tools/autograd/load_derivatives.py

# Parses derivatives.yaml into autograd functions
#
# Each autograd function is represented by `DifferentiabilityInfo` containing
# a list of `Derivative`. See `torchgen.api.autograd` for the data models.

from __future__ import annotations

import re
from collections import defaultdict
from typing import Any, Counter, Dict, Sequence, Set, Tuple

import yaml

from torchgen.api import cpp
from torchgen.api.autograd import (
    Derivative,
    DifferentiabilityInfo,
    ForwardDerivative,
    SavedAttribute,
)
from torchgen.api.types import (
    BaseCType,
    Binding,
    boolT,
    CppSignatureGroup,
    layoutT,
    longT,
    NamedCType,
    OptionalCType,
    scalarTypeT,
    SpecialArgName,
    stringT,
    symIntArrayRefT,
    SymIntT,
    tensorGeometryT,
    tensorOptionsT,
    typeAndSizeT,
    VectorCType,
)
from torchgen.context import with_native_function
from torchgen.gen import get_grouped_by_view_native_functions, parse_native_yaml
from torchgen.model import (
    AUTOGRAD_KEYS,
    FunctionSchema,
    NativeFunction,
    NativeFunctionsViewGroup,
    OperatorName,
    SchemaKind,
    Type,
    Variant,
)
from torchgen.utils import concatMap, IDENT_REGEX, split_name_params
from torchgen.yaml_utils import YamlLoader


DerivativeRet = Tuple[Dict[FunctionSchema, Dict[str, DifferentiabilityInfo]], Set[str]]

_GLOBAL_LOAD_DERIVATIVE_CACHE: dict[tuple[str, str], DerivativeRet] = {}

_VALID_AUTOGRAD_KEYS = set(AUTOGRAD_KEYS)


# This function directly adds per-dispatchkey derivative entries for {view}_copy variants of each view op.
# Since every {view} and {view}_copy op shares the same derivative formula,
# we generate them here instead of duplicating them in the yaml.
# See Note [Codegen'd {view}_copy Operators]
def add_view_copy_derivatives(
    infos: dict[FunctionSchema, dict[str, DifferentiabilityInfo]],
    view_groups: list[NativeFunctionsViewGroup],
) -> None:
    # Get the map from each view op's name to its corresponding view group
    view_name_to_group: dict[OperatorName, NativeFunctionsViewGroup] = {
        g.view.func.name: g for g in view_groups
    }

    view_infos = {}

    for info_dispatch_dict in infos.values():
        # maybe_view_group only needs to be calculated once per info_dispatch_dict
        maybe_view_group = None
        view_copy_differentiability_infos = {}
        for dispatch_key, info in info_dispatch_dict.items():
            maybe_view_group = view_name_to_group.get(info.func.func.name, None)
            if maybe_view_group is not None and maybe_view_group.view_copy is not None:
                view_copy_info = info.create_view_copy_from_view_derivative(
                    maybe_view_group
                )
                if view_copy_info is not None:
                    fn_schema = view_copy_info.func.func
                    view_copy_differentiability_infos[dispatch_key] = view_copy_info
            else:
                break
        # prefer manually-defined derivatives if any
        if len(view_copy_differentiability_infos) > 0 and fn_schema not in infos:
            assert fn_schema is not None
            view_infos[fn_schema] = view_copy_differentiability_infos

    infos.update(view_infos)


def load_derivatives(
    derivatives_yaml_path: str, native_yaml_path: str, tags_yaml_path: str
) -> DerivativeRet:
    # Do some caching as this is a deterministic function
    global _GLOBAL_LOAD_DERIVATIVE_CACHE
    key = (derivatives_yaml_path, native_yaml_path)
    if key not in _GLOBAL_LOAD_DERIVATIVE_CACHE:
        with open(derivatives_yaml_path) as f:
            definitions = yaml.load(f, Loader=YamlLoader)

        funcs = parse_native_yaml(native_yaml_path, tags_yaml_path).native_functions
        # From the parsed native functions, separate out the (generated) view_copy functions,
        # so we can generate derivatives for them separately.
        native_functions_with_view_groups = get_grouped_by_view_native_functions(funcs)
        native_functions = concatMap(
            lambda g: [g]
            if isinstance(g, NativeFunction)
            else list(g.functions(include_copy=True)),
            native_functions_with_view_groups,
        )
        view_groups = [
            g
            for g in native_functions_with_view_groups
            if isinstance(g, NativeFunctionsViewGroup)
        ]

