def test_ccode_sign():
    expr1, ref1 = sign(x) * y, 'y*(((x) > 0) - ((x) < 0))'
    expr2, ref2 = sign(cos(x)), '(((cos(x)) > 0) - ((cos(x)) < 0))'
    expr3, ref3 = sign(2 * x + x**2) * x + x**2, 'pow(x, 2) + x*(((pow(x, 2) + 2*x) > 0) - ((pow(x, 2) + 2*x) < 0))'
    assert ccode(expr1) == ref1
    assert ccode(expr1, 'z') == 'z = %s;' % ref1
    assert ccode(expr2) == ref2
    assert ccode(expr3) == ref3

def test_ccode_sign():

    expr = sign(x) * y
    assert ccode(expr) == 'y*(((x) > 0) - ((x) < 0))'
    assert ccode(expr, 'z') == 'z = y*(((x) > 0) - ((x) < 0));'

    assert ccode(sign(2 * x + x**2) * x + x**2) == \
        'pow(x, 2) + x*(((pow(x, 2) + 2*x) > 0) - ((pow(x, 2) + 2*x) < 0))'

    expr = sign(cos(x))
    assert ccode(expr) == '(((cos(x)) > 0) - ((cos(x)) < 0))'

def test_sign():
    expr = sign(x) * y
    assert rust_code(expr) == "y*x.signum()"
    assert rust_code(expr, assign_to='r') == "r = y*x.signum();"

    expr = sign(x + y) + 42
    assert rust_code(expr) == "(x + y).signum() + 42"
    assert rust_code(expr, assign_to='r') == "r = (x + y).signum() + 42;"

    expr = sign(cos(x))
    assert rust_code(expr) == "x.cos().signum()"

def test_rcode_sgn():

    expr = sign(x) * y
    assert rcode(expr) == 'y*sign(x)'
    p = rcode(expr, 'z')
    assert p  == 'z = y*sign(x);'

    p = rcode(sign(2 * x + x**2) * x + x**2)
    assert p  == "x^2 + x*sign(x^2 + 2*x)"

    expr = sign(cos(x))
    p = rcode(expr)
    assert p == 'sign(cos(x))'

def test_bounded():
    x, y = symbols('xy')
    assert ask(x, Q.bounded) == False
    assert ask(x, Q.bounded, Assume(x, Q.bounded)) == True
    assert ask(x, Q.bounded, Assume(y, Q.bounded)) == False
    assert ask(x, Q.bounded, Assume(x, Q.complex)) == False

    assert ask(x+1, Q.bounded) == False
    assert ask(x+1, Q.bounded, Assume(x, Q.bounded)) == True
    assert ask(x+y, Q.bounded) == None
    assert ask(x+y, Q.bounded, Assume(x, Q.bounded)) == False
    assert ask(x+1, Q.bounded, Assume(x, Q.bounded) & \
                Assume(y, Q.bounded)) == True

    assert ask(2*x, Q.bounded) == False
    assert ask(2*x, Q.bounded, Assume(x, Q.bounded)) == True
    assert ask(x*y, Q.bounded) == None
    assert ask(x*y, Q.bounded, Assume(x, Q.bounded)) == False
    assert ask(x*y, Q.bounded, Assume(x, Q.bounded) & \
                Assume(y, Q.bounded)) == True

    assert ask(x**2, Q.bounded) == False
    assert ask(2**x, Q.bounded) == False
    assert ask(2**x, Q.bounded, Assume(x, Q.bounded)) == True
    assert ask(x**x, Q.bounded) == False
    assert ask(Rational(1,2) ** x, Q.bounded) == True
    assert ask(x ** Rational(1,2), Q.bounded) == False

    # sign function
    assert ask(sign(x), Q.bounded) == True
    assert ask(sign(x), Q.bounded, Assume(x, Q.bounded, False)) == True

