Python exprtools.factor_terms函数代码示例

def heuristics(e, z, z0, dir):
    """Computes the limit of an expression term-wise.
    Parameters are the same as for the ``limit`` function.
    Works with the arguments of expression ``e`` one by one, computing
    the limit of each and then combining the results. This approach
    works only for simple limits, but it is fast.
    """

    from sympy.calculus.util import AccumBounds
    rv = None
    if abs(z0) is S.Infinity:
        rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
        if isinstance(rv, Limit):
            return
    elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
        r = []
        for a in e.args:
            l = limit(a, z, z0, dir)
            if l.has(S.Infinity) and l.is_finite is None:
                if isinstance(e, Add):
                    m = factor_terms(e)
                    if not isinstance(m, Mul): # try together
                        m = together(m)
                    if not isinstance(m, Mul): # try factor if the previous methods failed
                        m = factor(e)
                    if isinstance(m, Mul):
                        return heuristics(m, z, z0, dir)
                    return
                return
            elif isinstance(l, Limit):
                return
            elif l is S.NaN:
                return
            else:
                r.append(l)
        if r:
            rv = e.func(*r)
            if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
                r2 = []
                e2 = []
                for ii in range(len(r)):
                    if isinstance(r[ii], AccumBounds):
                        r2.append(r[ii])
                    else:
                        e2.append(e.args[ii])

                if len(e2) > 0:
                    e3 = Mul(*e2).simplify()
                    l = limit(e3, z, z0, dir)
                    rv = l * Mul(*r2)

            if rv is S.NaN:
                try:
                    rat_e = ratsimp(e)
                except PolynomialError:
                    return
                if rat_e is S.NaN or rat_e == e:
                    return
                return limit(rat_e, z, z0, dir)
    return rv

    def distance(self, o):
        """
        Finds the shortest distance between a line and a point.

        Raises
        ======

        NotImplementedError is raised if o is not a Point

        Examples
        ========

        >>> from sympy import Point, Line
        >>> p1, p2 = Point(0, 0), Point(1, 1)
        >>> s = Line(p1, p2)
        >>> s.distance(Point(-1, 1))
        sqrt(2)
        >>> s.distance((-1, 2))
        3*sqrt(2)/2
        """
        if not isinstance(o, Point):
            if is_sequence(o):
                o = Point(o)
        a, b, c = self.coefficients
        if 0 in (a, b):
            return self.perpendicular_segment(o).length
        m = self.slope
        x = o.x
        y = m*x - c/b
        return abs(factor_terms(o.y - y))/sqrt(1 + m**2)

    def _is_connected(cls, x, y):
        """
        Check if x and y are connected somehow.
        """
        from sympy.core.exprtools import factor_terms
        def hit(v, t, f):
            if not v.is_Relational:
                return t if v else f
        for i in range(2):
            if x == y:
                return True
            r = hit(x >= y, Max, Min)
            if r is not None:
                return r
            r = hit(y <= x, Max, Min)
            if r is not None:
                return r
            r = hit(x <= y, Min, Max)
            if r is not None:
                return r
            r = hit(y >= x, Min, Max)
            if r is not None:
                return r
            # simplification can be expensive, so be conservative
            # in what is attempted
            x = factor_terms(x - y)
            y = S.Zero

        return False

    def _eval_as_leading_term(self, x):
        from sympy import expand_mul, factor_terms

        old = self

        expr = expand_mul(self)
        if not expr.is_Add:
            return expr.as_leading_term(x)

        infinite = [t for t in expr.args if t.is_infinite]

        expr = expr.func(*[t.as_leading_term(x) for t in expr.args]).removeO()
        if not expr:
            # simple leading term analysis gave us 0 but we have to send
            # back a term, so compute the leading term (via series)
            return old.compute_leading_term(x)
        elif expr is S.NaN:
            return old.func._from_args(infinite)
        elif not expr.is_Add:
            return expr
        else:
            plain = expr.func(*[s for s, _ in expr.extract_leading_order(x)])
            rv = factor_terms(plain, fraction=False)
            rv_simplify = rv.simplify()
            # if it simplifies to an x-free expression, return that;
            # tests don't fail if we don't but it seems nicer to do this
            if x not in rv_simplify.free_symbols:
                if rv_simplify.is_zero and plain.is_zero is not True:
                    return (expr - plain)._eval_as_leading_term(x)
                return rv_simplify
            return rv

