def test_add():
a, b, c, d, x, y, t = symbols('a b c d x y t')
assert (a**2 - b - c).subs(a**2 - b, d) in [d - c, a**2 - b - c]
assert (a**2 - c).subs(a**2 - c, d) == d
assert (a**2 - b - c).subs(a**2 - c, d) in [d - b, a**2 - b - c]
assert (a**2 - x - c).subs(a**2 - c, d) in [d - x, a**2 - x - c]
assert (a**2 - b - sqrt(a)).subs(a**2 - sqrt(a), c) == c - b
assert (a + b + exp(a + b)).subs(a + b, c) == c + exp(c)
assert (c + b + exp(c + b)).subs(c + b, a) == a + exp(a)
assert (a + b + c + d).subs(b + c, x) == a + d + x
assert (a + b + c + d).subs(-b - c, x) == a + d - x
assert ((x + 1) * y).subs(x + 1, t) == t * y
assert ((-x - 1) * y).subs(x + 1, t) == -t * y
assert ((x - 1) * y).subs(x + 1, t) == y * (t - 2)
assert ((-x + 1) * y).subs(x + 1, t) == y * (-t + 2)
# this should work every time:
e = a**2 - b - c
assert e.subs(Add(*e.args[:2]), d) == d + e.args[2]
assert e.subs(a**2 - c, d) == d - b
# the fallback should recognize when a change has
# been made; while .1 == Rational(1, 10) they are not the same
# and the change should be made
assert (0.1 + a).subs(0.1, Rational(1, 10)) == Rational(1, 10) + a
e = (-x * (-y + 1) - y * (y - 1))
ans = (-x * (x) - y * (-x)).expand()
assert e.subs(-y + 1, x) == ans
#Test issue 18747
assert (exp(x) + cos(x)).subs(x, oo) == oo
assert Add(*[AccumBounds(-1, 1), oo]) == oo
assert Add(*[oo, AccumBounds(-1, 1)]) == oo
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 test_issue_18473():
assert limit(sin(x)**(1/x), x, oo) == Limit(sin(x)**(1/x), x, oo, dir='-')
assert limit(cos(x)**(1/x), x, oo) == Limit(cos(x)**(1/x), x, oo, dir='-')
assert limit(tan(x)**(1/x), x, oo) == Limit(tan(x)**(1/x), x, oo, dir='-')
assert limit((cos(x) + 2)**(1/x), x, oo) == 1
assert limit((sin(x) + 10)**(1/x), x, oo) == 1
assert limit((cos(x) - 2)**(1/x), x, oo) == Limit((cos(x) - 2)**(1/x), x, oo, dir='-')
assert limit((cos(x) + 1)**(1/x), x, oo) == AccumBounds(0, 1)
assert limit((tan(x)**2)**(2/x) , x, oo) == AccumBounds(0, oo)
assert limit((sin(x)**2)**(1/x), x, oo) == AccumBounds(0, 1)
def test_log_AccumBounds():
assert log(AccumBounds(1, E)) == AccumBounds(0, 1)
assert log(AccumBounds(0, E)) == AccumBounds(-oo, 1)
assert log(AccumBounds(-1, E)) == S.NaN
assert log(AccumBounds(0, oo)) == AccumBounds(-oo, oo)
assert log(AccumBounds(-oo, 0)) == S.NaN
assert log(AccumBounds(-oo, oo)) == S.NaN
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
lo = (-3 + cos(1)) / 2
hi = (1 + cos(1)) / 2
t1 = Mul(AccumBounds(lo, hi), 1 / (-1 + cos(1)), evaluate=False)
assert limit(
simplify(Sum(cos(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k,
oo) == t1
t2 = Mul(AccumBounds(-1 + sin(1) / 2, sin(1) / 2 + 1), 1 / (1 - cos(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
# https://github.com/sympy/sympy/issues/12312
e = 2**(-x) * (sin(x) + 1)**x
assert limit(e, x, oo) == AccumBounds(0, oo)