        # What's the difference between function schema v.s. signature?
        # function schema is the complete declaration including mutability annotation / default value and etc.
        # signature is the canonical schema for a group of functions (in-place/out/functional variants)
        # that are semantically related.
        functions_by_signature: dict[
            FunctionSchema, list[NativeFunction]
        ] = defaultdict(list)
        functions_by_schema: dict[str, NativeFunction] = {}
        for function in native_functions:
            functions_by_signature[function.func.signature()].append(function)
            assert str(function.func) not in functions_by_schema
            functions_by_schema[str(function.func)] = function

        # Keep track of how many of which ops we've seen so we can
        # disambiguate them with a numeric suffix.
        op_counter = Counter[str]()

        # infos is a dict that maps FunctionSchema -> a dict of per dispatch key DifferentiabilityInfos
        # this is useful because in tools/autograd/gen_autograd.py:match_differentiability_info
        # we ultimately need to categorize the DifferentiabilityInfos by FunctionSchema
        infos: dict[FunctionSchema, dict[str, DifferentiabilityInfo]] = {}
        used_dispatch_keys: set[str] = set()
        for defn_dict in definitions:
            # Ensure that the old derivatives.yaml schema with no dispatch key can be loaded.
            if "dispatch" not in defn_dict:
                specification = defn_dict.pop("name")
                output_differentiability = defn_dict.pop(
                    "output_differentiability", None
                )
                defn_dict = {"name": specification, "dispatch": {"Default": defn_dict}}
                if output_differentiability:
                    defn_dict["output_differentiability"] = output_differentiability
            name, per_dispatch_diffinfos = create_differentiability_info(
                defn_dict,
                functions_by_signature,
                functions_by_schema,
                op_counter,
                used_dispatch_keys,
            )
            infos[name] = per_dispatch_diffinfos

        add_view_copy_derivatives(infos, view_groups)

        # cache both loaded infos as well a a set of all the dispatch_keys/aliases
        # that appear in derivatives.yaml. used_dispatch_keys is useful for generating
        # VariableType.cpp where we need a TORCH_LIBRARY_IMPL for every autograd dispatch key used
        _GLOBAL_LOAD_DERIVATIVE_CACHE[key] = infos, used_dispatch_keys

    return _GLOBAL_LOAD_DERIVATIVE_CACHE[key]


# TODO: Why is this going through CppSignatureGroup, that doesn't make sense...
@with_native_function
def cpp_arguments(f: NativeFunction) -> Sequence[Binding]:
    sigs = CppSignatureGroup.from_native_function(f, method=False)
    if sigs.symint_signature is not None:
        return sigs.symint_signature.arguments()
    else:
        return sigs.signature.arguments()


def create_derivative(
    f: NativeFunction,
    formula: str,
    var_names: tuple[str, ...],
    available_named_gradients: Sequence[str],
) -> Derivative:
    original_formula = formula
    arguments: list[NamedCType] = [
        a.nctype.remove_const_ref() for a in cpp_arguments(f)
    ]

    return_names = tuple(n if n != "self" else "result" for n in cpp.return_names(f))
    return_types = tuple(
        cpp.return_type(r, symint=True).remove_const_ref() for r in f.func.returns
    )

    named_returns = [
        NamedCType(name, type) for name, type in zip(return_names, return_types)
    ]

    formula, saved_inputs = saved_variables(formula, arguments, var_names)
    formula, saved_outputs = saved_variables(formula, named_returns, var_names)

    used_named_gradients = {
        name
        for name in available_named_gradients
        if re.search(IDENT_REGEX.format(name), formula)
    }

    # Check that the referenced derivatives in the formula are in bounds
    for i in used_gradient_indices(formula):
        if i >= len(f.func.returns):
            raise RuntimeError(
                f"Out of bounds grads access: derivative formula for {cpp.name(f.func)} "
                f"used grads[{i}], but the forward only returns {len(f.func.returns)} outputs."
            )

    return Derivative(
        formula=formula,
        original_formula=original_formula,
        var_names=var_names,
        saved_inputs=saved_inputs,
        saved_outputs=saved_outputs,
        named_gradients=used_named_gradients,
    )


def create_forward_derivative(
    f: NativeFunction, formula: str, names: tuple[str, ...]
) -> ForwardDerivative:
    var_names = names
    var_types: tuple[Type, ...] | None = None
    for r in f.func.returns:
        if r.name in var_names:
            if var_types is None:
                var_types = ()
            var_types = var_types + (r.type,)