def heuristics(e, z, z0, dir):
    from sympy.calculus.util import AccumBounds
    rv = None
    if abs(z0) is S.Infinity:
        rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
        if isinstance(rv, Limit):
            return
    elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
        r = []
        for a in e.args:
            l = limit(a, z, z0, dir)
            if l.has(S.Infinity) and l.is_finite is None:
                if isinstance(e, Add):
                    m = factor_terms(e)
                    if not isinstance(m, Mul): # try together
                        m = together(m)
                    if not isinstance(m, Mul): # try factor if the previous methods failed
                        m = factor(e)
                    if isinstance(m, Mul):
                        return heuristics(m, z, z0, dir)
                    return
                return
            elif isinstance(l, Limit):
                return
            elif l is S.NaN:
                return
            else:
                r.append(l)
        if r:
            rv = e.func(*r)
            if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
                r2 = []
                e2 = []
                for ii in range(len(r)):
                    if isinstance(r[ii], AccumBounds):
                        r2.append(r[ii])
                    else:
                        e2.append(e.args[ii])

                if len(e2) > 0:
                    e3 = Mul(*e2).simplify()
                    l = limit(e3, z, z0, dir)
                    rv = l * Mul(*r2)

            if rv is S.NaN:
                try:
                    rat_e = ratsimp(e)
                except PolynomialError:
                    return
                if rat_e is S.NaN or rat_e == e:
                    return
                return limit(rat_e, z, z0, dir)
    return rv

 def eval(cls, arg):
     if not arg.is_Atom:
         c, arg_ = factor_terms(arg).as_coeff_Mul()
         if arg_.is_Mul:
             arg_ = Mul(*[a if (sign(a) not in (-1, 1)) else sign(a) for a in arg_.args])
         arg_ = sign(c) * arg_
     else:
         arg_ = arg
     x, y = re(arg_), im(arg_)
     rv = C.atan2(y, x)
     if rv.is_number and not rv.atoms(AppliedUndef):
         return rv
     if arg_ != arg:
         return cls(arg_, evaluate=False)

def test_factor_terms():
    A = Symbol('A', commutative=False)
    assert factor_terms(9*(x + x*y + 1) + (3*x + 3)**(2 + 2*x)) == \
        9*x*y + 9*x + _keep_coeff(S(3), x + 1)**_keep_coeff(S(2), x + 1) + 9
    assert factor_terms(9*(x + x*y + 1) + (3)**(2 + 2*x)) == \
        _keep_coeff(S(9), 3**(2*x) + x*y + x + 1)
    assert factor_terms(3**(2 + 2*x) + a*3**(2 + 2*x)) == \
        9*3**(2*x)*(a + 1)
    assert factor_terms(x + x*A) == \
        x*(1 + A)
    assert factor_terms(sin(x + x*A)) == \
        sin(x*(1 + A))
    assert factor_terms((3*x + 3)**((2 + 2*x)/3)) == \
        _keep_coeff(S(3), x + 1)**_keep_coeff(S(2)/3, x + 1)
    assert factor_terms(x + (x*y + x)**(3*x + 3)) == \
        x + (x*(y + 1))**_keep_coeff(S(3), x + 1)
    assert factor_terms(a*(x + x*y) + b*(x*2 + y*x*2)) == \
        x*(a + 2*b)*(y + 1)
    i = Integral(x, (x, 0, oo))
    assert factor_terms(i) == i

 def eval(cls, arg):
     if isinstance(arg, exp_polar):
         return periodic_argument(arg, oo)
     if not arg.is_Atom:
         c, arg_ = factor_terms(arg).as_coeff_Mul()
         if arg_.is_Mul:
             arg_ = Mul(*[a if (sign(a) not in (-1, 1)) else
                 sign(a) for a in arg_.args])
         arg_ = sign(c)*arg_
     else:
         arg_ = arg
     if arg_.atoms(AppliedUndef):
         return
     x, y = arg_.as_real_imag()
     rv = atan2(y, x)
     if rv.is_number:
         return rv
     if arg_ != arg:
         return cls(arg_, evaluate=False)

def calculate_series(e, x, logx=None):
    """ Calculates at least one term of the series of "e" in "x".