    # exponential functions
    assert ask(log(x), Q.bounded) == False
    assert ask(log(x), Q.bounded, Assume(x, Q.bounded)) == True
    assert ask(exp(x), Q.bounded) == False
    assert ask(exp(x), Q.bounded, Assume(x, Q.bounded)) == True
    assert ask(exp(2), Q.bounded) == True

    # trigonometric functions
    assert ask(sin(x), Q.bounded) == True
    assert ask(sin(x), Q.bounded, Assume(x, Q.bounded, False)) == True
    assert ask(cos(x), Q.bounded) == True
    assert ask(cos(x), Q.bounded, Assume(x, Q.bounded, False)) == True
    assert ask(2*sin(x), Q.bounded) == True
    assert ask(sin(x)**2, Q.bounded) == True
    assert ask(cos(x)**2, Q.bounded) == True
    assert ask(cos(x) + sin(x), Q.bounded) == True

def test_bounded():
    x, y = symbols('x,y')
    assert ask(Q.bounded(x)) == False
    assert ask(Q.bounded(x), Q.bounded(x)) == True
    assert ask(Q.bounded(x), Q.bounded(y)) == False
    assert ask(Q.bounded(x), Q.complex(x)) == False

    assert ask(Q.bounded(x+1)) == False
    assert ask(Q.bounded(x+1), Q.bounded(x)) == True
    assert ask(Q.bounded(x+y)) == None
    assert ask(Q.bounded(x+y), Q.bounded(x)) == False
    assert ask(Q.bounded(x+1), Q.bounded(x) & Q.bounded(y)) == True

    assert ask(Q.bounded(2*x)) == False
    assert ask(Q.bounded(2*x), Q.bounded(x)) == True
    assert ask(Q.bounded(x*y)) == None
    assert ask(Q.bounded(x*y), Q.bounded(x)) == False
    assert ask(Q.bounded(x*y), Q.bounded(x) & Q.bounded(y)) == True

    assert ask(Q.bounded(x**2)) == False
    assert ask(Q.bounded(2**x)) == False
    assert ask(Q.bounded(2**x), Q.bounded(x)) == True
    assert ask(Q.bounded(x**x)) == False
    assert ask(Q.bounded(Rational(1,2) ** x)) == None
    assert ask(Q.bounded(Rational(1,2) ** x), Q.positive(x)) == True
    assert ask(Q.bounded(Rational(1,2) ** x), Q.negative(x)) == False
    assert ask(Q.bounded(sqrt(x))) == False

    # sign function
    assert ask(Q.bounded(sign(x))) == True
    assert ask(Q.bounded(sign(x)), ~Q.bounded(x)) == True

    # exponential functions
    assert ask(Q.bounded(log(x))) == False
    assert ask(Q.bounded(log(x)), Q.bounded(x)) == True
    assert ask(Q.bounded(exp(x))) == False
    assert ask(Q.bounded(exp(x)), Q.bounded(x)) == True
    assert ask(Q.bounded(exp(2))) == True

    # trigonometric functions
    assert ask(Q.bounded(sin(x))) == True
    assert ask(Q.bounded(sin(x)), ~Q.bounded(x)) == True
    assert ask(Q.bounded(cos(x))) == True
    assert ask(Q.bounded(cos(x)), ~Q.bounded(x)) == True
    assert ask(Q.bounded(2*sin(x))) == True
    assert ask(Q.bounded(sin(x)**2)) == True
    assert ask(Q.bounded(cos(x)**2)) == True
    assert ask(Q.bounded(cos(x) + sin(x))) == True

 def calc_limit(a, b):
     """replace x with a, using subs if possible, otherwise limit
     where sign of b is considered"""
     wok = inverse_mapping.subs(x, a)
     if wok is S.NaN or wok.is_bounded is False and a.is_bounded:
         return limit(sign(b)*inverse_mapping, x, a)
     return wok

def refine_Pow(expr, assumptions):
    """
    Handler for instances of Pow.

