docstr: Simplify indexing expression with knowledge of the ranges of
  iteration variables.
code: def _simplify_with_ranges(
      self, expr: Expr, var_ranges: VarRanges
  ) -> Expr:
      """
      Simplify indexing expression with knowledge of the ranges of
      iteration variables.
      """
  
      expr = join_dimensions(self.simplify(expr))
      original_expr = expr
  
      def remove_zero_terms(base, divisor):
          """Symbols smaller than the divisor are zero"""
          for v in base.free_symbols:
              if v in var_ranges:
                  # var smaller than divisor can be removed
                  # if the rest is guaranteed to be multiple of divisor
                  rest = sympy.Wild("_rest", exclude=[v])
                  m = base.match(v + rest)
                  if m and v not in m[rest].free_symbols:
                      gcd = sympy.gcd(m[rest], divisor)
                      if gcd == divisor:
                          if self.statically_known_leq(
                              var_ranges[v], divisor
                          ):
                              base = m[rest]
          return base
  
      def visit_indexing_div(base, divisor):
          return FloorDiv(remove_zero_terms(base, divisor), divisor)
  
      def visit_modular_indexing(base, divisor, modulus):
          base = remove_zero_terms(base, divisor)
          base_pos = True
          if isinstance(base, ModularIndexing):
              # for modular indexing, biggest values from the ranges don't necessarily result in
              # the biggest result, the biggest result is modulus - 1
              base_s = base.args[2] - 1
          elif not base.has(ModularIndexing):
              # actual iteration range is to size-1
              iter_ranges_zero = {k: 0 for k, v in var_ranges.items()}
              base_lowest = sympy_subs(base, iter_ranges_zero)
              if self.statically_known_leq(0, base_lowest):
                  # can't replace with indexing div if base can be negative
                  base_pos = True
              else:
                  base_pos = False
              iter_ranges = {k: v - 1 for k, v in var_ranges.items()}
              base_s = sympy_subs(base, iter_ranges)
          else:
              base_s = base
          if (
              self.statically_known_lt(base_s, modulus * divisor)
              and base_pos
          ):
              return FloorDiv(base, divisor)
          return ModularIndexing(base, divisor, modulus)
  
      if expr.has(ModularIndexing):
          expr = expr.replace(
              ModularIndexing(
                  sympy.Wild("base"),
                  sympy.Wild("divisor"),
                  sympy.Wild("modulus"),
              ),
              visit_modular_indexing,
          )
  
      if expr.has(FloorDiv):
          expr = expr.replace(
              FloorDiv(
                  sympy.Wild("base"),
                  sympy.Wild("divisor"),
              ),
              visit_indexing_div,
          )
  
      if expr != original_expr:
          return self._simplify_with_ranges(expr, var_ranges)
      return expr
  
imports: import functools
  import itertools
  import logging
  from typing import Any, Callable, Dict, Iterable, List, Optional, Set, Tuple, Union
  import sympy
  from sympy import Expr
  from torch.fx.experimental.symbolic_shapes import ShapeEnv
  from torch.utils._sympy.functions import FloorDiv, ModularIndexing
  from torch.utils._sympy.value_ranges import bound_sympy
  from .utils import sympy_subs, sympy_symbol, VarRanges
  from .virtualized import V
label_desc: Use image node_id 5 for calling the SizeVarAllocator obj's underlying member method code with example usage: obj._simplify_with_ranges(expr, var_ranges) and returns: expr, base, FloorDiv, ModularIndexing, self, FloorDiv