code: def probe_den(nd):
      den = rand_digits(nd)
      count = 0
      for nn in range(nd, nd + 3000):
          num = rand_digits(nn)
          t0 = now()
          e1, e2 = divmod(num, den)
          t1 = now()
          f1, f2 = divmod_fast(num, den)
          t2 = now()
          s1 = t1 - t0
          s2 = t2 - t1
          assert e1 == f1
          assert e2 == f2
          if s2 < s1:
              count += 1
              if count >= 3:
                  print(
                      "for",
                      nd,
                      "denom digits,",
                      nn - nd,
                      "extra num digits is enough",
                  )
                  break
          else:
              count = 0
      else:
          print("for", nd, "denom digits, no num seems big enough")
  
imports: from random import randrange
  from time import perf_counter
  from _pylong import int_divmod
label_desc: Use image node_id 2 for calling a global function with example usage: probe_den(nd) without return types