derivatives.yaml

# Defines derivative formulas and Python signatures of methods on Variable
#
# Note about possibly confusing nomenclature: An 'output gradient' is the
# gradient of an output of a forward function. Output gradients are used as
# the inputs to backward functions. `grads` is a vector of output gradients,
# and `grad == grads[0]`, in all the derivative formulas in this file.
# An 'input gradient' is the gradient of an input to a forward function.
# Input gradients are the outputs of backward functions, corresponding to the
# input names included in the derivative formulas defined in this file.
# Also, every time we talk computing "gradient" we actually mean computing
# the vector jacobian product using the given 'output gradient' as the vector.
#
# Each entry consists of:
#   - A 'name', which specifies the ATen name of the function you
#     are defining derivatives for, and an argument specification.
#   - An optional 'dispatch' entry which can be used to specify
#     per-autograd dispatch key derivatives. If this entry is not
#     specified, then the gradient entries will be taken as the
#     default gradients (i.e. registered for every backward dispatch
#     key). (see _test_autograd_multiple_dispatch for an example
#     of how to register separate derivates for different dispatch keys).
#     The list of allowed dispatch keys (in addition to 'Default' which
#     represents the Autograd alias key) is torchgen/model.py:AUTOGRAD_KEYS.
#   - One or more gradients entries, mapping differentiable input
#     names to a formula specifying how to compute its gradient.
#     Note that a single gradient entry can specify the gradient
#     formula for multiple input names, by specifying a key
#     "input1, input2" (see atan2 for an example).
#   - An argument can be flagged as 'non_differentiable'.
#   - Optional entry with key 'output_differentiability' and value a list of the
#     same length as the number of outputs from the forward function. The list
#     should contain only booleans, specifying whether each of the output Tensor
#     is differentiable.
#     If it is not specified for a function that returns multiple elements but
#     uses `grad` instead of `grads[idx]`, then all but the first output will
#     be marked as non-differentiable.
#     If None of the output is differentiable, you can also add the function
#     name to `gen_variable_type.py`'s `DONT_REQUIRE_DERIVATIVE` list.
#
# There are two cases for Tensor and TensorList arguments here:
#   - If that argument is differentiable, in the sense that a gradient with respect
#     to that argument could exist. You should either:
#       - Specify the formula for that gradient
#       - Specify not_implemented("function_name") as a formula to say that this is not
#         implemented yet (but might be in the future and the user can request that on an issue)
#   - If that argument is not differentiable, because it is not a floating point dtype or the
#     function is not differentiable with respect to that argument  for
#     example. You should either:
#       - Do not specify any formula for this argument
#       - Specify explicitly that this argument is "non_differentiable". Note that in this case,
#         we trust you that this argument will never have requires_grad=True and it will be silently
#         ignored if it does.
#
# If a function has out-of-place and in-place variants, then the derivative
# definition for the in-place variant is optional. It will default to the
# definition for the out-of-place variant. Note that _out variants are never
# differentiable.
#
# Gradient expressions are standard C++ expressions operating on ATen
# variables.  In a gradient expression, the following variables/functions
# are in scope:
#
#   - 'grad', the gradient of the output (often spelled grad_output
#     in Python) which we are going to left-multiply.
#
#     When a function returns multiple *differentiable* outputs,
#     you can refer to the gradients of each outputs using 'grads',
#     e.g., 'grads[0]', 'grads[1]'.
#
#     When a function returns multiple *differentiable* outputs that
#     are named, you can refer to the gradients of each outputs using
#     'grad_{name}', e.g., 'grad_x', 'grad_y'.
