49]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [10]
assert solve(sqrt(x - 2) - 5) == [27]
assert solve(sqrt(17 * x - sqrt(x**2 - 5)) - 7) == [3]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
# don't posify the expession in unrad and 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) == [-S(1) / 11]
assert solve(p + 6 * I) == []
h1 is the horizontal distance from point 1 to point 0.
h is the horitzontal distance from point 1 to point 2.
h2 is the horizontal distance from point 0 to point 2.
n1 is refractive index of medium 1
n2 is refractive index of medium 2
z1 is height of point 1
z2 is depth of point 2
"""
from sympy import *
from sympy.solvers.solvers import unrad # tested with sympy==0.7.3
h1 = Symbol('h1', real=True, positive=True)
h = Symbol('h', real=True, positive=True)
h2 = h-h1
n1 = Symbol('n1', real=True, positive=True)
n2 = Symbol('n2', real=True, positive=True)
z1 = Symbol('z1', real=True, positive=True)
z2 = Symbol('z2', real=True, positive=True)
duration = n1*sqrt( h1*h1 + z1*z1 ) + n2*sqrt(z2*z2 + h2*h2)
ddur_dh1 = diff(duration,h1)
eq, cov, dens = unrad(ddur_dh1)
poly = Poly(eq,h1)
print poly