    # Handle default return names
    if var_types is None:
        if var_names == ("result",):
            assert len(f.func.returns) == 1
            var_types = (f.func.returns[0].type,)
        else:
            for var_name in var_names:
                res = re.findall(r"^result(\d+)$", var_name)
                if len(res) == 1:
                    if var_types is None:
                        var_types = ()
                    arg_idx = int(res[0])
                    var_types = var_types + (f.func.returns[arg_idx].type,)

    assert var_types is not None, "No matching output for forward derivative definition"
    return ForwardDerivative(
        formula=formula,
        var_names=var_names,
        var_types=var_types,
        required_inputs_fw_grad=None,
        required_inputs_primal=None,
        required_original_self_value=False,
        is_reusing_outplace_formula=False,
    )


def postprocess_forward_derivatives(
    f: NativeFunction,
    defn_name: str,
    all_arg_names: list[str],
    derivatives: list[Derivative],
    forward_derivatives: list[ForwardDerivative],
    args_with_derivatives: Sequence[Binding],
) -> list[ForwardDerivative]:
    def find_required_inputs(formula: str, postfix: str) -> tuple[str, ...]:
        is_foreach = f.func.name.name.base.startswith("_foreach_")
        required_inputs = set()
        for arg in args_with_derivatives:
            if (
                arg.type in ("at::TensorList", "const at::ITensorListRef &")
                and not is_foreach
            ):
                # The functions taking TensorList handle everything internally
                continue
            arg_name = arg.name

            found = re.search(IDENT_REGEX.format(arg_name), formula)
            if found:
                raise RuntimeError(
                    f"The forward formula for {defn_name} is using the base name of the {arg_name} "
                    f"argument which is ambiguous. You should use {arg_name}_p to access the primal "
                    f"value and {arg_name}_t to access the tangent."
                )

            found = re.search(IDENT_REGEX.format(arg_name + postfix), formula)
            if found:
                required_inputs.add(arg_name)

        return tuple(required_inputs)

    updated_derivatives: list[ForwardDerivative] = []

    for defn in forward_derivatives:
        formula = defn.formula
        required_inputs_tangent = find_required_inputs(formula, "_t")
        if formula == "auto_element_wise":
            assert (
                f.func.kind() != SchemaKind.inplace
            ), f"Cannot use auto_element_wise with {f.func.name} because it is an in-place variant"
            if (
                (not len(args_with_derivatives) == 1)
                or len(forward_derivatives) > 1
                or len(forward_derivatives[0].var_names) > 1
            ):
                raise RuntimeError(
                    f"Derivative definition of {defn_name} in derivatives.yaml defines the "
                    "forward definition of gradient as element_wise but this only "
                    "works for functions with a single differentiable input and a "
                    "single differentiable output."
                )
            if not len(derivatives) == 1:
                raise RuntimeError(
                    f"Derivative definition of {defn_name} in derivatives.yaml defines the "
                    "forward definition of gradient as element_wise but it does not "
                    "defines the gradient formula for its argument which is required."
                )
            # This transformation is based on the observation that for element-wise functions, the Jacobian
            # matrix is diagonal and thus doing J * v is the same as (v^T J)^T (in practice, we ignore the transpositions)
            # For the complex case, we use hermitian transpose and get (v.conj() J).conj()
            # So here we are going to re-use the backward formula and replace two things:
            # 1) all occurrences of "grad" with "foo_t.conj()", where foo is the name of the unique differentiable input.
            # 2) all usage of an original input "foo" with its primal value "foo_p".
            # 3) conjugate the final result
            # For example, for abs, the backward formula is:
            #   grad * self.sgn()
            # And this function generates a forward formula that is:
            #   (self_t.conj() * self_p.sgn()).conj()

            backward_formula = derivatives[0].original_formula
            input_name = args_with_derivatives[0].name

            # Do replacement 1) of the grad
            def repl(m: Any) -> str:
                return f"{m.group(1)}{input_name}_t.conj(){m.group(2)}"

            fw_formula = re.sub(IDENT_REGEX.format("grad"), repl, backward_formula)

            # Do replacement 2) of the input variables
            for arg in args_with_derivatives:
                arg_name = arg.name

                def repl(m: Any) -> str:
                    return f"{m.group(1)}{arg_name}_p{m.group(2)}"

                fw_formula = re.sub(IDENT_REGEX.format(arg_name), repl, fw_formula)