#
#     When a function returns *one* differentiable output (the
#     first output) and some more nondifferentiable outputs,
#     you MUST refer to the gradient of the differentiable output with
#     'grad' (this case is special-cased in our code generation).
#
#     Note that the number of differentiable outputs can be modified by the
#     'output_differentiability' entry (see above).
#
#     Across a differentiable function's derivatives set, it is not
#     permitted to mix the use of "grad", "grads", and
#     "grad_{name}". You must be consistent for that differentiable
#     function.
#
#   - Any of the input arguments, tensor or non-tensor, including
#     argument names that only appear in Declarations.yaml, e.g. 'output'.
#
#   - 'result', representing the result of evaluating the forward
#     expression for ATen native function declarations. If the forward
#     expression outputs a tuple, use 'resultX' instead to access the
#     X-th entry
#
#   - 'grad_input_mask', a std::array<bool, n>, specifies which input
#     gradients are actually needed.  For example, in the entry
#     `input0, input1: foo(grad_input_mask)`, `grad_input_mask` is a size
#     two array, where `grad_input_mask[0]` is true if `input0` requires
#     grad, and `grad_input_mask[1]` is true if `input1` requires grad.
#
#     (NB: if your function computes gradient for a list of tensors,
#     the `grad_input_mask` will only have a single entry for the list
#     specifying if either zero or at least one tensor from the list requires
#     grad.  If we want to support more fine-grained signalling,
#     we'll need some alternate variable which is not a std::array)
#
#   - 'retain_variables', a bool which is true if a user has specified
#     that saved variables should be retained in case the backwards is
#     run again later.  This allows an optimization where we can
#     destroy saved buffers if we know variables are not going to be retained,
#     e.g., it is used by _cudnn_rnn
#
#   - `wrap_opt_if`, is a 2-argument function that accepts a tensor
#     variable and a boolean condition that dictates whether to save that
#     variable in a graph. The result of this function is `c10::optional<Tensor>`,
#     and it is `::std::nullopt` when the condition evalutes to `false`,
#     otherwise it is the variable wrapped in `c10::optional<Tensor>`.
#     For example, wrap_opt_if(var_0, grad_input_mask[1] || grad_input_mask[2])
#     would mean that `var_0` is saved as long as the second (grad_input_mask[1])
#     or the third (grad_input_mask[2]) argument requires gradients.
#     Another interpretation of this expression would read as `var_0` is needed
#     in the backward computation of the second or the third argument.
#     NOTE: the usage of `var_i.requires_grad()` in the conditional expression
#     is not supported, use `grad_input_mask[i]` instead.
#     NOTE: `wrap_opt_if` could be used to prevent saving redundant variables
#     with multi-output backward formulas.
#     See https://github.com/pytorch/pytorch/issues/97575 for more details
#     on the issue.
#
# If you need a complex expression, e.g., with local variables,
# write a _backward function in torch/csrc/autograd/FunctionsManual.cpp
# and invoke it from here.  By the way, go read
# https://github.com/zdevito/ATen/issues/163; this describes an
# important hazard that occurs when porting backwards from Python to C++
#
# Double backwards gradient expressions can be somewhat confusing;
# the most important thing to remember is: (1) you need to define a
# derivative formula for every input, including inputs named things
# like 'grad_output', and (2) the gradient to multiply with is always
# called 'grad' (even though it really is a grad-grad).
#
# You can also add forward derivative definition by defining a formula for
# a returned value (in general "result" if the name is not specified). This
# formula works the same way as the backward one and advanced implementations
# should also be placed in the FunctionsManual file.
# This formula should compute a single Jacobian vector product using the (primal)
# value of the argument "foo_p", its forward grad "foo_t" and the result of the
# function as "result".
