code: def mixed_sequential_sum_product(
      sum_op, prod_op, trans, time, step, num_segments=None
  ):
      """
      For a funsor ``trans`` with dimensions ``time``, ``prev`` and ``curr``,
      computes a recursion equivalent to::
  
          tail_time = 1 + arange("time", trans.inputs["time"].size - 1)
          tail = sequential_sum_product(sum_op, prod_op,
                                        trans(time=tail_time),
                                        time, {"prev": "curr"})
          return prod_op(trans(time=0)(curr="drop"), tail(prev="drop")) \
             .reduce(sum_op, "drop")
  
      by mixing parallel and serial scan algorithms over ``num_segments`` segments.
  
      :param ~funsor.ops.AssociativeOp sum_op: A semiring sum operation.
      :param ~funsor.ops.AssociativeOp prod_op: A semiring product operation.
      :param ~funsor.terms.Funsor trans: A transition funsor.
      :param Variable time: The time input dimension.
      :param dict step: A dict mapping previous variables to current variables.
          This can contain multiple pairs of prev->curr variable names.
      :param int num_segments: number of segments for the first stage
      """
      time_var, time, duration = time, time.name, time.output.size
      num_segments = duration if num_segments is None else num_segments
      assert num_segments > 0 and duration > 0
  
      # handle unevenly sized segments by chopping off the final segment and calling mixed_sequential_sum_product again
      if (
          duration % num_segments
          and duration - duration % num_segments > 0
      ):
          remainder = trans(
              **{
                  time: Slice(
                      time,
                      duration - duration % num_segments,
                      duration,
                      1,
                      duration,
                  )
              }
          )
          initial = trans(
              **{
                  time: Slice(
                      time,
                      0,
                      duration - duration % num_segments,
                      1,
                      duration,
                  )
              }
          )
          initial_eliminated = mixed_sequential_sum_product(
              sum_op,
              prod_op,
              initial,
              Variable(time, Bint[duration - duration % num_segments]),
              step,
              num_segments=num_segments,
          )
          final = Cat(
              time, (Stack(time, (initial_eliminated,)), remainder)
          )
          final_eliminated = naive_sequential_sum_product(
              sum_op,
              prod_op,
              final,
              Variable(time, Bint[1 + duration % num_segments]),
              step,
          )
          return final_eliminated
  
      # handle degenerate cases that reduce to a single stage
      if num_segments == 1:
          return naive_sequential_sum_product(
              sum_op, prod_op, trans, time_var, step
          )
      if num_segments >= duration:
          return sequential_sum_product(
              sum_op, prod_op, trans, time_var, step
          )
  
      # break trans into num_segments segments of equal length
      segment_length = duration // num_segments
      segments = [
          trans(
              **{
                  time: Slice(
                      time,
                      i * segment_length,
                      (i + 1) * segment_length,
                      1,
                      duration,
                  )
              }
          )
          for i in range(num_segments)
      ]
  
      first_stage_result = naive_sequential_sum_product(
          sum_op,
          prod_op,
          Stack(time + "__SEGMENTED", tuple(segments)),
          Variable(time, Bint[segment_length]),
          step,
      )
  
      second_stage_result = sequential_sum_product(
          sum_op,
          prod_op,
          first_stage_result,
          Variable(time + "__SEGMENTED", Bint[num_segments]),
          step,
      )
  
      return second_stage_result
  
imports: import re
  from collections import OrderedDict, defaultdict
  from functools import reduce
  from math import gcd
  import funsor
  import funsor.ops
  from funsor.cnf import Contraction
  from funsor.domains import Bint, Reals
  from funsor.interpreter import gensym
  from funsor.ops import PRODUCT_TO_POWER, UNITS, AssociativeOp
  from funsor.terms import Cat, Funsor, FunsorMeta, Number, Slice, Stack, Subs, Variable, eager, substitute, to_funsor
  from funsor.util import quote
label_desc: Use image node_id 10 for calling a global function with example usage: mixed_sequential_sum_product(sum_op, prod_op, trans, time, step, num_segments) and returns: second_stage_result, final_eliminated, naive_sequential_sum_product, sequential_sum_product