def test_frac():
    assert isinstance(frac(x), frac)
    assert frac(oo) == AccumBounds(0, 1)
    assert frac(-oo) == AccumBounds(0, 1)

    assert frac(n) == 0
    assert frac(nan) == nan
    assert frac(Rational(4, 3)) == Rational(1, 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 frac(x + I*x) == 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))

def test_issue_8444_workingtests():
    x = symbols('x')
    assert Gt(x, floor(x)) == Gt(x, floor(x), evaluate=False)
    assert Ge(x, floor(x)) == Ge(x, floor(x), evaluate=False)
    assert Lt(x, ceiling(x)) == Lt(x, ceiling(x), evaluate=False)
    assert Le(x, ceiling(x)) == Le(x, ceiling(x), evaluate=False)
    i = symbols('i', integer=True)
    assert (i > floor(i)) == False
    assert (i < ceiling(i)) == False

def test_issue_8444_nonworkingtests():
    x = symbols('x', real=True)
    assert (x <= oo) == (x >= -oo) == True

    x = symbols('x')
    assert x >= floor(x)
    assert (x < floor(x)) == False
    assert x <= ceiling(x)
    assert (x > ceiling(x)) == False

def test_series():
    x, y = symbols('x,y')
    assert floor(x).nseries(x, y, 100) == floor(y)
    assert ceiling(x).nseries(x, y, 100) == ceiling(y)
    assert floor(x).nseries(x, pi, 100) == 3
    assert ceiling(x).nseries(x, pi, 100) == 4
    assert floor(x).nseries(x, 0, 100) == 0
    assert ceiling(x).nseries(x, 0, 100) == 1
    assert floor(-x).nseries(x, 0, 100) == -1
    assert ceiling(-x).nseries(x, 0, 100) == 0

def test_issue_8853():
    p = Symbol('x', even=True, positive=True)
    assert floor(-p - S.Half).is_even == False
    assert floor(-p + S.Half).is_even == True
    assert ceiling(p - S.Half).is_even == True
    assert ceiling(p + S.Half).is_even == False

    assert get_integer_part(S.Half, -1, {}, True) == (0, 0)
    assert get_integer_part(S.Half, 1, {}, True) == (1, 0)
    assert get_integer_part(-S.Half, -1, {}, True) == (-1, 0)
    assert get_integer_part(-S.Half, 1, {}, True) == (0, 0)

def test_upretty_ceiling():
    assert upretty(ceiling(x)) == u'⌈x⌉'

    u = upretty( ceiling(1 / (y - ceiling(x))) )
    s = \
u"""\
⎡   1   ⎤
⎢───────⎥
⎢y - ⌈x⌉⎥\
"""
    assert u == s

def test_evalf_integer_parts():
    a = floor(log(8)/log(2) - exp(-1000), evaluate=False)
    b = floor(log(8)/log(2), evaluate=False)
    assert a.evalf() == 3
    assert b.evalf() == 3
    # equals, as a fallback, can still fail but it might succeed as here
    assert ceiling(10*(sin(1)**2 + cos(1)**2)) == 10

    assert int(floor(factorial(50)/E, evaluate=False).evalf(70)) == \
        long(11188719610782480504630258070757734324011354208865721592720336800)
    assert int(ceiling(factorial(50)/E, evaluate=False).evalf(70)) == \
        long(11188719610782480504630258070757734324011354208865721592720336801)
    assert int(floor((GoldenRatio**999 / sqrt(5) + Rational(1, 2)))
               .evalf(1000)) == fibonacci(999)
    assert int(floor((GoldenRatio**1000 / sqrt(5) + Rational(1, 2)))
               .evalf(1000)) == fibonacci(1000)

