text: """
  There are three types of functions implemented in SymPy:
  
      1) defined functions (in the sense that they can be evaluated) like
         exp or sin; they have a name and a body:
             f = exp
      2) undefined function which have a name but no body. Undefined
         functions can be defined using a Function class as follows:
             f = Function('f')
         (the result will be a Function instance)
      3) anonymous function (or lambda function) which have a body (defined
         with dummy variables) but have no name:
             f = Lambda(x, exp(x)*x)
             f = Lambda((x, y), exp(x)*y)
      The fourth type of functions are composites, like (sin + cos)(x); these work in
      SymPy core, but are not yet part of SymPy.
  
      Examples
      ========
  
      >>> import sympy
      >>> f = sympy.Function("f")
      >>> from sympy.abc import x
      >>> f(x)
      f(x)
      >>> print(sympy.srepr(f(x).func))
      Function('f')
      >>> f(x).args
      (x,)
  
  """
  from __future__ import print_function, division
  
  from .add import Add
  from .assumptions import ManagedProperties, _assume_defined
  from .basic import Basic, _atomic
  from .cache import cacheit
  from .compatibility import iterable, is_sequence, as_int, ordered, Iterable
  from .decorators import _sympifyit
  from .expr import Expr, AtomicExpr
  from .numbers import Rational, Float
  from .operations import LatticeOp
  from .rules import Transform
  from .singleton import S
  from .sympify import sympify
  
  from sympy.core.containers import Tuple, Dict
  from sympy.core.logic import fuzzy_and
  from sympy.core.compatibility import string_types, with_metaclass, PY3, range
  from sympy.utilities import default_sort_key
  from sympy.utilities.misc import filldedent
  from sympy.utilities.iterables import has_dups, sift
  from sympy.core.evaluate import global_evaluate
  
  import mpmath
  import mpmath.libmp as mlib
  
  import inspect
  from collections import Counter
  
  def _coeff_isneg(a):
      """Return True if the leading Number is negative.
  
      Examples
      ========
  
      >>> from sympy.core.function import _coeff_isneg
      >>> from sympy import S, Symbol, oo, pi
      >>> _coeff_isneg(-3*pi)
      True
      >>> _coeff_isneg(S(3))
      False
      >>> _coeff_isneg(-oo)
      True
      >>> _coeff_isneg(Symbol('n', negative=True)) # coeff is 1
      False
  
      For matrix expressions:
  
      >>> from sympy import MatrixSymbol, sqrt
      >>> A = MatrixSymbol("A", 3, 3)
      >>> _coeff_isneg(-sqrt(2)*A)
      True
      >>> _coeff_isneg(sqrt(2)*A)
      False
      """
  
      if a.is_MatMul:
          a = a.args[0]
      if a.is_Mul:
          a = a.args[0]
      return a.is_Number and a.is_negative
  
  
  class PoleError(Exception):
      pass
  
  
  class ArgumentIndexError(ValueError):
      def __str__(self):
          return ("Invalid operation with argument number %s for Function %s" %
                 (self.args[1], self.args[0]))
  
  
  # Python 2 and 3 compatible version that do not raise a Deprecation warning.
  def arity(cls):
      """Return the arity of the function if it is known, else None.
  
      When default values are specified for some arguments, they are
      optional and the arity is reported as a tuple of possible values.
  
      Examples
      ========
  
      >>> from sympy.core.function import arity
      >>> from sympy import log
      >>> arity(lambda x: x)
      1
      >>> arity(log)
      (1, 2)
      >>> arity(lambda *x: sum(x)) is None
      True
      """
      eval_ = getattr(cls, 'eval', cls)
      if PY3:
          parameters = inspect.signature(eval_).parameters.items()
          if [p for _, p in parameters if p.kind == p.VAR_POSITIONAL]:
              return
          p_or_k = [p for _, p in parameters if p.kind == p.POSITIONAL_OR_KEYWORD]
          # how many have no default and how many have a default value
          no, yes = map(len, sift(p_or_k,
              lambda p:p.default == p.empty, binary=True))
          return no if not yes else tuple(range(no, no + yes + 1))
      else:
          cls_ = int(hasattr(cls, 'eval'))  # correction for cls arguments
          evalargspec = inspect.getargspec(eval_)
          if evalargspec.varargs:
              return
          else:
              evalargs = len(evalargspec.args) - cls_
              if evalargspec.defaults:
                  # if there are default args then they are optional; the
                  # fewest args will occur when all defaults are used and
                  # the most when none are used (i.e. all args are given)
                  fewest = evalargs - len(evalargspec.defaults)
                  return tuple(range(fewest, evalargs + 1))
              return evalargs
  
  
  class FunctionClass(ManagedProperties):
      """
      Base class for function classes. FunctionClass is a subclass of type.
  
      Use Function('<function name>' [ , signature ]) to create
      undefined function classes.
      """
      _new = type.__new__
  
      def __init__(cls, *args, **kwargs):
          # honor kwarg value or class-defined value before using
          # the number of arguments in the eval function (if present)
          nargs = kwargs.pop('nargs', cls.__dict__.get('nargs', arity(cls)))
  
          # Canonicalize nargs here; change to set in nargs.
          if is_sequence(nargs):
              if not nargs:
                  raise ValueError(filldedent('''
                      Incorrectly specified nargs as %s:
                      if there are no arguments, it should be
                      `nargs = 0`;
                      if there are any number of arguments,
                      it should be
                      `nargs = None`''' % str(nargs)))
              nargs = tuple(ordered(set(nargs)))
          elif nargs is not None:
              nargs = (as_int(nargs),)
          cls._nargs = nargs
  
          super(FunctionClass, cls).__init__(*args, **kwargs)
  
      @property
      def __signature__(self):
          """
          Allow Python 3's inspect.signature to give a useful signature for
          Function subclasses.
          """
          # Python 3 only, but backports (like the one in IPython) still might
          # call this.
          try:
              from inspect import signature
          except ImportError:
              return None
  
          # TODO: Look at nargs
          return signature(self.eval)
  
      @property
      def free_symbols(self):
          return set()
  
      @property
      def xreplace(self):
          # Function needs args so we define a property that returns
          # a function that takes args...and then use that function
          # to return the right value
          return lambda rule, **_: rule.get(self, self)
  
      @property
      def nargs(self):
          """Return a set of the allowed number of arguments for the function.
  
          Examples
          ========
  
          >>> from sympy.core.function import Function
          >>> from sympy.abc import x, y
          >>> f = Function('f')
  
          If the function can take any number of arguments, the set of whole
          numbers is returned:
  
          >>> Function('f').nargs
          Naturals0
  
          If the function was initialized to accept one or more arguments, a
          corresponding set will be returned:
  
          >>> Function('f', nargs=1).nargs
          {1}
          >>> Function('f', nargs=(2, 1)).nargs
          {1, 2}
  
          The undefined function, after application, also has the nargs
          attribute; the actual number of arguments is always available by
          checking the ``args`` attribute:
  
          >>> f = Function('f')
          >>> f(1).nargs
          Naturals0
          >>> len(f(1).args)
          1
          """
          from sympy.sets.sets import FiniteSet
          # XXX it would be nice to handle this in __init__ but there are import
          # problems with trying to import FiniteSet there
          return FiniteSet(*self._nargs) if self._nargs else S.Naturals0
  
      def __repr__(cls):
          return cls.__name__
  
  
  class Application(with_metaclass(FunctionClass, Basic)):
      """
      Base class for applied functions.
  
      Instances of Application represent the result of applying an application of
      any type to any object.
      """
  
      is_Function = True
  
      @cacheit
      def __new__(cls, *args, **options):
          from sympy.sets.fancysets import Naturals0
          from sympy.sets.sets import FiniteSet
  
          args = list(map(sympify, args))
          evaluate = options.pop('evaluate', global_evaluate[0])
          # WildFunction (and anything else like it) may have nargs defined
          # and we throw that value away here
          options.pop('nargs', None)
  
          if options:
              raise ValueError("Unknown options: %s" % options)
  
          if evaluate:
              evaluated = cls.eval(*args)
              if evaluated is not None:
                  return evaluated
  
          obj = super(Application, cls).__new__(cls, *args, **options)
  
          # make nargs uniform here
          sentinel = object()
          objnargs = getattr(obj, "nargs", sentinel)
          if objnargs is not sentinel:
              # things passing through here:
              #  - functions subclassed from Function (e.g. myfunc(1).nargs)
              #  - functions like cos(1).nargs
              #  - AppliedUndef with given nargs like Function('f', nargs=1)(1).nargs
              # Canonicalize nargs here
              if is_sequence(objnargs):
                  nargs = tuple(ordered(set(objnargs)))
              elif objnargs is not None:
                  nargs = (as_int(objnargs),)
              else:
                  nargs = None
          else:
              # things passing through here:
              #  - WildFunction('f').nargs
              #  - AppliedUndef with no nargs like Function('f')(1).nargs
              nargs = obj._nargs  # note the underscore here
          # convert to FiniteSet
          obj.nargs = FiniteSet(*nargs) if nargs else Naturals0()
          return obj
  
      @classmethod
      def eval(cls, *args):
          """
          Returns a canonical form of cls applied to arguments args.
  
          The eval() method is called when the class cls is about to be
          instantiated and it should return either some simplified instance
          (possible of some other class), or if the class cls should be
          unmodified, return None.
  
          Examples of eval() for the function "sign"
          ---------------------------------------------
  
          .. code-block:: python
  
              @classmethod
              def eval(cls, arg):
                  if arg is S.NaN:
                      return S.NaN
                  if arg is S.Zero: return S.Zero
                  if arg.is_positive: return S.One
                  if arg.is_negative: return S.NegativeOne
                  if isinstance(arg, Mul):
                      coeff, terms = arg.as_coeff_Mul(rational=True)
                      if coeff is not S.One:
                          return cls(coeff) * cls(terms)
  
          """
          return
  
      @property
      def func(self):
          return self.__class__
  
      def _eval_subs(self, old, new):
          if (old.is_Function and new.is_Function and
              callable(old) and callable(new) and
              old == self.func and len(self.args) in new.nargs):
              return new(*[i._subs(old, new) for i in self.args])
  
  
  class Function(Application, Expr):
      """
      Base class for applied mathematical functions.
  
      It also serves as a constructor for undefined function classes.
  
      Examples
      ========
  
      First example shows how to use Function as a constructor for undefined
      function classes:
  
      >>> from sympy import Function, Symbol
      >>> x = Symbol('x')
      >>> f = Function('f')
      >>> g = Function('g')(x)
      >>> f
      f
      >>> f(x)
      f(x)
      >>> g
      g(x)
      >>> f(x).diff(x)
      Derivative(f(x), x)
      >>> g.diff(x)
      Derivative(g(x), x)
  
      Assumptions can be passed to Function.
  
      >>> f_real = Function('f', real=True)
      >>> f_real(x).is_real
      True
  
      Note that assumptions on a function are unrelated to the assumptions on
      the variable it is called on. If you want to add a relationship, subclass
      Function and define the appropriate ``_eval_is_assumption`` methods.
  
      In the following example Function is used as a base class for
      ``my_func`` that represents a mathematical function *my_func*. Suppose
      that it is well known, that *my_func(0)* is *1* and *my_func* at infinity
      goes to *0*, so we want those two simplifications to occur automatically.
      Suppose also that *my_func(x)* is real exactly when *x* is real. Here is
      an implementation that honours those requirements:
  
      >>> from sympy import Function, S, oo, I, sin
      >>> class my_func(Function):
      ...
      ...     @classmethod
      ...     def eval(cls, x):
      ...         if x.is_Number:
      ...             if x is S.Zero:
      ...                 return S.One
      ...             elif x is S.Infinity:
      ...                 return S.Zero
      ...
      ...     def _eval_is_real(self):
      ...         return self.args[0].is_real
      ...
      >>> x = S('x')
      >>> my_func(0) + sin(0)
      1
      >>> my_func(oo)
      0
      >>> my_func(3.54).n() # Not yet implemented for my_func.
      my_func(3.54)
      >>> my_func(I).is_real
      False
  
      In order for ``my_func`` to become useful, several other methods would
      need to be implemented. See source code of some of the already
      implemented functions for more complete examples.
  
