def test_solver_23_reciprocal_lte():
for c in [1, 3]:
# Positive
# 1 / X < 10
solution = Interval.Ropen(-oo, 0) | Interval.Lopen(Rat(c, 10), oo)
event = (c / Y) < 10
assert event.solve() == solution
# 1 / X <= 10
solution = Interval.Ropen(-oo, 0) | Interval(Rat(c, 10), oo)
event = (c / Y) <= 10
assert event.solve() == solution
# 1 / X <= sqrt(2)
solution = Interval.Ropen(-oo, 0) | Interval(c / sympy.sqrt(2), oo)
event = (c / Y) <= sympy.sqrt(2)
assert event.solve() == solution
# Negative.
# 1 / X < -10
solution = Interval.open(-Rat(c, 10), 0)
event = (c / Y) < -10
assert event.solve() == solution
# 1 / X <= -10
solution = Interval.Ropen(-Rat(c, 10), 0)
event = (c / Y) <= -10
assert event.solve() == solution
# 1 / X <= -sqrt(2)
solution = Interval.Ropen(-c / sympy.sqrt(2), 0)
event = (c / Y) <= -sympy.sqrt(2)
assert event.solve() == solution
def test_solver_13():
# 2*sqrt(|x|**2) - 3 > 10
solution = Union(
Interval.open(-oo, -Rat(13, 2)),
Interval.open(Rat(13, 2), oo))
event = (2*Sqrt(abs(Y)**2) - 3) > 10
answer = event.solve()
assert answer == solution
def test_solver_9_closed():
# 2(log(x))**3 - log(x) -5 >= 0
solution = Interval(
sympy.exp(1/(6*(sympy.sqrt(2019)/36 + Rat(5,4))**(Rat(1, 3)))
+ (sympy.sqrt(2019)/36 + Rat(5,4))**(Rat(1,3))),
oo)
event = 2*(Log(Y))**3 - Log(Y) - 5 >= 0
answer = event.solve()
assert answer == solution
def test_solver_18():
# 3*(x**(1/7))**4 - 3*(x**(1/7))**2 <= 9
solution = Interval(0, (Rat(1, 2) + sympy.sqrt(13)/2)**(Rat(7, 2)))
Z = Y**(Rat(1, 7))
expr = 3*Z**4 - 3*Z**2
event = (expr <= 9)
answer = event.solve()
assert answer == solution
interval = (~event).solve()
assert interval == Interval.open(solution.right, oo)
def test_solver_27_piecewise_many():
expr = (Y < 0)*(Y**2) + (0 <= Y)*Y**(Rat(1, 2))
event = expr << {3}
assert sorted(event.solve()) == [-sympy.sqrt(3), 9]
event = 0 < expr
assert event.solve() == Union(
Interval.open(-oo, 0),
Interval.open(0, oo))
# TODO: Consider banning the restriction of a function
# to a segment outside of its domain.
expr = (Y < 0)*Y**(Rat(1, 2))
assert (expr < 1).solve() is EmptySet
def test_parse_17():
# https://www.wolframalpha.com/input/?i=Expand%5B%2810%2F7+%2B+X%29+%28-1%2F%285+Sqrt%5B2%5D%29+%2B+X%29+%28-Sqrt%5B5%5D+%2B+X%29%5D
for Z in [X, Log(X), Abs(1+X**2)]:
expr = (Z - Rat(1, 10) * sympy.sqrt(2)) \
* (Z + Rat(10, 7)) \
* (Z - sympy.sqrt(5))
coeffs = [
sympy.sqrt(10)/7,
1/sympy.sqrt(10) - (10*sympy.sqrt(5))/7 - sympy.sqrt(2)/7,
(-sympy.sqrt(5) - 1/(5 * sympy.sqrt(2))) + Rat(10)/7,
1,
]
expr_prime = Poly(Z, coeffs)
assert expr == expr_prime
def test_solver_9_open():
# 2(log(x))**3 - log(x) -5 > 0
solution = Interval.open(
sympy.exp(1/(6*(sympy.sqrt(2019)/36 + Rat(5,4))**(Rat(1, 3)))
+ (sympy.sqrt(2019)/36 + Rat(5,4))**(Rat(1,3))),
oo)