def test_atanh():
x = Symbol('x')
#at specific points
assert atanh(0) == 0
assert atanh(I) == I * pi / 4
assert atanh(-I) == -I * pi / 4
assert atanh(1) is oo
assert atanh(-1) is -oo
assert atanh(nan) is nan
# at infinites
assert atanh(oo) == -I * pi / 2
assert atanh(-oo) == I * pi / 2
assert atanh(I * oo) == I * pi / 2
assert atanh(-I * oo) == -I * pi / 2
assert atanh(zoo) == I * AccumBounds(-pi / 2, pi / 2)
#properties
assert atanh(-x) == -atanh(x)
assert atanh(I / sqrt(3)) == I * pi / 6
assert atanh(-I / sqrt(3)) == -I * pi / 6
assert atanh(I * sqrt(3)) == I * pi / 3
assert atanh(-I * sqrt(3)) == -I * pi / 3
assert atanh(I * (1 + sqrt(2))) == pi * I * Rational(3, 8)
assert atanh(I * (sqrt(2) - 1)) == pi * I / 8
assert atanh(I * (1 - sqrt(2))) == -pi * I / 8
assert atanh(-I * (1 + sqrt(2))) == pi * I * Rational(-3, 8)
assert atanh(I * sqrt(5 + 2 * sqrt(5))) == I * pi * Rational(2, 5)
assert atanh(-I * sqrt(5 + 2 * sqrt(5))) == I * pi * Rational(-2, 5)
assert atanh(I * (2 - sqrt(3))) == pi * I / 12
assert atanh(I * (sqrt(3) - 2)) == -pi * I / 12
assert atanh(oo) == -I * pi / 2
# Symmetry
assert atanh(Rational(-1, 2)) == -atanh(S.Half)
# inverse composition
assert unchanged(atanh, tanh(Symbol('v1')))
assert atanh(tanh(-5, evaluate=False)) == -5
assert atanh(tanh(0, evaluate=False)) == 0
assert atanh(tanh(7, evaluate=False)) == 7
assert atanh(tanh(I, evaluate=False)) == I
assert atanh(tanh(-I, evaluate=False)) == -I
assert atanh(tanh(-11 * I, evaluate=False)) == -11 * I + 4 * I * pi
assert atanh(tanh(3 + I)) == 3 + I
assert atanh(tanh(4 + 5 * I)) == 4 - 2 * I * pi + 5 * I
assert atanh(tanh(pi / 2)) == pi / 2
assert atanh(tanh(pi)) == pi
assert atanh(tanh(-3 + 7 * I)) == -3 - 2 * I * pi + 7 * I
assert atanh(tanh(9 - I * 2 / 3)) == 9 - I * 2 / 3
assert atanh(tanh(-32 - 123 * I)) == -32 - 123 * I + 39 * I * pi
def test_issues_14525():
assert limit(sin(x)**2 - cos(x) + tan(x)*csc(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(sin(x)**2 - cos(x) + sin(x)*cot(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(cot(x) - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(cos(x) - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.One)
assert limit(sin(x) - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.One)
assert limit(cos(x)**2 - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.One)
assert limit(tan(x)**2 + sin(x)**2 - cos(x), x, oo) == AccumBounds(-S.One, S.Infinity)
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 in (S.Infinity, 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_nseries(self, x, n, logx, cdir=0):
arg = self.args[0]
arg0 = arg.subs(x, 0)
if arg0.is_infinite:
from sympy.calculus.accumulationbounds 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_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(-S.Half, S(3)/2)
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(-S.Half, S(3)/2)