      Also, if the function can take more than one argument, then ``nargs``
      must be defined, e.g. if ``my_func`` can take one or two arguments
      then,
  
      >>> class my_func(Function):
      ...     nargs = (1, 2)
      ...
      >>>
  
      """
  
      @property
      def _diff_wrt(self):
          return False
  
      @cacheit
      def __new__(cls, *args, **options):
          # Handle calls like Function('f')
          if cls is Function:
              return UndefinedFunction(*args, **options)
  
          n = len(args)
          if n not in cls.nargs:
              # XXX: exception message must be in exactly this format to
              # make it work with NumPy's functions like vectorize(). See,
              # for example, https://github.com/numpy/numpy/issues/1697.
              # The ideal solution would be just to attach metadata to
              # the exception and change NumPy to take advantage of this.
              temp = ('%(name)s takes %(qual)s %(args)s '
                     'argument%(plural)s (%(given)s given)')
              raise TypeError(temp % {
                  'name': cls,
                  'qual': 'exactly' if len(cls.nargs) == 1 else 'at least',
                  'args': min(cls.nargs),
                  'plural': 's'*(min(cls.nargs) != 1),
                  'given': n})
  
          evaluate = options.get('evaluate', global_evaluate[0])
          result = super(Function, cls).__new__(cls, *args, **options)
          if evaluate and isinstance(result, cls) and result.args:
              pr2 = min(cls._should_evalf(a) for a in result.args)
              if pr2 > 0:
                  pr = max(cls._should_evalf(a) for a in result.args)
                  result = result.evalf(mlib.libmpf.prec_to_dps(pr))
  
          return result
  
      @classmethod
      def _should_evalf(cls, arg):
          """
          Decide if the function should automatically evalf().
  
          By default (in this implementation), this happens if (and only if) the
          ARG is a floating point number.
          This function is used by __new__.
  
          Returns the precision to evalf to, or -1 if it shouldn't evalf.
          """
          from sympy.core.evalf import pure_complex
          if arg.is_Float:
              return arg._prec
          if not arg.is_Add:
              return -1
          m = pure_complex(arg)
          if m is None or not (m[0].is_Float or m[1].is_Float):
              return -1
          l = [i._prec for i in m if i.is_Float]
          l.append(-1)
          return max(l)
  
      @classmethod
      def class_key(cls):
          from sympy.sets.fancysets import Naturals0
          funcs = {
              'exp': 10,
              'log': 11,
              'sin': 20,
              'cos': 21,
              'tan': 22,
              'cot': 23,
              'sinh': 30,
              'cosh': 31,
              'tanh': 32,
              'coth': 33,
              'conjugate': 40,
              're': 41,
              'im': 42,
              'arg': 43,
          }
          name = cls.__name__
  
          try:
              i = funcs[name]
          except KeyError:
              i = 0 if isinstance(cls.nargs, Naturals0) else 10000
  
          return 4, i, name
  
      @property
      def is_commutative(self):
          """
          Returns whether the function is commutative.
          """
          if all(getattr(t, 'is_commutative') for t in self.args):
              return True
          else:
              return False
  
      def _eval_evalf(self, prec):
  
          def _get_mpmath_func(fname):
              """Lookup mpmath function based on name"""
              if isinstance(self, AppliedUndef):
                  # Shouldn't lookup in mpmath but might have ._imp_
                  return None
  
              if not hasattr(mpmath, fname):
                  from sympy.utilities.lambdify import MPMATH_TRANSLATIONS
                  fname = MPMATH_TRANSLATIONS.get(fname, None)
                  if fname is None:
                      return None
              return getattr(mpmath, fname)
  
          func = _get_mpmath_func(self.func.__name__)
  
          # Fall-back evaluation
          if func is None:
              imp = getattr(self, '_imp_', None)
              if imp is None:
                  return None
              try:
                  return Float(imp(*[i.evalf(prec) for i in self.args]), prec)
              except (TypeError, ValueError) as e:
                  return None
  
          # Convert all args to mpf or mpc
          # Convert the arguments to *higher* precision than requested for the
          # final result.
          # XXX + 5 is a guess, it is similar to what is used in evalf.py. Should
          #     we be more intelligent about it?
          try:
              args = [arg._to_mpmath(prec + 5) for arg in self.args]
              def bad(m):
                  from mpmath import mpf, mpc
                  # the precision of an mpf value is the last element
                  # if that is 1 (and m[1] is not 1 which would indicate a
                  # power of 2), then the eval failed; so check that none of
                  # the arguments failed to compute to a finite precision.
                  # Note: An mpc value has two parts, the re and imag tuple;
                  # check each of those parts, too. Anything else is allowed to
                  # pass
                  if isinstance(m, mpf):
                      m = m._mpf_
                      return m[1] !=1 and m[-1] == 1
                  elif isinstance(m, mpc):
                      m, n = m._mpc_
                      return m[1] !=1 and m[-1] == 1 and \
                          n[1] !=1 and n[-1] == 1
                  else:
                      return False
              if any(bad(a) for a in args):
                  raise ValueError  # one or more args failed to compute with significance
          except ValueError:
              return
  
          with mpmath.workprec(prec):
              v = func(*args)
  
          return Expr._from_mpmath(v, prec)
  
      def _eval_derivative(self, s):
          # f(x).diff(s) -> x.diff(s) * f.fdiff(1)(s)
          i = 0
          l = []
          for a in self.args:
              i += 1
              da = a.diff(s)
              if da is S.Zero:
                  continue
              try:
                  df = self.fdiff(i)
              except ArgumentIndexError:
                  df = Function.fdiff(self, i)
              l.append(df * da)
          return Add(*l)
  
      def _eval_is_commutative(self):
          return fuzzy_and(a.is_commutative for a in self.args)
  
      def _eval_is_complex(self):
          return fuzzy_and(a.is_complex for a in self.args)
  
      def as_base_exp(self):
          """
          Returns the method as the 2-tuple (base, exponent).
          """
          return self, S.One
  
      def _eval_aseries(self, n, args0, x, logx):
          """
          Compute an asymptotic expansion around args0, in terms of self.args.
          This function is only used internally by _eval_nseries and should not
          be called directly; derived classes can overwrite this to implement
          asymptotic expansions.
          """
          from sympy.utilities.misc import filldedent
          raise PoleError(filldedent('''
              Asymptotic expansion of %s around %s is
              not implemented.''' % (type(self), args0)))
  
      def _eval_nseries(self, x, n, logx):
          """
          This function does compute series for multivariate functions,
          but the expansion is always in terms of *one* variable.
  
          Examples
          ========
  
          >>> from sympy import atan2
          >>> from sympy.abc import x, y
          >>> atan2(x, y).series(x, n=2)
          atan2(0, y) + x/y + O(x**2)
          >>> atan2(x, y).series(y, n=2)
          -y/x + atan2(x, 0) + O(y**2)
  
          This function also computes asymptotic expansions, if necessary
          and possible:
  
          >>> from sympy import loggamma
          >>> loggamma(1/x)._eval_nseries(x,0,None)
          -1/x - log(x)/x + log(x)/2 + O(1)
  
          """
          from sympy import Order
          from sympy.sets.sets import FiniteSet
          args = self.args
          args0 = [t.limit(x, 0) for t in args]
          if any(t.is_finite is False for t in args0):
              from sympy import oo, zoo, nan
              # XXX could use t.as_leading_term(x) here but it's a little
              # slower
              a = [t.compute_leading_term(x, logx=logx) for t in args]
              a0 = [t.limit(x, 0) for t in a]
              if any([t.has(oo, -oo, zoo, nan) for t in a0]):
                  return self._eval_aseries(n, args0, x, logx)
              # Careful: the argument goes to oo, but only logarithmically so. We
              # are supposed to do a power series expansion "around the
              # logarithmic term". e.g.
              #      f(1+x+log(x))
              #     -> f(1+logx) + x*f'(1+logx) + O(x**2)
              # where 'logx' is given in the argument
              a = [t._eval_nseries(x, n, logx) for t in args]
              z = [r - r0 for (r, r0) in zip(a, a0)]
              p = [Dummy() for _ in z]
              q = []
              v = None
              for ai, zi, pi in zip(a0, z, p):
                  if zi.has(x):
                      if v is not None:
                          raise NotImplementedError
                      q.append(ai + pi)
                      v = pi
                  else:
                      q.append(ai)
              e1 = self.func(*q)
              if v is None:
                  return e1
              s = e1._eval_nseries(v, n, logx)
              o = s.getO()
              s = s.removeO()
              s = s.subs(v, zi).expand() + Order(o.expr.subs(v, zi), x)
              return s
          if (self.func.nargs is S.Naturals0
                  or (self.func.nargs == FiniteSet(1) and args0[0])
                  or any(c > 1 for c in self.func.nargs)):
              e = self
              e1 = e.expand()
              if e == e1:
                  #for example when e = sin(x+1) or e = sin(cos(x))
                  #let's try the general algorithm
                  term = e.subs(x, S.Zero)
                  if term.is_finite is False or term is S.NaN:
                      raise PoleError("Cannot expand %s around 0" % (self))
                  series = term
                  fact = S.One
                  _x = Dummy('x')
                  e = e.subs(x, _x)
                  for i in range(n - 1):
                      i += 1
                      fact *= Rational(i)
                      e = e.diff(_x)
                      subs = e.subs(_x, S.Zero)
                      if subs is S.NaN:
                          # try to evaluate a limit if we have to
                          subs = e.limit(_x, S.Zero)
                      if subs.is_finite is False:
                          raise PoleError("Cannot expand %s around 0" % (self))
                      term = subs*(x**i)/fact
                      term = term.expand()
                      series += term
                  return series + Order(x**n, x)
              return e1.nseries(x, n=n, logx=logx)
          arg = self.args[0]
          l = []
          g = None
          # try to predict a number of terms needed
          nterms = n + 2
          cf = Order(arg.as_leading_term(x), x).getn()
          if cf != 0:
              nterms = int(nterms / cf)
          for i in range(nterms):
              g = self.taylor_term(i, arg, g)
              g = g.nseries(x, n=n, logx=logx)
              l.append(g)
          return Add(*l) + Order(x**n, x)
  
      def fdiff(self, argindex=1):
          """
          Returns the first derivative of the function.
          """
          if not (1 <= argindex <= len(self.args)):
              raise ArgumentIndexError(self, argindex)
          ix = argindex - 1
          A = self.args[ix]
          if A._diff_wrt:
              if len(self.args) == 1:
                  return Derivative(self, A)
              if A.is_Symbol:
                  for i, v in enumerate(self.args):
                      if i != ix and A in v.free_symbols:
                          # it can't be in any other argument's free symbols
                          # issue 8510
                          break
                  else:
                      return Derivative(self, A)
              else:
                  free = A.free_symbols
                  for i, a in enumerate(self.args):
                      if ix != i and a.free_symbols & free:
                          break
                  else:
                      # there is no possible interaction bewtween args
                      return Derivative(self, A)
          # See issue 4624 and issue 4719, 5600 and 8510
          D = Dummy('xi_%i' % argindex, dummy_index=hash(A))
          args = self.args[:ix] + (D,) + self.args[ix + 1:]
          return Subs(Derivative(self.func(*args), D), D, A)
  
      def _eval_as_leading_term(self, x):
          """Stub that should be overridden by new Functions to return
          the first non-zero term in a series if ever an x-dependent
          argument whose leading term vanishes as x -> 0 might be encountered.
          See, for example, cos._eval_as_leading_term.
          """
          from sympy import Order
          args = [a.as_leading_term(x) for a in self.args]
          o = Order(1, x)
          if any(x in a.free_symbols and o.contains(a) for a in args):
              # Whereas x and any finite number are contained in O(1, x),
              # expressions like 1/x are not. If any arg simplified to a
              # vanishing expression as x -> 0 (like x or x**2, but not
              # 3, 1/x, etc...) then the _eval_as_leading_term is needed
              # to supply the first non-zero term of the series,
              #
              # e.g. expression    leading term
              #      ----------    ------------
              #      cos(1/x)      cos(1/x)
              #      cos(cos(x))   cos(1)
              #      cos(x)        1        <- _eval_as_leading_term needed
              #      sin(x)        x        <- _eval_as_leading_term needed
              #
              raise NotImplementedError(
                  '%s has no _eval_as_leading_term routine' % self.func)
          else:
              return self.func(*args)
  
      def _sage_(self):
          import sage.all as sage
          fname = self.func.__name__
          func = getattr(sage, fname, None)
          args = [arg._sage_() for arg in self.args]
  
          # In the case the function is not known in sage:
          if func is None:
              import sympy
              if getattr(sympy, fname, None) is None:
                  # abstract function
                  return sage.function(fname)(*args)
  
              else:
                  # the function defined in sympy is not known in sage
                  # this exception is caught in sage
                  raise AttributeError
  
          return func(*args)
  
  
  class AppliedUndef(Function):
      """
      Base class for expressions resulting from the application of an undefined
      function.
      """
  
      is_number = False
  
      def __new__(cls, *args, **options):
          args = list(map(sympify, args))
          obj = super(AppliedUndef, cls).__new__(cls, *args, **options)
          return obj
  
      def _eval_as_leading_term(self, x):
          return self
  
      def _sage_(self):
          import sage.all as sage
          fname = str(self.func)
          args = [arg._sage_() for arg in self.args]
          func = sage.function(fname)(*args)
          return func
  
      @property
      def _diff_wrt(self):
          """
          Allow derivatives wrt to undefined functions.
  