    >>> from sympy import Symbol, Assume, Q
    >>> from sympy.assumptions.refine import refine_Pow
    >>> from sympy.abc import x
    >>> refine_Pow((-1)**x, Assume(x, Q.real))
    >>> refine_Pow((-1)**x, Assume(x, Q.even))
    1
    >>> refine_Pow((-1)**x, Assume(x, Q.odd))
    -1

    """
    from sympy.core import Pow, Rational
    from sympy.functions import sign

    if ask(expr.base, Q.real, assumptions):
        if expr.base.is_number:
            if ask(expr.exp, Q.even, assumptions):
                return abs(expr.base) ** expr.exp
            if ask(expr.exp, Q.odd, assumptions):
                return sign(expr.base) * abs(expr.base) ** expr.exp
        if isinstance(expr.exp, Rational):
            if type(expr.base) is Pow:
                return abs(expr.base.base) ** (expr.base.exp * expr.exp)

 def calc_limit(a, b):
     """replace x with a, using subs if possible, otherwise limit
     where sign of b is considered"""
     wok = inverse_mapping.subs(x, a)
     if not wok is S.NaN:
         return wok
     return limit(sign(b)*inverse_mapping, x, a)

def denester(nested):
    """
    Denests a list of expressions that contain nested square roots.
    This method should not be called directly - use 'sqrtdenest' instead.
    This algorithm is based on <http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf>.

    It is assumed that all of the elements of 'nested' share the same
    bottom-level radicand. (This is stated in the paper, on page 177, in
    the paragraph immediately preceding the algorithm.)

    When evaluating all of the arguments in parallel, the bottom-level
    radicand only needs to be denested once. This means that calling
    denester with x arguments results in a recursive invocation with x+1
    arguments; hence denester has polynomial complexity.

    However, if the arguments were evaluated separately, each call would
    result in two recursive invocations, and the algorithm would have
    exponential complexity.

    This is discussed in the paper in the middle paragraph of page 179.
    """
    if all((n**2).is_Number for n in nested): #If none of the arguments are nested
        for f in subsets(len(nested)): #Test subset 'f' of nested
            p = prod(nested[i]**2 for i in range(len(f)) if f[i]).expand()
            if 1 in f and f.count(1) > 1 and f[-1]:
                p = -p
            if sqrt(p).is_Number:
                return sqrt(p), f #If we got a perfect square, return its square root.
        return nested[-1], [0]*len(nested) #Otherwise, return the radicand from the previous invocation.
    else:
        a, b, r, R = Wild('a'), Wild('b'), Wild('r'), None
        values = [expr.match(sqrt(a + b * sqrt(r))) for expr in nested]
        if any(v is None for v in values): # this pattern is not recognized
            return nested[-1], [0]*len(nested) #Otherwise, return the radicand from the previous invocation.
        for v in values:
            if r in v: #Since if b=0, r is not defined
                if R is not None:
                    assert R == v[r] #All the 'r's should be the same.
                else:
                    R = v[r]
        if R is None:
            return nested[-1], [0]*len(nested) #return the radicand from the previous invocation.
        d, f = denester([sqrt((v[a]**2).expand()-(R*v[b]**2).expand()) for v in values] + [sqrt(R)])
        if all(fi == 0 for fi in f):
            v = values[-1]
            return sqrt(v[a] + v[b]*d), f
        else:
            v = prod(nested[i]**2 for i in range(len(nested)) if f[i]).expand().match(a+b*sqrt(r))
            if 1 in f and f.index(1) < len(nested) - 1 and f[len(nested)-1]:
                v[a] = -1 * v[a]
                v[b] = -1 * v[b]
            if not f[len(nested)]: #Solution denests with square roots
                vad = (v[a] + d).expand()
                if not vad:
                    return nested[-1], [0]*len(nested) #Otherwise, return the radicand from the previous invocation.
                return (sqrt(vad/2) + sign(v[b])*sqrt((v[b]**2*R/(2*vad)).expand())).expand(), f
            else: #Solution requires a fourth root
                FR, s = (R.expand()**Rational(1,4)), sqrt((v[b]*R).expand()+d)
                return (s/(sqrt(2)*FR) + v[a]*FR/(sqrt(2)*s)).expand(), f

 def _calc_limit_1(F, a, b):
     """
     replace d with a, using subs if possible, otherwise limit
     where sign of b is considered
     """
     wok = F.subs(d, a)
     if wok is S.NaN or wok.is_bounded is False and a.is_bounded:
         return limit(sign(b)*F, d, a)
     return wok

    def p_parameter(self):
        """P is a parameter of parabola.