            # Do the final conjugate 3)
            fw_formula = f"({fw_formula}).conj()"

            # Since there is a single differentiable inputs and we necessarily need its tangent we can
            # simply require all differentiable input's tangent.
            required_inputs_tangent = tuple(all_arg_names)
            formula = fw_formula
        elif formula == "auto_linear":
            if (
                len(forward_derivatives) > 1
                or len(forward_derivatives[0].var_names) > 1
            ):
                raise RuntimeError(
                    f"Derivative definition of {defn_name} in derivatives.yaml defines the "
                    "forward definition of gradient as linear but this only works "
                    "for functions with a single differentiable output."
                )
            # This transformation is based on the observation that linear functions can be written as:
            #   y = f(x) = A * x
            # For some matrix A and the Jacobian of the function f is also A.
            # So doing J * v = A * v = f(v).
            # Hence to do the jvp, we simply need to evaluate the function at the point v instead of x.
            # We do this by calling the forward again by replacing any occurrence of the differentiable
            # input "foo" by it's tangent "foo_t".
            # Note that multiple inputs are not a problem as long as the function is truly linear wrt to
            # the vector where all the differentiable inputs are stacked.

            diff_arg_names = [arg.name for arg in args_with_derivatives]
            assert len(diff_arg_names) > 0

            # Do replacement of input variables
            new_args = []
            for arg_name in all_arg_names:
                if arg_name in diff_arg_names:
                    arg_name = arg_name + "_t"
                new_args.append(arg_name)

            # TODO we are trolling
            if f.func.has_symint():
                defn_name += "_symint"

            # Call into the forward again. We need two cases here to handle both Tensor methods and at:: functions.
            if Variant.function in f.variants:
                fw_formula = f"at::{defn_name}({', '.join(new_args)})"
            else:
                assert Variant.method in f.variants
                fw_formula = f"{new_args[0]}.{defn_name}({', '.join(new_args[1:])})"

            # All of the input tangents are always used so all of them are required here.
            required_inputs_tangent = tuple(diff_arg_names)
            formula = fw_formula

        # At this point, the formula is final and is not modified anymore.

        # During forward formula, we use the primal instead of the input Tensors.
        # This call inspects the formula to find for which input's primal are used.
        required_inputs_primal = find_required_inputs(formula, "_p")

        updated_derivatives.append(
            ForwardDerivative(
                formula=formula,
                var_names=defn.var_names,
                var_types=defn.var_types,
                required_inputs_fw_grad=required_inputs_tangent,
                required_inputs_primal=required_inputs_primal,
                required_original_self_value=False,
                is_reusing_outplace_formula=False,
            )
        )

    return updated_derivatives


def is_forward_derivative_definition(
    all_arg_names: list[str], names: tuple[str, ...]
) -> bool:
    for name in names:
        return name not in all_arg_names
    raise RuntimeError("Expected `names` to be non-empty")


def create_differentiability_info(
    defn_dict: dict[Any, Any],
    functions_by_signature: dict[FunctionSchema, list[NativeFunction]],
    functions_by_schema: dict[str, NativeFunction],
    op_counter: Counter[str],
    used_dispatch_keys: set[str],
) -> tuple[FunctionSchema, dict[str, DifferentiabilityInfo]]:
    """Processes a single entry `defn` in derivatives.yaml"""

    def canonical_function(
        functions: Sequence[NativeFunction], name: str
    ) -> NativeFunction:
        for f in functions:
            if (
                not f.func.is_functional_fn()
                and not f.func.is_out_fn()
                and name == str(f.func.name.name)
            ):
                return f
        # some functions only have in-place variants
        assert name + "_" == cpp.name(functions[0].func)
        return functions[0]

    def split_names(raw_names: str) -> tuple[str, ...]:
        """Given "foo, bar", return ["foo", "bar"]."""
        return tuple(x.strip() for x in raw_names.split(","))

    def check_grad_usage(defn_name: str, derivatives: Sequence[Derivative]) -> None:
        """
        Check for some subtle mistakes one might make when writing derivatives.
        These mistakes will compile, but will be latent until a function is
        used with double backwards.
        """

        uses_grad = False  # true if any derivative uses "grad"
        num_grads_uses = 0  # count of uses of "grads" or "grads[INDEX]"
        uses_named_grads = False  # true if any derivative uses "grad_{name}"
        used_grads_indices: list[int] = []  # which indices of grads are used
        for d in derivatives:
            formula = d.formula
            uses_grad = uses_grad or bool(
                re.findall(IDENT_REGEX.format("grad"), formula)
            )
            num_grads_uses += len(re.findall(IDENT_REGEX.format("grads"), formula))
            uses_named_grads = uses_named_grads or bool(d.named_gradients)
            used_grads_indices.extend(used_gradient_indices(formula))
        # This is a basic sanity check: the number of places we see
        # "grads" should be no fewer than the number of indices we see
        # inside "grads". They may not be equal because we may use
        # "grads" without an index.
        assert num_grads_uses >= len(used_grads_indices)
        # Thus if the number is equal, every use of grads is also
        # indexed.
        only_used_grads_indices = num_grads_uses == len(used_grads_indices)

