def test_diff():
f = Function('f')
g = Function('g')
assert simplify(g(x).diff(x)*f(x).diff(x) - f(x).diff(x)*g(x).diff(x)) == 0
assert simplify(2*f(x)*f(x).diff(x) - diff(f(x)**2, x)) == 0
assert simplify(diff(1/f(x), x) + f(x).diff(x)/f(x)**2) == 0
assert simplify(f(x).diff(x, y) - f(x).diff(y, x)) == 0
def test_simplify_expr():
A = Symbol('A')
f = Function('f')
assert all(simplify(tmp) == tmp for tmp in [I, E, oo, x, -x, -oo, -E, -I])
e = 1 / x + 1 / y
assert e != (x + y) / (x * y)
assert simplify(e) == (x + y) / (x * y)
e = A**2 * s**4 / (4 * pi * k * m**3)
assert simplify(e) == e
e = (4 + 4 * x - 2 * (2 + 2 * x)) / (2 + 2 * x)
assert simplify(e) == 0
e = (-4 * x * y**2 - 2 * y**3 - 2 * x**2 * y) / (x + y)**2
assert simplify(e) == -2 * y
e = -x - y - (x + y)**(-1) * y**2 + (x + y)**(-1) * x**2
assert simplify(e) == -2 * y
e = (x + x * y) / x
assert simplify(e) == 1 + y
e = (f(x) + y * f(x)) / f(x)
assert simplify(e) == 1 + y
e = (2 * (1 / n - cos(n * pi) / n)) / pi
assert simplify(e) == (-cos(pi * n) + 1) / (pi * n) * 2
e = integrate(1 / (x**3 + 1), x).diff(x)
assert simplify(e) == 1 / (x**3 + 1)
e = integrate(x / (x**2 + 3 * x + 1), x).diff(x)
assert simplify(e) == x / (x**2 + 3 * x + 1)
f = Symbol('f')
A = Matrix([[2 * k - m * w**2, -k], [-k, k - m * w**2]]).inv()
assert simplify((A*Matrix([0, f]))[1]) == \
-f*(2*k - m*w**2)/(k**2 - (k - m*w**2)*(2*k - m*w**2))
f = -x + y / (z + t) + z * x / (z + t) + z * a / (z + t) + t * x / (z + t)
assert simplify(f) == (y + a * z) / (z + t)
A, B = symbols('A,B', commutative=False)
assert simplify(A * B - B * A) == A * B - B * A
assert simplify(A / (1 + y / x)) == x * A / (x + y)
assert simplify(A * (1 / x + 1 / y)) == A / x + A / y # (x + y)*A/(x*y)
assert simplify(log(2) + log(3)) == log(6)
assert simplify(log(2 * x) - log(2)) == log(x)
assert simplify(hyper([], [], x)) == exp(x)
def test_separatevars():
n = Symbol('n')
assert separatevars(2 * n * x * z + 2 * x * y * z) == 2 * x * z * (n + y)
assert separatevars(x * z + x * y * z) == x * z * (1 + y)
assert separatevars(pi * x * z + pi * x * y * z) == pi * x * z * (1 + y)
assert separatevars(x*y**2*sin(x) + x*sin(x)*sin(y)) == \
x*(sin(y) + y**2)*sin(x)
assert separatevars(x * exp(x + y) +
x * exp(x)) == x * (1 + exp(y)) * exp(x)
assert separatevars((x * (y + 1))**z).is_Pow # != x**z*(1 + y)**z
assert separatevars(1 + x + y + x * y) == (x + 1) * (y + 1)
assert separatevars(y/pi*exp(-(z - x)/cos(n))) == \
y*exp(x/cos(n))*exp(-z/cos(n))/pi
assert separatevars((x + y) * (x - y) + y**2 + 2 * x + 1) == (x + 1)**2
# issue sympy/sympy#4858
p = Symbol('p', positive=True)
assert separatevars(sqrt(p**2 + x * p**2)) == p * sqrt(1 + x)
assert separatevars(sqrt(y * (p**2 + x * p**2))) == p * sqrt(y * (1 + x))
assert separatevars(sqrt(y*(p**2 + x*p**2)), force=True) == \
p*sqrt(y)*sqrt(1 + x)
# issue sympy/sympy#4865
assert separatevars(sqrt(x * y)).is_Pow
assert separatevars(sqrt(x * y), force=True) == sqrt(x) * sqrt(y)
# issue sympy/sympy#4957
# any type sequence for symbols is fine
assert separatevars(((2*x + 2)*y), dict=True, symbols=()) == \
{'coeff': 1, x: 2*x + 2, y: y}
# separable
assert separatevars(((2*x + 2)*y), dict=True, symbols=[x]) == \
{'coeff': y, x: 2*x + 2}
assert separatevars(((2*x + 2)*y), dict=True, symbols=[]) == \
{'coeff': 1, x: 2*x + 2, y: y}
assert separatevars(((2*x + 2)*y), dict=True) == \
{'coeff': 1, x: 2*x + 2, y: y}
assert separatevars(((2*x + 2)*y), dict=True, symbols=None) == \
{'coeff': y*(2*x + 2)}
