code: def simplify(
      expr,
      ratio=1.7,
      measure=count_ops,
      rational=False,
      inverse=False,
      doit=True,
      **kwargs,
  ):
      """Simplifies the given expression.
  
      Explanation
      ===========
  
      Simplification is not a well defined term and the exact strategies
      this function tries can change in the future versions of SymPy. If
      your algorithm relies on "simplification" (whatever it is), try to
      determine what you need exactly  -  is it powsimp()?, radsimp()?,
      together()?, logcombine()?, or something else? And use this particular
      function directly, because those are well defined and thus your algorithm
      will be robust.
  
      Nonetheless, especially for interactive use, or when you do not know
      anything about the structure of the expression, simplify() tries to apply
      intelligent heuristics to make the input expression "simpler".  For
      example:
  
      >>> from sympy import simplify, cos, sin
      >>> from sympy.abc import x, y
      >>> a = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2)
      >>> a
      (x**2 + x)/(x*sin(y)**2 + x*cos(y)**2)
      >>> simplify(a)
      x + 1
  
      Note that we could have obtained the same result by using specific
      simplification functions:
  
      >>> from sympy import trigsimp, cancel
      >>> trigsimp(a)
      (x**2 + x)/x
      >>> cancel(_)
      x + 1
  
      In some cases, applying :func:`simplify` may actually result in some more
      complicated expression. The default ``ratio=1.7`` prevents more extreme
      cases: if (result length)/(input length) > ratio, then input is returned
      unmodified.  The ``measure`` parameter lets you specify the function used
      to determine how complex an expression is.  The function should take a
      single argument as an expression and return a number such that if
      expression ``a`` is more complex than expression ``b``, then
      ``measure(a) > measure(b)``.  The default measure function is
      :func:`~.count_ops`, which returns the total number of operations in the
      expression.
  
      For example, if ``ratio=1``, ``simplify`` output cannot be longer
      than input.
  
      ::
  
          >>> from sympy import sqrt, simplify, count_ops, oo
          >>> root = 1/(sqrt(2)+3)
  
      Since ``simplify(root)`` would result in a slightly longer expression,
      root is returned unchanged instead::
  
         >>> simplify(root, ratio=1) == root
         True
  
      If ``ratio=oo``, simplify will be applied anyway::
  
          >>> count_ops(simplify(root, ratio=oo)) > count_ops(root)
          True
  
      Note that the shortest expression is not necessary the simplest, so
      setting ``ratio`` to 1 may not be a good idea.
      Heuristically, the default value ``ratio=1.7`` seems like a reasonable
      choice.
  
      You can easily define your own measure function based on what you feel
      should represent the "size" or "complexity" of the input expression.  Note
      that some choices, such as ``lambda expr: len(str(expr))`` may appear to be
      good metrics, but have other problems (in this case, the measure function
      may slow down simplify too much for very large expressions).  If you do not
      know what a good metric would be, the default, ``count_ops``, is a good
      one.
  
      For example:
  
      >>> from sympy import symbols, log
      >>> a, b = symbols('a b', positive=True)
      >>> g = log(a) + log(b) + log(a)*log(1/b)
      >>> h = simplify(g)
      >>> h
      log(a*b**(1 - log(a)))
      >>> count_ops(g)
      8
      >>> count_ops(h)
      5
  
      So you can see that ``h`` is simpler than ``g`` using the count_ops metric.
      However, we may not like how ``simplify`` (in this case, using
      ``logcombine``) has created the ``b**(log(1/a) + 1)`` term.  A simple way
      to reduce this would be to give more weight to powers as operations in
      ``count_ops``.  We can do this by using the ``visual=True`` option:
  
      >>> print(count_ops(g, visual=True))
      2*ADD + DIV + 4*LOG + MUL
      >>> print(count_ops(h, visual=True))
      2*LOG + MUL + POW + SUB
  
      >>> from sympy import Symbol, S
      >>> def my_measure(expr):
      ...     POW = Symbol('POW')
      ...     # Discourage powers by giving POW a weight of 10
      ...     count = count_ops(expr, visual=True).subs(POW, 10)
      ...     # Every other operation gets a weight of 1 (the default)
      ...     count = count.replace(Symbol, type(S.One))
      ...     return count
      >>> my_measure(g)
      8
      >>> my_measure(h)
      14
      >>> 15./8 > 1.7 # 1.7 is the default ratio
      True
      >>> simplify(g, measure=my_measure)
      -log(a)*log(b) + log(a) + log(b)
  
      Note that because ``simplify()`` internally tries many different
      simplification strategies and then compares them using the measure
      function, we get a completely different result that is still different
      from the input expression by doing this.
  
      If ``rational=True``, Floats will be recast as Rationals before simplification.
      If ``rational=None``, Floats will be recast as Rationals but the result will
      be recast as Floats. If rational=False(default) then nothing will be done
      to the Floats.
  
      If ``inverse=True``, it will be assumed that a composition of inverse
      functions, such as sin and asin, can be cancelled in any order.
      For example, ``asin(sin(x))`` will yield ``x`` without checking whether
      x belongs to the set where this relation is true. The default is
      False.
  