        if uses_grad and num_grads_uses > 0:
            raise RuntimeError(
                f"Derivative definition of {defn_name} in derivatives.yaml illegally "
                "mixes use of 'grad' and 'grads'. Consider replacing "
                "occurrences of 'grad' with 'grads[0]'"
            )

        if only_used_grads_indices and set(used_grads_indices) == {0}:
            raise RuntimeError(
                f"Derivative definition of {defn_name} in derivatives.yaml solely "
                "refers to 'grads[0]'.  If the first output is indeed the "
                "only differentiable output, replace 'grads[0]' with 'grad'; "
                "otherwise, there is a likely error in your derivatives "
                "declaration."
            )

        if uses_named_grads and (uses_grad or num_grads_uses > 0):
            raise RuntimeError(
                f"Derivative definition of {defn_name} in derivatives.yaml illegally "
                'mixes use of "grad_RETURN_NAME" and "grad" or "grads[x]". Use '
                "only one method for identifying gradients."
            )

    @with_native_function
    def set_up_derivatives(
        f: NativeFunction,
    ) -> tuple[
        Sequence[Derivative],
        Sequence[ForwardDerivative],
        Sequence[Binding],
        Sequence[str],
        Sequence[str],
    ]:
        # Set up the derivative information
        derivatives: list[Derivative] = []
        forward_derivatives: list[ForwardDerivative] = []
        non_differentiable_arg_names: list[str] = []
        args_with_derivatives_set: set[str] = set()

        all_arg_names = [a.name for a in cpp_arguments(f)]
        all_ret_names = [
            r.name for r in f.func.returns
        ]  # only used for the assert below
        # output_differentiability is captured from the enclosed
        # scope. Don't modify it.
        #
        # If it is not present, then no output is explicitly
        # undifferentiable.
        #
        # It may be present and shorter than the length of return
        # values. If that's the case, any return value that does not
        # have a corresponding entry is considered not differentiable.
        differentiability = output_differentiability or [True] * len(f.func.returns)
        # A return is available as a named gradient ...
        available_named_gradients = [
            f"grad_{ret.name}"
            for ret, differentiable in zip(f.func.returns, differentiability)
            # if it has not been explicitly made undifferentiable
            if differentiable
            # and if it has a name
            and ret.name is not None
            # and if its type is differentiable
            and ret.type.is_tensor_like()
        ]

        for raw_names in sorted(defn.keys()):
            formula = defn[raw_names]
            names = split_names(raw_names)

            for name in names:
                assert not (name in all_arg_names and name in all_ret_names), (
                    f"While processing the derivative formula for '{f.func.name}' wrt '{name}', "
                    f"expected '{name}' to not be both an input arg and named return. "
                )

            if is_forward_derivative_definition(all_arg_names, names):
                forward_derivatives.append(create_forward_derivative(f, formula, names))
            else:
                if formula.lower().strip() == "non_differentiable":
                    non_differentiable_arg_names += names
                else:
                    derivative = create_derivative(
                        f, formula, names, available_named_gradients
                    )
                    derivatives.append(derivative)
                    args_with_derivatives_set |= set(names)

        overlap = args_with_derivatives_set.intersection(non_differentiable_arg_names)
        if overlap:
            raise RuntimeError(
                f"derivatives definition for {defn} have overlapped non_differentiable "
                f"and differentiable variables: {overlap}"
            )

        # Next, let us determine the list of inputs in order.
        # TODO: do we need eagerly calculate and save it here? Can it be derived
        # from NativeFunction and `derivatives` on callsites instead?
        args_with_derivatives = [
            a for a in cpp_arguments(f) if a.name in args_with_derivatives_set
        ]

        # Postprocess forward derivatives definitions now that we know the differentiable arguments
        forward_derivatives = postprocess_forward_derivatives(
            f,
            defn_name,
            all_arg_names,
            derivatives,
            forward_derivatives,
            args_with_derivatives,
        )

