def test_issue_7535():
assert limit(tan(x)/sin(tan(x)), x, pi/2) == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='+')
assert limit(tan(x)/sin(tan(x)), x, pi/2, dir='-') == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='-')
assert limit(tan(x)/sin(tan(x)), x, pi/2, dir='+-') == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='+-')
assert limit(sin(tan(x)),x,pi/2) == AccumBounds(-1, 1)
assert -oo*(1/sin(-oo)) == AccumBounds(-oo, oo)
assert oo*(1/sin(oo)) == AccumBounds(-oo, oo)
assert oo*(1/sin(-oo)) == AccumBounds(-oo, oo)
assert -oo*(1/sin(oo)) == AccumBounds(-oo, oo)
def eval(cls, arg):
from sympy import atan
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg is S.Infinity:
return -S.ImaginaryUnit * atan(arg)
elif arg is S.NegativeInfinity:
return S.ImaginaryUnit * atan(-arg)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
from sympy.calculus.util import AccumBounds
return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * atan(i_coeff)
else:
if arg._coeff_isneg():
return -cls(-arg)
def test_issue_18795():
r = Symbol('r', real=True)
a = AccumBounds(-1,1)
c = AccumBounds(7, oo)
b = AccumBounds(-oo, oo)
assert c - tan(r) == AccumBounds(7-tan(r), oo)
assert b + tan(r) == AccumBounds(-oo, oo)
assert (a + r)/a == AccumBounds(-oo, oo)*AccumBounds(r - 1, r + 1)
assert (b + a)/a == AccumBounds(-oo, oo)
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Pi * S.ImaginaryUnit / 2
elif arg is S.NegativeInfinity:
return S.Pi * S.ImaginaryUnit / 2
elif arg.is_zero:
return S.Infinity
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi * S.ImaginaryUnit
if arg.is_number:
cst_table = {
S.ImaginaryUnit:
-(S.Pi * S.ImaginaryUnit / 2) + log(1 + sqrt(2)),
-S.ImaginaryUnit:
(S.Pi * S.ImaginaryUnit / 2) + log(1 + sqrt(2)),
(sqrt(6) - sqrt(2)): S.Pi / 12,
(sqrt(2) - sqrt(6)): 11 * S.Pi / 12,
sqrt(2 - 2 / sqrt(5)): S.Pi / 10,
-sqrt(2 - 2 / sqrt(5)): 9 * S.Pi / 10,
2 / sqrt(2 + sqrt(2)): S.Pi / 8,
-2 / sqrt(2 + sqrt(2)): 7 * S.Pi / 8,
2 / sqrt(3): S.Pi / 6,
-2 / sqrt(3): 5 * S.Pi / 6,
(sqrt(5) - 1): S.Pi / 5,
(1 - sqrt(5)): 4 * S.Pi / 5,
sqrt(2): S.Pi / 4,
-sqrt(2): 3 * S.Pi / 4,
sqrt(2 + 2 / sqrt(5)): 3 * S.Pi / 10,
-sqrt(2 + 2 / sqrt(5)): 7 * S.Pi / 10,
S(2): S.Pi / 3,
-S(2): 2 * S.Pi / 3,
sqrt(2 * (2 + sqrt(2))): 3 * S.Pi / 8,
-sqrt(2 * (2 + sqrt(2))): 5 * S.Pi / 8,
(1 + sqrt(5)): 2 * S.Pi / 5,
(-1 - sqrt(5)): 3 * S.Pi / 5,
(sqrt(6) + sqrt(2)): 5 * S.Pi / 12,
(-sqrt(6) - sqrt(2)): 7 * S.Pi / 12,
}
if arg in cst_table:
if arg.is_extended_real:
return cst_table[arg] * S.ImaginaryUnit
return cst_table[arg]
if arg is S.ComplexInfinity:
from sympy.calculus.util import AccumBounds
return S.ImaginaryUnit * AccumBounds(-S.Pi / 2, S.Pi / 2)
if arg.is_zero:
return S.Infinity
def test_contains_AccumBounds():
assert (1 in AccumBounds(1, 2)) == S.true
raises(TypeError, lambda: a in AccumBounds(1, 2))
assert 0 in AccumBounds(-1, 0)
raises(TypeError, lambda:
(cos(1)**2 + sin(1)**2 - 1) in AccumBounds(-1, 0))
assert (-oo in AccumBounds(1, oo)) == S.true