      Note that ``simplify()`` automatically calls ``doit()`` on the final
      expression. You can avoid this behavior by passing ``doit=False`` as
      an argument.
  
      Also, it should be noted that simplifying a boolean expression is not
      well defined. If the expression prefers automatic evaluation (such as
      :obj:`~.Eq()` or :obj:`~.Or()`), simplification will return ``True`` or
      ``False`` if truth value can be determined. If the expression is not
      evaluated by default (such as :obj:`~.Predicate()`), simplification will
      not reduce it and you should use :func:`~.refine()` or :func:`~.ask()`
      function. This inconsistency will be resolved in future version.
  
      See Also
      ========
  
      sympy.assumptions.refine.refine : Simplification using assumptions.
      sympy.assumptions.ask.ask : Query for boolean expressions using assumptions.
      """
  
      def shorter(*choices):
          """
          Return the choice that has the fewest ops. In case of a tie,
          the expression listed first is selected.
          """
          if not has_variety(choices):
              return choices[0]
          return min(choices, key=measure)
  
      def done(e):
          rv = e.doit() if doit else e
          return shorter(rv, collect_abs(rv))
  
      expr = sympify(expr, rational=rational)
      kwargs = {
          "ratio": kwargs.get("ratio", ratio),
          "measure": kwargs.get("measure", measure),
          "rational": kwargs.get("rational", rational),
          "inverse": kwargs.get("inverse", inverse),
          "doit": kwargs.get("doit", doit),
      }
      # no routine for Expr needs to check for is_zero
      if isinstance(expr, Expr) and expr.is_zero:
          return S.Zero if not expr.is_Number else expr
  
      _eval_simplify = getattr(expr, "_eval_simplify", None)
      if _eval_simplify is not None:
          return _eval_simplify(**kwargs)
  
      original_expr = expr = collect_abs(signsimp(expr))
  
      if (
          not isinstance(expr, Basic) or not expr.args
      ):  # XXX: temporary hack
          return expr
  
      if inverse and expr.has(Function):
          expr = inversecombine(expr)
          if not expr.args:  # simplified to atomic
              return expr
  
      # do deep simplification
      handled = Add, Mul, Pow, ExpBase
      expr = expr.replace(
          # here, checking for x.args is not enough because Basic has
          # args but Basic does not always play well with replace, e.g.
          # when simultaneous is True found expressions will be masked
          # off with a Dummy but not all Basic objects in an expression
          # can be replaced with a Dummy
          lambda x: isinstance(x, Expr)
          and x.args
          and not isinstance(x, handled),
          lambda x: x.func(*[simplify(i, **kwargs) for i in x.args]),
          simultaneous=False,
      )
      if not isinstance(expr, handled):
          return done(expr)
  
      if not expr.is_commutative:
          expr = nc_simplify(expr)
  
      # TODO: Apply different strategies, considering expression pattern:
      # is it a purely rational function? Is there any trigonometric function?...
      # See also https://github.com/sympy/sympy/pull/185.
  
      # rationalize Floats
      floats = False
      if rational is not False and expr.has(Float):
          floats = True
          expr = nsimplify(expr, rational=True)
  
      expr = _bottom_up(
          expr, lambda w: getattr(w, "normal", lambda: w)()
      )
      expr = Mul(*powsimp(expr).as_content_primitive())
      _e = cancel(expr)
      expr1 = shorter(_e, _mexpand(_e).cancel())  # issue 6829
      expr2 = shorter(
          together(expr, deep=True), together(expr1, deep=True)
      )
  
      if ratio is S.Infinity:
          expr = expr2
      else:
          expr = shorter(expr2, expr1, expr)
      if not isinstance(expr, Basic):  # XXX: temporary hack
          return expr
  
      expr = factor_terms(expr, sign=False)
  
      # must come before `Piecewise` since this introduces more `Piecewise` terms
      if expr.has(sign):
          expr = expr.rewrite(Abs)
  
      # Deal with Piecewise separately to avoid recursive growth of expressions
      if expr.has(Piecewise):
          # Fold into a single Piecewise
          expr = piecewise_fold(expr)
          # Apply doit, if doit=True
          expr = done(expr)
          # Still a Piecewise?
          if expr.has(Piecewise):
              # Fold into a single Piecewise, in case doit lead to some
              # expressions being Piecewise
              expr = piecewise_fold(expr)
              # kroneckersimp also affects Piecewise
              if expr.has(KroneckerDelta):
                  expr = kroneckersimp(expr)
              # Still a Piecewise?
              if expr.has(Piecewise):
                  # Do not apply doit on the segments as it has already
                  # been done above, but simplify
                  expr = piecewise_simplify(expr, deep=True, doit=False)
                  # Still a Piecewise?
                  if expr.has(Piecewise):
                      # Try factor common terms
                      expr = shorter(expr, factor_terms(expr))
                      # As all expressions have been simplified above with the
                      # complete simplify, nothing more needs to be done here
                      return expr
  
