code: def match_differentiability_info(
      native_functions: List[NativeFunction],
      differentiability_infos: Dict[
          FunctionSchema, Dict[str, DifferentiabilityInfo]
      ],
  ) -> List[NativeFunctionWithDifferentiabilityInfo]:
      """Sets the "derivative" key on declarations to matching autograd function
      In-place functions will use the out-of-place derivative definition if there
      is no in-place specific derivative.
      """
  
      functional_info_by_signature = {
          schema.signature(strip_default=True): info_dict
          for schema, info_dict in differentiability_infos.items()
          if schema.kind() == SchemaKind.functional
      }
      non_functional_info_by_signature = {
          schema.signature(strip_default=True): info_dict
          for schema, info_dict in differentiability_infos.items()
          if schema.kind() != SchemaKind.functional
      }
  
      def find_info(
          f: NativeFunction,
      ) -> Tuple[Optional[Dict[str, DifferentiabilityInfo]], bool]:
          # Don't bother matching info to generated out= variants
          if "generated" in f.tags and f.func.kind() == SchemaKind.out:
              return None, False
  
          # (1) Check for an exact match
          if f.func in differentiability_infos:
              return differentiability_infos[f.func], True
  
          # (2) If no exact match, check if the out-of-place variant
          # of this operator has a match.
          # i.e mul() for mul_() or mul_out()
          # note(crcrpar): Check foreach or not because in-place foreach functions use backward defined for the existing
          # native functions instead of the out-place counterparts.
          f_sig = f.func.signature(strip_default=True)
          if (
              f_sig in functional_info_by_signature
              and not is_foreach_func(f)
          ):
              return functional_info_by_signature[f_sig], False
  
          # (3) Some operators have a derivative explicitly defined for the mutable
          # variant, but get a code-generated out-of-place variant which does *not*
          # come with a derivative formula.
          # For the generated out-of-place variant, use the mutable variant's formula
          # if it exists.
          if (
              "generated" in f.tags
              and f_sig in non_functional_info_by_signature
          ):
              info_dict = non_functional_info_by_signature[f_sig]
              # See https://github.com/pytorch/pytorch/pull/76320/files#r874816389
              assert not any(
                  any(
                      "self" in str(inpt.nctype.name)
                      for inpt in info.all_saved_inputs
                  )
                  for info in info_dict.values()
              ), f"""\
  Attempted to convert a derivative formula for a mutable operator
   to be used by automatically by its functional variant ("{str(f.func)}").
   this is not currently supported (we'd need to fix up the formula in the codegen)."""
              return info_dict, False
  
          # (4) Generate derivative information of foreach functions if none is defined in `derivatives.yaml`
          if is_foreach_func(f):
              assert f.func not in differentiability_infos
              diff_info, is_generated = gen_foreach_derivativeinfo(
                  f,
                  functional_info_by_signature,
                  non_functional_info_by_signature,
              )
              if diff_info is None:
                  return None, False
              # TODO(crcrpar): Avoid hard coding "Default" ideally.
              diff_info_dict = {"Default": diff_info}
              if is_generated:
                  differentiability_infos[f.func] = diff_info_dict
                  functional_info_by_signature[f.func] = diff_info_dict
              return diff_info_dict, is_generated
  
          return None, False
  
      result: List[NativeFunctionWithDifferentiabilityInfo] = []
      for f in native_functions:
          info_dict, is_exact_match = find_info(f)
  
          # Currently, the '.strides()' to 'strides_or_error' replacement does not support
          # 'self' derivatives of an inplace function, so we must check for this case.
          if f.func.kind() == SchemaKind.inplace and (
              info_dict is not None
          ):
              for info in info_dict.values():
                  for derivative in info.derivatives:
                      if "self" in derivative.var_names:
                          for saved_input in derivative.saved_inputs:
                              assert (
                                  "strides_or_error"
                                  not in saved_input.expr
                              ), (
                                  "Calling '.strides()' in the 'self' derivative formula of an "
                                  f"in-place function is not supported: {f.func}"
                              )
  
          if not info_dict:
              result.append(
                  NativeFunctionWithDifferentiabilityInfo(
                      func=f, info=None, fw_derivatives=None
                  )
              )
              continue
  
          fw_derivative_dict: Dict[
              str, Sequence[ForwardDerivative]
          ] = {}
          for key, info in info_dict.items():
              if not info.forward_derivatives:
                  fw_derivative_dict[key] = []
                  continue
  
              forward_derivatives = info.forward_derivatives
  
              # For functions that have a single def for out-of-place and inplace (like abs())
              if f.func.kind() == SchemaKind.inplace:
                  # For inplace functions there is a little bit of work to do:
                  #  1) Validate the formula and make sure the input that is modified in not used:
                  #    - If there is a formula for the inplace variant of the function (is_exact_match == True) then
                  #      we make sure that the original value of the input that is being modified inplace (self_p) is
                  #      not used in the formula. Note that the formula can use "original_self_p" here and that would
                  #      trigger a clone of the original input.
                  #    - If we are re-using the out of place formula (is_exact_match == False) then we replace every
                  #      occurrence of self_p and self_t by original_self_p and original_self_t. These will be
                  #      populated by cloned version of the original input (either the clone done by the backward AD
                  #      logic if self is also used in a backward formula or a special clone that we add).
                  #  2) At this point, there cannot be a self_p in the formula.
                  #  3) Change "result" into "self_p" as by design, in the inplace function codegen, the result is
                  #     simply called self (as it is modified inplace).
                  #  4) Update the required primals data in case it used to contain "result" but should now contain
                  #     "self"
                  #  5) If it is not an exact match, the user formula is not modifying the existing forward grad
                  #     inplace as it should. So add some code that makes sure that we do so if the forward grad
                  #     already exists.
  