assert (oo in AccumBounds(-oo, 0)) == S.true
# issue 13159
assert Mul(0, AccumBounds(-1, 1)) == Mul(AccumBounds(-1, 1), 0) == 0
import itertools
for perm in itertools.permutations([0, AccumBounds(-1, 1), x]):
assert Mul(*perm) == 0
def test_contains_AccumBounds():
assert (1 in AccumBounds(1, 2)) == S.true
raises(TypeError, lambda: a in AccumBounds(1, 2))
assert 0 in AccumBounds(-1, 0)
raises(TypeError, lambda:
(cos(1)**2 + sin(1)**2 - 1) in AccumBounds(-1, 0))
assert (-oo in AccumBounds(1, oo)) == S.true
assert (oo in AccumBounds(-oo, 0)) == S.true
# issue 13159
assert Mul(0, AccumBounds(-1, 1)) == Mul(AccumBounds(-1, 1), 0) == 0
def test_issue_19770():
m = Symbol('m')
# the result is not 0 for non-real m
assert limit(cos(m*x)/x, x, oo) == Limit(cos(m*x)/x, x, oo, dir='-')
m = Symbol('m', real=True)
# can be improved to give the correct result 0
assert limit(cos(m*x)/x, x, oo) == Limit(cos(m*x)/x, x, oo, dir='-')
m = Symbol('m', nonzero=True)
assert limit(cos(m*x), x, oo) == AccumBounds(-1, 1)
assert limit(cos(m*x)/x, x, oo) == 0
def _eval(arg):
if arg is S.Infinity or arg is S.NegativeInfinity:
return AccumBounds(0, 1)
if arg.is_integer:
return S.Zero
if arg.is_number:
if arg is S.NaN:
return S.NaN
elif arg is S.ComplexInfinity:
return S.NaN
else:
return arg - floor(arg)
return cls(arg, evaluate=False)
def eval(cls, arg):
from sympy import atan
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.Zero
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg is S.Infinity:
return -S.ImaginaryUnit * atan(arg)
elif arg is S.NegativeInfinity:
return S.ImaginaryUnit * atan(-arg)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
from sympy.calculus.util import AccumBounds
return S.ImaginaryUnit * AccumBounds(-S.Pi / 2, S.Pi / 2)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * atan(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.is_zero:
return S.Zero
if isinstance(arg, tanh) and arg.args[0].is_number:
z = arg.args[0]
if z.is_real:
return z
r, i = match_real_imag(z)
if r is not None and i is not None:
f = floor(2 * i / pi)
even = f.is_even
m = z - I * f * pi / 2
if even is True:
return m
elif even is False:
return m - I * pi / 2
def _eval_nseries(self, x, n, logx, cdir=0):
arg = self.args[0]
arg0 = arg.subs(x, 0)
if arg0.is_infinite:
from sympy.calculus.util import AccumBounds
from sympy.series.order import Order
s = arg._eval_nseries(x, n, logx, cdir)
o = Order(1, (x, 0)) if n <= 0 else AccumBounds(0, 1)
return s + o
r = self.subs(x, 0)
if arg0 == r:
ndir = arg.dir(x, cdir=cdir if cdir != 0 else 1)
return r if ndir.is_negative else r + 1
else:
return r
def test_ceiling():
assert limit(ceiling(x), x, -2, "+") == -1
assert limit(ceiling(x), x, -2, "-") == -2
assert limit(ceiling(x), x, -1, "+") == 0
assert limit(ceiling(x), x, -1, "-") == -1
assert limit(ceiling(x), x, 0, "+") == 1
assert limit(ceiling(x), x, 0, "-") == 0
assert limit(ceiling(x), x, 1, "+") == 2
assert limit(ceiling(x), x, 1, "-") == 1
