def test_piecewise_interval():
p1 = Piecewise((x, Interval(0, 1).contains(x)), (0, True))
assert p1.subs({x: -0.5}) == 0
assert p1.subs({x: 0.5}) == 0.5
assert p1.diff(x) == Piecewise((1, Interval(0, 1).contains(x)), (0, True))
assert integrate(
p1, x) == Piecewise((x**2/2, Interval(0, 1).contains(x)), (0, True))
def test_piecewise_interval():
p1 = Piecewise((x, Interval(0, 1).contains(x)), (0, True))
assert p1.subs(x, -0.5) == 0
assert p1.subs(x, 0.5) == 0.5
assert p1.diff(x) == Piecewise((1, Interval(0, 1).contains(x)), (0, True))
assert integrate(p1, x) == Piecewise(
(x**2 / 2, Interval(0, 1).contains(x)), (0, True))
def test_piecewise_solve():
abs2 = Piecewise((-x, x <= 0), (x, x > 0))
f = abs2.subs(x, x - 2)
assert solve(f, x) == [2]
assert solve(f - 1, x) == [1, 3]
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert solve(f, x) == [2]
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert solve(g, x) == [2, 5]
g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
assert solve(g, x) == [2, 5]
g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (f, True))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (-x + 2, x - 2 <= 0),
(x - 2, x - 2 > 0))
assert solve(g, x) == [5]
def test_piecewise_solve():
abs2 = Piecewise((-x, x <= 0), (x, x > 0))
f = abs2.subs({x: x - 2})
assert solve(f, x) == [{x: 2}]
assert solve(f - 1, x) == [{x: 1}, {x: 3}]
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert solve(f, x) == [{x: 2}]
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert solve(g, x) == [{x: 2}, {x: 5}]
g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
assert solve(g, x) == [{x: 2}, {x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, True))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (-x + 2, x - 2 <= 0),
(x - 2, x - 2 > 0))
assert solve(g, x) == [{x: 5}]
def test_piecewise_solve():
abs2 = Piecewise((-x, x <= 0), (x, x > 0))
f = abs2.subs({x: x - 2})
assert solve(f, x) == [{x: 2}]
assert solve(f - 1, x) == [{x: 1}, {x: 3}]
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert solve(f, x) == [{x: 2}]
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert solve(g, x) == [{x: 2}, {x: 5}]
g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
assert solve(g, x) == [{x: 2}, {x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, True))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2),
(-x + 2, x - 2 <= 0), (x - 2, x - 2 > 0))
assert solve(g, x) == [{x: 5}]
def test_piecewise_fold_piecewise_in_cond():
p1 = Piecewise((cos(x), x < 0), (0, True))
p2 = Piecewise((0, Eq(p1, 0)), (p1 / Abs(p1), True))
p3 = piecewise_fold(p2)
assert (p2.subs(x, -pi / 2) == 0.0)
assert (p2.subs(x, 1) == 0.0)
assert (p2.subs(x, -pi / 4) == 1.0)
p4 = Piecewise((0, Eq(p1, 0)), (1, True))
assert (piecewise_fold(p4) == Piecewise(
(0, Or(And(Eq(cos(x), 0), x < 0), Not(x < 0))), (1, True)))
r1 = 1 < Piecewise((1, x < 1), (3, True))
assert (piecewise_fold(r1) == Not(x < 1))
p5 = Piecewise((1, x < 0), (3, True))
p6 = Piecewise((1, x < 1), (3, True))
p7 = piecewise_fold(Piecewise((1, p5 < p6), (0, True)))
assert (Piecewise((1, And(Not(x < 1), x < 0)), (0, True)))
def test_piecewise_fold_piecewise_in_cond():
p1 = Piecewise((cos(x), x < 0), (0, True))
p2 = Piecewise((0, Eq(p1, 0)), (p1 / Abs(p1), True))