    This is a place that fails most often, so it is in its own function.
    """
    from sympy.core.exprtools import factor_terms

    n = 1
    while 1:
        series = e.nseries(x, n=n, logx=logx)
        if not series.has(Order):
            # The series expansion is locally exact.
            return series

        series = series.removeO()
        series = factor_terms(series, fraction=True)
        if series:
            return series
        n *= 2

def heuristics(e, z, z0, dir):
    rv = None
    if abs(z0) is S.Infinity:
        rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
        if isinstance(rv, Limit):
            return
    elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
        r = []
        for a in e.args:
            l = limit(a, z, z0, dir)
            if l.has(S.Infinity) and l.is_finite is None:
                if isinstance(e, Add):
                    m = factor_terms(e)
                    if not isinstance(m, Mul): # try together
                        m = together(m)
                    if not isinstance(m, Mul): # try factor if the previous methods failed
                        m = factor(e)
                    if isinstance(m, Mul):
                        return heuristics(m, z, z0, dir)
                    return
                return
            elif isinstance(l, Limit):
                return
            elif l is S.NaN:
                return
            else:
                r.append(l)
        if r:
            rv = e.func(*r)
            if rv is S.NaN:
                try:
                    rat_e = ratsimp(e)
                except PolynomialError:
                    return
                if rat_e is S.NaN or rat_e == e:
                    return
                return limit(rat_e, z, z0, dir)
    return rv

 def _eval_power(self, e):
     if e.is_Rational and self.is_number:
         from sympy.core.evalf import pure_complex
         from sympy.core.mul import _unevaluated_Mul
         from sympy.core.exprtools import factor_terms
         from sympy.core.function import expand_multinomial
         from sympy.functions.elementary.complexes import sign
         from sympy.functions.elementary.miscellaneous import sqrt
         ri = pure_complex(self)
         if ri:
             r, i = ri
             if e.q == 2:
                 D = sqrt(r**2 + i**2)
                 if D.is_Rational:
                     # (r, i, D) is a Pythagorean triple
                     root = sqrt(factor_terms((D - r)/2))**e.p
                     return root*expand_multinomial((
                         # principle value
                         (D + r)/abs(i) + sign(i)*S.ImaginaryUnit)**e.p)
             elif e == -1:
                 return _unevaluated_Mul(
                     r - i*S.ImaginaryUnit,
                     1/(r**2 + i**2))

    def _eval_product(self, term, limits):
        from sympy.concrete.delta import deltaproduct, _has_simple_delta
        from sympy.concrete.summations import summation
        from sympy.functions import KroneckerDelta, RisingFactorial

        (k, a, n) = limits

        if k not in term.free_symbols:
            if (term - 1).is_zero:
                return S.One
            return term**(n - a + 1)

        if a == n:
            return term.subs(k, a)

        if term.has(KroneckerDelta) and _has_simple_delta(term, limits[0]):
            return deltaproduct(term, limits)

        dif = n - a
        if dif.is_Integer:
            return Mul(*[term.subs(k, a + i) for i in range(dif + 1)])

        elif term.is_polynomial(k):
            poly = term.as_poly(k)

            A = B = Q = S.One

            all_roots = roots(poly)

            M = 0
            for r, m in all_roots.items():
                M += m
                A *= RisingFactorial(a - r, n - a + 1)**m
                Q *= (n - r)**m

            if M < poly.degree():
                arg = quo(poly, Q.as_poly(k))
                B = self.func(arg, (k, a, n)).doit()

            return poly.LC()**(n - a + 1) * A * B

        elif term.is_Add:
            factored = factor_terms(term, fraction=True)
            if factored.is_Mul:
                return self._eval_product(factored, (k, a, n))

        elif term.is_Mul:
            exclude, include = [], []

            for t in term.args:
                p = self._eval_product(t, (k, a, n))

                if p is not None:
                    exclude.append(p)
                else:
                    include.append(t)

            if not exclude:
                return None
            else:
                arg = term._new_rawargs(*include)
                A = Mul(*exclude)
                B = self.func(arg, (k, a, n)).doit()
                return A * B

        elif term.is_Pow:
            if not term.base.has(k):
                s = summation(term.exp, (k, a, n))

                return term.base**s
            elif not term.exp.has(k):
                p = self._eval_product(term.base, (k, a, n))

                if p is not None:
                    return p**term.exp

        elif isinstance(term, Product):
            evaluated = term.doit()
            f = self._eval_product(evaluated, limits)
            if f is None:
                return self.func(evaluated, limits)
            else:
                return f

def test_issue_3261():
    a, b = symbols("a b")
    apb = a + b
    eq = apb + apb**2*(-2*a - 2*b)
    assert factor_terms(sub_pre(eq)) == a + b - 2*(a + b)**3