          Examples
          ========
  
          >>> from sympy import Function, Symbol
          >>> f = Function('f')
          >>> x = Symbol('x')
          >>> f(x)._diff_wrt
          True
          >>> f(x).diff(x)
          Derivative(f(x), x)
          """
          return True
  
  
  class UndefinedFunction(FunctionClass):
      """
      The (meta)class of undefined functions.
      """
      def __new__(mcl, name, bases=(AppliedUndef,), __dict__=None, **kwargs):
          __dict__ = __dict__ or {}
          # Allow Function('f', real=True)
          __dict__.update({'is_' + arg: val for arg, val in kwargs.items() if arg in _assume_defined})
          # You can add other attributes, although they do have to be hashable
          # (but seriously, if you want to add anything other than assumptions,
          # just subclass Function)
          __dict__.update(kwargs)
          # Save these for __eq__
          __dict__.update({'_extra_kwargs': kwargs})
          __dict__['__module__'] = None # For pickling
          ret = super(UndefinedFunction, mcl).__new__(mcl, name, bases, __dict__)
          ret.name = name
          return ret
  
      def __instancecheck__(cls, instance):
          return cls in type(instance).__mro__
  
      _extra_kwargs = {}
  
      def __hash__(self):
          return hash((self.class_key(), frozenset(self._extra_kwargs.items())))
  
      def __eq__(self, other):
          return (isinstance(other, self.__class__) and
              self.class_key() == other.class_key() and
              self._extra_kwargs == other._extra_kwargs)
  
      def __ne__(self, other):
          return not self == other
  
  
  class WildFunction(Function, AtomicExpr):
      """
      A WildFunction function matches any function (with its arguments).
  
      Examples
      ========
  
      >>> from sympy import WildFunction, Function, cos
      >>> from sympy.abc import x, y
      >>> F = WildFunction('F')
      >>> f = Function('f')
      >>> F.nargs
      Naturals0
      >>> x.match(F)
      >>> F.match(F)
      {F_: F_}
      >>> f(x).match(F)
      {F_: f(x)}
      >>> cos(x).match(F)
      {F_: cos(x)}
      >>> f(x, y).match(F)
      {F_: f(x, y)}
  
      To match functions with a given number of arguments, set ``nargs`` to the
      desired value at instantiation:
  
      >>> F = WildFunction('F', nargs=2)
      >>> F.nargs
      {2}
      >>> f(x).match(F)
      >>> f(x, y).match(F)
      {F_: f(x, y)}
  
      To match functions with a range of arguments, set ``nargs`` to a tuple
      containing the desired number of arguments, e.g. if ``nargs = (1, 2)``
      then functions with 1 or 2 arguments will be matched.
  
      >>> F = WildFunction('F', nargs=(1, 2))
      >>> F.nargs
      {1, 2}
      >>> f(x).match(F)
      {F_: f(x)}
      >>> f(x, y).match(F)
      {F_: f(x, y)}
      >>> f(x, y, 1).match(F)
  
      """
  
      include = set()
  
      def __init__(cls, name, **assumptions):
          from sympy.sets.sets import Set, FiniteSet
          cls.name = name
          nargs = assumptions.pop('nargs', S.Naturals0)
          if not isinstance(nargs, Set):
              # Canonicalize nargs here.  See also FunctionClass.
              if is_sequence(nargs):
                  nargs = tuple(ordered(set(nargs)))
              elif nargs is not None:
                  nargs = (as_int(nargs),)
              nargs = FiniteSet(*nargs)
          cls.nargs = nargs
  
      def matches(self, expr, repl_dict={}, old=False):
          if not isinstance(expr, (AppliedUndef, Function)):
              return None
          if len(expr.args) not in self.nargs:
              return None
  
          repl_dict = repl_dict.copy()
          repl_dict[self] = expr
          return repl_dict
  
  
  class Derivative(Expr):
      """
      Carries out differentiation of the given expression with respect to symbols.
  
      Examples
      ========
  
      >>> from sympy import Derivative, Function, symbols, Subs
      >>> from sympy.abc import x, y
      >>> f, g = symbols('f g', cls=Function)
  
      >>> Derivative(x**2, x, evaluate=True)
      2*x
  
      Denesting of derivatives retains the ordering of variables:
  
          >>> Derivative(Derivative(f(x, y), y), x)
          Derivative(f(x, y), y, x)
  
      Contiguously identical symbols are merged into a tuple giving
      the symbol and the count:
  
          >>> Derivative(f(x), x, x, y, x)
          Derivative(f(x), (x, 2), y, x)
  
      If the derivative cannot be performed, and evaluate is True, the
      order of the variables of differentiation will be made canonical:
  
          >>> Derivative(f(x, y), y, x, evaluate=True)
          Derivative(f(x, y), x, y)
  
      Derivatives with respect to undefined functions can be calculated:
  
          >>> Derivative(f(x)**2, f(x), evaluate=True)
          2*f(x)
  
      Such derivatives will show up when the chain rule is used to
      evalulate a derivative:
  
          >>> f(g(x)).diff(x)
          Derivative(f(g(x)), g(x))*Derivative(g(x), x)
  
      Substitution is used to represent derivatives of functions with
      arguments that are not symbols or functions:
  
          >>> f(2*x + 3).diff(x) == 2*Subs(f(y).diff(y), y, 2*x + 3)
          True
  
      Notes
      =====
  
      Simplification of high-order derivatives:
  
      Because there can be a significant amount of simplification that can be
      done when multiple differentiations are performed, results will be
      automatically simplified in a fairly conservative fashion unless the
      keyword ``simplify`` is set to False.
  
          >>> from sympy import cos, sin, sqrt, diff, Function, symbols
          >>> from sympy.abc import x, y, z
          >>> f, g = symbols('f,g', cls=Function)
  
          >>> e = sqrt((x + 1)**2 + x)
          >>> diff(e, (x, 5), simplify=False).count_ops()
          136
          >>> diff(e, (x, 5)).count_ops()
          30
  
      Ordering of variables:
  
      If evaluate is set to True and the expression cannot be evaluated, the
      list of differentiation symbols will be sorted, that is, the expression is
      assumed to have continuous derivatives up to the order asked.
  
      Derivative wrt non-Symbols:
  
      For the most part, one may not differentiate wrt non-symbols.
      For example, we do not allow differentiation wrt `x*y` because
      there are multiple ways of structurally defining where x*y appears
      in an expression: a very strict definition would make
      (x*y*z).diff(x*y) == 0. Derivatives wrt defined functions (like
      cos(x)) are not allowed, either:
  
          >>> (x*y*z).diff(x*y)
          Traceback (most recent call last):
          ...
          ValueError: Can't calculate derivative wrt x*y.
  
      To make it easier to work with variational calculus, however,
      derivatives wrt AppliedUndef and Derivatives are allowed.
      For example, in the Euler-Lagrange method one may write
      F(t, u, v) where u = f(t) and v = f'(t). These variables can be
      written explicity as functions of time::
  
          >>> from sympy.abc import t
          >>> F = Function('F')
          >>> U = f(t)
          >>> V = U.diff(t)
  
      The derivative wrt f(t) can be obtained directly:
  
          >>> direct = F(t, U, V).diff(U)
  
      When differentiation wrt a non-Symbol is attempted, the non-Symbol
      is temporarily converted to a Symbol while the differentiation
      is performed and the same answer is obtained:
  
          >>> indirect = F(t, U, V).subs(U, x).diff(x).subs(x, U)
          >>> assert direct == indirect
  
      The implication of this non-symbol replacement is that all
      functions are treated as independent of other functions and the
      symbols are independent of the functions that contain them::
  
          >>> x.diff(f(x))
          0
          >>> g(x).diff(f(x))
          0
  
      It also means that derivatives are assumed to depend only
      on the variables of differentiation, not on anything contained
      within the expression being differentiated::
  
          >>> F = f(x)
          >>> Fx = F.diff(x)
          >>> Fx.diff(F)  # derivative depends on x, not F
          0
          >>> Fxx = Fx.diff(x)
          >>> Fxx.diff(Fx)  # derivative depends on x, not Fx
          0
  
      The last example can be made explicit by showing the replacement
      of Fx in Fxx with y:
  
          >>> Fxx.subs(Fx, y)
          Derivative(y, x)
  
          Since that in itself will evaluate to zero, differentiating
          wrt Fx will also be zero:
  
          >>> _.doit()
          0
  
      Replacing undefined functions with concrete expressions
  
      One must be careful to replace undefined functions with expressions
      that contain variables consistent with the function definition and
      the variables of differentiation or else insconsistent result will
      be obtained. Consider the following example:
  
      >>> eq = f(x)*g(y)
      >>> eq.subs(f(x), x*y).diff(x, y).doit()
      y*Derivative(g(y), y) + g(y)
      >>> eq.diff(x, y).subs(f(x), x*y).doit()
      y*Derivative(g(y), y)
  
      The results differ because `f(x)` was replaced with an expression
      that involved both variables of differentiation. In the abstract
      case, differentiation of `f(x)` by `y` is 0; in the concrete case,
      the presence of `y` made that derivative nonvanishing and produced
      the extra `g(y)` term.
  
      Defining differentiation for an object
  
      An object must define ._eval_derivative(symbol) method that returns
      the differentiation result. This function only needs to consider the
      non-trivial case where expr contains symbol and it should call the diff()
      method internally (not _eval_derivative); Derivative should be the only
      one to call _eval_derivative.
  
      Any class can allow derivatives to be taken with respect to
      itself (while indicating its scalar nature). See the
      docstring of Expr._diff_wrt.
  
      See Also
      ========
      _sort_variable_count
      """
  
      is_Derivative = True
  
      @property
      def _diff_wrt(self):
          """An expression may be differentiated wrt a Derivative if
          it is in elementary form.
  
          Examples
          ========
  
          >>> from sympy import Function, Derivative, cos
          >>> from sympy.abc import x
          >>> f = Function('f')
  
          >>> Derivative(f(x), x)._diff_wrt
          True
          >>> Derivative(cos(x), x)._diff_wrt
          False
          >>> Derivative(x + 1, x)._diff_wrt
          False
  
          A Derivative might be an unevaluated form of what will not be
          a valid variable of differentiation if evaluated. For example,
  
          >>> Derivative(f(f(x)), x).doit()
          Derivative(f(x), x)*Derivative(f(f(x)), f(x))
  
          Such an expression will present the same ambiguities as arise
          when dealing with any other product, like `2*x`, so `_diff_wrt`
          is False:
  
          >>> Derivative(f(f(x)), x)._diff_wrt
          False
          """
          return self.expr._diff_wrt and isinstance(self.doit(), Derivative)
  
      def __new__(cls, expr, *variables, **kwargs):
  
          from sympy.matrices.common import MatrixCommon
          from sympy import Integer, MatrixExpr
          from sympy.tensor.array import Array, NDimArray, derive_by_array
          from sympy.utilities.misc import filldedent
  
          expr = sympify(expr)
          symbols_or_none = getattr(expr, "free_symbols", None)
          has_symbol_set = isinstance(symbols_or_none, set)
  
          if not has_symbol_set:
              raise ValueError(filldedent('''
                  Since there are no variables in the expression %s,
                  it cannot be differentiated.''' % expr))
  
          # determine value for variables if it wasn't given
          if not variables:
              variables = expr.free_symbols
              if len(variables) != 1:
                  if expr.is_number:
                      return S.Zero
                  if len(variables) == 0:
                      raise ValueError(filldedent('''
                          Since there are no variables in the expression,
                          the variable(s) of differentiation must be supplied
                          to differentiate %s''' % expr))
                  else:
                      raise ValueError(filldedent('''
                          Since there is more than one variable in the
                          expression, the variable(s) of differentiation
                          must be supplied to differentiate %s''' % expr))
  