def test_subs_AccumBounds():
e = x
e = e.subs(x, AccumBounds(1, 3))
assert e == AccumBounds(1, 3)
e = 2 * x
e = e.subs(x, AccumBounds(1, 3))
assert e == AccumBounds(2, 6)
e = x + x**2
e = e.subs(x, AccumBounds(-1, 1))
assert e == AccumBounds(-1, 2)
def test_issue_18473():
assert exp(x * log(cos(1 / x))).as_leading_term(x) == S.NaN
assert exp(x * log(tan(1 / x))).as_leading_term(x) == S.NaN
assert log(cos(1 / x)).as_leading_term(x) == S.NaN
assert log(tan(1 / x)).as_leading_term(x) == S.NaN
assert log(cos(1 / x) + 2).as_leading_term(x) == AccumBounds(0, log(3))
assert exp(x * log(cos(1 / x) + 2)).as_leading_term(x) == 1
assert log(cos(1 / x) - 2).as_leading_term(x) == S.NaN
assert exp(x * log(cos(1 / x) - 2)).as_leading_term(x) == S.NaN
assert log(cos(1 / x) + 1).as_leading_term(x) == AccumBounds(-oo, log(2))
assert exp(x * log(cos(1 / x) + 1)).as_leading_term(x) == AccumBounds(0, 1)
assert log(sin(1 / x)**2).as_leading_term(x) == AccumBounds(-oo, 0)
assert exp(x * log(sin(1 / x)**2)).as_leading_term(x) == AccumBounds(0, 1)
assert log(tan(1 / x)**2).as_leading_term(x) == AccumBounds(-oo, oo)
assert exp(2 * x * (log(tan(1 / x)**2))).as_leading_term(x) == AccumBounds(
0, oo)
def _eval_nseries(self, x, n, logx, cdir=0):
arg = self.args[0]
arg0 = arg.subs(x, 0)
r = self.subs(x, 0)
if arg0 is S.NaN:
arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
r = floor(arg0)
if arg0.is_infinite:
from sympy.calculus.accumulationbounds 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(-1, 0)
return s + o
if arg0 == r:
ndir = arg.dir(x, cdir=cdir if cdir != 0 else 1)
return r - 1 if ndir.is_negative else r
else:
return r
def test_frac():
assert isinstance(frac(x), frac)
assert frac(oo) == AccumBounds(0, 1)
assert frac(-oo) == AccumBounds(0, 1)
assert frac(zoo) is nan
assert frac(n) == 0
assert frac(nan) is nan
assert frac(Rational(4, 3)) == Rational(1, 3)
assert frac(-Rational(4, 3)) == Rational(2, 3)
assert frac(Rational(-4, 3)) == Rational(2, 3)
r = Symbol('r', real=True)
assert frac(I*r) == I*frac(r)
assert frac(1 + I*r) == I*frac(r)
assert frac(0.5 + I*r) == 0.5 + I*frac(r)
assert frac(n + I*r) == I*frac(r)
assert frac(n + I*k) == 0
assert unchanged(frac, x + I*x)
assert frac(x + I*n) == frac(x)
assert frac(x).rewrite(floor) == x - floor(x)
assert frac(x).rewrite(ceiling) == x + ceiling(-x)
assert frac(y).rewrite(floor).subs(y, pi) == frac(pi)
assert frac(y).rewrite(floor).subs(y, -E) == frac(-E)
assert frac(y).rewrite(ceiling).subs(y, -pi) == frac(-pi)
assert frac(y).rewrite(ceiling).subs(y, E) == frac(E)
assert Eq(frac(y), y - floor(y))
assert Eq(frac(y), y + ceiling(-y))
r = Symbol('r', real=True)
p_i = Symbol('p_i', integer=True, positive=True)
n_i = Symbol('p_i', integer=True, negative=True)
np_i = Symbol('np_i', integer=True, nonpositive=True)