# Note that the forward derivative can be automatically generated in two cases:
#     - if your function is linear (NOT affine or multi-linear), then you can
#       specify so by just using the string "auto_linear" for the formula.
#     - if your function is applied element wise (and has a single input), you
#       can specify so by just using the string "auto_element_wise" for the formula.
#
# Note that to avoid unpacking overhead, functions taking TensorList as inputs
# will always have their forward grad formula called. This function is responsible
# to check if any computation is needed and should return an undefined Tensor when
# there is nothing to do. You can check "cat_forward" for a full example.
#
# NB: There are a number of gradient definitions in here which are bogus
# (implemented using zeros_like).  These gradients are (hopefully) not
# used by our frontend.  You MUST check the frontend code; search for
# OpName.apply to see if it's still using a legacy Python style API.
#
# Note: Returning views.
# The following cases exist:
#     - If a function returns no view, it can have arbitrary outputs.
#     - If a function return at least one Tensor that is a differentiable view
#       of one of its input:
#         - If there is only one differentiable output, this Tensor is marked as a
#           differentiable view. (alias or transpose for example)
#         - If there are more than one differentiable output, by default all the views are
#           marked as differentiable views and created with allow_rebase_history=false.
#           Meaning that any inplace operation on it will raise an error. (unbind for example)
#
#  Notes about undefined output gradients:
#     All backward functions must support all combinations of undefined output
#     gradient Tensors, where `grad[i].defined() == false`. Depending on the
#     number of input and output grads your derivative formula uses, code
#     generation may automatically add some level of undefined grad support,
#     according to these three cases:
#
#       * 1 input grad and 1 output grad:
#           Complete undefined grad support is automatically added, so you
#           shouldn't have to think about it, unless there is a bug in the code
#           generation.
#
#       * 1 input grad and multiple output grads:
#           Undefined grad support is automatically added ONLY in the case where
#           all output grads are undefined. You will have to add explicit support
#           for cases where a subset of output grads is undefined.
#
#       * multiple input grads:
#           No automatic support, so you will need to add it.
#
#     If your derivative formula uses more than one output grad, it is usually
#     preferable to add undefined grad support in the backward function itself
#     (if you're using one), rather than in the derivative formula in this file.
#
#     Undefined Tensors are created with the default constructor `at::Tensor()`.
#     It is an efficient way to represent a Tensor filled with zeros because
#     the Tensor holds no sizing information and no Storage data is allocated.
#     But consequentially, Tensor operations cannot be performed on them.
#     Therefore, your backward function should treat an undefined output grad as
#     a zero, and it needs to be a special case.
#
#     If all output grads are undefined, then it should be correct for the
#     backward function to return undefined input grads. Since we use the chain
#     rule, output grads equal to zero should result in input grads equal to zero,
#     unless there is some rare special case.
#
#     If a subset of output grads is undefined, then it may be acceptable for
#     the backward function to return undefined input grads--it depends on the
#     specific function, so you'll have to determine that yourself. If returning
#     an undefined Tensor is correct for a given input grad, it is also logically
#     correct to return a defined grad full of zeros, but that would not be
#     preferable since it would be less efficient.
#
# NB: The parameter names here MUST be consistent with the parameter names
# in native_functions.yaml
- name: abs(Tensor self) -> Tensor
  self: grad * self.sgn()
  result: handle_r_to_c(result.scalar_type(), self_t.conj() * self_p.sgn())

