code: def dotprodsimp(expr, withsimp=False):
      """Simplification for a sum of products targeted at the kind of blowup that
      occurs during summation of products. Intended to reduce expression blowup
      during matrix multiplication or other similar operations. Only works with
      algebraic expressions and does not recurse into non.
  
      Parameters
      ==========
  
      withsimp : bool, optional
          Specifies whether a flag should be returned along with the expression
          to indicate roughly whether simplification was successful. It is used
          in ``MatrixArithmetic._eval_pow_by_recursion`` to avoid attempting to
          simplify an expression repetitively which does not simplify.
      """
  
      def count_ops_alg(expr):
          """Optimized count algebraic operations with no recursion into
          non-algebraic args that ``core.function.count_ops`` does. Also returns
          whether rational functions may be present according to negative
          exponents of powers or non-number fractions.
  
          Returns
          =======
  
          ops, ratfunc : int, bool
              ``ops`` is the number of algebraic operations starting at the top
              level expression (not recursing into non-alg children). ``ratfunc``
              specifies whether the expression MAY contain rational functions
              which ``cancel`` MIGHT optimize.
          """
  
          ops = 0
          args = [expr]
          ratfunc = False
  
          while args:
              a = args.pop()
  
              if not isinstance(a, Basic):
                  continue
  
              if a.is_Rational:
                  if a is not S.One:  # -1/3 = NEG + DIV
                      ops += bool(a.p < 0) + bool(a.q != 1)
  
              elif a.is_Mul:
                  if a.could_extract_minus_sign():
                      ops += 1
                      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 += 1 + bool(n < 0)
                      args.append(d)  # won't be -Mul but could be Add
  
                  elif d is not S.One:
                      if not d.is_Integer:
                          args.append(d)
                          ratfunc = True
  
                      ops += 1
                      args.append(n)  # could be -Mul
  
                  else:
                      ops += len(a.args) - 1
                      args.extend(a.args)
  
              elif a.is_Add:
                  laargs = len(a.args)
                  negs = 0
  
                  for ai in a.args:
                      if ai.could_extract_minus_sign():
                          negs += 1
                          ai = -ai
                      args.append(ai)
  
                  ops += laargs - (negs != laargs)  # -x - y = NEG + SUB
  
              elif a.is_Pow:
                  ops += 1
                  args.append(a.base)
  
                  if not ratfunc:
                      ratfunc = a.exp.is_negative is not False
  
          return ops, ratfunc
  
      def nonalg_subs_dummies(expr, dummies):
          """Substitute dummy variables for non-algebraic expressions to avoid
          evaluation of non-algebraic terms that ``polys.polytools.cancel`` does.
          """
  
          if not expr.args:
              return expr
  
          if expr.is_Add or expr.is_Mul or expr.is_Pow:
              args = None
  
              for i, a in enumerate(expr.args):
                  c = nonalg_subs_dummies(a, dummies)
  
                  if c is a:
                      continue
  
                  if args is None:
                      args = list(expr.args)
  
                  args[i] = c
  
              if args is None:
                  return expr
  
              return expr.func(*args)
  
          return dummies.setdefault(expr, Dummy())
  
      simplified = False  # doesn't really mean simplified, rather "can simplify again"
  
      if isinstance(expr, Basic) and (
          expr.is_Add or expr.is_Mul or expr.is_Pow
      ):
          expr2 = expr.expand(
              deep=True,
              modulus=None,
              power_base=False,
              power_exp=False,
              mul=True,
              log=False,
              multinomial=True,
              basic=False,
          )
  
          if expr2 != expr:
              expr = expr2
              simplified = True
  
          exprops, ratfunc = count_ops_alg(expr)
  
          if (
              exprops >= 6
          ):  # empirically tested cutoff for expensive simplification
              if ratfunc:
                  dummies = {}
                  expr2 = nonalg_subs_dummies(expr, dummies)
  
                  if (
                      expr2 is expr or count_ops_alg(expr2)[0] >= 6
                  ):  # check again after substitution
                      expr3 = cancel(expr2)
  
                      if expr3 != expr2:
                          expr = expr3.subs(
                              [(d, e) for e, d in dummies.items()]
                          )
                          simplified = True
  
          # very special case: x/(x-1) - 1/(x-1) -> 1
          elif (
              exprops == 5
              and expr.is_Add
              and expr.args[0].is_Mul
              and expr.args[1].is_Mul
              and expr.args[0].args[-1].is_Pow
              and expr.args[1].args[-1].is_Pow
              and expr.args[0].args[-1].exp is S.NegativeOne
              and expr.args[1].args[-1].exp is S.NegativeOne
          ):
              expr2 = together(expr)
              expr2ops = count_ops_alg(expr2)[0]
  
              if expr2ops < exprops:
                  expr = expr2
                  simplified = True
  
          else:
              simplified = True
  
      return (expr, simplified) if withsimp else 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 25 for calling a global function with example usage: dotprodsimp(expr, withsimp) and returns: unknown, ops,  ratfunc, dummies, expr, expr, expr