assert limit(ceiling(x), x, 2, "+") == 3
assert limit(ceiling(x), x, 2, "-") == 2
assert limit(ceiling(x), x, 248, "+") == 249
assert limit(ceiling(x), x, 248, "-") == 248
# https://github.com/sympy/sympy/issues/14478
assert limit(x*ceiling(3/x)/2, x, 0, '+') == Rational(3, 2)
assert limit(ceiling(x + 1/2) - ceiling(x), x, oo) == AccumBounds(-0.5, 1.5)
def test_floor():
assert limit(floor(x), x, -2, "+") == -2
assert limit(floor(x), x, -2, "-") == -3
assert limit(floor(x), x, -1, "+") == -1
assert limit(floor(x), x, -1, "-") == -2
assert limit(floor(x), x, 0, "+") == 0
assert limit(floor(x), x, 0, "-") == -1
assert limit(floor(x), x, 1, "+") == 1
assert limit(floor(x), x, 1, "-") == 0
assert limit(floor(x), x, 2, "+") == 2
assert limit(floor(x), x, 2, "-") == 1
assert limit(floor(x), x, 248, "+") == 248
assert limit(floor(x), x, 248, "-") == 247
# https://github.com/sympy/sympy/issues/14478
assert limit(x*floor(3/x)/2, x, 0, '+') == Rational(3, 2)
assert limit(floor(x + 1/2) - floor(x), x, oo) == AccumBounds(-0.5, 1.5)
def test_series_AccumBounds():
assert limit(sin(k) - sin(k + 1), k, oo) == AccumBounds(-2, 2)
assert limit(cos(k) - cos(k + 1) + 1, k, oo) == AccumBounds(-1, 3)
# not the exact bound
assert limit(sin(k) - sin(k)*cos(k), k, oo) == AccumBounds(-2, 2)
# test for issue #9934
t1 = Mul(S.Half, 1/(-1 + cos(1)), Add(AccumBounds(-3, 1), cos(1)))
assert limit(simplify(Sum(cos(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k, oo) == t1
t2 = Mul(S.Half, Add(AccumBounds(-2, 2), sin(1)), 1/(-cos(1) + 1))
assert limit(simplify(Sum(sin(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k, oo) == t2
assert limit(frac(x)**x, x, oo) == AccumBounds(0, oo) # wolfram gives (0, 1)
assert limit(((sin(x) + 1)/2)**x, x, oo) == AccumBounds(0, oo) # wolfram says 0
def _eval_nseries(self, x, n, logx, cdir=0):
arg = self.args[0]
arg0 = arg.subs(x, 0)
if arg0.is_infinite:
from sympy.calculus.util import AccumBounds
from sympy.series.order import Order
s = arg._eval_nseries(x, n, logx, cdir)
o = Order(1, (x, 0)) if n <= 0 else AccumBounds(0, 1)
return s + o
r = self.subs(x, 0)
if arg0 == r:
direction = (arg - arg0).leadterm(x)[0]
if direction.is_positive:
return r + 1
else:
return r
else:
return r
def test_AccumBounds_pow():
assert AccumBounds(0, 2)**2 == AccumBounds(0, 4)
assert AccumBounds(-1, 1)**2 == AccumBounds(0, 1)
assert AccumBounds(1, 2)**2 == AccumBounds(1, 4)
assert AccumBounds(-1, 2)**3 == AccumBounds(-1, 8)
assert AccumBounds(-1, 1)**0 == 1
assert AccumBounds(1, 2)**Rational(5, 2) == AccumBounds(1, 4*sqrt(2))
assert AccumBounds(-1, 2)**Rational(1, 3) == AccumBounds(-1, 2**Rational(1, 3))
assert AccumBounds(0, 2)**S.Half == AccumBounds(0, sqrt(2))
assert AccumBounds(-4, 2)**Rational(2, 3) == AccumBounds(0, 2*2**Rational(1, 3))
assert AccumBounds(-1, 5)**S.Half == AccumBounds(0, sqrt(5))
assert AccumBounds(-oo, 2)**S.Half == AccumBounds(0, sqrt(2))
assert AccumBounds(-2, 3)**Rational(-1, 4) == AccumBounds(0, oo)