          # Standardize the variables by sympifying them:
          variables = list(sympify(variables))
  
          # Split the list of variables into a list of the variables we are diff
          # wrt, where each element of the list has the form (s, count) where
          # s is the entity to diff wrt and count is the order of the
          # derivative.
          variable_count = []
          array_likes = (tuple, list, Tuple)
  
          for i, v in enumerate(variables):
              if isinstance(v, Integer):
                  if i == 0:
                      raise ValueError("First variable cannot be a number: %i" % v)
                  count = v
                  prev, prevcount = variable_count[-1]
                  if prevcount != 1:
                      raise TypeError("tuple {0} followed by number {1}".format((prev, prevcount), v))
                  if count == 0:
                      variable_count.pop()
                  else:
                      variable_count[-1] = Tuple(prev, count)
              else:
                  if isinstance(v, array_likes):
                      if len(v) == 0:
                          # Ignore empty tuples: Derivative(expr, ... , (), ... )
                          continue
                      if isinstance(v[0], array_likes):
                          # Derive by array: Derivative(expr, ... , [[x, y, z]], ... )
                          if len(v) == 1:
                              v = Array(v[0])
                              count = 1
                          else:
                              v, count = v
                              v = Array(v)
                      else:
                          v, count = v
                      if count == 0:
                          continue
                  else:
                      count = 1
                  variable_count.append(Tuple(v, count))
  
          # light evaluation of contiguous, identical
          # items: (x, 1), (x, 1) -> (x, 2)
          merged = []
          for t in variable_count:
              v, c = t
              if c.is_negative:
                  raise ValueError(
                      'order of differentiation must be nonnegative')
              if merged and merged[-1][0] == v:
                  c += merged[-1][1]
                  if not c:
                      merged.pop()
                  else:
                      merged[-1] = Tuple(v, c)
              else:
                  merged.append(t)
          variable_count = merged
  
          # sanity check of variables of differentation; we waited
          # until the counts were computed since some variables may
          # have been removed because the count was 0
          for v, c in variable_count:
              # v must have _diff_wrt True
              if not v._diff_wrt:
                  __ = ''  # filler to make error message neater
                  raise ValueError(filldedent('''
                      Can't calculate derivative wrt %s.%s''' % (v,
                      __)))
  
          # We make a special case for 0th derivative, because there is no
          # good way to unambiguously print this.
          if len(variable_count) == 0:
              return expr
  
          evaluate = kwargs.get('evaluate', False)
  
          if evaluate:
              if isinstance(expr, Derivative):
                  expr = expr.canonical
              variable_count = [
                  (v.canonical if isinstance(v, Derivative) else v, c)
                  for v, c in variable_count]
  
              # Look for a quick exit if there are symbols that don't appear in
              # expression at all. Note, this cannot check non-symbols like
              # Derivatives as those can be created by intermediate
              # derivatives.
              zero = False
              free = expr.free_symbols
              for v, c in variable_count:
                  vfree = v.free_symbols
                  if c.is_positive and vfree:
                      if isinstance(v, AppliedUndef):
                          # these match exactly since
                          # x.diff(f(x)) == g(x).diff(f(x)) == 0
                          # and are not created by differentiation
                          D = Dummy()
                          if not expr.xreplace({v: D}).has(D):
                              zero = True
                              break
                      elif isinstance(v, MatrixExpr):
                          zero = False
                          break
                      elif isinstance(v, Symbol) and v not in free:
                          zero = True
                          break
                      else:
                          if not free & vfree:
                              # e.g. v is IndexedBase or Matrix
                              zero = True
                              break
              if zero:
                  if isinstance(expr, (MatrixCommon, NDimArray)):
                      return expr.zeros(*expr.shape)
                  else:
                      return S.Zero
  
              # make the order of symbols canonical
              #TODO: check if assumption of discontinuous derivatives exist
              variable_count = cls._sort_variable_count(variable_count)
  
          # denest
          if isinstance(expr, Derivative):
              variable_count = list(expr.variable_count) + variable_count
              expr = expr.expr
              return Derivative(expr, *variable_count, **kwargs)
  
          # we return here if evaluate is False or if there is no
          # _eval_derivative method
          if not evaluate or not hasattr(expr, '_eval_derivative'):
              # return an unevaluated Derivative
              if evaluate and variable_count == [(expr, 1)] and expr.is_scalar:
                  # special hack providing evaluation for classes
                  # that have defined is_scalar=True but have no
                  # _eval_derivative defined
                  return S.One
              return Expr.__new__(cls, expr, *variable_count)
  
          # evaluate the derivative by calling _eval_derivative method
          # of expr for each variable
          # -------------------------------------------------------------
          nderivs = 0  # how many derivatives were performed
          unhandled = []
          for i, (v, count) in enumerate(variable_count):
  
              old_expr = expr
              old_v = None
  
              is_symbol = v.is_symbol or isinstance(v,
                  (Iterable, Tuple, MatrixCommon, NDimArray))
  
              if not is_symbol:
                  old_v = v
                  v = Dummy('xi')
                  expr = expr.xreplace({old_v: v})
                  # Derivatives and UndefinedFunctions are independent
                  # of all others
                  clashing = not (isinstance(old_v, Derivative) or \
                      isinstance(old_v, AppliedUndef))
                  if not v in expr.free_symbols and not clashing:
                      return expr.diff(v)  # expr's version of 0
                  if not old_v.is_scalar and not hasattr(
                          old_v, '_eval_derivative'):
                      # special hack providing evaluation for classes
                      # that have defined is_scalar=True but have no
                      # _eval_derivative defined
                      expr *= old_v.diff(old_v)
  
              # Evaluate the derivative `n` times.  If
              # `_eval_derivative_n_times` is not overridden by the current
              # object, the default in `Basic` will call a loop over
              # `_eval_derivative`:
              obj = expr._eval_derivative_n_times(v, count)
              if obj is not None and obj.is_zero:
                  return obj
  
              nderivs += count
  
              if old_v is not None:
                  if obj is not None:
                      # remove the dummy that was used
                      obj = obj.subs(v, old_v)
                  # restore expr
                  expr = old_expr
  
              if obj is None:
                  # we've already checked for quick-exit conditions
                  # that give 0 so the remaining variables
                  # are contained in the expression but the expression
                  # did not compute a derivative so we stop taking
                  # derivatives
                  unhandled = variable_count[i:]
                  break
  
              expr = obj
  
          # what we have so far can be made canonical
          expr = expr.replace(
              lambda x: isinstance(x, Derivative),
              lambda x: x.canonical)
  
          if unhandled:
              if isinstance(expr, Derivative):
                  unhandled = list(expr.variable_count) + unhandled
                  expr = expr.expr
              expr = Expr.__new__(cls, expr, *unhandled)
  
          if (nderivs > 1) == True and kwargs.get('simplify', True):
              from sympy.core.exprtools import factor_terms
              from sympy.simplify.simplify import signsimp
              expr = factor_terms(signsimp(expr))
          return expr
  
      @property
      def canonical(cls):
          return cls.func(cls.expr,
              *Derivative._sort_variable_count(cls.variable_count))
  
      @classmethod
      def _sort_variable_count(cls, vc):
          """
          Sort (variable, count) pairs into canonical order while
          retaining order of variables that do not commute during
          differentiation:
  
          * symbols and functions commute with each other
          * derivatives commute with each other
          * a derivative doesn't commute with anything it contains
          * any other object is not allowed to commute if it has
            free symbols in common with another object
  
          Examples
          ========
  
          >>> from sympy import Derivative, Function, symbols, cos
          >>> vsort = Derivative._sort_variable_count
          >>> x, y, z = symbols('x y z')
          >>> f, g, h = symbols('f g h', cls=Function)
  
          Contiguous items are collapsed into one pair:
  
          >>> vsort([(x, 1), (x, 1)])
          [(x, 2)]
          >>> vsort([(y, 1), (f(x), 1), (y, 1), (f(x), 1)])
          [(y, 2), (f(x), 2)]
  
          Ordering is canonical.
  
          >>> def vsort0(*v):
          ...     # docstring helper to
          ...     # change vi -> (vi, 0), sort, and return vi vals
          ...     return [i[0] for i in vsort([(i, 0) for i in v])]
  
          >>> vsort0(y, x)
          [x, y]
          >>> vsort0(g(y), g(x), f(y))
          [f(y), g(x), g(y)]
  
          Symbols are sorted as far to the left as possible but never
          move to the left of a derivative having the same symbol in
          its variables; the same applies to AppliedUndef which are
          always sorted after Symbols:
  
          >>> dfx = f(x).diff(x)
          >>> assert vsort0(dfx, y) == [y, dfx]
          >>> assert vsort0(dfx, x) == [dfx, x]
          """
          from sympy.utilities.iterables import uniq, topological_sort
          if not vc:
              return []
          vc = list(vc)
          if len(vc) == 1:
              return [Tuple(*vc[0])]
          V = list(range(len(vc)))
          E = []
          v = lambda i: vc[i][0]
          D = Dummy()
          def _block(d, v, wrt=False):
              # return True if v should not come before d else False
              if d == v:
                  return wrt
              if d.is_Symbol:
                  return False
              if isinstance(d, Derivative):
                  # a derivative blocks if any of it's variables contain
                  # v; the wrt flag will return True for an exact match
                  # and will cause an AppliedUndef to block if v is in
                  # the arguments
                  if any(_block(k, v, wrt=True)
                          for k in d._wrt_variables):
                      return True
                  return False
              if not wrt and isinstance(d, AppliedUndef):
                  return False
              if v.is_Symbol:
                  return v in d.free_symbols
              if isinstance(v, AppliedUndef):
                  return _block(d.xreplace({v: D}), D)
              return d.free_symbols & v.free_symbols
          for i in range(len(vc)):
              for j in range(i):
                  if _block(v(j), v(i)):
                      E.append((j,i))
          # this is the default ordering to use in case of ties
          O = dict(zip(ordered(uniq([i for i, c in vc])), range(len(vc))))
          ix = topological_sort((V, E), key=lambda i: O[v(i)])
          # merge counts of contiguously identical items
          merged = []
          for v, c in [vc[i] for i in ix]:
              if merged and merged[-1][0] == v:
                  merged[-1][1] += c
              else:
                  merged.append([v, c])
          return [Tuple(*i) for i in merged]
  
      def _eval_is_commutative(self):
          return self.expr.is_commutative
  
      def _eval_derivative(self, v):
          # If v (the variable of differentiation) is not in
          # self.variables, we might be able to take the derivative.
          if v not in self._wrt_variables:
              dedv = self.expr.diff(v)
              if isinstance(dedv, Derivative):
                  return dedv.func(dedv.expr, *(self.variable_count + dedv.variable_count))
              # dedv (d(self.expr)/dv) could have simplified things such that the
              # derivative wrt things in self.variables can now be done. Thus,
              # we set evaluate=True to see if there are any other derivatives
              # that can be done. The most common case is when dedv is a simple
              # number so that the derivative wrt anything else will vanish.
              return self.func(dedv, *self.variables, evaluate=True)
          # In this case v was in self.variables so the derivative wrt v has
          # already been attempted and was not computed, either because it
          # couldn't be or evaluate=False originally.
          variable_count = list(self.variable_count)
          variable_count.append((v, 1))
          return self.func(self.expr, *variable_count, evaluate=False)
  
      def doit(self, **hints):
          expr = self.expr
          if hints.get('deep', True):
              expr = expr.doit(**hints)
          hints['evaluate'] = True
          rv = self.func(expr, *self.variable_count, **hints)
          if rv!= self and rv.has(Derivative):
              rv =  rv.doit(**hints)
          return rv
  
      @_sympifyit('z0', NotImplementedError)
      def doit_numerically(self, z0):
          """
          Evaluate the derivative at z numerically.
  