        # Test to see if the use of 'grads' makes sense.
        check_grad_usage(defn_name, derivatives)

        return (
            derivatives,
            forward_derivatives,
            args_with_derivatives,
            non_differentiable_arg_names,
            available_named_gradients,
        )

    # NB: Removes 'name' from defn dictionary
    specification = defn_dict.pop("name")
    defn_name, _ = split_name_params(specification)
    # NB: Removes 'output_differentiability' from defn dictionary
    #     `None` means all differentiable.
    output_differentiability = defn_dict.pop("output_differentiability", None)
    output_differentiability_conditions = None
    if output_differentiability and any(
        isinstance(diff, str) for diff in output_differentiability
    ):
        if len(output_differentiability) != 1:
            raise RuntimeError(
                f"Not supported: for {specification},"
                f"output_differentiability must either be "
                f"List[bool] or a List[str] where each str is a "
                f"condition. In the case where it is a condition, "
                f"we only support single-output functions. "
                f"Please file us an issue. "
            )
        output_differentiability_conditions = output_differentiability
        output_differentiability = [True]

    schema_function = functions_by_schema.get(specification)
    if not schema_function:
        avail = "\n".join(
            k for k, v in functions_by_schema.items() if cpp.name(v.func) == defn_name
        )
        raise RuntimeError(
            f"could not find ATen function for schema: {specification} "
            f".  Available signatures:\n{avail}"
        )

    # now map this to the legacy schema; this isn't technically necessary, but we'd need some logic here
    # to map in-place schemas to the out-of-place variants.
    # TODO: maybe the logic to handle the legacy schema is no longer necessary?
    signature = schema_function.func.signature()
    functions = functions_by_signature[signature]
    if len(functions) == 0:
        avail = "\n".join(
            str(k)
            for k, v in functions_by_signature.items()
            if cpp.name(k) == defn_name
        )
        raise RuntimeError(
            f"could not find ATen function for legacy signature: {signature} "
            f"corresponding to schema {specification}.  Please report a bug to PyTorch. "
            f"Available signatures:\n{avail}"
        )

    canonical = canonical_function(functions, defn_name)
    if "grad_input_mask" in (a.name for a in cpp_arguments(canonical)):
        raise RuntimeError(
            f"Schema for {defn_name} has an argument named grad_input_mask, "
            "but this name would be shadowed by our codegen. "
            "Please use a different name in native_functions.yaml."
        )

    if "result" in (a.name for a in cpp_arguments(canonical)):
        raise RuntimeError(
            f"Schema for {defn_name} has an argument named result, "
            "but this is only allowed for outputs."
            "Please use a different name in native_functions.yaml."
        )

    diffinfo_dict = {}
    for key, defn in defn_dict["dispatch"].items():
        if key != "Default" and key not in _VALID_AUTOGRAD_KEYS:
            raise RuntimeError(
                f"Invalid dispatch key {key} in derivatives.yaml for {specification},"
                f" expected key to be one of {_VALID_AUTOGRAD_KEYS}"
            )
        if key not in used_dispatch_keys:
            used_dispatch_keys.add(key)

        (
            derivatives,
            forward_derivatives,
            args_with_derivatives,
            non_differentiable_arg_names,
            available_named_gradients,
        ) = set_up_derivatives(canonical)

        used_named_gradients: set[str] = set()
        for d in derivatives:
            used_named_gradients |= d.named_gradients

        # only assign an op name if we are actually going to calculate a derivative
        op = None
        if args_with_derivatives:
            op_prefix = _create_op_prefix(defn_name)
            if key != "Default":
                op_prefix = op_prefix + key
            op = f"{op_prefix}{op_counter[op_prefix]}"
            op_counter[op_prefix] += 1

        diffinfo_dict[key] = DifferentiabilityInfo(
            name=defn_name,
            func=canonical,
            op=op,
            derivatives=derivatives,
            forward_derivatives=forward_derivatives,
            all_saved_inputs=dedup_vars(
                [v for d in derivatives for v in d.saved_inputs]
            ),
            all_saved_outputs=dedup_vars(
                [v for d in derivatives for v in d.saved_outputs]
            ),
            available_named_gradients=available_named_gradients,
            used_named_gradients=used_named_gradients,
            args_with_derivatives=args_with_derivatives,
            non_differentiable_arg_names=non_differentiable_arg_names,
            output_differentiability=output_differentiability,
            output_differentiability_conditions=output_differentiability_conditions,
        )

    return canonical.func, diffinfo_dict


GRAD_INDEX_REGEX = r"(?:^|\W)grads\[(\d+)\]"


def used_gradient_indices(formula: str) -> list[int]:
    """Determine a list of gradient indices (the i in grads[i]) that
    are used by the formula.