assert AccumBounds(1, 5)**(-2) == AccumBounds(Rational(1, 25), 1)
assert AccumBounds(-1, 3)**(-2) == AccumBounds(0, oo)
assert AccumBounds(0, 2)**(-2) == AccumBounds(Rational(1, 4), oo)
assert AccumBounds(-1, 2)**(-3) == AccumBounds(-oo, oo)
assert AccumBounds(-3, -2)**(-3) == AccumBounds(Rational(-1, 8), Rational(-1, 27))
assert AccumBounds(-3, -2)**(-2) == AccumBounds(Rational(1, 9), Rational(1, 4))
assert AccumBounds(0, oo)**S.Half == AccumBounds(0, oo)
assert AccumBounds(-oo, -1)**Rational(1, 3) == AccumBounds(-oo, -1)
assert AccumBounds(-2, 3)**(Rational(-1, 3)) == AccumBounds(-oo, oo)
assert AccumBounds(-oo, 0)**(-2) == AccumBounds(0, oo)
assert AccumBounds(-2, 0)**(-2) == AccumBounds(Rational(1, 4), oo)
assert AccumBounds(Rational(1, 3), S.Half)**oo is S.Zero
assert AccumBounds(0, S.Half)**oo is S.Zero
assert AccumBounds(S.Half, 1)**oo == AccumBounds(0, oo)
assert AccumBounds(0, 1)**oo == AccumBounds(0, oo)
assert AccumBounds(2, 3)**oo is oo
assert AccumBounds(1, 2)**oo == AccumBounds(0, oo)
assert AccumBounds(S.Half, 3)**oo == AccumBounds(0, oo)
assert AccumBounds(Rational(-1, 3), Rational(-1, 4))**oo is S.Zero
assert AccumBounds(-1, Rational(-1, 2))**oo == AccumBounds(-oo, oo)
assert AccumBounds(-3, -2)**oo == FiniteSet(-oo, oo)
assert AccumBounds(-2, -1)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-2, Rational(-1, 2))**oo == AccumBounds(-oo, oo)
assert AccumBounds(Rational(-1, 2), S.Half)**oo is S.Zero
assert AccumBounds(Rational(-1, 2), 1)**oo == AccumBounds(0, oo)
assert AccumBounds(Rational(-2, 3), 2)**oo == AccumBounds(0, oo)
assert AccumBounds(-1, 1)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-1, S.Half)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-1, 2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-2, S.Half)**oo == AccumBounds(-oo, oo)
assert AccumBounds(1, 2)**x == Pow(AccumBounds(1, 2), x)
assert AccumBounds(2, 3)**(-oo) is S.Zero
assert AccumBounds(0, 2)**(-oo) == AccumBounds(0, oo)
assert AccumBounds(-1, 2)**(-oo) == AccumBounds(-oo, oo)
assert (tan(x)**sin(2*x)).subs(x, AccumBounds(0, pi/2)) == \
Pow(AccumBounds(-oo, oo), AccumBounds(0, 1))
def test_AccumBounds_func():
assert (x**2 + 2*x + 1).subs(x, AccumBounds(-1, 1)) == AccumBounds(-1, 4)
assert exp(AccumBounds(0, 1)) == AccumBounds(1, E)
assert exp(AccumBounds(-oo, oo)) == AccumBounds(0, oo)
assert log(AccumBounds(3, 6)) == AccumBounds(log(3), log(6))
def test_AccumBounds_div():
assert AccumBounds(-1, 3)/AccumBounds(3, 4) == AccumBounds(Rational(-1, 3), 1)
assert AccumBounds(-2, 4)/AccumBounds(-3, 4) == AccumBounds(-oo, oo)
assert AccumBounds(-3, -2)/AccumBounds(-4, 0) == AccumBounds(S.Half, oo)
# these two tests can have a better answer
# after Union of AccumBounds is improved
assert AccumBounds(-3, -2)/AccumBounds(-2, 1) == AccumBounds(-oo, oo)
assert AccumBounds(2, 3)/AccumBounds(-2, 2) == AccumBounds(-oo, oo)
assert AccumBounds(-3, -2)/AccumBounds(0, 4) == AccumBounds(-oo, Rational(-1, 2))
assert AccumBounds(2, 4)/AccumBounds(-3, 0) == AccumBounds(-oo, Rational(-2, 3))