nn_i = Symbol('nn_i', integer=True, nonnegative=True)
p_r = Symbol('p_r', positive=True)
n_r = Symbol('n_r', negative=True)
np_r = Symbol('np_r', real=True, nonpositive=True)
nn_r = Symbol('nn_r', real=True, nonnegative=True)
# Real frac argument, integer rhs
assert frac(r) <= p_i
assert not frac(r) <= n_i
assert (frac(r) <= np_i).has(Le)
assert (frac(r) <= nn_i).has(Le)
assert frac(r) < p_i
assert not frac(r) < n_i
assert not frac(r) < np_i
assert (frac(r) < nn_i).has(Lt)
assert not frac(r) >= p_i
assert frac(r) >= n_i
assert frac(r) >= np_i
assert (frac(r) >= nn_i).has(Ge)
assert not frac(r) > p_i
assert frac(r) > n_i
assert (frac(r) > np_i).has(Gt)
assert (frac(r) > nn_i).has(Gt)
assert not Eq(frac(r), p_i)
assert not Eq(frac(r), n_i)
assert Eq(frac(r), np_i).has(Eq)
assert Eq(frac(r), nn_i).has(Eq)
assert Ne(frac(r), p_i)
assert Ne(frac(r), n_i)
assert Ne(frac(r), np_i).has(Ne)
assert Ne(frac(r), nn_i).has(Ne)
# Real frac argument, real rhs
assert (frac(r) <= p_r).has(Le)
assert not frac(r) <= n_r
assert (frac(r) <= np_r).has(Le)
assert (frac(r) <= nn_r).has(Le)
assert (frac(r) < p_r).has(Lt)
assert not frac(r) < n_r
assert not frac(r) < np_r
assert (frac(r) < nn_r).has(Lt)
assert (frac(r) >= p_r).has(Ge)
assert frac(r) >= n_r
assert frac(r) >= np_r
assert (frac(r) >= nn_r).has(Ge)
assert (frac(r) > p_r).has(Gt)
assert frac(r) > n_r
assert (frac(r) > np_r).has(Gt)
assert (frac(r) > nn_r).has(Gt)
assert not Eq(frac(r), n_r)
assert Eq(frac(r), p_r).has(Eq)
assert Eq(frac(r), np_r).has(Eq)
assert Eq(frac(r), nn_r).has(Eq)
assert Ne(frac(r), p_r).has(Ne)
assert Ne(frac(r), n_r)
assert Ne(frac(r), np_r).has(Ne)
assert Ne(frac(r), nn_r).has(Ne)
# Real frac argument, +/- oo rhs
assert frac(r) < oo
assert frac(r) <= oo
assert not frac(r) > oo
assert not frac(r) >= oo
assert not frac(r) < -oo
assert not frac(r) <= -oo
assert frac(r) > -oo
assert frac(r) >= -oo
assert frac(r) < 1
assert frac(r) <= 1
assert not frac(r) > 1
assert not frac(r) >= 1
assert not frac(r) < 0
assert (frac(r) <= 0).has(Le)
assert (frac(r) > 0).has(Gt)
assert frac(r) >= 0
# Some test for numbers
assert frac(r) <= sqrt(2)
assert (frac(r) <= sqrt(3) - sqrt(2)).has(Le)
assert not frac(r) <= sqrt(2) - sqrt(3)
assert not frac(r) >= sqrt(2)
assert (frac(r) >= sqrt(3) - sqrt(2)).has(Ge)
assert frac(r) >= sqrt(2) - sqrt(3)
assert not Eq(frac(r), sqrt(2))
assert Eq(frac(r), sqrt(3) - sqrt(2)).has(Eq)
assert not Eq(frac(r), sqrt(2) - sqrt(3))
assert Ne(frac(r), sqrt(2))
assert Ne(frac(r), sqrt(3) - sqrt(2)).has(Ne)
assert Ne(frac(r), sqrt(2) - sqrt(3))
assert frac(p_i, evaluate=False).is_zero
assert frac(p_i, evaluate=False).is_finite
assert frac(p_i, evaluate=False).is_integer
assert frac(p_i, evaluate=False).is_real
assert frac(r).is_finite
assert frac(r).is_real
assert frac(r).is_zero is None
assert frac(r).is_integer is None
assert frac(oo).is_finite