          When we can represent derivatives at a point, this should be folded
          into the normal evalf. For now, we need a special method.
          """
          if len(self.free_symbols) != 1 or len(self.variables) != 1:
              raise NotImplementedError('partials and higher order derivatives')
          z = list(self.free_symbols)[0]
  
          def eval(x):
              f0 = self.expr.subs(z, Expr._from_mpmath(x, prec=mpmath.mp.prec))
              f0 = f0.evalf(mlib.libmpf.prec_to_dps(mpmath.mp.prec))
              return f0._to_mpmath(mpmath.mp.prec)
          return Expr._from_mpmath(mpmath.diff(eval,
                                               z0._to_mpmath(mpmath.mp.prec)),
                                   mpmath.mp.prec)
  
      @property
      def expr(self):
          return self._args[0]
  
      @property
      def _wrt_variables(self):
          # return the variables of differentiation without
          # respect to the type of count (int or symbolic)
          return [i[0] for i in self.variable_count]
  
      @property
      def variables(self):
          # TODO: deprecate?  YES, make this 'enumerated_variables' and
          #       name _wrt_variables as variables
          # TODO: support for `d^n`?
          rv = []
          for v, count in self.variable_count:
              if not count.is_Integer:
                  raise TypeError(filldedent('''
                  Cannot give expansion for symbolic count. If you just
                  want a list of all variables of differentiation, use
                  _wrt_variables.'''))
              rv.extend([v]*count)
          return tuple(rv)
  
      @property
      def variable_count(self):
          return self._args[1:]
  
      @property
      def derivative_count(self):
          return sum([count for var, count in self.variable_count], 0)
  
      @property
      def free_symbols(self):
          return self.expr.free_symbols
  
      def _eval_subs(self, old, new):
          # The substitution (old, new) cannot be done inside
          # Derivative(expr, vars) for a variety of reasons
          # as handled below.
          if old in self._wrt_variables:
              # first handle the counts
              expr = self.func(self.expr, *[(v, c.subs(old, new))
                  for v, c in self.variable_count])
              if expr != self:
                  return expr._eval_subs(old, new)
              # quick exit case
              if not getattr(new, '_diff_wrt', False):
                  # case (0): new is not a valid variable of
                  # differentiation
                  if isinstance(old, Symbol):
                      # don't introduce a new symbol if the old will do
                      return Subs(self, old, new)
                  else:
                      xi = Dummy('xi')
                      return Subs(self.xreplace({old: xi}), xi, new)
  
          # If both are Derivatives with the same expr, check if old is
          # equivalent to self or if old is a subderivative of self.
          if old.is_Derivative and old.expr == self.expr:
              if self.canonical == old.canonical:
                  return new
  
              # collections.Counter doesn't have __le__
              def _subset(a, b):
                  return all((a[i] <= b[i]) == True for i in a)
  
              old_vars = Counter(dict(reversed(old.variable_count)))
              self_vars = Counter(dict(reversed(self.variable_count)))
              if _subset(old_vars, self_vars):
                  return Derivative(new, *(self_vars - old_vars).items()).canonical
  
          args = list(self.args)
          newargs = list(x._subs(old, new) for x in args)
          if args[0] == old:
              # complete replacement of self.expr
              # we already checked that the new is valid so we know
              # it won't be a problem should it appear in variables
              return Derivative(*newargs)
  
          if newargs[0] != args[0]:
              # case (1) can't change expr by introducing something that is in
              # the _wrt_variables if it was already in the expr
              # e.g.
              # for Derivative(f(x, g(y)), y), x cannot be replaced with
              # anything that has y in it; for f(g(x), g(y)).diff(g(y))
              # g(x) cannot be replaced with anything that has g(y)
              syms = {vi: Dummy() for vi in self._wrt_variables
                  if not vi.is_Symbol}
              wrt = set(syms.get(vi, vi) for vi in self._wrt_variables)
              forbidden = args[0].xreplace(syms).free_symbols & wrt
              nfree = new.xreplace(syms).free_symbols
              ofree = old.xreplace(syms).free_symbols
              if (nfree - ofree) & forbidden:
                  return Subs(self, old, new)
  
          viter = ((i, j) for ((i, _), (j, _)) in zip(newargs[1:], args[1:]))
          if any(i != j for i, j in viter):  # a wrt-variable change
              # case (2) can't change vars by introducing a variable
              # that is contained in expr, e.g.
              # for Derivative(f(z, g(h(x), y)), y), y cannot be changed to
              # x, h(x), or g(h(x), y)
              for a in _atomic(self.expr, recursive=True):
                  for i in range(1, len(newargs)):
                      vi, _ = newargs[i]
                      if a == vi and vi != args[i][0]:
                          return Subs(self, old, new)
              # more arg-wise checks
              vc = newargs[1:]
              oldv = self._wrt_variables
              newe = self.expr
              subs = []
              for i, (vi, ci) in enumerate(vc):
                  if not vi._diff_wrt:
                      # case (3) invalid differentiation expression so
                      # create a replacement dummy
                      xi = Dummy('xi_%i' % i)
                      # replace the old valid variable with the dummy
                      # in the expression
                      newe = newe.xreplace({oldv[i]: xi})
                      # and replace the bad variable with the dummy
                      vc[i] = (xi, ci)
                      # and record the dummy with the new (invalid)
                      # differentiation expression
                      subs.append((xi, vi))
  
              if subs:
                  # handle any residual substitution in the expression
                  newe = newe._subs(old, new)
                  # return the Subs-wrapped derivative
                  return Subs(Derivative(newe, *vc), *zip(*subs))
  
          # everything was ok
          return Derivative(*newargs)
  
      def _eval_lseries(self, x, logx):
          dx = self.variables
          for term in self.expr.lseries(x, logx=logx):
              yield self.func(term, *dx)
  
      def _eval_nseries(self, x, n, logx):
          arg = self.expr.nseries(x, n=n, logx=logx)
          o = arg.getO()
          dx = self.variables
          rv = [self.func(a, *dx) for a in Add.make_args(arg.removeO())]
          if o:
              rv.append(o/x)
          return Add(*rv)
  
      def _eval_as_leading_term(self, x):
          series_gen = self.expr.lseries(x)
          d = S.Zero
          for leading_term in series_gen:
              d = diff(leading_term, *self.variables)
              if d != 0:
                  break
          return d
  
      def _sage_(self):
          import sage.all as sage
          args = [arg._sage_() for arg in self.args]
          return sage.derivative(*args)
  
      def as_finite_difference(self, points=1, x0=None, wrt=None):
          """ Expresses a Derivative instance as a finite difference.
  
          Parameters
          ==========
          points : sequence or coefficient, optional
              If sequence: discrete values (length >= order+1) of the
              independent variable used for generating the finite
              difference weights.
              If it is a coefficient, it will be used as the step-size
              for generating an equidistant sequence of length order+1
              centered around ``x0``. Default: 1 (step-size 1)
          x0 : number or Symbol, optional
              the value of the independent variable (``wrt``) at which the
              derivative is to be approximated. Default: same as ``wrt``.
          wrt : Symbol, optional
              "with respect to" the variable for which the (partial)
              derivative is to be approximated for. If not provided it
              is required that the derivative is ordinary. Default: ``None``.
  
  
          Examples
          ========
          >>> from sympy import symbols, Function, exp, sqrt, Symbol
          >>> x, h = symbols('x h')
          >>> f = Function('f')
          >>> f(x).diff(x).as_finite_difference()
          -f(x - 1/2) + f(x + 1/2)
  
          The default step size and number of points are 1 and
          ``order + 1`` respectively. We can change the step size by
          passing a symbol as a parameter:
  
          >>> f(x).diff(x).as_finite_difference(h)
          -f(-h/2 + x)/h + f(h/2 + x)/h
  
          We can also specify the discretized values to be used in a
          sequence:
  
          >>> f(x).diff(x).as_finite_difference([x, x+h, x+2*h])
          -3*f(x)/(2*h) + 2*f(h + x)/h - f(2*h + x)/(2*h)
  
          The algorithm is not restricted to use equidistant spacing, nor
          do we need to make the approximation around ``x0``, but we can get
          an expression estimating the derivative at an offset:
  
          >>> e, sq2 = exp(1), sqrt(2)
          >>> xl = [x-h, x+h, x+e*h]
          >>> f(x).diff(x, 1).as_finite_difference(xl, x+h*sq2)  # doctest: +ELLIPSIS
          2*h*((h + sqrt(2)*h)/(2*h) - (-sqrt(2)*h + h)/(2*h))*f(E*h + x)/...
  
          Partial derivatives are also supported:
  
          >>> y = Symbol('y')
          >>> d2fdxdy=f(x,y).diff(x,y)
          >>> d2fdxdy.as_finite_difference(wrt=x)
          -Derivative(f(x - 1/2, y), y) + Derivative(f(x + 1/2, y), y)
  
          We can apply ``as_finite_difference`` to ``Derivative`` instances in
          compound expressions using ``replace``:
  
          >>> (1 + 42**f(x).diff(x)).replace(lambda arg: arg.is_Derivative,
          ...     lambda arg: arg.as_finite_difference())
          42**(-f(x - 1/2) + f(x + 1/2)) + 1
  
  
          See also
          ========
  
          sympy.calculus.finite_diff.apply_finite_diff
          sympy.calculus.finite_diff.differentiate_finite
          sympy.calculus.finite_diff.finite_diff_weights
  
          """
          from ..calculus.finite_diff import _as_finite_diff
          return _as_finite_diff(self, points, x0, wrt)
  
  
  class Lambda(Expr):
      """
      Lambda(x, expr) represents a lambda function similar to Python's
      'lambda x: expr'. A function of several variables is written as
      Lambda((x, y, ...), expr).
  
      A simple example:
  
      >>> from sympy import Lambda
      >>> from sympy.abc import x
      >>> f = Lambda(x, x**2)
      >>> f(4)
      16
  
      For multivariate functions, use:
  
      >>> from sympy.abc import y, z, t
      >>> f2 = Lambda((x, y, z, t), x + y**z + t**z)
      >>> f2(1, 2, 3, 4)
      73
  
      A handy shortcut for lots of arguments:
  
      >>> p = x, y, z
      >>> f = Lambda(p, x + y*z)
      >>> f(*p)
      x + y*z
  
      """
      is_Function = True
  
      def __new__(cls, variables, expr):
          from sympy.sets.sets import FiniteSet
          v = list(variables) if iterable(variables) else [variables]
          for i in v:
              if not getattr(i, 'is_symbol', False):
                  raise TypeError('variable is not a symbol: %s' % i)
          if len(v) == 1 and v[0] == expr:
              return S.IdentityFunction
  
          obj = Expr.__new__(cls, Tuple(*v), sympify(expr))
          obj.nargs = FiniteSet(len(v))
          return obj
  
      @property
      def variables(self):
          """The variables used in the internal representation of the function"""
          return self._args[0]
  
      bound_symbols = variables
  
      @property
      def expr(self):
          """The return value of the function"""
          return self._args[1]
  
      @property
      def free_symbols(self):
          return self.expr.free_symbols - set(self.variables)
  
      def __call__(self, *args):
          n = len(args)
          if n not in self.nargs:  # Lambda only ever has 1 value in nargs
              # XXX: exception message must be in exactly this format to
              # make it work with NumPy's functions like vectorize(). See,
              # for example, https://github.com/numpy/numpy/issues/1697.
              # The ideal solution would be just to attach metadata to
              # the exception and change NumPy to take advantage of this.
              ## XXX does this apply to Lambda? If not, remove this comment.
              temp = ('%(name)s takes exactly %(args)s '
                     'argument%(plural)s (%(given)s given)')
              raise TypeError(temp % {
                  'name': self,
                  'args': list(self.nargs)[0],
                  'plural': 's'*(list(self.nargs)[0] != 1),
                  'given': n})
          return self.expr.xreplace(dict(list(zip(self.variables, args))))
  
      def __eq__(self, other):
          if not isinstance(other, Lambda):
              return False
          if self.nargs != other.nargs:
              return False
  
          selfexpr = self.args[1]
          otherexpr = other.args[1]
          otherexpr = otherexpr.xreplace(dict(list(zip(other.args[0], self.args[0]))))
          return selfexpr == otherexpr
  
      def __ne__(self, other):
          return not(self == other)
  
      def __hash__(self):
          return super(Lambda, self).__hash__()
  
      def _hashable_content(self):
          return (self.expr.xreplace(self.canonical_variables),)
  
      @property
      def is_identity(self):
          """Return ``True`` if this ``Lambda`` is an identity function. """
          if len(self.args) == 2:
              return self.args[0] == self.args[1]
          else:
              return None
  
  
  class Subs(Expr):
      """
      Represents unevaluated substitutions of an expression.
  
      ``Subs(expr, x, x0)`` receives 3 arguments: an expression, a variable or
      list of distinct variables and a point or list of evaluation points
      corresponding to those variables.
  
      ``Subs`` objects are generally useful to represent unevaluated derivatives
      calculated at a point.
  
      The variables may be expressions, but they are subjected to the limitations
      of subs(), so it is usually a good practice to use only symbols for
      variables, since in that case there can be no ambiguity.
  
      There's no automatic expansion - use the method .doit() to effect all
      possible substitutions of the object and also of objects inside the
      expression.
  
      When evaluating derivatives at a point that is not a symbol, a Subs object
      is returned. One is also able to calculate derivatives of Subs objects - in
      this case the expression is always expanded (for the unevaluated form, use
      Derivative()).
  