assert(p2.subs({x: -pi/2}) == 0.0)
assert(p2.subs({x: 1}) == 0.0)
assert(p2.subs({x: -pi/4}) == 1.0)
p4 = Piecewise((0, Eq(p1, 0)), (1, True))
assert(piecewise_fold(p4) == Piecewise(
(0, Or(And(Eq(cos(x), 0), x < 0), Not(x < 0))), (1, True)))
r1 = 1 < Piecewise((1, x < 1), (3, True))
assert(piecewise_fold(r1) == Not(x < 1))
p5 = Piecewise((1, x < 0), (3, True))
p6 = Piecewise((1, x < 1), (3, True))
p7 = piecewise_fold(Piecewise((1, p5 < p6), (0, True)))
assert p7
assert Piecewise((1, And(Not(x < 1), x < 0)), (0, True))
def test_piecewise_fold_piecewise_in_cond():
p1 = Piecewise((cos(x), x < 0), (0, True))
p2 = Piecewise((0, Eq(p1, 0)), (p1 / abs(p1), True))
assert p2.subs({x: -pi / 2}) == 0.0
assert p2.subs({x: 1}) == 0.0
assert p2.subs({x: -pi / 4}) == 1.0
p4 = Piecewise((0, Eq(p1, 0)), (1, True))
assert (piecewise_fold(p4) == Piecewise(
(0, Or(And(Eq(cos(x), 0), x < 0), Not(x < 0))), (1, True)))
r1 = 1 < Piecewise((1, x < 1), (3, True))
assert (piecewise_fold(r1) == Not(x < 1))
p5 = Piecewise((1, x < 0), (3, True))
p6 = Piecewise((1, x < 1), (3, True))
p7 = piecewise_fold(Piecewise((1, p5 < p6), (0, True)))
assert p7
assert Piecewise((1, And(Not(x < 1), x < 0)), (0, True))
def test_Function_subs():
f, g, h, i = symbols('f g h i', cls=Function)
p = Piecewise((g(f(x, y)), x < -1), (g(x), x <= 1))
assert p.subs({g: h}) == Piecewise((h(f(x, y)), x < -1), (h(x), x <= 1))
assert (f(y) + g(x)).subs({f: h, g: i}) == i(x) + h(y)
def test_Function_subs():
from diofant.abc import x, y
f, g, h, i = symbols('f g h i', cls=Function)
p = Piecewise((g(f(x, y)), x < -1), (g(x), x <= 1))
assert p.subs(g, h) == Piecewise((h(f(x, y)), x < -1), (h(x), x <= 1))
assert (f(y) + g(x)).subs({f: h, g: i}) == i(x) + h(y)
def test_piecewise():
# Test canonization
assert Piecewise((x, x < 1), (0, True)) == Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, True), (1, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, False), (-1, 1 > 2)) == \
Piecewise((x, x < 1))
assert Piecewise((x, x < 1), (0, x < 1), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, x < 2), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (0, True)) == \
Piecewise((x, Or(x < 1, x < 2)), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (x, True)) == x
assert Piecewise((x, True)) == x
pytest.raises(TypeError, lambda: Piecewise(x))
pytest.raises(TypeError, lambda: Piecewise((x, x**2)))
# Test subs
p = Piecewise((-1, x < -1), (x**2, x < 0), (log(x), x >= 0))
p_x2 = Piecewise((-1, x**2 < -1), (x**4, x**2 < 0), (log(x**2), x**2 >= 0))
assert p.subs(x, x**2) == p_x2
assert p.subs(x, -5) == -1
assert p.subs(x, -1) == 1
assert p.subs(x, 1) == log(1)
# More subs tests
p2 = Piecewise((1, x < pi), (-1, x < 2 * pi), (0, x > 2 * pi))
p3 = Piecewise((1, Eq(x, 0)), (1 / x, True))
p4 = Piecewise((1, Eq(x, 0)), (2, 1 / x > 2))
assert p2.subs(x, 2) == 1
assert p2.subs(x, 4) == -1
assert p2.subs(x, 10) == 0
assert p3.subs(x, 0.0) == 1
assert p4.subs(x, 0.0) == 1
f, g, h = symbols('f,g,h', cls=Function)
pf = Piecewise((f(x), x < -1), (f(x) + h(x) + 2, x <= 1))
pg = Piecewise((g(x), x < -1), (g(x) + h(x) + 2, x <= 1))
assert pg.subs(g, f) == pf
assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 0) == 1
assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 1) == 0
