def test_cse_single():
# Simple substitution.
e = Add(Pow(x + y, 2), sqrt(x + y))
substs, reduced = cse([e])
assert substs == [(x0, x + y)]
assert reduced == [sqrt(x0) + x0**2]
assert cse([e], order='none') == cse([e])
def test_non_commutative_order():
A, B, C = symbols('A B C', commutative=False)
x0 = symbols('x0', commutative=False)
l = [B+C, A*(B+C)]
assert cse(l) == ([(x0, B+C)], [x0, A*x0])
l = [(A - B)**2 + A - B]
assert cse(l) == ([(x0, A - B)], [x0**2 + x0])
def test_sympyissue_7840():
# daveknippers' example
C393 = sympify(
'Piecewise((C391 - 1.65, C390 < 0.5), (Piecewise((C391 - 1.65, \
C391 > 2.35), (C392, True)), True))'
)
C391 = sympify(
'Piecewise((2.05*C390**(-1.03), C390 < 0.5), (2.5*C390**(-0.625), True))'
)
C393 = C393.subs({'C391': C391})
# simple substitution
sub = {}
sub['C390'] = 0.703451854
sub['C392'] = 1.01417794
ss_answer = C393.subs(sub)
# cse
substitutions, new_eqn = cse(C393)
for pair in substitutions:
sub[pair[0].name] = pair[1].subs(sub)
cse_answer = new_eqn[0].subs(sub)
# both methods should be the same
assert ss_answer == cse_answer
# GitRay's example
expr = Piecewise((Symbol('ON'), Eq(Symbol('mode'), Symbol('ON'))),
(Piecewise((Piecewise((Symbol('OFF'), Symbol('x') < Symbol('threshold')),
(Symbol('ON'), true)), Eq(Symbol('mode'), Symbol('AUTO'))),
(Symbol('OFF'), true)), true))
substitutions, new_eqn = cse(expr)
# this Piecewise should be exactly the same
assert new_eqn[0] == expr
# there should not be any replacements
assert len(substitutions) < 1
def test_cse_single2():
# Simple substitution, test for being able to pass the expression directly
e = Add(Pow(x + y, 2), sqrt(x + y))
substs, reduced = cse(e)
assert substs == [(x0, x + y)]
assert reduced == [sqrt(x0) + x0**2]
substs, reduced = cse(Matrix([[1]]))
assert isinstance(reduced[0], Matrix)
def test_bypass_non_commutatives():
A, B, C = symbols('A B C', commutative=False)
l = [A*B*C, A*C]
assert cse(l) == ([], l)
l = [A*B*C, A*B]
assert cse(l) == ([], l)
l = [B*C, A*B*C]
assert cse(l) == ([], l)
def test_cse_not_possible():
# No substitution possible.
e = Add(x, y)
substs, reduced = cse([e])
assert substs == []
assert reduced == [x + y]
# issue sympy/sympy#6329
eq = (meijerg((1, 2), (y, 4), (5,), [], x) +
meijerg((1, 3), (y, 4), (5,), [], x))
assert cse(eq) == ([], [eq])
def test_derivative_subs():
y = Symbol('y')
f = Function('f')
assert Derivative(f(x), x).subs({f(x): y}) != 0
assert Derivative(f(x), x).subs({f(x): y}).subs({y: f(x)}) == \
Derivative(f(x), x)
# issues sympy/sympy#5085, sympy/sympy#5037
assert cse(Derivative(f(x), x) + f(x))[1][0].has(Derivative)
assert cse(Derivative(f(x, y), x) +
Derivative(f(x, y), y))[1][0].has(Derivative)
def test_dont_cse_tuples():
f = Function("f")
g = Function("g")
name_val, (expr,) = cse(Subs(f(x, y), (x, 0), (y, 1)) +
Subs(g(x, y), (x, 0), (y, 1)))
assert name_val == []
assert expr == (Subs(f(x, y), (x, 0), (y, 1))
+ Subs(g(x, y), (x, 0), (y, 1)))
name_val, (expr,) = cse(Subs(f(x, y), (x, 0), (y, x + y)) +
Subs(g(x, y), (x, 0), (y, x + y)))
assert name_val == [(x0, x + y)]
assert expr == Subs(f(x, y), (x, 0), (y, x0)) + Subs(g(x, y), (x, 0), (y, x0))
def test_sympyissue_8891():
for cls in (MutableDenseMatrix, MutableSparseMatrix,
ImmutableDenseMatrix, ImmutableSparseMatrix):
m = cls(2, 2, [x + y, 0, 0, 0])
res = cse([x + y, m])
ans = ([(x0, x + y)], [x0, cls([[x0, 0], [0, 0]])])
assert res == ans
assert isinstance(res[1][-1], cls)
def test_sympyissue_6559():
assert (-12*x + y).subs({-x: 1}) == 12 + y
# though this involves cse it generated a failure in Mul._eval_subs
x0, x1 = symbols('x0 x1')