        Returns
        =======

        p : number or symbolic expression

        Notes
        =====

        The absolute value of p is the focal length. The sign on p tells
        which way the parabola faces. Vertical parabolas that open up
        and horizontal that open right, give a positive value for p.
        Vertical parabolas that open down and horizontal that open left,
        give a negative value for p.


        See Also
        ========

        http://www.sparknotes.com/math/precalc/conicsections/section2.rhtml

        Examples
        ========

        >>> from sympy import Parabola, Point, Line
        >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
        >>> p1.p_parameter
        -4

        """
        if self.axis_of_symmetry.slope == 0:
            x = self.directrix.coefficients[2]
            p = sign(self.focus.args[0] + x)
        else:
            y = self.directrix.coefficients[2]
            p = sign(self.focus.args[1] + y)
        return p * self.focal_length

def test_Function():
    assert mcode(sin(x) ** cos(x)) == "sin(x).^cos(x)"
    assert mcode(sign(x)) == "sign(x)"
    assert mcode(exp(x)) == "exp(x)"
    assert mcode(log(x)) == "log(x)"
    assert mcode(factorial(x)) == "factorial(x)"
    assert mcode(floor(x)) == "floor(x)"
    assert mcode(atan2(y, x)) == "atan2(y, x)"
    assert mcode(beta(x, y)) == 'beta(x, y)'
    assert mcode(polylog(x, y)) == 'polylog(x, y)'
    assert mcode(harmonic(x)) == 'harmonic(x)'
    assert mcode(bernoulli(x)) == "bernoulli(x)"
    assert mcode(bernoulli(x, y)) == "bernoulli(x, y)"

def _sqrt_numeric_denest(a, b, r, d2):
    """Helper that denest expr = a + b*sqrt(r), with d2 = a**2 - b**2*r > 0
    or returns None if not denested.
    """
    from sympy.simplify.simplify import radsimp
    depthr = sqrt_depth(r)
    d = sqrt(d2)
    vad = a + d
    # sqrt_depth(res) <= sqrt_depth(vad) + 1
    # sqrt_depth(expr) = depthr + 2
    # there is denesting if sqrt_depth(vad)+1 < depthr + 2
    # if vad**2 is Number there is a fourth root
    if sqrt_depth(vad) < depthr + 1 or (vad**2).is_Rational:
        vad1 = radsimp(1/vad)
        return (sqrt(vad/2) + sign(b)*sqrt((b**2*r*vad1/2).expand())).expand()

def test_PythonCodePrinter():
    prntr = PythonCodePrinter()
    assert not prntr.module_imports
    assert prntr.doprint(x**y) == 'x**y'
    assert prntr.doprint(Mod(x, 2)) == 'x % 2'
    assert prntr.doprint(And(x, y)) == 'x and y'
    assert prntr.doprint(Or(x, y)) == 'x or y'
    assert not prntr.module_imports
    assert prntr.doprint(pi) == 'math.pi'
    assert prntr.module_imports == {'math': {'pi'}}
    assert prntr.doprint(acos(x)) == 'math.acos(x)'
    assert prntr.doprint(Assignment(x, 2)) == 'x = 2'
    assert prntr.doprint(Piecewise((1, Eq(x, 0)),
                        (2, x>6))) == '((1) if (x == 0) else (2) if (x > 6) else None)'
    assert prntr.doprint(Piecewise((2, Le(x, 0)),
                        (3, Gt(x, 0)), evaluate=False)) == '((2) if (x <= 0) else'\
                                                        ' (3) if (x > 0) else None)'
    assert prntr.doprint(sign(x)) == '(0.0 if x == 0 else math.copysign(1, x))'

def _denester(nested, av0, h, max_depth_level):
    """Denests a list of expressions that contain nested square roots.