# not separable
assert separatevars(3, dict=True) is None
assert separatevars(2 * x + y, dict=True, symbols=()) is None
assert separatevars(2 * x + y, dict=True) is None
assert separatevars(2 * x + y, dict=True, symbols=None) == {
'coeff': 2 * x + y
}
# issue sympy/sympy#4808
n, m = symbols('n,m', commutative=False)
assert separatevars(m + n * m) == (1 + n) * m
assert separatevars(x + x * n) == x * (1 + n)
# issue sympy/sympy#4910
f = Function('f')
assert separatevars(f(x) + x * f(x)) == f(x) + x * f(x)
# a noncommutable object present
eq = x * (1 + hyper((), (), y * z))
assert separatevars(eq) == eq
def test_sympyissue_6920():
e = [cos(x) + I*sin(x), cos(x) - I*sin(x),
cosh(x) - sinh(x), cosh(x) + sinh(x)]
ok = [exp(I*x), exp(-I*x), exp(-x), exp(x)]
# wrap in f to show that the change happens wherever ei occurs
f = Function('f')
assert [simplify(f(ei)).args[0] for ei in e] == ok
def test_collect_D_0():
D = Derivative
f = Function('f')
fxx = D(f(x), x, x)
# collect does not distinguish nested derivatives, so it returns
# -- (a + b)*D(D(f, x), x)
assert collect(a * fxx + b * fxx, fxx) == (a + b) * fxx
def test_collect_3():
"""Collect with respect to a product"""
f = Function('f')
assert collect(-x / 8 + x * y, -x) == x * (y - Rational(1, 8))
assert collect(1 + x * (y**2), x * y) == 1 + x * (y**2)
assert collect(x * y + a * x * y, x * y) == x * y * (1 + a)
assert collect(1 + x * y + a * x * y, x * y) == 1 + x * y * (1 + a)
assert collect(a * x * f(x) + b * (x * f(x)),
x * f(x)) == x * (a + b) * f(x)
assert collect(a * x * log(x) + b * (x * log(x)),
x * log(x)) == x * (a + b) * log(x)
assert collect(a*x**2*log(x)**2 + b*(x*log(x))**2, x*log(x)) == \
x**2*log(x)**2*(a + b)
# with respect to a product of three symbols
assert collect(y * x * z + a * x * y * z, x * y * z) == (1 + a) * x * y * z
def test_sympyissue_4194():
# simplify should call cancel
f = Function('f')
assert simplify((4 * x + 6 * f(y)) / (2 * x + 3 * f(y))) == 2
def test_collect_Wild():
"""Collect with respect to functions with Wild argument"""
f = Function('f')
w1 = Wild('.1')
w2 = Wild('.2')
assert collect(f(x) + a * f(x), f(w1)) == (1 + a) * f(x)
assert collect(f(x, y) + a * f(x, y), f(w1)) == f(x, y) + a * f(x, y)
assert collect(f(x, y) + a * f(x, y), f(w1, w2)) == (1 + a) * f(x, y)
assert collect(f(x, y) + a * f(x, y), f(w1, w1)) == f(x, y) + a * f(x, y)
assert collect(f(x, x) + a * f(x, x), f(w1, w1)) == (1 + a) * f(x, x)
assert collect(a * (x + 1)**y + (x + 1)**y, w1**y) == (1 + a) * (x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, w1**b) == \
a*(x + 1)**y + (x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, (x + 1)**w2) == \
(1 + a)*(x + 1)**y
assert collect(a * (x + 1)**y + (x + 1)**y, w1**w2) == (1 + a) * (x + 1)**y
def test_collect_issues():
D = Derivative
f = Function('f')
e = (1 + x * D(f(x), x) + D(f(x), x)) / f(x)
assert collect(e.expand(), f(x).diff(x)) != e
def test_collect_D():
D = Derivative
f = Function('f')
fx = D(f(x), x)
fxx = D(f(x), x, x)
assert collect(a * fx + b * fx, fx) == (a + b) * fx
assert collect(a * D(fx, x) + b * D(fx, x), fx) == (a + b) * D(fx, x)
assert collect(a * fxx + b * fxx, fx) == (a + b) * D(fx, x)
# issue sympy/sympy#4784
assert collect(5 * f(x) + 3 * fx, fx) == 5 * f(x) + 3 * fx
assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x), exact=True) == \
(x*f(x) + f(x))*D(f(x), x) + f(x)
assert collect(1/f(x) + 1/f(x)*diff(f(x), x) + x*diff(f(x), x)/f(x), f(x).diff(x), exact=True) == \
(1/f(x) + x/f(x))*D(f(x), x) + 1/f(x)