    >>> used_gradient_indices("foo(grads[0], grads[1])")
    [0, 1]
    """
    return [int(i) for i in re.findall(GRAD_INDEX_REGEX, formula)]


def saved_variables(
    formula: str,
    nctypes: list[NamedCType],
    var_names: tuple[str, ...],
) -> tuple[str, tuple[SavedAttribute, ...]]:
    def stride_expr(name: str) -> str:
        assert var_names == (name,), (
            'Replacement for ".strides()" is currently only supported for single derivatives of the same tensor '
            'that ".strides()" is being called on.'
        )
        return f'strides_or_error({name}, "{name}")'

    REPLACEMENTS: list[tuple[str, dict[str, Any]]] = [
        # replace self.sym_sizes() with self_sym_sizes
        (
            r"{}.sym_sizes\(\)",
            {
                "suffix": "_sym_sizes",
                "nctype": lambda name: NamedCType(name, BaseCType(symIntArrayRefT)),
            },
        ),
        # replace self->sym_sizes() with self_sym_sizes_opt
        (
            r"{}->sym_sizes\(\)",
            {
                "suffix": "_sym_sizes_opt",
                "nctype": lambda name: NamedCType(
                    name, OptionalCType(BaseCType(symIntArrayRefT))
                ),
                "expr": lambda name: f"{name}.has_value() ? std::optional<c10::SymIntArrayRef>({name}->sym_sizes()) : std::nullopt",
            },
        ),
        # replace self.sym_blocksize() with self_sym_blocksize_opt
        (
            r"{}.sym_blocksize\(\)",
            {
                "suffix": "_self_sym_blocksize_opt",
                "nctype": lambda name: NamedCType(
                    name, OptionalCType(BaseCType(symIntArrayRefT))
                ),
                "expr": lambda name: f"at::sparse_csr::getSymIntBlockSize({name})",
            },
        ),
        # replace self.options() with self_options
        (
            r"{}.options\(\)",
            {
                "suffix": "_options",
                "nctype": lambda name: NamedCType(name, BaseCType(tensorOptionsT)),
            },
        ),
        # replace zeros_like(self) with self_info
        (
            r"zeros_like\({}\)",
            {
                "suffix": "_info",
                "nctype": lambda name: NamedCType(name, BaseCType(typeAndSizeT)),
                "expr": lambda name: name,  # at save-time
                "res": lambda name: name + "_info.zeros()",  # at eval-time
            },
        ),
        # replace self.sym_size(2) with self_sym_size_2
        (
            r"{}.sym_size\((-?\w+)\)",
            {
                "suffix": lambda m: f"_sym_argsize_{m.groups()[0].replace('-', 'minus_')}",
                "nctype": lambda name: NamedCType(name, BaseCType(SymIntT)),
            },
        ),
        # replace self.numel() with self_numel
        (
            r"{}.numel\(\)",
            {
                "suffix": "_numel",
                "nctype": lambda name: NamedCType(name, BaseCType(longT)),
            },
        ),
        # replace self.sym_numel() with self_sym_numel
        (
            r"{}.sym_numel\(\)",
            {
                "suffix": "_sym_numel",
                "nctype": lambda name: NamedCType(name, BaseCType(SymIntT)),
            },
        ),
        # replace to_args_sizes(self) with self_args_sizes
        (
            r"to_args_sizes\({}\)",
            {
                "suffix": "_args_sizes",
                "nctype": lambda name: NamedCType(
                    name, VectorCType(VectorCType(BaseCType(longT)))
                ),
            },
        ),
        # replace to_args_sizes_symint(self) with self_args_sizes
        (
            r"to_args_sizes_symint\({}\)",
            {
                "suffix": "_args_sizes_symint",
                "nctype": lambda name: NamedCType(
                    name, VectorCType(VectorCType(BaseCType(SymIntT)))
                ),
            },
        ),
        # replace to_args_scalartypes(self) with self_args_scalartypes
        (
            r"to_args_scalartypes\({}\)",
            {
                "suffix": "_args_scalartypes",
                "nctype": lambda name: NamedCType(
                    name, VectorCType(BaseCType(scalarTypeT))
                ),
            },
        ),
        # replace TensorGeometry(self) with self_geometry
        (
            r"TensorGeometry\({}\)",
            {
                "suffix": "_geometry",
                "nctype": lambda name: NamedCType(name, BaseCType(tensorGeometryT)),
            },
        ),
        (
            r"{}.scalar_type\(\)",
            {
                "suffix": "_scalar_type",
                "nctype": lambda name: NamedCType(name, BaseCType(scalarTypeT)),
            },
        ),
        # replace self.dim() with self_dim
        (
            r"{}.dim\(\)",
            {
                "suffix": "_dim",
                "nctype": lambda name: NamedCType(name, BaseCType(longT)),
            },
        ),
        # replace self.sym_strides() with self_sym_strides
        (
            r"{}.sym_strides\(\)",
            {
                "suffix": "_sym_strides",
                "nctype": lambda name: NamedCType(name, BaseCType(symIntArrayRefT)),
                "expr": stride_expr,
            },
        ),
        # replace self.layout() with self_layout
        (
            r"{}.layout\(\)",
            {
                "suffix": "_layout",
                "nctype": lambda name: NamedCType(name, BaseCType(layoutT)),
            },
        ),
        # replace self.is_conj() with self_conjugate
        (
            r"{}.is_conj\(\)",
            {
                "suffix": "_conjugate",
                "nctype": lambda name: NamedCType(name, BaseCType(boolT)),
            },
        ),
    ]