e = -log(-12*sqrt(2) + 17)/24 - log(-2*sqrt(2) + 3)/12 + sqrt(2)/3
# XXX modify cse so x1 is eliminated and x0 = -sqrt(2)?
assert cse(e) == (
[(x0, sqrt(2))], [x0/3 - log(-12*x0 + 17)/24 - log(-2*x0 + 3)/12])
def test_subtraction_opt():
# Make sure subtraction is optimized.
e = (x - y)*(z - y) + exp((x - y)*(z - y))
substs, reduced = cse(
[e], optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)])
assert substs == [(x0, (x - y)*(y - z))]
assert reduced == [-x0 + exp(-x0)]
e = -(x - y)*(z - y) + exp(-(x - y)*(z - y))
substs, reduced = cse(
[e], optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)])
assert substs == [(x0, (x - y)*(y - z))]
assert reduced == [x0 + exp(x0)]
# issue sympy/sympy#4077
n = -1 + 1/x
e = n/x/(-n)**2 - 1/n/x
assert cse(e, optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)]) == \
([], [0])
def test_cse_MatrixSymbol():
A = MatrixSymbol('A', 3, 3)
y = MatrixSymbol('y', 3, 1)
expr1 = (A.T*A).inverse() * A * y
expr2 = (A.T*A) * A * y
replacements, reduced_exprs = cse([expr1, expr2])
assert len(replacements) > 0
def test_cse_Indexed():
len_y = 5
y = IndexedBase('y', shape=(len_y,))
x = IndexedBase('x', shape=(len_y,))
i = Idx('i', len_y-1)
expr1 = (y[i+1]-y[i])/(x[i+1]-x[i])
expr2 = 1/(x[i+1]-x[i])
replacements, reduced_exprs = cse([expr1, expr2])
assert len(replacements) > 0
def test_pow_invpow():
assert cse(1/x**2 + x**2) == \
([(x0, x**2)], [x0 + 1/x0])
assert cse(x**2 + (1 + 1/x**2)/x**2) == \
([(x0, x**2), (x1, 1/x0)], [x0 + x1*(x1 + 1)])
assert cse(1/x**2 + (1 + 1/x**2)*x**2) == \
([(x0, x**2), (x1, 1/x0)], [x0*(x1 + 1) + x1])
assert cse(cos(1/x**2) + sin(1/x**2)) == \
([(x0, x**(-2))], [sin(x0) + cos(x0)])
assert cse(cos(x**2) + sin(x**2)) == \
([(x0, x**2)], [sin(x0) + cos(x0)])
assert cse(y/(2 + x**2) + z/x**2/y) == \
([(x0, x**2)], [y/(x0 + 2) + z/(x0*y)])
assert cse(exp(x**2) + x**2*cos(1/x**2)) == \
([(x0, x**2)], [x0*cos(1/x0) + exp(x0)])
assert cse((1 + 1/x**2)/x**2) == \
([(x0, x**(-2))], [x0*(x0 + 1)])
assert cse(x**(2*y) + x**(-2*y)) == \
([(x0, x**(2*y))], [x0 + 1/x0])
def test_sympyissue_4499():
# previously, this gave 16 constants
B = Function('B')
G = Function('G')
t = Tuple(*
(a, a + Rational(1, 2), 2*a, b, 2*a - b + 1, (sqrt(z)/2)**(-2*a + 1)*B(2*a -
b, sqrt(z))*B(b - 1, sqrt(z))*G(b)*G(2*a - b + 1),