- name: acos(Tensor self) -> Tensor
  self: grad * -((-self * self + 1).rsqrt()).conj()
  result: auto_element_wise

- name: add.Tensor(Tensor self, Tensor other, *, Scalar alpha=1) -> Tensor
  self: handle_r_to_c(self.scalar_type(), grad)
  other: handle_r_to_c(other.scalar_type(), maybe_multiply(grad, alpha.conj()))
  result: self_t + maybe_multiply(other_t, alpha)

- name: add.Scalar(Tensor self, Scalar other, Scalar alpha=1) -> Tensor
  self: handle_r_to_c(self.scalar_type(), grad)
  result: self_t.clone()

- name: addbmm(Tensor self, Tensor batch1, Tensor batch2, *, Scalar beta=1, Scalar alpha=1) -> Tensor
  self: maybe_multiply(grad, beta.conj())
  batch1: maybe_multiply(grad.unsqueeze(0).expand_symint({ batch1.sym_size(0), batch1.sym_size(1), batch2.sym_size(2) }).bmm(batch2.transpose(1, 2).conj()), alpha.conj())
  batch2: maybe_multiply(batch1.transpose(1, 2).conj().bmm(grad.unsqueeze(0).expand_symint({ batch1.sym_size(0), batch1.sym_size(1), batch2.sym_size(2) })), alpha.conj())
  result: maybe_multiply(self_t, beta) + maybe_multiply(batch1_t.bmm(batch2_p).sum(0), alpha) + maybe_multiply(batch1_p.bmm(batch2_t).sum(0), alpha)

- name: addcdiv(Tensor self, Tensor tensor1, Tensor tensor2, *, Scalar value=1) -> Tensor
  self: handle_r_to_c(self.scalar_type(), grad)
  tensor1: handle_r_to_c(tensor1.scalar_type(), grad * (value / tensor2).conj())
  tensor2: handle_r_to_c(tensor2.scalar_type(), -grad * (value * tensor1 / (tensor2 * tensor2)).conj())
  result: self_t + maybe_multiply(tensor1_t / tensor2_p, value) - maybe_multiply(tensor2_t * (tensor1_p / tensor2_p) / tensor2_p, value)

- name: addcmul(Tensor self, Tensor tensor1, Tensor tensor2, *, Scalar value=1) -> Tensor
  self: handle_r_to_c(self.scalar_type(), grad)
  tensor1: handle_r_to_c(tensor1.scalar_type(), grad * (tensor2 * value).conj())
  tensor2: handle_r_to_c(tensor2.scalar_type(), grad * (tensor1 * value).conj())
  result: self_t + maybe_multiply(tensor1_t * tensor2_p, value) + maybe_multiply(tensor2_t * tensor1_p, value)

- name: addmm(Tensor self, Tensor mat1, Tensor mat2, *, Scalar beta=1, Scalar alpha=1) -> Tensor
  self: maybe_multiply(grad, beta.conj())
  mat1: mm_mat1_backward(grad, mat2, mat1.sym_sizes(), mat1.sym_strides(), mat1.layout(), alpha)
  mat2: mm_mat2_backward(grad, mat1, mat2.sym_sizes(), mat2.sym_strides(), mat2.layout(), alpha)
  result: maybe_multiply(self_t, beta) + maybe_multiply(mat1_t.mm(mat2_p), alpha) + maybe_multiply(mat1_p.mm(mat2_t), alpha)

- name: _sparse_addmm(Tensor self, Tensor mat1, Tensor mat2, *, Scalar beta=1, Scalar alpha=1) -> Tensor
  self: maybe_multiply(grad, beta)
  mat1: mm_mat1_sparse_backward(grad, mat1, mat2, alpha)
  mat2: mm_mat2_backward(grad, mat1, mat2.sym_sizes(), mat2.sym_strides(), mat2.layout(), alpha)

- name: addmv(Tensor self, Tensor mat, Tensor vec, *, Scalar beta=1, Scalar alpha=1) -> Tensor
  self: maybe_multiply(grad, beta.conj())
  mat: maybe_multiply(grad.ger(vec.conj()), alpha.conj())
  vec: maybe_multiply(mat.t().conj().mv(grad), alpha.conj())
  result: maybe_multiply(self_t, beta) + maybe_multiply(mat_t.mv(vec_p), alpha) + maybe_multiply(mat_p.mv(vec_t), alpha)

- name: addr(Tensor self, Tensor vec1, Tensor vec2, *, Scalar beta=1, Scalar alpha=1) -> Tensor
  self: maybe_multiply(grad, beta.conj())
  vec1: maybe_multiply(grad.mv(vec2.conj()), alpha.conj())
  vec2: maybe_multiply(grad.t().mv(vec1.conj()), alpha.conj())
  result: maybe_multiply(self_t, beta) + maybe_multiply(vec1_t.outer(vec2_p), alpha) + maybe_multiply(vec1_p.outer(vec2_t), alpha)