    # find which arguments need to be saved
    saved: list[SavedAttribute] = []

    if ".sizes()" in formula or "->sizes()" in formula:
        raise RuntimeError(
            ".sizes() is not supported in derivative formulas. Instead, please use the SymInt version,"
            + f".sym_sizes(), which returned a c10::SymIntArrayRef. formula={formula}"
        )
    if re.search(r"\.size\([-]?\d+\)", formula) or re.search(
        r"->size\([-]?\d+\)", formula
    ):
        raise RuntimeError(
            ".size(int) is not supported in derivative formulas. Instead, please use the SymInt version,"
            + f".sym_size(int), which returned a c10::SymIntArrayRef. formula={formula}"
        )
    if ".strides()" in formula or "->strides()" in formula:
        raise RuntimeError(
            ".strides() is not supported in derivative formulas. Instead, please use the SymInt version,"
            + f".sym_strides(), which returned a c10::SymIntArrayRef. formula={formula}"
        )
    for nctype in nctypes:
        name = (
            nctype.name.name if isinstance(nctype.name, SpecialArgName) else nctype.name
        )
        # First search the formula for expressions which can be evaluated
        # when the autograd Function is created to avoid saving variables
        for regex, info in REPLACEMENTS:

            def repl(m: re.Match[str]) -> str:
                suffix: str = (
                    info["suffix"](m) if callable(info["suffix"]) else info["suffix"]
                )
                expr: str = info["expr"](name) if "expr" in info else m.group(0)
                saved.append(
                    SavedAttribute(
                        nctype=info["nctype"](name + suffix),
                        expr=expr,
                    )
                )
                if "res" in info:
                    replacement: str = info["res"](name)
                    return replacement
                return name + suffix

            formula = re.sub(regex.format(name), repl, formula)

        # std::optional<std::string> types stored in Backward nodes must be
        # converted to std::optional<std::string_view> before being passed into
        # the backward function
        if nctype.type == OptionalCType(BaseCType(stringT)):
            formula = re.sub(
                rf"\b{name}\b",
                f"{name}.has_value() ? std::optional<c10::string_view>({name}.value()) : std::nullopt",
                formula,
            )

        # Find any variables which remain in the formula and save them
        if re.search(IDENT_REGEX.format(name), formula):
            saved.append(
                SavedAttribute(
                    nctype=nctype,
                    expr=name,
                )
            )

    return formula, tuple(saved)


def _create_op_prefix(name: str) -> str:
    """Takes a native function name converts to a op prefix name.

    Note that the "name" parameter must be the native function name
    without the optional variant suffix, so "add" instead of
    "add.out".

    OP names correspond to classes, hence the change to title case.

    Example::
    >>> _create_op_prefix('add')
    'AddBackward'
    """
    camel_case = "".join([p.title() for p in name.split("_")])
    return (camel_case + "Backward").replace("ForwardBackward", "Backward")


def dedup_vars(vars: Sequence[SavedAttribute]) -> Sequence[SavedAttribute]:
    seen: set[str] = set()
    saved: list[SavedAttribute] = []
    for var in vars:
        name = (
            var.nctype.name.name
            if isinstance(var.nctype.name, SpecialArgName)
            else var.nctype.name
        )
        if name in seen:
            continue
        seen.add(name)
        saved.append(var)
    return saved