sqrt(z)*(sqrt(z)/2)**(-2*a + 1)*B(b, sqrt(z))*B(2*a - b,
sqrt(z))*G(b)*G(2*a - b + 1), sqrt(z)*(sqrt(z)/2)**(-2*a + 1)*B(b - 1,
sqrt(z))*B(2*a - b + 1, sqrt(z))*G(b)*G(2*a - b + 1),
(sqrt(z)/2)**(-2*a + 1)*B(b, sqrt(z))*B(2*a - b + 1,
sqrt(z))*G(b)*G(2*a - b + 1), 1, 0, Rational(1, 2), z/2, -b + 1, -2*a + b,
-2*a))
c = cse(t)
ans = (
[(x0, 2*a), (x1, -b), (x2, x1 + 1), (x3, x0 + x2), (x4, sqrt(z)), (x5,
B(x0 + x1, x4)), (x6, G(b)), (x7, G(x3)), (x8, -x0), (x9,
(x4/2)**(x8 + 1)), (x10, x6*x7*x9*B(b - 1, x4)), (x11, x6*x7*x9*B(b,
x4)), (x12, B(x3, x4))], [(a, a + Rational(1, 2), x0, b, x3, x10*x5,
x11*x4*x5, x10*x12*x4, x11*x12, 1, 0, Rational(1, 2), z/2, x2, b + x8, x8)])
assert ans == c
def test_postprocess():
eq = (x + 1 + exp((x + 1) / (y + 1)) + cos(y + 1))
assert cse([eq, Eq(x, z + 1), z - 2, (z + 1)*(x + 1)],
postprocess=cse_main.cse_separate) == \
[[(x1, y + 1), (x2, z + 1), (x, x2), (x0, x + 1)],
[x0 + exp(x0/x1) + cos(x1), z - 2, x0*x2]]
def test_sympyissue_4203():
assert cse(sin(x**x) / x**x) == ([(x0, x**x)], [sin(x0) / x0])
def test_sympyissue_4498():
assert cse(w/(x - y) + z/(y - x), optimizations='basic') == \
([], [(w - z)/(x - y)])
def test_non_commutative_cse_mul():
x0 = symbols('x0', commutative=False)
A, B, C = symbols('A B C', commutative=False)
l = [A * B * C, A * B]
assert cse(l) == ([(x0, A * B)], [x0 * C, x0])
def test_multiple_expressions():
e1 = (x + y) * z
e2 = (x + y) * w
substs, reduced = cse([e1, e2])
assert substs == [(x0, x + y)]
assert reduced == [x0 * z, x0 * w]
l = [w * x * y + z, w * y]
substs, reduced = cse(l)
rsubsts, _ = cse(reversed(l))
assert substs == rsubsts
assert reduced == [z + x * x0, x0]
l = [w * x * y, w * x * y + z, w * y]
substs, reduced = cse(l)
rsubsts, _ = cse(reversed(l))
assert substs == rsubsts
assert reduced == [x1, x1 + z, x0]
l = [(x - z) * (y - z), x - z, y - z]
substs, reduced = cse(l)
rsubsts, _ = cse(reversed(l))
assert substs == [(x0, -z), (x1, x + x0), (x2, x0 + y)]
assert rsubsts == [(x0, -z), (x1, x0 + y), (x2, x + x0)]
assert reduced == [x1 * x2, x1, x2]
l = [w * y + w + x + y + z, w * x * y]
assert cse(l) == ([(x0, w * y)], [w + x + x0 + y + z, x * x0])
assert cse([x + y, x + y + z]) == ([(x0, x + y)], [x0, z + x0])
assert cse([x + y, x + z]) == ([], [x + y, x + z])
assert cse([x*y, z + x*y, x*y*z + 3]) == \
([(x0, x*y)], [x0, z + x0, 3 + x0*z])