      Examples
      ========
  
      >>> from sympy import Subs, Function, sin, cos
      >>> from sympy.abc import x, y, z
      >>> f = Function('f')
  
      Subs are created when a particular substitution cannot be made. The
      x in the derivative cannot be replaced with 0 because 0 is not a
      valid variables of differentiation:
  
      >>> f(x).diff(x).subs(x, 0)
      Subs(Derivative(f(x), x), x, 0)
  
      Once f is known, the derivative and evaluation at 0 can be done:
  
      >>> _.subs(f, sin).doit() == sin(x).diff(x).subs(x, 0) == cos(0)
      True
  
      Subs can also be created directly with one or more variables:
  
      >>> Subs(f(x)*sin(y) + z, (x, y), (0, 1))
      Subs(z + f(x)*sin(y), (x, y), (0, 1))
      >>> _.doit()
      z + f(0)*sin(1)
  
      Notes
      =====
  
      In order to allow expressions to combine before doit is done, a
      representation of the Subs expression is used internally to make
      expressions that are superficially different compare the same:
  
      >>> a, b = Subs(x, x, 0), Subs(y, y, 0)
      >>> a + b
      2*Subs(x, x, 0)
  
      This can lead to unexpected consequences when using methods
      like `has` that are cached:
  
      >>> s = Subs(x, x, 0)
      >>> s.has(x), s.has(y)
      (True, False)
      >>> ss = s.subs(x, y)
      >>> ss.has(x), ss.has(y)
      (True, False)
      >>> s, ss
      (Subs(x, x, 0), Subs(y, y, 0))
      """
      def __new__(cls, expr, variables, point, **assumptions):
          from sympy import Symbol
  
          if not is_sequence(variables, Tuple):
              variables = [variables]
          variables = Tuple(*variables)
  
          if has_dups(variables):
              repeated = [str(v) for v, i in Counter(variables).items() if i > 1]
              __ = ', '.join(repeated)
              raise ValueError(filldedent('''
                  The following expressions appear more than once: %s
                  ''' % __))
  
          point = Tuple(*(point if is_sequence(point, Tuple) else [point]))
  
          if len(point) != len(variables):
              raise ValueError('Number of point values must be the same as '
                               'the number of variables.')
  
          if not point:
              return sympify(expr)
  
          # denest
          if isinstance(expr, Subs):
              variables = expr.variables + variables
              point = expr.point + point
              expr = expr.expr
          else:
              expr = sympify(expr)
  
          # use symbols with names equal to the point value (with preppended _)
          # to give a variable-independent expression
          pre = "_"
          pts = sorted(set(point), key=default_sort_key)
          from sympy.printing import StrPrinter
          class CustomStrPrinter(StrPrinter):
              def _print_Dummy(self, expr):
                  return str(expr) + str(expr.dummy_index)
          def mystr(expr, **settings):
              p = CustomStrPrinter(settings)
              return p.doprint(expr)
          while 1:
              s_pts = {p: Symbol(pre + mystr(p)) for p in pts}
              reps = [(v, s_pts[p])
                  for v, p in zip(variables, point)]
              # if any underscore-preppended symbol is already a free symbol
              # and is a variable with a different point value, then there
              # is a clash, e.g. _0 clashes in Subs(_0 + _1, (_0, _1), (1, 0))
              # because the new symbol that would be created is _1 but _1
              # is already mapped to 0 so __0 and __1 are used for the new
              # symbols
              if any(r in expr.free_symbols and
                     r in variables and
                     Symbol(pre + mystr(point[variables.index(r)])) != r
                     for _, r in reps):
                  pre += "_"
                  continue
              break
  
          obj = Expr.__new__(cls, expr, Tuple(*variables), point)
          obj._expr = expr.xreplace(dict(reps))
          return obj
  
      def _eval_is_commutative(self):
          return self.expr.is_commutative
  
      def doit(self, **hints):
          e, v, p = self.args
  
          # remove self mappings
          for i, (vi, pi) in enumerate(zip(v, p)):
              if vi == pi:
                  v = v[:i] + v[i + 1:]
                  p = p[:i] + p[i + 1:]
          if not v:
              return self.expr
  
          if isinstance(e, Derivative):
              # apply functions first, e.g. f -> cos
              undone = []
              for i, vi in enumerate(v):
                  if isinstance(vi, FunctionClass):
                      e = e.subs(vi, p[i])
                  else:
                      undone.append((vi, p[i]))
              if not isinstance(e, Derivative):
                  e = e.doit()
              if isinstance(e, Derivative):
                  # do Subs that aren't related to differentiation
                  undone2 = []
                  D = Dummy()
                  for vi, pi in undone:
                      if D not in e.xreplace({vi: D}).free_symbols:
                          e = e.subs(vi, pi)
                      else:
                          undone2.append((vi, pi))
                  undone = undone2
                  # differentiate wrt variables that are present
                  wrt = []
                  D = Dummy()
                  expr = e.expr
                  free = expr.free_symbols
                  for vi, ci in e.variable_count:
                      if isinstance(vi, Symbol) and vi in free:
                          expr = expr.diff((vi, ci))
                      elif D in expr.subs(vi, D).free_symbols:
                          expr = expr.diff((vi, ci))
                      else:
                          wrt.append((vi, ci))
                  # inject remaining subs
                  rv = expr.subs(undone)
                  # do remaining differentiation *in order given*
                  for vc in wrt:
                      rv = rv.diff(vc)
              else:
                  # inject remaining subs
                  rv = e.subs(undone)
          else:
              rv = e.doit(**hints).subs(list(zip(v, p)))
  
          if hints.get('deep', True) and rv != self:
              rv = rv.doit(**hints)
          return rv
  
      def evalf(self, prec=None, **options):
          return self.doit().evalf(prec, **options)
  
      n = evalf
  
      @property
      def variables(self):
          """The variables to be evaluated"""
          return self._args[1]
  
      bound_symbols = variables
  
      @property
      def expr(self):
          """The expression on which the substitution operates"""
          return self._args[0]
  
      @property
      def point(self):
          """The values for which the variables are to be substituted"""
          return self._args[2]
  
      @property
      def free_symbols(self):
          return (self.expr.free_symbols - set(self.variables) |
              set(self.point.free_symbols))
  
      @property
      def expr_free_symbols(self):
          return (self.expr.expr_free_symbols - set(self.variables) |
              set(self.point.expr_free_symbols))
  
      def __eq__(self, other):
          if not isinstance(other, Subs):
              return False
          return self._hashable_content() == other._hashable_content()
  
      def __ne__(self, other):
          return not(self == other)
  
      def __hash__(self):
          return super(Subs, self).__hash__()
  
      def _hashable_content(self):
          return (self._expr.xreplace(self.canonical_variables),
              ) + tuple(ordered([(v, p) for v, p in
              zip(self.variables, self.point) if not self.expr.has(v)]))
  
      def _eval_subs(self, old, new):
          # Subs doit will do the variables in order; the semantics
          # of subs for Subs is have the following invariant for
          # Subs object foo:
          #    foo.doit().subs(reps) == foo.subs(reps).doit()
          pt = list(self.point)
          if old in self.variables:
              if _atomic(new) == set([new]) and not any(
                      i.has(new) for i in self.args):
                  # the substitution is neutral
                  return self.xreplace({old: new})
              # any occurance of old before this point will get
              # handled by replacements from here on
              i = self.variables.index(old)
              for j in range(i, len(self.variables)):
                  pt[j] = pt[j]._subs(old, new)
              return self.func(self.expr, self.variables, pt)
          v = [i._subs(old, new) for i in self.variables]
          if v != list(self.variables):
              return self.func(self.expr, self.variables + (old,), pt + [new])
          expr = self.expr._subs(old, new)
          pt = [i._subs(old, new) for i in self.point]
          return self.func(expr, v, pt)
  
      def _eval_derivative(self, s):
          # Apply the chain rule of the derivative on the substitution variables:
          val = Add.fromiter(p.diff(s) * Subs(self.expr.diff(v), self.variables, self.point).doit() for v, p in zip(self.variables, self.point))
  
          # Check if there are free symbols in `self.expr`:
          # First get the `expr_free_symbols`, which returns the free symbols
          # that are directly contained in an expression node (i.e. stop
          # searching if the node isn't an expression). At this point turn the
          # expressions into `free_symbols` and check if there are common free
          # symbols in `self.expr` and the deriving factor.
          fs1 = {j for i in self.expr_free_symbols for j in i.free_symbols}
          if len(fs1 & s.free_symbols) > 0:
              val += Subs(self.expr.diff(s), self.variables, self.point).doit()
          return val
  
      def _eval_nseries(self, x, n, logx):
          if x in self.point:
              # x is the variable being substituted into
              apos = self.point.index(x)
              other = self.variables[apos]
          else:
              other = x
          arg = self.expr.nseries(other, n=n, logx=logx)
          o = arg.getO()
          terms = Add.make_args(arg.removeO())
          rv = Add(*[self.func(a, *self.args[1:]) for a in terms])
          if o:
              rv += o.subs(other, x)
          return rv
  
      def _eval_as_leading_term(self, x):
          if x in self.point:
              ipos = self.point.index(x)
              xvar = self.variables[ipos]
              return self.expr.as_leading_term(xvar)
          if x in self.variables:
              # if `x` is a dummy variable, it means it won't exist after the
              # substitution has been performed:
              return self
          # The variable is independent of the substitution:
          return self.expr.as_leading_term(x)
  
  
  def diff(f, *symbols, **kwargs):
      """
      Differentiate f with respect to symbols.
  
      This is just a wrapper to unify .diff() and the Derivative class; its
      interface is similar to that of integrate().  You can use the same
      shortcuts for multiple variables as with Derivative.  For example,
      diff(f(x), x, x, x) and diff(f(x), x, 3) both return the third derivative
      of f(x).
  
      You can pass evaluate=False to get an unevaluated Derivative class.  Note
      that if there are 0 symbols (such as diff(f(x), x, 0), then the result will
      be the function (the zeroth derivative), even if evaluate=False.
  
      Examples
      ========
  
      >>> from sympy import sin, cos, Function, diff
      >>> from sympy.abc import x, y
      >>> f = Function('f')
  
      >>> diff(sin(x), x)
      cos(x)
      >>> diff(f(x), x, x, x)
      Derivative(f(x), (x, 3))
      >>> diff(f(x), x, 3)
      Derivative(f(x), (x, 3))
      >>> diff(sin(x)*cos(y), x, 2, y, 2)
      sin(x)*cos(y)
  
      >>> type(diff(sin(x), x))
      cos
      >>> type(diff(sin(x), x, evaluate=False))
      <class 'sympy.core.function.Derivative'>
      >>> type(diff(sin(x), x, 0))
      sin
      >>> type(diff(sin(x), x, 0, evaluate=False))
      sin
  
      >>> diff(sin(x))
      cos(x)
      >>> diff(sin(x*y))
      Traceback (most recent call last):
      ...
      ValueError: specify differentiation variables to differentiate sin(x*y)
  
      Note that ``diff(sin(x))`` syntax is meant only for convenience
      in interactive sessions and should be avoided in library code.
  
      References
      ==========
  
      http://reference.wolfram.com/legacy/v5_2/Built-inFunctions/AlgebraicComputation/Calculus/D.html
  
      See Also
      ========
  
      Derivative
      sympy.geometry.util.idiff: computes the derivative implicitly
  
      """
      if hasattr(f, 'diff'):
          return f.diff(*symbols, **kwargs)
      kwargs.setdefault('evaluate', True)
      return Derivative(f, *symbols, **kwargs)
  
  
  def expand(e, deep=True, modulus=None, power_base=True, power_exp=True,
          mul=True, log=True, multinomial=True, basic=True, **hints):
      r"""
      Expand an expression using methods given as hints.
  
      Hints evaluated unless explicitly set to False are:  ``basic``, ``log``,
      ``multinomial``, ``mul``, ``power_base``, and ``power_exp`` The following
      hints are supported but not applied unless set to True:  ``complex``,
      ``func``, and ``trig``.  In addition, the following meta-hints are
      supported by some or all of the other hints:  ``frac``, ``numer``,
      ``denom``, ``modulus``, and ``force``.  ``deep`` is supported by all
      hints.  Additionally, subclasses of Expr may define their own hints or
      meta-hints.
  
      The ``basic`` hint is used for any special rewriting of an object that
      should be done automatically (along with the other hints like ``mul``)
      when expand is called. This is a catch-all hint to handle any sort of
      expansion that may not be described by the existing hint names. To use
      this hint an object should override the ``_eval_expand_basic`` method.
      Objects may also define their own expand methods, which are not run by
      default.  See the API section below.
  
      If ``deep`` is set to ``True`` (the default), things like arguments of
      functions are recursively expanded.  Use ``deep=False`` to only expand on
      the top level.
  
      If the ``force`` hint is used, assumptions about variables will be ignored
      in making the expansion.
  
      Hints
      =====
  
      These hints are run by default
  
      mul
      ---
  
      Distributes multiplication over addition:
  
      >>> from sympy import cos, exp, sin
      >>> from sympy.abc import x, y, z
      >>> (y*(x + z)).expand(mul=True)
      x*y + y*z
  
      multinomial
      -----------
  
      Expand (x + y + ...)**n where n is a positive integer.
  