    assert ceiling(x).evalf(subs={x: 3}) == 3
    assert ceiling(x).evalf(subs={x: 3*I}) == 3*I
    assert ceiling(x).evalf(subs={x: 2 + 3*I}) == 2 + 3*I
    assert ceiling(x).evalf(subs={x: 3.}) == 3
    assert ceiling(x).evalf(subs={x: 3.*I}) == 3*I
    assert ceiling(x).evalf(subs={x: 2. + 3*I}) == 2 + 3*I

    assert float((floor(1.5, evaluate=False)+1/9).evalf()) == 1 + 1/9
    assert float((floor(0.5, evaluate=False)+20).evalf()) == 20

def test_latex_functions():
    assert latex(exp(x)) == "$e^{x}$"
    assert latex(exp(1) + exp(2)) == "$e + e^{2}$"

    f = Function("f")
    assert latex(f(x)) == "$\\operatorname{f}\\left(x\\right)$"

    beta = Function("beta")

    assert latex(beta(x)) == r"$\operatorname{beta}\left(x\right)$"
    assert latex(sin(x)) == r"$\operatorname{sin}\left(x\right)$"
    assert latex(sin(x), fold_func_brackets=True) == r"$\operatorname{sin}x$"
    assert latex(sin(2 * x ** 2), fold_func_brackets=True) == r"$\operatorname{sin}2 x^{2}$"
    assert latex(sin(x ** 2), fold_func_brackets=True) == r"$\operatorname{sin}x^{2}$"

    assert latex(asin(x) ** 2) == r"$\operatorname{asin}^{2}\left(x\right)$"
    assert latex(asin(x) ** 2, inv_trig_style="full") == r"$\operatorname{arcsin}^{2}\left(x\right)$"
    assert latex(asin(x) ** 2, inv_trig_style="power") == r"$\operatorname{sin}^{-1}\left(x\right)^{2}$"
    assert latex(asin(x ** 2), inv_trig_style="power", fold_func_brackets=True) == r"$\operatorname{sin}^{-1}x^{2}$"

    assert latex(factorial(k)) == r"$k!$"
    assert latex(factorial(-k)) == r"$\left(- k\right)!$"

    assert latex(floor(x)) == r"$\lfloor{x}\rfloor$"
    assert latex(ceiling(x)) == r"$\lceil{x}\rceil$"
    assert latex(abs(x)) == r"$\lvert{x}\rvert$"
    assert latex(re(x)) == r"$\Re{x}$"
    assert latex(im(x)) == r"$\Im{x}$"
    assert latex(conjugate(x)) == r"$\overline{x}$"
    assert latex(gamma(x)) == r"$\operatorname{\Gamma}\left(x\right)$"
    assert latex(Order(x)) == r"$\operatorname{\mathcal{O}}\left(x\right)$"

def test_ansi_math1_codegen():
    # not included: log10
    from sympy import acos, asin, atan, ceiling, cos, cosh, floor, log, ln, \
        sin, sinh, sqrt, tan, tanh, N
    x = symbols('x')
    name_expr = [
        ("test_fabs", abs(x)),
        ("test_acos", acos(x)),
        ("test_asin", asin(x)),
        ("test_atan", atan(x)),
        ("test_ceil", ceiling(x)),
        ("test_cos", cos(x)),
        ("test_cosh", cosh(x)),
        ("test_floor", floor(x)),
        ("test_log", log(x)),
        ("test_ln", ln(x)),
        ("test_sin", sin(x)),
        ("test_sinh", sinh(x)),
        ("test_sqrt", sqrt(x)),
        ("test_tan", tan(x)),
        ("test_tanh", tanh(x)),
    ]
    numerical_tests = []
    for name, expr in name_expr:
        for xval in 0.2, 0.5, 0.8:
            expected = N(expr.subs(x, xval))
            numerical_tests.append((name, (xval,), expected, 1e-14))
    run_cc_test("ansi_math1", name_expr, numerical_tests)

def test_latex_functions():
    assert latex(exp(x)) == "$e^{x}$"
    assert latex(exp(1)+exp(2)) == "$e + e^{2}$"