assert Piecewise((1, Eq(x, y)), (0, True)).subs(x, y) == 1
assert Piecewise((1, Eq(x, z)), (0, True)).subs(x, z) == 1
assert Piecewise((1, Eq(exp(x), cos(z))), (0, True)).subs(x, z) == \
Piecewise((1, Eq(exp(z), cos(z))), (0, True))
assert Piecewise((1, Eq(x, y * (y + 1))),
(0, True)).subs(x, y**2 + y).simplify() == 1
p5 = Piecewise((0, Eq(cos(x) + y, 0)), (1, True))
assert p5.subs(y, 0) == Piecewise((0, Eq(cos(x), 0)), (1, True))
# Test evalf
assert p.evalf() == p
assert p.evalf(subs={x: -2}) == -1
assert p.evalf(subs={x: -1}) == 1
assert p.evalf(subs={x: 1}) == log(1)
# Test doit
f_int = Piecewise((Integral(x, (x, 0, 1)), x < 1))
assert f_int.doit() == Piecewise((1.0 / 2.0, x < 1))
# Test differentiation
f = x
fp = x * p
dp = Piecewise((0, x < -1), (2 * x, x < 0), (1 / x, x >= 0))
fp_dx = x * dp + p
assert diff(p, x) == dp
assert diff(f * p, x) == fp_dx
# Test simple arithmetic
assert x * p == fp
assert x * p + p == p + x * p
assert p + f == f + p
assert p + dp == dp + p
assert p - dp == -(dp - p)
# Test power
dp2 = Piecewise((0, x < -1), (4 * x**2, x < 0), (1 / x**2, x >= 0))
assert dp**2 == dp2
# Test _eval_interval
f1 = x * y + 2
f2 = x * y**2 + 3
peval = Piecewise((f1, x < 0), (f2, x > 0))
peval_interval = f1.subs(x, 0) - f1.subs(x, -1) + f2.subs(x, 1) - f2.subs(
x, 0)
assert peval._eval_interval(x, 0, 0) == 0
assert peval._eval_interval(x, -1, 1) == peval_interval
peval2 = Piecewise((f1, x < 0), (f2, True))
assert peval2._eval_interval(x, 0, 0) == 0
assert peval2._eval_interval(x, 1, -1) == -peval_interval
assert peval2._eval_interval(x, -1, -2) == f1.subs(x, -2) - f1.subs(x, -1)
assert peval2._eval_interval(x, -1, 1) == peval_interval
assert peval2._eval_interval(x, None, 0) == peval2.subs(x, 0)
assert peval2._eval_interval(x, -1, None) == -peval2.subs(x, -1)
# Test integration
p_int = Piecewise((-x, x < -1), (x**3 / 3.0, x < 0),
(-x + x * log(x), x >= 0))
assert integrate(p, x) == p_int
p = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
assert integrate(p, (x, -2, 2)) == 5.0 / 6.0
assert integrate(p, (x, 2, -2)) == -5.0 / 6.0
p = Piecewise((0, x < 0), (1, x < 1), (0, x < 2), (1, x < 3), (0, True))
assert integrate(p, (x, -oo, oo)) == 2
p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
pytest.raises(ValueError, lambda: integrate(p, (x, -2, 2)))
# Test commutativity
assert p.is_commutative is True
def test_piecewise():
# Test canonization
assert Piecewise((x, x < 1), (0, True)) == Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, True), (1, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, False), (-1, 1 > 2)) == \
Piecewise((x, x < 1))
assert Piecewise((x, x < 1), (0, x < 1), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, x < 2), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (0, True)) == \
Piecewise((x, Or(x < 1, x < 2)), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (x, True)) == x
assert Piecewise((x, True)) == x
pytest.raises(TypeError, lambda: Piecewise(x))
pytest.raises(TypeError, lambda: Piecewise((x, x**2)))
assert Piecewise((0, Eq(z, 0, evaluate=False)), (1, True)) == 1
# Test subs
p = Piecewise((-1, x < -1), (x**2, x < 0), (log(x), x >= 0))
p_x2 = Piecewise((-1, x**2 < -1), (x**4, x**2 < 0), (log(x**2), x**2 >= 0))
assert p.subs({x: x**2}) == p_x2
assert p.subs({x: -5}) == -1