      >>> ((x + y + z)**2).expand(multinomial=True)
      x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2
  
      power_exp
      ---------
  
      Expand addition in exponents into multiplied bases.
  
      >>> exp(x + y).expand(power_exp=True)
      exp(x)*exp(y)
      >>> (2**(x + y)).expand(power_exp=True)
      2**x*2**y
  
      power_base
      ----------
  
      Split powers of multiplied bases.
  
      This only happens by default if assumptions allow, or if the
      ``force`` meta-hint is used:
  
      >>> ((x*y)**z).expand(power_base=True)
      (x*y)**z
      >>> ((x*y)**z).expand(power_base=True, force=True)
      x**z*y**z
      >>> ((2*y)**z).expand(power_base=True)
      2**z*y**z
  
      Note that in some cases where this expansion always holds, SymPy performs
      it automatically:
  
      >>> (x*y)**2
      x**2*y**2
  
      log
      ---
  
      Pull out power of an argument as a coefficient and split logs products
      into sums of logs.
  
      Note that these only work if the arguments of the log function have the
      proper assumptions--the arguments must be positive and the exponents must
      be real--or else the ``force`` hint must be True:
  
      >>> from sympy import log, symbols
      >>> log(x**2*y).expand(log=True)
      log(x**2*y)
      >>> log(x**2*y).expand(log=True, force=True)
      2*log(x) + log(y)
      >>> x, y = symbols('x,y', positive=True)
      >>> log(x**2*y).expand(log=True)
      2*log(x) + log(y)
  
      basic
      -----
  
      This hint is intended primarily as a way for custom subclasses to enable
      expansion by default.
  
      These hints are not run by default:
  
      complex
      -------
  
      Split an expression into real and imaginary parts.
  
      >>> x, y = symbols('x,y')
      >>> (x + y).expand(complex=True)
      re(x) + re(y) + I*im(x) + I*im(y)
      >>> cos(x).expand(complex=True)
      -I*sin(re(x))*sinh(im(x)) + cos(re(x))*cosh(im(x))
  
      Note that this is just a wrapper around ``as_real_imag()``.  Most objects
      that wish to redefine ``_eval_expand_complex()`` should consider
      redefining ``as_real_imag()`` instead.
  
      func
      ----
  
      Expand other functions.
  
      >>> from sympy import gamma
      >>> gamma(x + 1).expand(func=True)
      x*gamma(x)
  
      trig
      ----
  
      Do trigonometric expansions.
  
      >>> cos(x + y).expand(trig=True)
      -sin(x)*sin(y) + cos(x)*cos(y)
      >>> sin(2*x).expand(trig=True)
      2*sin(x)*cos(x)
  
      Note that the forms of ``sin(n*x)`` and ``cos(n*x)`` in terms of ``sin(x)``
      and ``cos(x)`` are not unique, due to the identity `\sin^2(x) + \cos^2(x)
      = 1`.  The current implementation uses the form obtained from Chebyshev
      polynomials, but this may change.  See `this MathWorld article
      <http://mathworld.wolfram.com/Multiple-AngleFormulas.html>`_ for more
      information.
  
      Notes
      =====
  
      - You can shut off unwanted methods::
  
          >>> (exp(x + y)*(x + y)).expand()
          x*exp(x)*exp(y) + y*exp(x)*exp(y)
          >>> (exp(x + y)*(x + y)).expand(power_exp=False)
          x*exp(x + y) + y*exp(x + y)
          >>> (exp(x + y)*(x + y)).expand(mul=False)
          (x + y)*exp(x)*exp(y)
  
      - Use deep=False to only expand on the top level::
  
          >>> exp(x + exp(x + y)).expand()
          exp(x)*exp(exp(x)*exp(y))
          >>> exp(x + exp(x + y)).expand(deep=False)
          exp(x)*exp(exp(x + y))
  
      - Hints are applied in an arbitrary, but consistent order (in the current
        implementation, they are applied in alphabetical order, except
        multinomial comes before mul, but this may change).  Because of this,
        some hints may prevent expansion by other hints if they are applied
        first. For example, ``mul`` may distribute multiplications and prevent
        ``log`` and ``power_base`` from expanding them. Also, if ``mul`` is
        applied before ``multinomial`, the expression might not be fully
        distributed. The solution is to use the various ``expand_hint`` helper
        functions or to use ``hint=False`` to this function to finely control
        which hints are applied. Here are some examples::
  
          >>> from sympy import expand, expand_mul, expand_power_base
          >>> x, y, z = symbols('x,y,z', positive=True)
  
          >>> expand(log(x*(y + z)))
          log(x) + log(y + z)
  
        Here, we see that ``log`` was applied before ``mul``.  To get the mul
        expanded form, either of the following will work::
  
          >>> expand_mul(log(x*(y + z)))
          log(x*y + x*z)
          >>> expand(log(x*(y + z)), log=False)
          log(x*y + x*z)
  
        A similar thing can happen with the ``power_base`` hint::
  
          >>> expand((x*(y + z))**x)
          (x*y + x*z)**x
  
        To get the ``power_base`` expanded form, either of the following will
        work::
  
          >>> expand((x*(y + z))**x, mul=False)
          x**x*(y + z)**x
          >>> expand_power_base((x*(y + z))**x)
          x**x*(y + z)**x
  
          >>> expand((x + y)*y/x)
          y + y**2/x
  
        The parts of a rational expression can be targeted::
  
          >>> expand((x + y)*y/x/(x + 1), frac=True)
          (x*y + y**2)/(x**2 + x)
          >>> expand((x + y)*y/x/(x + 1), numer=True)
          (x*y + y**2)/(x*(x + 1))
          >>> expand((x + y)*y/x/(x + 1), denom=True)
          y*(x + y)/(x**2 + x)
  
      - The ``modulus`` meta-hint can be used to reduce the coefficients of an
        expression post-expansion::
  
          >>> expand((3*x + 1)**2)
          9*x**2 + 6*x + 1
          >>> expand((3*x + 1)**2, modulus=5)
          4*x**2 + x + 1
  
      - Either ``expand()`` the function or ``.expand()`` the method can be
        used.  Both are equivalent::
  
          >>> expand((x + 1)**2)
          x**2 + 2*x + 1
          >>> ((x + 1)**2).expand()
          x**2 + 2*x + 1
  
      API
      ===
  
      Objects can define their own expand hints by defining
      ``_eval_expand_hint()``.  The function should take the form::
  
          def _eval_expand_hint(self, **hints):
              # Only apply the method to the top-level expression
              ...
  
      See also the example below.  Objects should define ``_eval_expand_hint()``
      methods only if ``hint`` applies to that specific object.  The generic
      ``_eval_expand_hint()`` method defined in Expr will handle the no-op case.
  
      Each hint should be responsible for expanding that hint only.
      Furthermore, the expansion should be applied to the top-level expression
      only.  ``expand()`` takes care of the recursion that happens when
      ``deep=True``.
  
      You should only call ``_eval_expand_hint()`` methods directly if you are
      100% sure that the object has the method, as otherwise you are liable to
      get unexpected ``AttributeError``s.  Note, again, that you do not need to
      recursively apply the hint to args of your object: this is handled
      automatically by ``expand()``.  ``_eval_expand_hint()`` should
      generally not be used at all outside of an ``_eval_expand_hint()`` method.
      If you want to apply a specific expansion from within another method, use
      the public ``expand()`` function, method, or ``expand_hint()`` functions.
  
      In order for expand to work, objects must be rebuildable by their args,
      i.e., ``obj.func(*obj.args) == obj`` must hold.
  
      Expand methods are passed ``**hints`` so that expand hints may use
      'metahints'--hints that control how different expand methods are applied.
      For example, the ``force=True`` hint described above that causes
      ``expand(log=True)`` to ignore assumptions is such a metahint.  The
      ``deep`` meta-hint is handled exclusively by ``expand()`` and is not
      passed to ``_eval_expand_hint()`` methods.
  
      Note that expansion hints should generally be methods that perform some
      kind of 'expansion'.  For hints that simply rewrite an expression, use the
      .rewrite() API.
  
      Examples
      ========
  
      >>> from sympy import Expr, sympify
      >>> class MyClass(Expr):
      ...     def __new__(cls, *args):
      ...         args = sympify(args)
      ...         return Expr.__new__(cls, *args)
      ...
      ...     def _eval_expand_double(self, **hints):
      ...         '''
      ...         Doubles the args of MyClass.
      ...
      ...         If there more than four args, doubling is not performed,
      ...         unless force=True is also used (False by default).
      ...         '''
      ...         force = hints.pop('force', False)
      ...         if not force and len(self.args) > 4:
      ...             return self
      ...         return self.func(*(self.args + self.args))
      ...
      >>> a = MyClass(1, 2, MyClass(3, 4))
      >>> a
      MyClass(1, 2, MyClass(3, 4))
      >>> a.expand(double=True)
      MyClass(1, 2, MyClass(3, 4, 3, 4), 1, 2, MyClass(3, 4, 3, 4))
      >>> a.expand(double=True, deep=False)
      MyClass(1, 2, MyClass(3, 4), 1, 2, MyClass(3, 4))
  
      >>> b = MyClass(1, 2, 3, 4, 5)
      >>> b.expand(double=True)
      MyClass(1, 2, 3, 4, 5)
      >>> b.expand(double=True, force=True)
      MyClass(1, 2, 3, 4, 5, 1, 2, 3, 4, 5)
  
      See Also
      ========
  
      expand_log, expand_mul, expand_multinomial, expand_complex, expand_trig,
      expand_power_base, expand_power_exp, expand_func, hyperexpand
  
      """
      # don't modify this; modify the Expr.expand method
      hints['power_base'] = power_base
      hints['power_exp'] = power_exp
      hints['mul'] = mul
      hints['log'] = log
      hints['multinomial'] = multinomial
      hints['basic'] = basic
      return sympify(e).expand(deep=deep, modulus=modulus, **hints)
  
  # This is a special application of two hints
  
  def _mexpand(expr, recursive=False):
      # expand multinomials and then expand products; this may not always
      # be sufficient to give a fully expanded expression (see
      # test_issue_8247_8354 in test_arit)
      if expr is None:
          return
      was = None
      while was != expr:
          was, expr = expr, expand_mul(expand_multinomial(expr))
          if not recursive:
              break
      return expr
  
  
  # These are simple wrappers around single hints.
  
  
  def expand_mul(expr, deep=True):
      """
      Wrapper around expand that only uses the mul hint.  See the expand
      docstring for more information.
  
      Examples
      ========
  
      >>> from sympy import symbols, expand_mul, exp, log
      >>> x, y = symbols('x,y', positive=True)
      >>> expand_mul(exp(x+y)*(x+y)*log(x*y**2))
      x*exp(x + y)*log(x*y**2) + y*exp(x + y)*log(x*y**2)
  
      """
      return sympify(expr).expand(deep=deep, mul=True, power_exp=False,
      power_base=False, basic=False, multinomial=False, log=False)
  
  
  def expand_multinomial(expr, deep=True):
      """
      Wrapper around expand that only uses the multinomial hint.  See the expand
      docstring for more information.
  
      Examples
      ========
  
      >>> from sympy import symbols, expand_multinomial, exp
      >>> x, y = symbols('x y', positive=True)
      >>> expand_multinomial((x + exp(x + 1))**2)
      x**2 + 2*x*exp(x + 1) + exp(2*x + 2)
  
      """
      return sympify(expr).expand(deep=deep, mul=False, power_exp=False,
      power_base=False, basic=False, multinomial=True, log=False)
  
  
  def expand_log(expr, deep=True, force=False):
      """
      Wrapper around expand that only uses the log hint.  See the expand
      docstring for more information.
  
      Examples
      ========
  
      >>> from sympy import symbols, expand_log, exp, log
      >>> x, y = symbols('x,y', positive=True)
      >>> expand_log(exp(x+y)*(x+y)*log(x*y**2))
      (x + y)*(log(x) + 2*log(y))*exp(x + y)
  
      """
      return sympify(expr).expand(deep=deep, log=True, mul=False,
          power_exp=False, power_base=False, multinomial=False,
          basic=False, force=force)
  
  
  def expand_func(expr, deep=True):
      """
      Wrapper around expand that only uses the func hint.  See the expand
      docstring for more information.
  
