code: def constant_renumber(expr, variables=None, newconstants=None):
      r"""
      Renumber arbitrary constants in ``expr`` to use the symbol names as given
      in ``newconstants``. In the process, this reorders expression terms in a
      standard way.
  
      If ``newconstants`` is not provided then the new constant names will be
      ``C1``, ``C2`` etc. Otherwise ``newconstants`` should be an iterable
      giving the new symbols to use for the constants in order.
  
      The ``variables`` argument is a list of non-constant symbols. All other
      free symbols found in ``expr`` are assumed to be constants and will be
      renumbered. If ``variables`` is not given then any numbered symbol
      beginning with ``C`` (e.g. ``C1``) is assumed to be a constant.
  
      Symbols are renumbered based on ``.sort_key()``, so they should be
      numbered roughly in the order that they appear in the final, printed
      expression.  Note that this ordering is based in part on hashes, so it can
      produce different results on different machines.
  
      The structure of this function is very similar to that of
      :py:meth:`~sympy.solvers.ode.constantsimp`.
  
      Examples
      ========
  
      >>> from sympy import symbols
      >>> from sympy.solvers.ode.ode import constant_renumber
      >>> x, C1, C2, C3 = symbols('x,C1:4')
      >>> expr = C3 + C2*x + C1*x**2
      >>> expr
      C1*x**2  + C2*x + C3
      >>> constant_renumber(expr)
      C1 + C2*x + C3*x**2
  
      The ``variables`` argument specifies which are constants so that the
      other symbols will not be renumbered:
  
      >>> constant_renumber(expr, [C1, x])
      C1*x**2  + C2 + C3*x
  
      The ``newconstants`` argument is used to specify what symbols to use when
      replacing the constants:
  
      >>> constant_renumber(expr, [x], newconstants=symbols('E1:4'))
      E1 + E2*x + E3*x**2
  
      """
  
      # System of expressions
      if isinstance(expr, (set, list, tuple)):
          return type(expr)(
              constant_renumber(
                  Tuple(*expr),
                  variables=variables,
                  newconstants=newconstants,
              )
          )
  
      # Symbols in solution but not ODE are constants
      if variables is not None:
          variables = set(variables)
          free_symbols = expr.free_symbols
          constantsymbols = list(free_symbols - variables)
      # Any Cn is a constant...
      else:
          variables = set()
          isconstant = lambda s: s.startswith("C") and s[1:].isdigit()
          constantsymbols = [
              sym for sym in expr.free_symbols if isconstant(sym.name)
          ]
  
      # Find new constants checking that they aren't already in the ODE
      if newconstants is None:
          iter_constants = numbered_symbols(
              start=1, prefix="C", exclude=variables
          )
      else:
          iter_constants = (
              sym for sym in newconstants if sym not in variables
          )
  
      constants_found = []
  
      # make a mapping to send all constantsymbols to S.One and use
      # that to make sure that term ordering is not dependent on
      # the indexed value of C
      C_1 = [(ci, S.One) for ci in constantsymbols]
      sort_key = lambda arg: default_sort_key(arg.subs(C_1))
  
      def _constant_renumber(expr):
          r"""
          We need to have an internal recursive function
          """
  
          # For system of expressions
          if isinstance(expr, Tuple):
              renumbered = [_constant_renumber(e) for e in expr]
              return Tuple(*renumbered)
  
          if isinstance(expr, Equality):
              return Eq(
                  _constant_renumber(expr.lhs),
                  _constant_renumber(expr.rhs),
              )
  
          if (
              type(expr) not in (Mul, Add, Pow)
              and not expr.is_Function
              and not expr.has(*constantsymbols)
          ):
              # Base case, as above.  Hope there aren't constants inside
              # of some other class, because they won't be renumbered.
              return expr
          elif expr.is_Piecewise:
              return expr
          elif expr in constantsymbols:
              if expr not in constants_found:
                  constants_found.append(expr)
              return expr
          elif expr.is_Function or expr.is_Pow:
              return expr.func(
                  *[_constant_renumber(x) for x in expr.args]
              )
          else:
              sortedargs = list(expr.args)
              sortedargs.sort(key=sort_key)
              return expr.func(
                  *[_constant_renumber(x) for x in sortedargs]
              )
  
      expr = _constant_renumber(expr)
  
      # Don't renumber symbols present in the ODE.
      constants_found = [
          c for c in constants_found if c not in variables
      ]
  
      # Renumbering happens here
      subs_dict = dict(zip(constants_found, iter_constants))
      expr = expr.subs(subs_dict, simultaneous=True)
  
      return expr
  
imports: from sympy.core import Add, S, Mul, Pow, oo
  from sympy.core.containers import Tuple
  from sympy.core.expr import AtomicExpr, Expr
  from sympy.core.function import Function, Derivative, AppliedUndef, diff, expand, expand_mul, Subs
  from sympy.core.multidimensional import vectorize
  from sympy.core.numbers import nan, zoo, Number
  from sympy.core.relational import Equality, Eq
  from sympy.core.sorting import default_sort_key, ordered
  from sympy.core.symbol import Symbol, Wild, Dummy, symbols
  from sympy.core.sympify import sympify
  from sympy.core.traversal import preorder_traversal
  from sympy.logic.boolalg import BooleanAtom, BooleanTrue, BooleanFalse
  from sympy.functions import exp, log, sqrt
  from sympy.functions.combinatorial.factorials import factorial
  from sympy.integrals.integrals import Integral
  from sympy.polys import Poly, terms_gcd, PolynomialError, lcm
  from sympy.polys.polytools import cancel
  from sympy.series import Order
  from sympy.series.series import series
  from sympy.simplify import collect, logcombine, powsimp, separatevars, simplify, cse
  from sympy.simplify.radsimp import collect_const
  from sympy.solvers import checksol, solve
  from sympy.utilities import numbered_symbols
  from sympy.utilities.iterables import uniq, sift, iterable
  from sympy.solvers.deutils import _preprocess, ode_order, _desolve
  from .single import SingleODEProblem, SingleODESolver, solver_map
label_desc: Use image node_id 19 for calling a global function with example usage: constant_renumber(expr, variables, newconstants) and returns: expr, type, Tuple, Eq, expr, expr, expr, expr, expr