def test_multiple_expressions():
e1 = (x + y)*z
e2 = (x + y)*w
substs, reduced = cse([e1, e2])
assert substs == [(x0, x + y)]
assert reduced == [x0*z, x0*w]
l = [w*x*y + z, w*y]
substs, reduced = cse(l)
rsubsts, _ = cse(reversed(l))
assert substs == rsubsts
assert reduced == [z + x*x0, x0]
l = [w*x*y, w*x*y + z, w*y]
substs, reduced = cse(l)
rsubsts, _ = cse(reversed(l))
assert substs == rsubsts
assert reduced == [x1, x1 + z, x0]
l = [(x - z)*(y - z), x - z, y - z]
substs, reduced = cse(l)
rsubsts, _ = cse(reversed(l))
assert substs == [(x0, -z), (x1, x + x0), (x2, x0 + y)]
assert rsubsts == [(x0, -z), (x1, x0 + y), (x2, x + x0)]
assert reduced == [x1*x2, x1, x2]
l = [w*y + w + x + y + z, w*x*y]
assert cse(l) == ([(x0, w*y)], [w + x + x0 + y + z, x*x0])
assert cse([x + y, x + y + z]) == ([(x0, x + y)], [x0, z + x0])
assert cse([x + y, x + z]) == ([], [x + y, x + z])
assert cse([x*y, z + x*y, x*y*z + 3]) == \
([(x0, x*y)], [x0, z + x0, 3 + x0*z])
def test_ignore_order_terms():
eq = exp(x).series(x, 0, 3) + sin(y + x**3) - 1
assert cse(eq) == ([], [sin(x**3 + y) + x + x**2/2 + O(x**3)])
def test_Piecewise():
f = Piecewise((-z + x*y, Eq(y, 0)), (-z - x*y, True))
ans = cse(f)
actual_ans = ([(x0, -z), (x1, x*y)], [Piecewise((x0+x1, Eq(y, 0)), (x0 - x1, True))])
assert ans == actual_ans
def test_sympyissue_6169():
r = RootOf(x**6 - 4*x**5 - 2, 1)
assert cse(r) == ([], [r])
# and a check that the right thing is done with the new
# mechanism
assert sub_post(sub_pre((-x - y)*z - x - y)) == -z*(x + y) - x - y
def test_postprocess():
eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1))
assert cse([eq, Eq(x, z + 1), z - 2, (z + 1)*(x + 1)],
postprocess=cse_main.cse_separate) == \
[[(x1, y + 1), (x2, z + 1), (x, x2), (x0, x + 1)],
[x0 + exp(x0/x1) + cos(x1), z - 2, x0*x2]]
def test_Piecewise():
f = Piecewise((-z + x * y, Eq(y, 0)), (-z - x * y, True))
ans = cse(f)
actual_ans = ([(x0, -z), (x1, x * y)],
[Piecewise((x0 + x1, Eq(y, 0)), (x0 - x1, True))])
assert ans == actual_ans
def test_non_commutative_order():
A, B, C = symbols('A B C', commutative=False)
x0 = symbols('x0', commutative=False)
l = [B+C, A*(B+C)]
assert cse(l) == ([(x0, B+C)], [x0, A*x0])
def test_sympyissue_6263():
e = Eq(x*(-x + 1) + x*(x - 1), 0)
assert cse(e, optimizations='basic') == ([], [True])
def test_cse_single():