      Examples
      ========
  
      >>> from sympy import expand_func, gamma
      >>> from sympy.abc import x
      >>> expand_func(gamma(x + 2))
      x*(x + 1)*gamma(x)
  
      """
      return sympify(expr).expand(deep=deep, func=True, basic=False,
      log=False, mul=False, power_exp=False, power_base=False, multinomial=False)
  
  
  def expand_trig(expr, deep=True):
      """
      Wrapper around expand that only uses the trig hint.  See the expand
      docstring for more information.
  
      Examples
      ========
  
      >>> from sympy import expand_trig, sin
      >>> from sympy.abc import x, y
      >>> expand_trig(sin(x+y)*(x+y))
      (x + y)*(sin(x)*cos(y) + sin(y)*cos(x))
  
      """
      return sympify(expr).expand(deep=deep, trig=True, basic=False,
      log=False, mul=False, power_exp=False, power_base=False, multinomial=False)
  
  
  def expand_complex(expr, deep=True):
      """
      Wrapper around expand that only uses the complex hint.  See the expand
      docstring for more information.
  
      Examples
      ========
  
      >>> from sympy import expand_complex, exp, sqrt, I
      >>> from sympy.abc import z
      >>> expand_complex(exp(z))
      I*exp(re(z))*sin(im(z)) + exp(re(z))*cos(im(z))
      >>> expand_complex(sqrt(I))
      sqrt(2)/2 + sqrt(2)*I/2
  
      See Also
      ========
      Expr.as_real_imag
      """
      return sympify(expr).expand(deep=deep, complex=True, basic=False,
      log=False, mul=False, power_exp=False, power_base=False, multinomial=False)
  
  
  def expand_power_base(expr, deep=True, force=False):
      """
      Wrapper around expand that only uses the power_base hint.
  
      See the expand docstring for more information.
  
      A wrapper to expand(power_base=True) which separates a power with a base
      that is a Mul into a product of powers, without performing any other
      expansions, provided that assumptions about the power's base and exponent
      allow.
  
      deep=False (default is True) will only apply to the top-level expression.
  
      force=True (default is False) will cause the expansion to ignore
      assumptions about the base and exponent. When False, the expansion will
      only happen if the base is non-negative or the exponent is an integer.
  
      >>> from sympy.abc import x, y, z
      >>> from sympy import expand_power_base, sin, cos, exp
  
      >>> (x*y)**2
      x**2*y**2
  
      >>> (2*x)**y
      (2*x)**y
      >>> expand_power_base(_)
      2**y*x**y
  
      >>> expand_power_base((x*y)**z)
      (x*y)**z
      >>> expand_power_base((x*y)**z, force=True)
      x**z*y**z
      >>> expand_power_base(sin((x*y)**z), deep=False)
      sin((x*y)**z)
      >>> expand_power_base(sin((x*y)**z), force=True)
      sin(x**z*y**z)
  
      >>> expand_power_base((2*sin(x))**y + (2*cos(x))**y)
      2**y*sin(x)**y + 2**y*cos(x)**y
  
      >>> expand_power_base((2*exp(y))**x)
      2**x*exp(y)**x
  
      >>> expand_power_base((2*cos(x))**y)
      2**y*cos(x)**y
  
      Notice that sums are left untouched. If this is not the desired behavior,
      apply full ``expand()`` to the expression:
  
      >>> expand_power_base(((x+y)*z)**2)
      z**2*(x + y)**2
      >>> (((x+y)*z)**2).expand()
      x**2*z**2 + 2*x*y*z**2 + y**2*z**2
  
      >>> expand_power_base((2*y)**(1+z))
      2**(z + 1)*y**(z + 1)
      >>> ((2*y)**(1+z)).expand()
      2*2**z*y*y**z
  
      """
      return sympify(expr).expand(deep=deep, log=False, mul=False,
          power_exp=False, power_base=True, multinomial=False,
          basic=False, force=force)
  
  
  def expand_power_exp(expr, deep=True):
      """
      Wrapper around expand that only uses the power_exp hint.
  
      See the expand docstring for more information.
  
      Examples
      ========
  
      >>> from sympy import expand_power_exp
      >>> from sympy.abc import x, y
      >>> expand_power_exp(x**(y + 2))
      x**2*x**y
      """
      return sympify(expr).expand(deep=deep, complex=False, basic=False,
      log=False, mul=False, power_exp=True, power_base=False, multinomial=False)
  
  
  def count_ops(expr, visual=False):
      """
      Return a representation (integer or expression) of the operations in expr.
  
      If ``visual`` is ``False`` (default) then the sum of the coefficients of the
      visual expression will be returned.
  
      If ``visual`` is ``True`` then the number of each type of operation is shown
      with the core class types (or their virtual equivalent) multiplied by the
      number of times they occur.
  
      If expr is an iterable, the sum of the op counts of the
      items will be returned.
  
      Examples
      ========
  
      >>> from sympy.abc import a, b, x, y
      >>> from sympy import sin, count_ops
  
      Although there isn't a SUB object, minus signs are interpreted as
      either negations or subtractions:
  
      >>> (x - y).count_ops(visual=True)
      SUB
      >>> (-x).count_ops(visual=True)
      NEG
  
      Here, there are two Adds and a Pow:
  
      >>> (1 + a + b**2).count_ops(visual=True)
      2*ADD + POW
  
      In the following, an Add, Mul, Pow and two functions:
  
      >>> (sin(x)*x + sin(x)**2).count_ops(visual=True)
      ADD + MUL + POW + 2*SIN
  
      for a total of 5:
  
      >>> (sin(x)*x + sin(x)**2).count_ops(visual=False)
      5
  
      Note that "what you type" is not always what you get. The expression
      1/x/y is translated by sympy into 1/(x*y) so it gives a DIV and MUL rather
      than two DIVs:
  
      >>> (1/x/y).count_ops(visual=True)
      DIV + MUL
  
      The visual option can be used to demonstrate the difference in
      operations for expressions in different forms. Here, the Horner
      representation is compared with the expanded form of a polynomial:
  
      >>> eq=x*(1 + x*(2 + x*(3 + x)))
      >>> count_ops(eq.expand(), visual=True) - count_ops(eq, visual=True)
      -MUL + 3*POW
  
      The count_ops function also handles iterables:
  
      >>> count_ops([x, sin(x), None, True, x + 2], visual=False)
      2
      >>> count_ops([x, sin(x), None, True, x + 2], visual=True)
      ADD + SIN
      >>> count_ops({x: sin(x), x + 2: y + 1}, visual=True)
      2*ADD + SIN
  
      """
      from sympy import Integral, Symbol
      from sympy.core.relational import Relational
      from sympy.simplify.radsimp import fraction
      from sympy.logic.boolalg import BooleanFunction
      from sympy.utilities.misc import func_name
  
      expr = sympify(expr)
      if isinstance(expr, Expr) and not expr.is_Relational:
  
          ops = []
          args = [expr]
          NEG = Symbol('NEG')
          DIV = Symbol('DIV')
          SUB = Symbol('SUB')
          ADD = Symbol('ADD')
          while args:
              a = args.pop()
  
              # XXX: This is a hack to support non-Basic args
              if isinstance(a, string_types):
                  continue
  
              if a.is_Rational:
                  #-1/3 = NEG + DIV
                  if a is not S.One:
                      if a.p < 0:
                          ops.append(NEG)
                      if a.q != 1:
                          ops.append(DIV)
                      continue
              elif a.is_Mul or a.is_MatMul:
                  if _coeff_isneg(a):
                      ops.append(NEG)
                      if a.args[0] is S.NegativeOne:
                          a = a.as_two_terms()[1]
                      else:
                          a = -a
                  n, d = fraction(a)
                  if n.is_Integer:
                      ops.append(DIV)
                      if n < 0:
                          ops.append(NEG)
                      args.append(d)
                      continue  # won't be -Mul but could be Add
                  elif d is not S.One:
                      if not d.is_Integer:
                          args.append(d)
                      ops.append(DIV)
                      args.append(n)
                      continue  # could be -Mul
              elif a.is_Add or a.is_MatAdd:
                  aargs = list(a.args)
                  negs = 0
                  for i, ai in enumerate(aargs):
                      if _coeff_isneg(ai):
                          negs += 1
                          args.append(-ai)
                          if i > 0:
                              ops.append(SUB)
                      else:
                          args.append(ai)
                          if i > 0:
                              ops.append(ADD)
                  if negs == len(aargs):  # -x - y = NEG + SUB
                      ops.append(NEG)
                  elif _coeff_isneg(aargs[0]):  # -x + y = SUB, but already recorded ADD
                      ops.append(SUB - ADD)
                  continue
              if a.is_Pow and a.exp is S.NegativeOne:
                  ops.append(DIV)
                  args.append(a.base)  # won't be -Mul but could be Add
                  continue
              if (a.is_Mul or
                  a.is_Pow or
                  a.is_Function or
                  isinstance(a, Derivative) or
                      isinstance(a, Integral)):
  
                  o = Symbol(a.func.__name__.upper())
                  # count the args
                  if (a.is_Mul or isinstance(a, LatticeOp)):
                      ops.append(o*(len(a.args) - 1))
                  else:
                      ops.append(o)
              if not a.is_Symbol:
                  args.extend(a.args)
  
      elif type(expr) is dict:
          ops = [count_ops(k, visual=visual) +
                 count_ops(v, visual=visual) for k, v in expr.items()]
      elif iterable(expr):
          ops = [count_ops(i, visual=visual) for i in expr]
      elif isinstance(expr, (Relational, BooleanFunction)):
          ops = []
          for arg in expr.args:
              ops.append(count_ops(arg, visual=True))
          o = Symbol(func_name(expr, short=True).upper())
          ops.append(o)
      elif not isinstance(expr, Basic):
          ops = []
      else:  # it's Basic not isinstance(expr, Expr):
          if not isinstance(expr, Basic):
              raise TypeError("Invalid type of expr")
          else:
              ops = []
              args = [expr]
              while args:
                  a = args.pop()
  
                  # XXX: This is a hack to support non-Basic args
                  if isinstance(a, string_types):
                      continue
  
                  if a.args:
                      o = Symbol(a.func.__name__.upper())
                      if a.is_Boolean:
                          ops.append(o*(len(a.args)-1))
                      else:
                          ops.append(o)
                      args.extend(a.args)
  
      if not ops:
          if visual:
              return S.Zero
          return 0
  
      ops = Add(*ops)
  
      if visual:
          return ops
  
      if ops.is_Number:
          return int(ops)
  
      return sum(int((a.args or [1])[0]) for a in Add.make_args(ops))
  
  
  def nfloat(expr, n=15, exponent=False):
      """Make all Rationals in expr Floats except those in exponents
      (unless the exponents flag is set to True).
  
      Examples
      ========
  
      >>> from sympy.core.function import nfloat
      >>> from sympy.abc import x, y
      >>> from sympy import cos, pi, sqrt
      >>> nfloat(x**4 + x/2 + cos(pi/3) + 1 + sqrt(y))
      x**4 + 0.5*x + sqrt(y) + 1.5
      >>> nfloat(x**4 + sqrt(y), exponent=True)
      x**4.0 + y**0.5
  
      """
      from sympy.core.power import Pow
      from sympy.polys.rootoftools import RootOf
  
      if iterable(expr, exclude=string_types):
          if isinstance(expr, (dict, Dict)):
              return type(expr)([(k, nfloat(v, n, exponent)) for k, v in
                                 list(expr.items())])
          return type(expr)([nfloat(a, n, exponent) for a in expr])
      rv = sympify(expr)
  
      if rv.is_Number:
          return Float(rv, n)
      elif rv.is_number:
          # evalf doesn't always set the precision
          rv = rv.n(n)
          if rv.is_Number:
              rv = Float(rv.n(n), n)
          else:
              pass  # pure_complex(rv) is likely True
          return rv
  
      # watch out for RootOf instances that don't like to have
      # their exponents replaced with Dummies and also sometimes have
      # problems with evaluating at low precision (issue 6393)
      rv = rv.xreplace({ro: ro.n(n) for ro in rv.atoms(RootOf)})
  
      if not exponent:
          reps = [(p, Pow(p.base, Dummy())) for p in rv.atoms(Pow)]
          rv = rv.xreplace(dict(reps))
      rv = rv.n(n)
      if not exponent:
          rv = rv.xreplace({d.exp: p.exp for p, d in reps})
      else:
          # Pow._eval_evalf special cases Integer exponents so if
          # exponent is suppose to be handled we have to do so here
          rv = rv.xreplace(Transform(
              lambda x: Pow(x.base, Float(x.exp, n)),
              lambda x: x.is_Pow and x.exp.is_Integer))
  
      return rv.xreplace(Transform(
          lambda x: x.func(*nfloat(x.args, n, exponent)),
          lambda x: isinstance(x, Function)))
  
  
  from sympy.core.symbol import Dummy, Symbol
  