# Our solver handles this case as follows
# expr' = 2*Z**3 - Z - 5 > 0 [[subst. Z=log(X)]]
# [Z_low, Z_high] = sympy_solver(expr')
# Z_low < Z iff Z_low < log(X) iff exp(Z_low) < X
# Z < Z_high iff log(X) < Z_high iff X < exp(Z_high)
# sympy_solver(expr) = [exp(Z_low), exp(Z_high)]
# For F invertible, can thus solve Poly(coeffs, F) > 0 using this method.
event = 2*(Log(Y))**3 - Log(Y) - 5 > 0
answer = event.solve()
assert answer == solution
def test_solver_finite_injective():
sqrt3 = sympy.sqrt(3)
# Identity.
solution = FiniteReal(2, 4, -10, sqrt3)
event = Y << {2, 4, -10, sqrt3}
assert event.solve() == solution
# Exp.
solution = FiniteReal(sympy.log(10), sympy.log(3), sympy.log(sqrt3))
event = Exp(Y) << {10, 3, sqrt3}
assert event.solve() == solution
# Exp2.
solution = FiniteReal(sympy.log(10, 2), 4, sympy.log(sqrt3, 2))
event = (2**Y) << {10, 16, sqrt3}
assert event.solve() == solution
# Log.
solution = FiniteReal(sympy.exp(10), sympy.exp(-3), sympy.exp(sqrt3))
event = Log(Y) << {10, -3, sqrt3}
assert event.solve() == solution
# Log2
solution = FiniteReal(sympy.Pow(2, 10), sympy.Pow(2, -3), sympy.Pow(2, sqrt3))
event = Logarithm(Y, 2) << {10, -3, sqrt3}
assert event.solve() == solution
# Radical.
solution = FiniteReal(7**4, 12**4, sqrt3**4)
event = Y**Rat(1, 4) << {7, 12, sqrt3}
assert event.solve() == solution
def test_solver_24_negative_power_Rat():
# Case 1.
event = Y**Rat(-1, 3) < 6
assert event.solve() == Interval.Lopen(Rat(1, 216), oo)
# Case 2.
event = (-1 < Y**Rat(-1, 3)) < 6
assert event.solve() == Interval.Lopen(Rat(1, 216), oo)
# Case 3.
event = 5 <= Y**Rat(-1, 3)
assert event.solve() == Interval.Lopen(0, Rat(1, 125))
# Case 4.
event = (5 <= Y**Rat(-1, 3)) < 6
assert event.solve() == Interval.Lopen(Rat(1, 216), Rat(1, 125))
def test_parse_15():
# ((x**4)**(1/7)) < 9
expr = ((X**4))**Rat(1, 7)
expr_prime = Radical(Pow(Y, 4), 7)
assert expr == expr_prime
event = EventInterval(expr_prime, Interval.open(-oo, 9))
assert (expr < 9) == event
def test_solver_24_negative_power_integer():
# Case 1.
event = Y**(-3) < 6
assert event.solve() == Union(
Interval.open(-oo, 0),
Interval.open(6**Rat(-1, 3), oo))
# Case 2.
event = (-1 < Y**(-3)) < 6
assert event.solve() == Union(
Interval.open(-oo, -1),
Interval.open(6**Rat(-1, 3), oo))
# Case 3.
event = 5 <= Y**(-3)
assert event.solve() == Interval.Lopen(0, 5**Rat(-1, 3))