assert AccumBounds(2, 4)/AccumBounds(0, 3) == AccumBounds(Rational(2, 3), oo)
assert AccumBounds(0, 1)/AccumBounds(0, 1) == AccumBounds(0, oo)
assert AccumBounds(-1, 0)/AccumBounds(0, 1) == AccumBounds(-oo, 0)
assert AccumBounds(-1, 2)/AccumBounds(-2, 2) == AccumBounds(-oo, oo)
assert 1/AccumBounds(-1, 2) == AccumBounds(-oo, oo)
assert 1/AccumBounds(0, 2) == AccumBounds(S.Half, oo)
assert (-1)/AccumBounds(0, 2) == AccumBounds(-oo, Rational(-1, 2))
assert 1/AccumBounds(-oo, 0) == AccumBounds(-oo, 0)
assert 1/AccumBounds(-1, 0) == AccumBounds(-oo, -1)
assert (-2)/AccumBounds(-oo, 0) == AccumBounds(0, oo)
assert 1/AccumBounds(-oo, -1) == AccumBounds(-1, 0)
assert AccumBounds(1, 2)/a == Mul(AccumBounds(1, 2), 1/a, evaluate=False)
assert AccumBounds(1, 2)/0 == AccumBounds(1, 2)*zoo
assert AccumBounds(1, oo)/oo == AccumBounds(0, oo)
assert AccumBounds(1, oo)/(-oo) == AccumBounds(-oo, 0)
assert AccumBounds(-oo, -1)/oo == AccumBounds(-oo, 0)
assert AccumBounds(-oo, -1)/(-oo) == AccumBounds(0, oo)
assert AccumBounds(-oo, oo)/oo == AccumBounds(-oo, oo)
assert AccumBounds(-oo, oo)/(-oo) == AccumBounds(-oo, oo)
assert AccumBounds(-1, oo)/oo == AccumBounds(0, oo)
assert AccumBounds(-1, oo)/(-oo) == AccumBounds(-oo, 0)
assert AccumBounds(-oo, 1)/oo == AccumBounds(-oo, 0)
assert AccumBounds(-oo, 1)/(-oo) == AccumBounds(0, oo)
def test_AccumBounds_pow():
assert AccumBounds(0, 2)**2 == AccumBounds(0, 4)
assert AccumBounds(-1, 1)**2 == AccumBounds(0, 1)
assert AccumBounds(1, 2)**2 == AccumBounds(1, 4)
assert AccumBounds(-1, 2)**3 == AccumBounds(-1, 8)
assert AccumBounds(-1, 1)**0 == 1
assert AccumBounds(1, 2)**(S(5) / 2) == AccumBounds(1, 4 * sqrt(2))
assert AccumBounds(-1, 2)**(S(1) / 3) == AccumBounds(-1, 2**(S(1) / 3))
assert AccumBounds(0, 2)**(S(1) / 2) == AccumBounds(0, sqrt(2))
assert AccumBounds(-4, 2)**(S(2) / 3) == AccumBounds(0, 2 * 2**(S(1) / 3))
assert AccumBounds(-1, 5)**(S(1) / 2) == AccumBounds(0, sqrt(5))
assert AccumBounds(-oo, 2)**(S(1) / 2) == AccumBounds(0, sqrt(2))
assert AccumBounds(-2, 3)**(S(-1) / 4) == AccumBounds(0, oo)
assert AccumBounds(1, 5)**(-2) == AccumBounds(S(1) / 25, 1)
assert AccumBounds(-1, 3)**(-2) == AccumBounds(0, oo)
assert AccumBounds(0, 2)**(-2) == AccumBounds(S(1) / 4, oo)
assert AccumBounds(-1, 2)**(-3) == AccumBounds(-oo, oo)
assert AccumBounds(-3, -2)**(-3) == AccumBounds(S(-1) / 8, -S(1) / 27)
assert AccumBounds(-3, -2)**(-2) == AccumBounds(S(1) / 9, S(1) / 4)
assert AccumBounds(0, oo)**(S(1) / 2) == AccumBounds(0, oo)
assert AccumBounds(-oo, -1)**(S(1) / 3) == AccumBounds(-oo, -1)
assert AccumBounds(-2, 3)**(-S(1) / 3) == AccumBounds(-oo, oo)
assert AccumBounds(-oo, 0)**(-2) == AccumBounds(0, oo)
assert AccumBounds(-2, 0)**(-2) == AccumBounds(S(1) / 4, oo)
assert AccumBounds(S(1) / 3, S(1) / 2)**oo == S(0)
assert AccumBounds(0, S(1) / 2)**oo == S(0)
assert AccumBounds(S(1) / 2, 1)**oo == AccumBounds(0, oo)
assert AccumBounds(0, 1)**oo == AccumBounds(0, oo)
assert AccumBounds(2, 3)**oo == oo