                  assert (
                      len(info.forward_derivatives) == 1
                  )  # Only single output inplace should exist
                  fw_info = info.forward_derivatives[0]
                  formula = fw_info.formula
  
                  def replace_self_with_original_self(
                      formula: str, postfix: str
                  ) -> str:
                      def repl(m: Match[str]) -> str:
                          return f"{m.group(1)}original_self{postfix}{m.group(2)}"
  
                      return re.sub(
                          IDENT_REGEX.format(f"self{postfix}"),
                          repl,
                          formula,
                      )
  
                  if re.search(IDENT_REGEX.format("self_p"), formula):
                      if is_exact_match:
                          # For manually defined formulas, don't allow the original value to be used
                          raise RuntimeError(
                              f'The formula for "{f.func.name}" is using the original value of self '
                              "that is being modified inplace. This would lead to wrong forward gradients. "
                              'Please use "result" in the formula only.'
                          )
                      else:
                          # When the original formula is out of place, we save a clone of the primal
                          # value to be able to access this value if needed
                          # replace "self_p"/"self_t" from the formula by "original_self_p"/"original_self_t"
                          formula = replace_self_with_original_self(
                              formula, "_p"
                          )
                          formula = replace_self_with_original_self(
                              formula, "_t"
                          )
  
                  # replace "result" from the formula by "self_p"
                  def repl(m: Match[str]) -> str:
                      return f"{m.group(1)}self_p{m.group(2)}"
  
                  formula = re.sub(
                      IDENT_REGEX.format("result"), repl, formula
                  )
  
                  required_primals = fw_info.required_inputs_primal
                  if re.search(IDENT_REGEX.format("self_p"), formula):
                      required_primals = (
                          required_primals + ("self",)
                          if required_primals
                          else ("self",)
                      )
  
                  if not is_exact_match:
                      # NOTE [In-place forward AD formula Optimization]
                      #
                      # This optimization transforms the formula to directly do inplace, i.e.
                      # instead of self_t.copy_(self_t.op()) we do self_t.op_() when the following are met:
                      #
                      # 1) the formula satisfies the pattern: "self_t.op(*args)"
                      # 2) "op" in (1) needs to be the same as the op the derivative is for
                      #
                      # (2) may seem too strict, but currently the only ops that satisfy (1) also satisfy (2)
                      # If there is a need, we can relax (2) to allow any op that has an in-place variant
                      is_single_method_on_self_t = False
                      directly_do_inplace = False
                      op_name: Optional[str] = None
                      between_parens: Optional[str] = None
                      match = re.fullmatch(
                          r"self_t.([\w]*)\((.*)\)", formula
                      )
                      if match:
                          op_name, between_parens = match.group(
                              1
                          ), match.group(2)
  
                          # We want to...
                          #   Match: self_t.op1(other_p.op2(arg))
                          #   Avoid: self_t.op1(args) + self_t.op2(args)
                          #   Avoid: self_t.op1(other_p.op2(arg)) + self_t.op2(args)
                          def check_parens_nest_level_gt_zero(
                              s: str,
                          ) -> bool:
                              level = 1
                              for ch in s:
                                  if ch == ")":
                                      level -= 1
                                      if level == 0:
                                          return False
                                  if ch == "(":
                                      level += 1
                              return True
  
                          is_single_method_on_self_t = (
                              check_parens_nest_level_gt_zero(
                                  between_parens
                              )
                          )
                          directly_do_inplace = (
                              is_single_method_on_self_t
                              and op_name == info.name
                          )
  
                      if directly_do_inplace:
                          assert op_name is not None
                          assert between_parens is not None
                          formula = f"self_t_raw.defined() ? self_t_raw.{op_name}_({between_parens}) : {formula}"
                      else:
                          # Make sure that the forward grad is modified inplace when the original formula
                          # is out of place
                          formula = f"self_t_raw.defined() ? self_t_raw.copy_({formula}) : {formula}"
  
                  required_original_self_value = bool(
                      re.search(
                          IDENT_REGEX.format("original_self_p"), formula
                      )
                  ) or bool(
                      re.search(
                          IDENT_REGEX.format("original_self_t"), formula
                      )
                  )
  
                  forward_derivatives = [
                      ForwardDerivative(
                          formula=formula,
                          var_names=("self",),
                          var_types=fw_info.var_types,
                          required_inputs_fw_grad=fw_info.required_inputs_fw_grad,
                          required_inputs_primal=required_primals,
                          required_original_self_value=required_original_self_value,
                          is_reusing_outplace_formula=not is_exact_match,
                      ),
                  ]
  
              fw_derivative_dict[key] = forward_derivatives
  
          result.append(
              NativeFunctionWithDifferentiabilityInfo(
                  func=f,
                  info=info_dict,
                  fw_derivatives=fw_derivative_dict,
              )
          )
  
      return result
  
imports: import re
  from dataclasses import dataclass
  from typing import cast, Dict, List, Match, Optional, Sequence, Set, Tuple
  from torchgen import local
  from torchgen.api import cpp
  from torchgen.api.types import BaseCType, Binding, NamedCType, tensorListT
  from torchgen.model import BaseTy, BaseType, FunctionSchema, ListType, NativeFunction, NativeFunctionsViewGroup, SchemaKind, Type
  from torchgen.utils import IDENT_REGEX
label_desc: Use image node_id 8 for calling a global function with example usage: match_differentiability_info(native_functions, differentiability_infos) and returns: result, None,  False, None,  False, differentiability_infos,  True, functional_info_by_signature,  False, info_dict,  False, diff_info_dict,  is_generated, None,  False, re, m, str, m, m, str, postfix, m, True, False