    Algorithm based on <http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf>.

    It is assumed that all of the elements of 'nested' share the same
    bottom-level radicand. (This is stated in the paper, on page 177, in
    the paragraph immediately preceding the algorithm.)

    When evaluating all of the arguments in parallel, the bottom-level
    radicand only needs to be denested once. This means that calling
    _denester with x arguments results in a recursive invocation with x+1
    arguments; hence _denester has polynomial complexity.

    However, if the arguments were evaluated separately, each call would
    result in two recursive invocations, and the algorithm would have
    exponential complexity.

    This is discussed in the paper in the middle paragraph of page 179.
    """
    from sympy.simplify.simplify import radsimp
    if h > max_depth_level:
        return None, None
    if av0[1] is None:
        return None, None
    if (av0[0] is None and
        all(n.is_Number for n in nested)): # no arguments are nested
        for f in subsets(len(nested)): # test subset 'f' of nested
            p = _mexpand(Mul(*[nested[i] for i in range(len(f)) if f[i]]))
            if f.count(1) > 1 and f[-1]:
                p = -p
            sqp = sqrt(p)
            if sqp.is_Rational:
                return sqp, f # got a perfect square so return its square root.
        # Otherwise, return the radicand from the previous invocation.
        return sqrt(nested[-1]), [0]*len(nested)
    else:
        R = None
        if av0[0] is not None:
            values = [av0[:2]]
            R = av0[2]
            nested2 = [av0[3], R]
            av0[0] = None
        else:
            values = filter(None, [_sqrt_match(expr) for expr in nested])
            for v in values:
                if v[2]: #Since if b=0, r is not defined
                    if R is not None:
                        if R != v[2]:
                            av0[1] = None
                            return None, None
                    else:
                        R = v[2]
            if R is None:
                # return the radicand from the previous invocation
                return sqrt(nested[-1]), [0]*len(nested)
            nested2 = [_mexpand(v[0]**2) -
                       _mexpand(R*v[1]**2) for v in values] + [R]
        d, f = _denester(nested2, av0, h + 1, max_depth_level)
        if not f:
            return None, None
        if not any(f[i] for i in range(len(nested))):
            v = values[-1]
            return sqrt(v[0] + v[1]*d), f
        else:
            p = Mul(*[nested[i] for i in range(len(nested)) if f[i]])
            v = _sqrt_match(p)
            if 1 in f and f.index(1) < len(nested) - 1 and f[len(nested) - 1]:
                v[0] = -v[0]
                v[1] = -v[1]
            if not f[len(nested)]: #Solution denests with square roots
                vad = _mexpand(v[0] + d)
                if vad <= 0:
                    # return the radicand from the previous invocation.
                    return sqrt(nested[-1]), [0]*len(nested)
                if not(sqrt_depth(vad) < sqrt_depth(R) + 1 or
                       (vad**2).is_Number):
                    av0[1] = None
                    return None, None

                vad1 = radsimp(1/vad)
                return _mexpand(sqrt(vad/2) +
                                sign(v[1])*sqrt(_mexpand(v[1]**2*R*vad1/2))), f
            else: #Solution requires a fourth root
                s2 = _mexpand(v[1]*R) + d
                if s2 <= 0:
                    return sqrt(nested[-1]), [0]*len(nested)
                FR, s = root(_mexpand(R), 4), sqrt(s2)
                return _mexpand(s/(sqrt(2)*FR) + v[0]*FR/(sqrt(2)*s)), f

def Jump(x,x_i,Xi):
    return 1./2.*(sign(x-Xi)-sign(x_i-Xi))

def refine_Pow(expr, assumptions):
    """
    Handler for instances of Pow.