assert AccumBounds(1, 2)**oo == AccumBounds(0, oo)
assert AccumBounds(S(1) / 2, 3)**oo == AccumBounds(0, oo)
assert AccumBounds(-S(1) / 3, -S(1) / 4)**oo == S(0)
assert AccumBounds(-1, -S(1) / 2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-3, -2)**oo == FiniteSet(-oo, oo)
assert AccumBounds(-2, -1)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-2, -S(1) / 2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-S(1) / 2, S(1) / 2)**oo == S(0)
assert AccumBounds(-S(1) / 2, 1)**oo == AccumBounds(0, oo)
assert AccumBounds(-S(2) / 3, 2)**oo == AccumBounds(0, oo)
assert AccumBounds(-1, 1)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-1, S(1) / 2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-1, 2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-2, S(1) / 2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(1, 2)**x == Pow(AccumBounds(1, 2), x, evaluate=False)
assert AccumBounds(2, 3)**(-oo) == S(0)
assert AccumBounds(0, 2)**(-oo) == AccumBounds(0, oo)
assert AccumBounds(-1, 2)**(-oo) == AccumBounds(-oo, oo)
assert (tan(x)**sin(2*x)).subs(x, AccumBounds(0, pi/2)) == \
Pow(AccumBounds(-oo, oo), AccumBounds(0, 1), evaluate=False)
def test_heuristic():
x = Symbol("x", real=True)
assert heuristics(sin(1 / x) + atan(x), x, 0, "+") == AccumBounds(-1, 1)
assert limit(log(2 + sqrt(atan(x)) * sqrt(sin(1 / x))), x, 0) == log(2)
def test_intersection_AccumBounds():
assert AccumBounds(0, 3).intersection(AccumBounds(1, 2)) == AccumBounds(1, 2)
assert AccumBounds(0, 3).intersection(AccumBounds(1, 4)) == AccumBounds(1, 3)
assert AccumBounds(0, 3).intersection(AccumBounds(-1, 2)) == AccumBounds(0, 2)
assert AccumBounds(0, 3).intersection(AccumBounds(-1, 4)) == AccumBounds(0, 3)
assert AccumBounds(0, 1).intersection(AccumBounds(2, 3)) == S.EmptySet
raises(TypeError, lambda: AccumBounds(0, 3).intersection(1))
def test_contains_AccumBounds():
assert (1 in AccumBounds(1, 2)) == S.true
raises(TypeError, lambda: a in AccumBounds(1, 2))
assert (-oo in AccumBounds(1, oo)) == S.true
assert (oo in AccumBounds(-oo, 0)) == S.true
def test_comparison_AccumBounds():
assert (AccumBounds(1, 3) < 4) == S.true
assert (AccumBounds(1, 3) < -1) == S.false
assert (AccumBounds(1, 3) < 2) is None
assert (AccumBounds(1, 3) > 4) == S.false
assert (AccumBounds(1, 3) > -1) == S.true
assert (AccumBounds(1, 3) > 2) is None
assert (AccumBounds(1, 3) < AccumBounds(4, 6)) == S.true
assert (AccumBounds(1, 3) < AccumBounds(2, 4)) is None
assert (AccumBounds(1, 3) < AccumBounds(-2, 0)) == S.false
def test_comparison_AccumBounds():
assert (AccumBounds(1, 3) < 4) == S.true
assert (AccumBounds(1, 3) < -1) == S.false
assert (AccumBounds(1, 3) < 2).rel_op == '<'
assert (AccumBounds(1, 3) <= 2).rel_op == '<='
assert (AccumBounds(1, 3) > 4) == S.false
assert (AccumBounds(1, 3) > -1) == S.true
assert (AccumBounds(1, 3) > 2).rel_op == '>'
assert (AccumBounds(1, 3) >= 2).rel_op == '>='
assert (AccumBounds(1, 3) < AccumBounds(4, 6)) == S.true