# Case 4.
event = (5 <= Y**(-3)) < 6
assert event.solve() == Interval.Lopen(6**Rat(-1, 3), 5**Rat(-1, 3))
def test_solver_19():
# 3*(x**(1/7))**4 - 3*(x**(1/7))**2 <= 9
# or || 3*(x**(1/7))**4 - 3*(x**(1/7))**2 > 11
solution = Union(
Interval(0, (Rat(1, 2) + sympy.sqrt(13)/2)**(Rat(7, 2))),
Interval.open((Rat(1,2) + sympy.sqrt(141)/6)**(Rat(7, 2)), oo))
Z = Y**(Rat(1, 7))
expr = 3*Z**4 - 3*Z**2
event = (expr <= 9) | (expr > 11)
answer = event.solve()
assert answer == solution
interval = (~event).solve()
assert interval == Interval.Lopen(
solution.args[0].right,
solution.args[1].left)
def test_parse_16():
# (x**(1/7))**4 < 9
for expr in [((X**Rat(1,7)))**4, (X**(1,7))**4]:
expr_prime = Pow(Radical(Y, 7), 4)
assert expr == expr_prime
event = EventInterval(expr, Interval.open(-oo, 9))
assert (expr < 9) == event
def test_solver_23_reciprocal_range():
solution = Interval.Ropen(-1, -Rat(1, 3))
event = ((-3 < 1/Y) <= -1)
assert event.solve() == solution
solution = Interval.open(0, Rat(1, 3))
event = ((-3 < 1/(2*Y-1)) < -1)
assert event.solve() == solution
solution = Interval.open(-1 / sympy.sqrt(3), 1 / sympy.sqrt(3))
event = ((-3 < 1/(2*(abs(Y)**2)-1)) <= -1)
assert event.solve() == solution
solution = Union(
Interval.open(-1 / sympy.sqrt(3), 0),
Interval.open(0, 1 / sympy.sqrt(3)))
event = ((-3 < 1/(2*(abs(Y)**2)-1)) < -1)
assert event.solve() == solution
def test_solver_23_reciprocal_gte():
for c in [1, 3]:
# Positive
# 10 < 1 / X
solution = Interval.open(0, Rat(c, 10))
event = 10 < (c / Y)
assert event.solve() == solution
# 10 <= 1 / X
solution = Interval.Lopen(0, Rat(c, 10))
event = 10 <= (c / Y)
assert event.solve() == solution
# Negative
# -10 < 1 / X
solution = Interval.Lopen(0, oo) | Interval.open(-oo, -Rat(c, 10))
event = -10 < (c / Y)
assert event.solve() == solution
# -10 <= 1 / X
solution = Interval.Lopen(0, oo) | Interval.Lopen(-oo, -Rat(c, 10))
event = -10 <= (c / Y)
assert event.solve() == solution
def test_solver_finite_non_injective():
sqrt2 = sympy.sqrt(2)
# Abs.
solution = FiniteReal(-10, -3, 3, 10)
event = abs(Y) << {10, 3}
assert event.solve() == solution
# Abs(Poly).
solution = FiniteReal(-5, -Rat(3,2), Rat(3,2), 5)
event = abs(2*Y) << {10, 3}
assert event.solve() == solution
# Poly order 2.
solution = FiniteReal(-sqrt2, sqrt2)
event = (Y**2) << {2}
assert event.solve() == solution
# Poly order 3.
solution = FiniteReal(1, 3)
event = Y**3 << {1, 27}
assert event.solve() == solution
# Poly Abs.
solution = FiniteReal(-3, -1, 1, 3)
event = (abs(Y))**3 << {1, 27}
assert event.solve() == solution
# Abs Not.
solution = Union(
Interval.open(-oo, -1),
Interval.open(-1, 1),
Interval.open(1, oo))
event = ~(abs(Y) << {1})
assert event.solve() == solution
# Abs in EmptySet.
solution = EmptySet
event = (abs(Y))**3 << set()
assert event.solve() == solution
# Abs Not in EmptySet (yields all reals).
solution = Interval(-oo, oo)
event = ~(((abs(Y))**3) << set())