# Simple substitution.
e = Add(Pow(x + y, 2), sqrt(x + y))
substs, reduced = cse([e])
assert substs == [(x0, x + y)]
assert reduced == [sqrt(x0) + x0**2]
def test_name_conflict():
z1 = x0 + y
z2 = x2 + x3
l = [cos(z1) + z1, cos(z2) + z2, x0 + x2]
substs, reduced = cse(l)
assert [e.subs(reversed(substs)) for e in reduced] == l
def test_sympyissue_4020():
assert cse(x**5 + x**4 + x**3 + x**2, optimizations='basic') \
== ([(x0, x**2)], [x0*(x**3 + x + x0 + 1)])
def test_symbols_exhausted_error():
l = cos(x+y)+x+y+cos(w+y)+sin(w+y)
sym = [x, y, z]
with pytest.raises(ValueError) as excinfo:
cse(l, symbols=sym)
def test_name_conflict_cust_symbols():
z1 = x0 + y
z2 = x2 + x3
l = [cos(z1) + z1, cos(z2) + z2, x0 + x2]
substs, reduced = cse(l, symbols("x:10"))
assert [e.subs(reversed(substs)) for e in reduced] == l
def test_nested_substitution():
# Substitution within a substitution.
e = Add(Pow(w*x + y, 2), sqrt(w*x + y))
substs, reduced = cse([e])
assert substs == [(x0, w*x + y)]
assert reduced == [sqrt(x0) + x0**2]
def test_symbols_exhausted_error():
l = cos(x+y)+x+y+cos(w+y)+sin(w+y)
sym = [x, y, z]
with pytest.raises(ValueError):
cse(l, symbols=sym)
def test_non_commutative_cse():
A, B, C = symbols('A B C', commutative=False)
l = [A * B * C, A * C]
assert cse(l) == ([], l)
def test_name_conflict():
z1 = x0 + y
z2 = x2 + x3
l = [cos(z1) + z1, cos(z2) + z2, x0 + x2]
substs, reduced = cse(l)
assert [e.subs(reversed(substs)) for e in reduced] == l
def test_powers():
assert cse(x * y**2 + x * y) == ([(x0, x * y)], [x0 * y + x0])
def test_sympyissue_4498():
assert cse(w/(x - y) + z/(y - x), optimizations='basic') == \
([], [(w - z)/(x - y)])
def test_sympyissue_4020():
assert cse(x**5 + x**4 + x**3 + x**2, optimizations='basic') \
== ([(x0, x**2)], [x0*(x**3 + x + x0 + 1)])
def test_powers():
assert cse(x*y**2 + x*y) == ([(x0, x*y)], [x0*y + x0])
def test_sympyissue_6263():
e = Eq(x * (-x + 1) + x * (x - 1), 0)
assert cse(e, optimizations='basic') == ([], [True])
def test_non_commutative_cse_mul():
x0 = symbols('x0', commutative=False)
A, B, C = symbols('A B C', commutative=False)
l = [A*B*C, A*B]
assert cse(l) == ([(x0, A*B)], [x0*C, x0])
def test_sympyissue_6169():
r = RootOf(x**6 - 4 * x**5 - 2, 1)
assert cse(r) == ([], [r])
# and a check that the right thing is done with the new
# mechanism
assert sub_post(sub_pre((-x - y) * z - x - y)) == -z * (x + y) - x - y
def test_nested_substitution():
# Substitution within a substitution.
e = Add(Pow(w*x + y, 2), sqrt(w*x + y))
substs, reduced = cse([e])
assert substs == [(x0, w*x + y)]
assert reduced == [sqrt(x0) + x0**2]
def test_ignore_order_terms():
eq = exp(x).series(x, 0, 3) + sin(y + x**3) - 1
assert cse(eq) == ([], [sin(x**3 + y) + x + x**2 / 2 + O(x**3)])
def test_sympyissue_4203():
assert cse(sin(x**x)/x**x) == ([(x0, x**x)], [sin(x0)/x0])
def test_name_conflict_cust_symbols():
z1 = x0 + y
z2 = x2 + x3
l = [cos(z1) + z1, cos(z2) + z2, x0 + x2]
substs, reduced = cse(l, symbols('x:10'))
assert [e.subs(reversed(substs)) for e in reduced] == l
def test_basic_optimization():
# issue sympy/sympy#4498
assert cse(w/(x - y) + z/(y - x), optimizations='basic') == \
([], [(w - z)/(x - y)])