    f = Function('f')
    assert latex(f(x)) == '$\\operatorname{f}\\left(x\\right)$'

    beta = Function('beta')

    assert latex(beta(x)) == r"$\operatorname{beta}\left(x\right)$"
    assert latex(sin(x)) == r"$\operatorname{sin}\left(x\right)$"
    assert latex(sin(x), fold_func_brackets=True) == r"$\operatorname{sin}x$"
    assert latex(sin(2*x**2), fold_func_brackets=True) == \
    r"$\operatorname{sin}2 x^{2}$"

    assert latex(factorial(k)) == r"$k!$"
    assert latex(factorial(-k)) == r"$\left(- k\right)!$"

    assert latex(floor(x)) == r"$\lfloor{x}\rfloor$"
    assert latex(ceiling(x)) == r"$\lceil{x}\rceil$"
    assert latex(abs(x)) == r"$\lvert{x}\rvert$"
    assert latex(re(x)) == r"$\Re{x}$"
    assert latex(im(x)) == r"$\Im{x}$"
    assert latex(conjugate(x)) == r"$\overline{x}$"
    assert latex(gamma(x)) == r"$\operatorname{\Gamma}\left(x\right)$"
    assert latex(Order(x)) == r"$\operatorname{\mathcal{O}}\left(x\right)$"

 def eval(cls, ar, period):
     # Our strategy is to evaluate the argument on the riemann surface of the
     # logarithm, and then reduce.
     # NOTE evidently this means it is a rather bad idea to use this with
     # period != 2*pi and non-polar numbers.
     from sympy import ceiling, oo, atan2, atan, polar_lift, pi, Mul
     if not period.is_positive:
         return None
     if period == oo and isinstance(ar, principal_branch):
         return periodic_argument(*ar.args)
     if ar.func is polar_lift and period >= 2*pi:
         return periodic_argument(ar.args[0], period)
     if ar.is_Mul:
         newargs = [x for x in ar.args if not x.is_positive]
         if len(newargs) != len(ar.args):
             return periodic_argument(Mul(*newargs), period)
     unbranched = cls._getunbranched(ar)
     if unbranched is None:
         return None
     if unbranched.has(periodic_argument, atan2, arg, atan):
         return None
     if period == oo:
         return unbranched
     if period != oo:
         n = ceiling(unbranched/period - S(1)/2)*period
         if not n.has(ceiling):
             return unbranched - n

def test_bng_rounding():
    """ Test Ceiling and Floor match BNG implementation of rounding """
    x_sym = sympy.symbols('x')
    for x in (-1.5, -1, -0.5, 0, 0.5, 1.0, 1.5):
        for expr in (sympy.ceiling(x_sym), sympy.floor(x_sym)):
            assert expr.subs({'x': x}) == eval(
                _bng_print(expr), {}, {'rint': _rint, 'x': x})

def test_evalf_integer_parts():
    a = floor(log(8)/log(2) - exp(-1000), evaluate=False)
    b = floor(log(8)/log(2), evaluate=False)
    raises(PrecisionExhausted, "a.evalf()")
    assert a.evalf(chop=True) == 3
    assert a.evalf(maxn=500) == 2
    assert b.evalf() == 3
    # equals, as a fallback, can still fail but it might succeed as here
    assert ceiling(10*(sin(1)**2 + cos(1)**2)) == 10

    assert int(floor(factorial(50)/E,evaluate=False).evalf()) == \
        11188719610782480504630258070757734324011354208865721592720336800L
    assert int(ceiling(factorial(50)/E,evaluate=False).evalf()) == \
        11188719610782480504630258070757734324011354208865721592720336801L
    assert int(floor((GoldenRatio**999 / sqrt(5) + Rational(1,2))).evalf(1000)) == fibonacci(999)
    assert int(floor((GoldenRatio**1000 / sqrt(5) + Rational(1,2))).evalf(1000)) == fibonacci(1000)