assert event.solve() == solution
# Log in Reals (yields positive reals).
solution = Interval.open(0, oo)
event = ~((Log(Y))**3 << set())
assert event.solve() == solution
def test_parse_18():
# 3*(x**(1/7))**4 - 3*(x**(1/7))**2 <= 9
Z = X**Rat(1, 7)
expr = 3*Z**4 - 3*Z**2
expr_prime = Poly(Radical(Y, 7), [0, 0, -3, 0, 3])
assert expr == expr_prime
event = EventInterval(expr_prime, Interval(-oo, 9))
assert (expr <= 9) == event
event_not = EventInterval(expr_prime, Interval.open(9, oo))
assert ~(expr <= 9) == event_not
expr = (3*Abs(Z))**4 - (3*Abs(Z))**2
expr_prime = Poly(Poly(Abs(Z), [0, 3]), [0, 0, -1, 0, 3])
def test_errors():
with pytest.raises(ValueError):
1 + Log(X) - Exp(X)
with pytest.raises(TypeError):
Log(X) ** Exp(X)
with pytest.raises(ValueError):
Abs(X) ** sympy.sqrt(10)
with pytest.raises(ValueError):
Log(X) * X
with pytest.raises(ValueError):
(2*Log(X)) - Rat(1, 10) * Abs(X)
with pytest.raises(ValueError):
X**(1.71)
with pytest.raises(ValueError):
Abs(X)**(1.1, 8)
with pytest.raises(ValueError):
Abs(X)**(7, 8)
with pytest.raises(ValueError):
(-3)**X
with pytest.raises(ValueError):
(Identity('Z')**2)*(Y > 0)
with pytest.raises(ValueError):
(Y > 0) * (Identity('Z')**2)
with pytest.raises(ValueError):
((Y > 0) | (Identity('Z') < 3)) * (Identity('Z')**2)
with pytest.raises(ValueError):
Y**2 + (0 <= Y) * Y
with pytest.raises(ValueError):
(Y <= 0)*(Y**2) + (0 <= Y)*Y**((1, 2))
with pytest.raises(ValueError):
(Y <= 0)*(Y**2) + (0 <= Identity('Z'))*Y**((1, 2))
with pytest.raises(ValueError):
(Y <= 0)*(Y**2) + (0 <= Identity('Z'))*Identity('Z')**((1, 2))
# TypeErrors from 'return NotImplemented'.
with pytest.raises(TypeError):
X + 'a'
with pytest.raises(TypeError):
X * 'a'
with pytest.raises(TypeError):
X / 'a'
with pytest.raises(TypeError):
X**'s'
def test_parse_19():
# 3*(x**(1/7))**4 - 3*(x**(1/7))**2 <= 9
# or || 3*(x**(1/7))**4 - 3*(x**(1/7))**2 > 11
Z = X**Rat(1, 7)
expr = 3*Z**4 - 3*Z**2
event = (expr <= 9) | (expr > 11)
event_prime = EventOr([
EventInterval(expr, Interval(-oo, 9)),
EventInterval(expr, Interval.open(11, oo)),
])
assert event == event_prime
event = ((expr <= 9) | (expr > 11)) & (~(expr < 10))
event_prime = EventAnd([
EventOr([
EventInterval(expr, Interval(-oo, 9)),
EventInterval(expr, Interval.open(11, oo))]),
EventInterval(expr, Interval(10, oo))
])
assert event == event_prime
def test_solver_15():
# ((x**4)**(1/7)) < 9
solution = Interval.open(-27*sympy.sqrt(3), 27*sympy.sqrt(3))
event = ((Y**4))**(Rat(1, 7)) < 9
answer = event.solve()
assert answer == solution
def test_parse_24_negative_power():
assert X**(-3) == Reciprocal(Pow(X, 3))
assert X**((-3, 1)) == Reciprocal(Pow(X, 3))
assert X**(-Rat(1, 3)) == Reciprocal(Radical(X, 3))
with pytest.raises(ValueError):
X**0
def test_solver_17():
p = sympy.Poly(
(X - sympy.sqrt(2)/10) * (X+Rat(10, 7)) * (X - sympy.sqrt(5)),
X)
expr = p.args[0] < 1
sympy_solver(expr)
def test_divide_multiplication():
assert (X**2 + 2) / 2 == Poly(X, [1, 0, Rat(1, 2)])
e = (n + nbar) / 2
ebar = n - e
I = e1 * e2 * e * ebar
def intersect_lines(L1, L2):
global I
'''
Computes the intersection bivector of two conformal lines L1, L2
'''
C = I * ((I * L1) ^ (I * L2))
return C
A = F(Rat(1, 2) * e1)
B = F(2 * e1)
C = F(Rat(4, 5) * e1 + Rat(3, 5) * e2)
D = F(Rat(4, 5) * e1 - Rat(3, 5) * e2)
print('A =', A)
print('B =', B)
print('C =', C)
print('D =', D)
T = A ^ B ^ C
print('T =', T)
U = F(e1) ^ (F(e2)) ^ F(-1 * e1)
print('U =', U)
inter = intersect_lines(U, T)
print('inter =', inter)
def test_solver_16():
# (x**(1/7))**4 < 9
solution = Interval.Ropen(0, 27*sympy.sqrt(3))
event = ((Y**Rat(1,7)))**4 < 9
answer = event.solve()
assert answer == solution