assert p.subs({x: -1}) == 1
assert p.subs({x: 1}) == log(1)
# More subs tests
p2 = Piecewise((1, x < pi), (-1, x < 2*pi), (0, x > 2*pi))
p3 = Piecewise((1, Eq(x, 0)), (1/x, True))
p4 = Piecewise((1, Eq(x, 0)), (2, 1/x > 2))
assert p2.subs({x: 2}) == 1
assert p2.subs({x: 4}) == -1
assert p2.subs({x: 10}) == 0
assert p3.subs({x: 0.0}) == 1
assert p4.subs({x: 0.0}) == 1
f, g, h = symbols('f,g,h', cls=Function)
pf = Piecewise((f(x), x < -1), (f(x) + h(x) + 2, x <= 1))
pg = Piecewise((g(x), x < -1), (g(x) + h(x) + 2, x <= 1))
assert pg.subs({g: f}) == pf
assert Piecewise((1, Eq(x, 0)), (0, True)).subs({x: 0}) == 1
assert Piecewise((1, Eq(x, 0)), (0, True)).subs({x: 1}) == 0
assert Piecewise((1, Eq(x, y)), (0, True)).subs({x: y}) == 1
assert Piecewise((1, Eq(x, z)), (0, True)).subs({x: z}) == 1
assert Piecewise((1, Eq(exp(x), cos(z))), (0, True)).subs({x: z}) == \
Piecewise((1, Eq(exp(z), cos(z))), (0, True))
assert Piecewise((1, Eq(x, y*(y + 1))),
(0, True)).subs({x: y**2 + y}).simplify() == 1
p5 = Piecewise( (0, Eq(cos(x) + y, 0)), (1, True))
assert p5.subs({y: 0}) == Piecewise( (0, Eq(cos(x), 0)), (1, True))
# Test evalf
assert p.evalf() == p
assert p.evalf(subs={x: -2}) == -1
assert p.evalf(subs={x: -1}) == 1
assert p.evalf(subs={x: 1}) == log(1)
# Test doit
f_int = Piecewise((Integral(x, (x, 0, 1)), x < 1))
assert f_int.doit() == Piecewise( (1.0/2.0, x < 1) )
# Test differentiation
f = x
fp = x*p
dp = Piecewise((0, x < -1), (2*x, x < 0), (1/x, x >= 0))
fp_dx = x*dp + p
assert diff(p, x) == dp
assert diff(f*p, x) == fp_dx
# Test simple arithmetic
assert x*p == fp
assert x*p + p == p + x*p
assert p + f == f + p
assert p + dp == dp + p
assert p - dp == -(dp - p)
# Test power
dp2 = Piecewise((0, x < -1), (4*x**2, x < 0), (1/x**2, x >= 0))
assert dp**2 == dp2
# Test _eval_interval
f1 = x*y + 2
f2 = x*y**2 + 3
peval = Piecewise( (f1, x < 0), (f2, x > 0))
peval_interval = (f1.subs({x: 0}) - f1.subs({x: -1}) +
f2.subs({x: 1}) - f2.subs({x: 0}))
assert peval._eval_interval(x, 0, 0) == 0
assert peval._eval_interval(x, -1, 1) == peval_interval
peval2 = Piecewise((f1, x < 0), (f2, True))
assert peval2._eval_interval(x, 0, 0) == 0
assert peval2._eval_interval(x, 1, -1) == -peval_interval
assert peval2._eval_interval(x, -1, -2) == (f1.subs({x: -2}) -
f1.subs({x: -1}))
assert peval2._eval_interval(x, -1, 1) == peval_interval
assert peval2._eval_interval(x, None, 0) == peval2.subs({x: 0})
assert peval2._eval_interval(x, -1, None) == -peval2.subs({x: -1})
# Test integration
p_int = Piecewise((-x, x < -1), (x**3/3.0, x < 0), (-x + x*log(x), x >= 0))
assert integrate(p, x) == p_int
p = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
assert integrate(p, (x, -2, 2)) == 5.0/6.0
assert integrate(p, (x, 2, -2)) == -5.0/6.0
p = Piecewise((0, x < 0), (1, x < 1), (0, x < 2), (1, x < 3), (0, True))
assert integrate(p, (x, -oo, oo)) == 2
p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
pytest.raises(ValueError, lambda: integrate(p, (x, -2, 2)))
# Test commutativity
assert p.is_commutative is True
def test_Function_subs():
f, g, h, i = symbols('f g h i', cls=Function)
p = Piecewise((g(f(x, y)), x < -1), (g(x), x <= 1))
assert p.subs({g: h}) == Piecewise((h(f(x, y)), x < -1), (h(x), x <= 1))
assert (f(y) + g(x)).subs({f: h, g: i}) == i(x) + h(y)