def test_factor_terms():
    A = Symbol('A', commutative=False)
    assert factor_terms(9*(x + x*y + 1) + (3*x + 3)**(2 + 2*x)) == \
        9*x*y + 9*x + _keep_coeff(S(3), x + 1)**_keep_coeff(S(2), x + 1) + 9
    assert factor_terms(9*(x + x*y + 1) + (3)**(2 + 2*x)) == \
        _keep_coeff(S(9), 3**(2*x) + x*y + x + 1)
    assert factor_terms(3**(2 + 2*x) + a*3**(2 + 2*x)) == \
        9*3**(2*x)*(a + 1)
    assert factor_terms(x + x*A) == \
        x*(1 + A)
    assert factor_terms(sin(x + x*A)) == \
        sin(x*(1 + A))
    assert factor_terms((3*x + 3)**((2 + 2*x)/3)) == \
        _keep_coeff(S(3), x + 1)**_keep_coeff(S(2)/3, x + 1)
    assert factor_terms(x + (x*y + x)**(3*x + 3)) == \
        x + (x*(y + 1))**_keep_coeff(S(3), x + 1)
    assert factor_terms(a*(x + x*y) + b*(x*2 + y*x*2)) == \
        x*(a + 2*b)*(y + 1)
    i = Integral(x, (x, 0, oo))
    assert factor_terms(i) == i
    eq = sqrt(2) + sqrt(10)
    assert factor_terms(eq) == eq
    assert factor_terms(eq, radical=True) == sqrt(2)*(1 + sqrt(5))
    eq = [x + x*y]
    ans = [x*(y + 1)]
    for c in [list, tuple, set]:
        assert factor_terms(c(eq)) == c(ans)
    assert factor_terms(Tuple(x + x*y)) == Tuple(x*(y + 1))
    assert factor_terms(Interval(0, 1)) == Interval(0, 1)
    e = 1/sqrt(a/2 + 1)
    assert factor_terms(e, clear=False) == 1/sqrt(a/2 + 1)
    assert factor_terms(e, clear=True) == sqrt(2)/sqrt(a + 2)

    eq = x/(x + 1/x) + 1/(x**2 + 1)
    assert factor_terms(eq, fraction=False) == eq
    assert factor_terms(eq, fraction=True) == 1

    assert factor_terms((1/(x**3 + x**2) + 2/x**2)*y) == \
        y*(2 + 1/(x + 1))/x**2

    assert factor_terms(-x - y) == Mul(-1, x + y, evaluate=False)
    # if not True, then processes for this in factor_terms is not necessary
    assert gcd_terms(-x - y) == -x - y

    def doit(self, **hints):
        """Evaluates limit"""
        from sympy.series.limitseq import limit_seq
        from sympy.functions import RisingFactorial

        e, z, z0, dir = self.args

        if hints.get('deep', True):
            e = e.doit(**hints)
            z = z.doit(**hints)
            z0 = z0.doit(**hints)

        if e == z:
            return z0

        if not e.has(z):
            return e

        # gruntz fails on factorials but works with the gamma function
        # If no factorial term is present, e should remain unchanged.
        # factorial is defined to be zero for negative inputs (which
        # differs from gamma) so only rewrite for positive z0.
        if z0.is_positive:
            e = e.rewrite([factorial, RisingFactorial], gamma)

        if e.is_Mul:
            if abs(z0) is S.Infinity:
                e = factor_terms(e)
                e = e.rewrite(fibonacci, GoldenRatio)
                ok = lambda w: (z in w.free_symbols and
                                any(a.is_polynomial(z) or
                                    any(z in m.free_symbols and m.is_polynomial(z)
                                        for m in Mul.make_args(a))
                                    for a in Add.make_args(w)))
                if all(ok(w) for w in e.as_numer_denom()):
                    u = Dummy(positive=(z0 is S.Infinity))
                    inve = e.subs(z, 1/u)
                    r = limit(inve.as_leading_term(u), u,
                              S.Zero, "+" if z0 is S.Infinity else "-")
                    if isinstance(r, Limit):
                        return self
                    else:
                        return r

        if e.is_Order:
            return Order(limit(e.expr, z, z0), *e.args[1:])

        try:
            r = gruntz(e, z, z0, dir)
            if r is S.NaN:
                raise PoleError()
        except (PoleError, ValueError):
            r = heuristics(e, z, z0, dir)
            if r is None:
                return self
        except NotImplementedError:
            # Trying finding limits of sequences
            if hints.get('sequence', True) and z0 is S.Infinity:
                trials = hints.get('trials', 5)
                r = limit_seq(e, z, trials)
                if r is None:
                    raise NotImplementedError()
            else:
                raise NotImplementedError()

        return r