assert (AccumBounds(1, 3) < AccumBounds(2, 4)).rel_op == '<'
assert (AccumBounds(1, 3) < AccumBounds(-2, 0)) == S.false
assert (AccumBounds(1, 3) <= AccumBounds(4, 6)) == S.true
assert (AccumBounds(1, 3) <= AccumBounds(-2, 0)) == S.false
assert (AccumBounds(1, 3) > AccumBounds(4, 6)) == S.false
assert (AccumBounds(1, 3) > AccumBounds(-2, 0)) == S.true
assert (AccumBounds(1, 3) >= AccumBounds(4, 6)) == S.false
assert (AccumBounds(1, 3) >= AccumBounds(-2, 0)) == S.true
# issue 13499
assert (cos(x) > 0).subs(x, oo) == (AccumBounds(-1, 1) > 0)
c = Symbol('c')
raises(TypeError, lambda: (AccumBounds(0, 1) < c))
raises(TypeError, lambda: (AccumBounds(0, 1) <= c))
raises(TypeError, lambda: (AccumBounds(0, 1) > c))
raises(TypeError, lambda: (AccumBounds(0, 1) >= c))
def test_comparison_AccumBounds():
assert (AccumBounds(1, 3) < 4) == S.true
assert (AccumBounds(1, 3) < -1) == S.false
assert (AccumBounds(1, 3) < 2).rel_op == '<'
assert (AccumBounds(1, 3) <= 2).rel_op == '<='
assert (AccumBounds(1, 3) > 4) == S.false
assert (AccumBounds(1, 3) > -1) == S.true
assert (AccumBounds(1, 3) > 2).rel_op == '>'
assert (AccumBounds(1, 3) >= 2).rel_op == '>='
assert (AccumBounds(1, 3) < AccumBounds(4, 6)) == S.true
assert (AccumBounds(1, 3) < AccumBounds(2, 4)).rel_op == '<'
assert (AccumBounds(1, 3) < AccumBounds(-2, 0)) == S.false
# issue 13499
assert (cos(x) > 0).subs(x, oo) == (AccumBounds(-1, 1) > 0)
def test_AccumBounds():
assert AccumBounds(1, 2).args == (1, 2)
assert AccumBounds(1, 2).delta is S.One
assert AccumBounds(1, 2).mid == Rational(3, 2)
assert AccumBounds(1, 3).is_real == True
assert AccumBounds(1, 1) is S.One
assert AccumBounds(1, 2) + 1 == AccumBounds(2, 3)
assert 1 + AccumBounds(1, 2) == AccumBounds(2, 3)
assert AccumBounds(1, 2) + AccumBounds(2, 3) == AccumBounds(3, 5)
assert -AccumBounds(1, 2) == AccumBounds(-2, -1)
assert AccumBounds(1, 2) - 1 == AccumBounds(0, 1)
assert 1 - AccumBounds(1, 2) == AccumBounds(-1, 0)
assert AccumBounds(2, 3) - AccumBounds(1, 2) == AccumBounds(0, 2)
assert x + AccumBounds(1, 2) == Add(AccumBounds(1, 2), x)
assert a + AccumBounds(1, 2) == AccumBounds(1 + a, 2 + a)
assert AccumBounds(1, 2) - x == Add(AccumBounds(1, 2), -x)
assert AccumBounds(-oo, 1) + oo == AccumBounds(-oo, oo)
assert AccumBounds(1, oo) + oo is oo
assert AccumBounds(1, oo) - oo == AccumBounds(-oo, oo)
assert (-oo - AccumBounds(-1, oo)) is -oo
assert AccumBounds(-oo, 1) - oo is -oo
assert AccumBounds(1, oo) - oo == AccumBounds(-oo, oo)
assert AccumBounds(-oo, 1) - (-oo) == AccumBounds(-oo, oo)
assert (oo - AccumBounds(1, oo)) == AccumBounds(-oo, oo)
assert (-oo - AccumBounds(1, oo)) is -oo
assert AccumBounds(1, 2)/2 == AccumBounds(S.Half, 1)
assert 2/AccumBounds(2, 3) == AccumBounds(Rational(2, 3), 1)
assert 1/AccumBounds(-1, 1) == AccumBounds(-oo, oo)
assert abs(AccumBounds(1, 2)) == AccumBounds(1, 2)
assert abs(AccumBounds(-2, -1)) == AccumBounds(1, 2)
assert abs(AccumBounds(-2, 1)) == AccumBounds(0, 2)
assert abs(AccumBounds(-1, 2)) == AccumBounds(0, 2)
c = Symbol('c')