def test_intrinsic_math1_codegen():
    # not included: log10
    from sympy import acos, asin, atan, ceiling, cos, cosh, floor, log, ln, \
        sin, sinh, sqrt, tan, tanh, N
    x = symbols('x')
    name_expr = [
        ("test_fabs", abs(x)),
        ("test_acos", acos(x)),
        ("test_asin", asin(x)),
        ("test_atan", atan(x)),
        ("test_cos", cos(x)),
        ("test_cosh", cosh(x)),
        ("test_log", log(x)),
        ("test_ln", ln(x)),
        ("test_sin", sin(x)),
        ("test_sinh", sinh(x)),
        ("test_sqrt", sqrt(x)),
        ("test_tan", tan(x)),
        ("test_tanh", tanh(x)),
    ]
    numerical_tests = []
    for name, expr in name_expr:
        for xval in 0.2, 0.5, 0.8:
            expected = N(expr.subs(x, xval))
            numerical_tests.append((name, (xval,), expected, 1e-14))
    for lang, commands in valid_lang_commands:
        if lang == "C":
            name_expr_C = [("test_floor", floor(x)), ("test_ceil", ceiling(x))]
        else:
            name_expr_C = []
        run_test("intrinsic_math1", name_expr + name_expr_C, numerical_tests, lang, commands)

def test_ceiling_requires_robust_assumptions():
    assert limit(ceiling(sin(x)), x, 0, "+") == 1
    assert limit(ceiling(sin(x)), x, 0, "-") == 0
    assert limit(ceiling(cos(x)), x, 0, "+") == 1
    assert limit(ceiling(cos(x)), x, 0, "-") == 1
    assert limit(ceiling(5 + sin(x)), x, 0, "+") == 6
    assert limit(ceiling(5 + sin(x)), x, 0, "-") == 5
    assert limit(ceiling(5 + cos(x)), x, 0, "+") == 6
    assert limit(ceiling(5 + cos(x)), x, 0, "-") == 6

def test_ceiling():
    assert ceiling(x).series(x) == 1
    assert ceiling(-x).series(x) == 0
    assert ceiling(sin(x)).series(x) == 1
    assert ceiling(sin(-x)).series(x) == 0
    assert ceiling(1 - cos(x)).series(x) == 1
    assert ceiling(1 - cos(-x)).series(x) == 1
    assert ceiling(x).series(x, 2) == 3
    assert ceiling(-x).series(x, 2) == -2

def test_dir():
    assert abs(x).series(x, 0, dir="+") == x
    assert abs(x).series(x, 0, dir="-") == -x
    assert floor(x + 2).series(x, 0, dir='+') == 2
    assert floor(x + 2).series(x, 0, dir='-') == 1
    assert floor(x + 2.2).series(x, 0, dir='-') == 2
    assert ceiling(x + 2.2).series(x, 0, dir='-') == 3
    assert sin(x + y).series(x, 0, dir='-') == sin(x + y).series(x, 0, dir='+')

def test_ceiling():
    x = Symbol('x')
    assert ceiling(x).series(x) == 1
    assert ceiling(-x).series(x) == 0
    assert ceiling(sin(x)).series(x) == 1
    assert ceiling(sin(-x)).series(x) == 0
    assert ceiling(1-cos(x)).series(x) == 1
    assert ceiling(1-cos(-x)).series(x) == 1
    assert ceiling(x).series(x, 2) == 3
    assert ceiling(-x).series(x, 2) == -2

def solve(a, b):
    s = sympy.fraction(sympy.Rational(a,b))
    if sympy.log(s[1],2) % 1 != 0:
        return 'impossible'
    y = sympy.ceiling(sympy.log(s[1]/s[0],2))
    if y > 40:
        return 'impossible'
    else:
        return str(y)

def test_own_module():
    f = lambdify(x, sin(x), math)
    assert f(0)==0.0
    f = lambdify(x, sympy.ceiling(x), math)
    try:
        f(4.5)
        assert False
    except NameError:
        pass