assert frac(oo).is_real
def test_basic1():
assert limit(x, x, oo) is oo
assert limit(x, x, -oo) is -oo
assert limit(-x, x, oo) is -oo
assert limit(x**2, x, -oo) is oo
assert limit(-x**2, x, oo) is -oo
assert limit(x*log(x), x, 0, dir="+") == 0
assert limit(1/x, x, oo) == 0
assert limit(exp(x), x, oo) is oo
assert limit(-exp(x), x, oo) is -oo
assert limit(exp(x)/x, x, oo) is oo
assert limit(1/x - exp(-x), x, oo) == 0
assert limit(x + 1/x, x, oo) is oo
assert limit(x - x**2, x, oo) is -oo
assert limit((1 + x)**(1 + sqrt(2)), x, 0) == 1
assert limit((1 + x)**oo, x, 0) == Limit((x + 1)**oo, x, 0)
assert limit((1 + x)**oo, x, 0, dir='-') == Limit((x + 1)**oo, x, 0, dir='-')
assert limit((1 + x + y)**oo, x, 0, dir='-') == Limit((1 + x + y)**oo, x, 0, dir='-')
assert limit(y/x/log(x), x, 0) == -oo*sign(y)
assert limit(cos(x + y)/x, x, 0) == sign(cos(y))*oo
assert limit(gamma(1/x + 3), x, oo) == 2
assert limit(S.NaN, x, -oo) is S.NaN
assert limit(Order(2)*x, x, S.NaN) is S.NaN
assert limit(1/(x - 1), x, 1, dir="+") is oo
assert limit(1/(x - 1), x, 1, dir="-") is -oo
assert limit(1/(5 - x)**3, x, 5, dir="+") is -oo
assert limit(1/(5 - x)**3, x, 5, dir="-") is oo
assert limit(1/sin(x), x, pi, dir="+") is -oo
assert limit(1/sin(x), x, pi, dir="-") is oo
assert limit(1/cos(x), x, pi/2, dir="+") is -oo
assert limit(1/cos(x), x, pi/2, dir="-") is oo
assert limit(1/tan(x**3), x, (2*pi)**Rational(1, 3), dir="+") is oo
assert limit(1/tan(x**3), x, (2*pi)**Rational(1, 3), dir="-") is -oo
assert limit(1/cot(x)**3, x, (pi*Rational(3, 2)), dir="+") is -oo
assert limit(1/cot(x)**3, x, (pi*Rational(3, 2)), dir="-") is oo
assert limit(tan(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(cot(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(sec(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(csc(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
# test bi-directional limits
assert limit(sin(x)/x, x, 0, dir="+-") == 1
assert limit(x**2, x, 0, dir="+-") == 0
assert limit(1/x**2, x, 0, dir="+-") is oo
# test failing bi-directional limits
assert limit(1/x, x, 0, dir="+-") is zoo
# approaching 0
# from dir="+"
assert limit(1 + 1/x, x, 0) is oo
# from dir='-'
# Add
assert limit(1 + 1/x, x, 0, dir='-') is -oo
# Pow
assert limit(x**(-2), x, 0, dir='-') is oo
assert limit(x**(-3), x, 0, dir='-') is -oo
assert limit(1/sqrt(x), x, 0, dir='-') == (-oo)*I
assert limit(x**2, x, 0, dir='-') == 0
assert limit(sqrt(x), x, 0, dir='-') == 0
assert limit(x**-pi, x, 0, dir='-') == oo/(-1)**pi
assert limit((1 + cos(x))**oo, x, 0) == Limit((cos(x) + 1)**oo, x, 0)
# test pull request 22491
assert limit(1/asin(x), x, 0, dir = '+') == oo
assert limit(1/asin(x), x, 0, dir = '-') == -oo
assert limit(1/sinh(x), x, 0, dir = '+') == oo
assert limit(1/sinh(x), x, 0, dir = '-') == -oo
assert limit(log(1/x) + 1/sin(x), x, 0, dir = '+') == oo