raises(ValueError, lambda: AccumBounds(0, c))
raises(ValueError, lambda: AccumBounds(1, -1))
def test_union_AccumBounds():
assert AccumBounds(0, 3).union(AccumBounds(1, 2)) == AccumBounds(0, 3)
assert AccumBounds(0, 3).union(AccumBounds(1, 4)) == AccumBounds(0, 4)
assert AccumBounds(0, 3).union(AccumBounds(-1, 2)) == AccumBounds(-1, 3)
assert AccumBounds(0, 3).union(AccumBounds(-1, 4)) == AccumBounds(-1, 4)
raises(TypeError, lambda: AccumBounds(0, 3).union(1))
def test_AccumBounds_mul():
assert AccumBounds(1, 2)*2 == AccumBounds(2, 4)
assert 2*AccumBounds(1, 2) == AccumBounds(2, 4)
assert AccumBounds(1, 2)*AccumBounds(2, 3) == AccumBounds(2, 6)
assert AccumBounds(1, 2)*0 == 0
assert AccumBounds(1, oo)*0 == AccumBounds(0, oo)
assert AccumBounds(-oo, 1)*0 == AccumBounds(-oo, 0)
assert AccumBounds(-oo, oo)*0 == AccumBounds(-oo, oo)
assert AccumBounds(1, 2)*x == Mul(AccumBounds(1, 2), x, evaluate=False)
assert AccumBounds(0, 2)*oo == AccumBounds(0, oo)
assert AccumBounds(-2, 0)*oo == AccumBounds(-oo, 0)
assert AccumBounds(0, 2)*(-oo) == AccumBounds(-oo, 0)
assert AccumBounds(-2, 0)*(-oo) == AccumBounds(0, oo)
assert AccumBounds(-1, 1)*oo == AccumBounds(-oo, oo)
assert AccumBounds(-1, 1)*(-oo) == AccumBounds(-oo, oo)
assert AccumBounds(-oo, oo)*oo == AccumBounds(-oo, oo)
def test_issue_18442():
assert limit(tan(x)**(2**(sqrt(pi))), x, oo,
dir="-") == AccumBounds(-oo, oo)
def test_AccumBounds():
assert AccumBounds(1, 2).args == (1, 2)
assert AccumBounds(1, 2).delta == S(1)
assert AccumBounds(1, 2).mid == S(3) / 2
assert AccumBounds(1, 3).is_real == True
assert AccumBounds(1, 1) == S(1)
assert AccumBounds(1, 2) + 1 == AccumBounds(2, 3)
assert 1 + AccumBounds(1, 2) == AccumBounds(2, 3)
assert AccumBounds(1, 2) + AccumBounds(2, 3) == AccumBounds(3, 5)
assert -AccumBounds(1, 2) == AccumBounds(-2, -1)
assert AccumBounds(1, 2) - 1 == AccumBounds(0, 1)
assert 1 - AccumBounds(1, 2) == AccumBounds(-1, 0)
assert AccumBounds(2, 3) - AccumBounds(1, 2) == AccumBounds(0, 2)
assert x + AccumBounds(1, 2) == Add(AccumBounds(1, 2), x)
assert a + AccumBounds(1, 2) == AccumBounds(1 + a, 2 + a)
assert AccumBounds(1, 2) - x == Add(AccumBounds(1, 2), -x)
assert AccumBounds(-oo, 1) + oo == AccumBounds(-oo, oo)
assert AccumBounds(1, oo) + oo == oo
assert AccumBounds(1, oo) - oo == AccumBounds(-oo, oo)
assert (-oo - AccumBounds(-1, oo)) == -oo
assert AccumBounds(-oo, 1) - oo == -oo
assert AccumBounds(1, oo) - oo == AccumBounds(-oo, oo)
assert AccumBounds(-oo, 1) - (-oo) == AccumBounds(-oo, oo)
assert (oo - AccumBounds(1, oo)) == AccumBounds(-oo, oo)
assert (-oo - AccumBounds(1, oo)) == -oo
assert AccumBounds(1, 2) / 2 == AccumBounds(S(1) / 2, 1)
assert 2 / AccumBounds(2, 3) == AccumBounds(S(2) / 3, 1)
assert 1 / AccumBounds(-1, 1) == AccumBounds(-oo, oo)
assert abs(AccumBounds(1, 2)) == AccumBounds(1, 2)
assert abs(AccumBounds(-2, -1)) == AccumBounds(1, 2)
assert abs(AccumBounds(-2, 1)) == AccumBounds(0, 2)
assert abs(AccumBounds(-1, 2)) == AccumBounds(0, 2)