      # hyperexpand automatically only works on hypergeometric terms
      # Do this after the Piecewise part to avoid recursive expansion
      expr = hyperexpand(expr)
  
      if expr.has(KroneckerDelta):
          expr = kroneckersimp(expr)
  
      if expr.has(BesselBase):
          expr = besselsimp(expr)
  
      if expr.has(TrigonometricFunction, HyperbolicFunction):
          expr = trigsimp(expr, deep=True)
  
      if expr.has(log):
          expr = shorter(expand_log(expr, deep=True), logcombine(expr))
  
      if expr.has(CombinatorialFunction, gamma):
          # expression with gamma functions or non-integer arguments is
          # automatically passed to gammasimp
          expr = combsimp(expr)
  
      if expr.has(Sum):
          expr = sum_simplify(expr, **kwargs)
  
      if expr.has(Integral):
          expr = expr.xreplace(
              {i: factor_terms(i) for i in expr.atoms(Integral)}
          )
  
      if expr.has(Product):
          expr = product_simplify(expr, **kwargs)
  
      from sympy.physics.units import Quantity
  
      if expr.has(Quantity):
          from sympy.physics.units.util import quantity_simplify
  
          expr = quantity_simplify(expr)
  
      short = shorter(
          powsimp(expr, combine="exp", deep=True), powsimp(expr), expr
      )
      short = shorter(short, cancel(short))
      short = shorter(
          short,
          factor_terms(short),
          expand_power_exp(expand_mul(short)),
      )
      if short.has(
          TrigonometricFunction, HyperbolicFunction, ExpBase, exp
      ):
          short = exptrigsimp(short)
  
      # get rid of hollow 2-arg Mul factorization
      hollow_mul = Transform(
          lambda x: Mul(*x.args),
          lambda x: x.is_Mul
          and len(x.args) == 2
          and x.args[0].is_Number
          and x.args[1].is_Add
          and x.is_commutative,
      )
      expr = short.xreplace(hollow_mul)
  
      numer, denom = expr.as_numer_denom()
      if denom.is_Add:
          n, d = fraction(
              radsimp(1 / denom, symbolic=False, max_terms=1)
          )
          if n is not S.One:
              expr = (numer * n).expand() / d
  
      if expr.could_extract_minus_sign():
          n, d = fraction(expr)
          if d != 0:
              expr = signsimp(-n / (-d))
  
      if measure(expr) > ratio * measure(original_expr):
          expr = original_expr
  
      # restore floats
      if floats and rational is None:
          expr = nfloat(expr, exponent=False)
  
      return done(expr)
  
imports: from collections import defaultdict
  from sympy.concrete.products import Product
  from sympy.concrete.summations import Sum
  from sympy.core import Basic, S, Add, Mul, Pow, Symbol, sympify, expand_func, Function, Dummy, Expr, factor_terms, expand_power_exp, Eq
  from sympy.core.exprtools import factor_nc
  from sympy.core.parameters import global_parameters
  from sympy.core.function import expand_log, count_ops, _mexpand, nfloat, expand_mul, expand
  from sympy.core.numbers import Float, I, pi, Rational
  from sympy.core.relational import Relational
  from sympy.core.rules import Transform
  from sympy.core.sorting import ordered
  from sympy.core.sympify import _sympify
  from sympy.core.traversal import bottom_up, walk
  from sympy.functions import gamma, exp, sqrt, log, exp_polar, re
  from sympy.functions.combinatorial.factorials import CombinatorialFunction
  from sympy.functions.elementary.complexes import unpolarify, Abs, sign
  from sympy.functions.elementary.exponential import ExpBase
  from sympy.functions.elementary.hyperbolic import HyperbolicFunction
  from sympy.functions.elementary.integers import ceiling
  from sympy.functions.elementary.piecewise import Piecewise, piecewise_fold, piecewise_simplify
  from sympy.functions.elementary.trigonometric import TrigonometricFunction
  from sympy.functions.special.bessel import BesselBase, besselj, besseli, besselk, bessely, jn
  from sympy.functions.special.tensor_functions import KroneckerDelta
  from sympy.integrals.integrals import Integral
  from sympy.matrices.expressions import MatrixExpr, MatAdd, MatMul, MatPow, MatrixSymbol
  from sympy.polys import together, cancel, factor
  from sympy.polys.numberfields.minpoly import _is_sum_surds, _minimal_polynomial_sq
  from sympy.simplify.combsimp import combsimp
  from sympy.simplify.cse_opts import sub_pre, sub_post
  from sympy.simplify.hyperexpand import hyperexpand
  from sympy.simplify.powsimp import powsimp
  from sympy.simplify.radsimp import radsimp, fraction, collect_abs
  from sympy.simplify.sqrtdenest import sqrtdenest
  from sympy.simplify.trigsimp import trigsimp, exptrigsimp
  from sympy.utilities.decorator import deprecated
  from sympy.utilities.iterables import has_variety, sift, subsets, iterable
  from sympy.utilities.misc import as_int
  import mpmath
label_desc: Use image node_id 8 for calling a global function with example usage: simplify(expr, ratio, measure, rational, inverse, doit) and returns: done, min, shorter, unknown, _eval_simplify, expr, done, expr, choices, expr, expr