assert limit(log(1/x) + 1/x, x, 0, dir = '+') == oo
def test_issue_19558():
e = (7*x*cos(x) - 12*log(x)**3)*(-log(x)**4 + 2*sin(x) + 1)**2/ \
(2*(x*cos(x) - 2*log(x)**3)*(3*log(x)**4 - 7*sin(x) + 3)**2)
assert e.subs(x, oo) == AccumBounds(-oo, oo)
assert (sin(x) + cos(x)).subs(x, oo) == AccumBounds(-2, 2)
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_asech():
x = Symbol('x')
assert unchanged(asech, -x)
# values at fixed points
assert asech(1) == 0
assert asech(-1) == pi * I
assert asech(0) is oo
assert asech(2) == I * pi / 3
assert asech(-2) == 2 * I * pi / 3
assert asech(nan) is nan
# at infinites
assert asech(oo) == I * pi / 2
assert asech(-oo) == I * pi / 2
assert asech(zoo) == I * AccumBounds(-pi / 2, pi / 2)
assert asech(I) == log(1 + sqrt(2)) - I * pi / 2
assert asech(-I) == log(1 + sqrt(2)) + I * pi / 2
assert asech(sqrt(2) - sqrt(6)) == 11 * I * pi / 12
assert asech(sqrt(2 - 2 / sqrt(5))) == I * pi / 10
assert asech(-sqrt(2 - 2 / sqrt(5))) == 9 * I * pi / 10
assert asech(2 / sqrt(2 + sqrt(2))) == I * pi / 8
assert asech(-2 / sqrt(2 + sqrt(2))) == 7 * I * pi / 8
assert asech(sqrt(5) - 1) == I * pi / 5
assert asech(1 - sqrt(5)) == 4 * I * pi / 5
assert asech(-sqrt(2 * (2 + sqrt(2)))) == 5 * I * pi / 8
# properties
# asech(x) == acosh(1/x)
assert asech(sqrt(2)) == acosh(1 / sqrt(2))
assert asech(2 / sqrt(3)) == acosh(sqrt(3) / 2)
assert asech(2 / sqrt(2 + sqrt(2))) == acosh(sqrt(2 + sqrt(2)) / 2)
assert asech(2) == acosh(S.Half)
# asech(x) == I*acos(1/x)
# (Note: the exact formula is asech(x) == +/- I*acos(1/x))
assert asech(-sqrt(2)) == I * acos(-1 / sqrt(2))
assert asech(-2 / sqrt(3)) == I * acos(-sqrt(3) / 2)
assert asech(-S(2)) == I * acos(Rational(-1, 2))
assert asech(-2 / sqrt(2)) == I * acos(-sqrt(2) / 2)
# sech(asech(x)) / x == 1
assert expand_mul(sech(asech(sqrt(6) - sqrt(2))) /
(sqrt(6) - sqrt(2))) == 1
assert expand_mul(sech(asech(sqrt(6) + sqrt(2))) /
(sqrt(6) + sqrt(2))) == 1
assert (sech(asech(sqrt(2 + 2 / sqrt(5)))) /
(sqrt(2 + 2 / sqrt(5)))).simplify() == 1
assert (sech(asech(-sqrt(2 + 2 / sqrt(5)))) /
(-sqrt(2 + 2 / sqrt(5)))).simplify() == 1
assert (sech(asech(sqrt(2 * (2 + sqrt(2))))) /
(sqrt(2 * (2 + sqrt(2))))).simplify() == 1
assert expand_mul(sech(asech(1 + sqrt(5))) / (1 + sqrt(5))) == 1
assert expand_mul(sech(asech(-1 - sqrt(5))) / (-1 - sqrt(5))) == 1
assert expand_mul(sech(asech(-sqrt(6) - sqrt(2))) /
(-sqrt(6) - sqrt(2))) == 1
# numerical evaluation
assert str(asech(5 * I).n(6)) == '0.19869 - 1.5708*I'
assert str(asech(-5 * I).n(6)) == '0.19869 + 1.5708*I'
def test_log_AccumBounds():
assert log(AccumBounds(1, E)) == AccumBounds(0, 1)
def test_AccumBounds():
a = Symbol('a', real=True)
assert str(AccumBounds(0, a)) == "AccumBounds(0, a)"
assert str(AccumBounds(0, 1)) == "AccumBounds(0, 1)"
def test_exp_AccumBounds():
assert exp(AccumBounds(1, 2)) == AccumBounds(E, E**2)