    >>> from sympy import Symbol, Q
    >>> from sympy.assumptions.refine import refine_Pow
    >>> from sympy.abc import x,y,z
    >>> refine_Pow((-1)**x, Q.real(x))
    >>> refine_Pow((-1)**x, Q.even(x))
    1
    >>> refine_Pow((-1)**x, Q.odd(x))
    -1

    For powers of -1, even parts of the exponent can be simplified:

    >>> refine_Pow((-1)**(x+y), Q.even(x))
    (-1)**y
    >>> refine_Pow((-1)**(x+y+z), Q.odd(x) & Q.odd(z))
    (-1)**y
    >>> refine_Pow((-1)**(x+y+2), Q.odd(x))
    (-1)**(y + 1)
    >>> refine_Pow((-1)**(x+3), True)
    (-1)**(x + 1)

    """
    from sympy.core import Pow, Rational
    from sympy.functions.elementary.complexes import Abs
    from sympy.functions import sign
    if isinstance(expr.base, Abs):
        if ask(Q.real(expr.base.args[0]), assumptions) and \
                ask(Q.even(expr.exp), assumptions):
            return expr.base.args[0] ** expr.exp
    if ask(Q.real(expr.base), assumptions):
        if expr.base.is_number:
            if ask(Q.even(expr.exp), assumptions):
                return abs(expr.base) ** expr.exp
            if ask(Q.odd(expr.exp), assumptions):
                return sign(expr.base) * abs(expr.base) ** expr.exp
        if isinstance(expr.exp, Rational):
            if type(expr.base) is Pow:
                return abs(expr.base.base) ** (expr.base.exp * expr.exp)

        if expr.base is S.NegativeOne:
            if expr.exp.is_Add:

                old = expr

                # For powers of (-1) we can remove
                #  - even terms
                #  - pairs of odd terms
                #  - a single odd term + 1
                #  - A numerical constant N can be replaced with mod(N,2)

                coeff, terms = expr.exp.as_coeff_add()
                terms = set(terms)
                even_terms = set([])
                odd_terms = set([])
                initial_number_of_terms = len(terms)

                for t in terms:
                    if ask(Q.even(t), assumptions):
                        even_terms.add(t)
                    elif ask(Q.odd(t), assumptions):
                        odd_terms.add(t)

                terms -= even_terms
                if len(odd_terms) % 2:
                    terms -= odd_terms
                    new_coeff = (coeff + S.One) % 2
                else:
                    terms -= odd_terms
                    new_coeff = coeff % 2

                if new_coeff != coeff or len(terms) < initial_number_of_terms:
                    terms.add(new_coeff)
                    expr = expr.base**(Add(*terms))

                # Handle (-1)**((-1)**n/2 + m/2)
                e2 = 2*expr.exp
                if ask(Q.even(e2), assumptions):
                    if e2.could_extract_minus_sign():
                        e2 *= expr.base
                if e2.is_Add:
                    i, p = e2.as_two_terms()
                    if p.is_Pow and p.base is S.NegativeOne:
                        if ask(Q.integer(p.exp), assumptions):
                            i = (i + 1)/2
                            if ask(Q.even(i), assumptions):
                                return expr.base**p.exp
                            elif ask(Q.odd(i), assumptions):
                                return expr.base**(p.exp + 1)
                            else:
                                return expr.base**(p.exp + i)

                if old != expr:
                    return expr

def test_NumPyPrinter():
    p = NumPyPrinter()
    assert p.doprint(sign(x)) == 'numpy.sign(x)'
    A = MatrixSymbol("A", 2, 2)
    assert p.doprint(A**(-1)) == "numpy.linalg.inv(A)"
    assert p.doprint(A**5) == "numpy.linalg.matrix_power(A, 5)"

def test_MpmathPrinter():
    p = MpmathPrinter()
    assert p.doprint(sign(x)) == 'mpmath.sign(x)'
    assert p.doprint(Rational(1, 2)) == 'mpmath.mpf(1)/mpmath.mpf(2)'