- name: affine_grid_generator(Tensor theta, SymInt[] size, bool align_corners) -> Tensor
  theta: affine_grid_generator_backward_symint(grad, size, align_corners)
  result: auto_linear

- name: alias(Tensor(a) self) -> Tensor(a)
  self: grad
  result: self_t

- name: angle(Tensor self) -> Tensor
  self: angle_backward(grad, self)
  result: handle_r_to_c(result.scalar_type(), angle_backward(self_t.conj(), self_p).conj())

# The four items below are necessary because TensorIterator doesn't work on
# Variables (codegen does not unwrap the input Tensor for all() and any() ).
- name: any(Tensor self) -> Tensor
  output_differentiability: [False]

- name: any.dim(Tensor self, int dim, bool keepdim=False) -> Tensor
  output_differentiability: [False]

- name: any.dims(Tensor self, int[]? dim=None, bool keepdim=False) -> Tensor
  output_differentiability: [False]

- name: _is_all_true(Tensor self) -> Tensor
  self: non_differentiable

- name: _is_any_true(Tensor self) -> Tensor
  self: non_differentiable

- name: all(Tensor self) -> Tensor
  output_differentiability: [False]

- name: all.dim(Tensor self, int dim, bool keepdim=False) -> Tensor
  output_differentiability: [False]

- name: all.dims(Tensor self, int[]? dim=None, bool keepdim=False) -> Tensor
  output_differentiability: [False]

- name: acosh(Tensor self) -> Tensor
# Save one rsqrt in the real case by using that for x real and positive sqrt(x*y) = sqrt(x)*sqrt(y) (not true in the complex case)
  self: "self.is_complex() ? grad * ((self + 1).rsqrt() * (self - 1).rsqrt()).conj() : grad * (self * self - 1).rsqrt()"
  result: auto_element_wise

- name: acosh_(Tensor(a!) self) -> Tensor(a!)
  self: not_implemented("inplace version of acosh")

- name: asinh(Tensor self) -> Tensor
  self: grad * (self.pow(2) + 1).rsqrt().conj()
  result: auto_element_wise

- name: asinh_(Tensor(a!) self) -> Tensor(a!)
  self: not_implemented("inplace version of asinh")

- name: atanh(Tensor self) -> Tensor
  self: grad * 1 / (1 - self.pow(2)).conj()
  result: auto_element_wise

- name: atanh_(Tensor(a!) self) -> Tensor(a!)
  self: not_implemented("inplace version of atanh")

- name: as_strided(Tensor(a) self, SymInt[] size, SymInt[] stride, SymInt? storage_offset=None) -> Tensor(a)
  self: as_strided_backward(grad, TensorGeometry(self), size, stride, storage_offset)
  result: auto_linear

- name: as_strided_(Tensor(a!) self, SymInt[] size, SymInt[] stride, SymInt? storage_offset=None) -> Tensor(a!)
  self: as_strided_backward(grad, TensorGeometry(self), size, stride, storage_offset)
  result: auto_linear

- name: asin(Tensor self) -> Tensor
  self: grad * (-self * self + 1).rsqrt().conj()
  result: auto_element_wise

- name: atan(Tensor self) -> Tensor
  self: grad / (self * self + 1).conj()
  result: auto_element_wise

- name: atan2(Tensor self, Tensor other) -> Tensor
  self, other: atan2_backward(grad, self, other, grad_input_mask)
  result: (-self_p * other_t + other_p * self_t) / (self_p.pow(2) + other_p.pow(2))

- name: baddbmm(Tensor self, Tensor batch1, Tensor batch2, *, Scalar beta=1, Scalar alpha=1) -> Tensor
  self: maybe_multiply(grad, beta.conj())
  batch1: maybe_multiply(grad.bmm(batch2.transpose(1, 2).conj()), alpha.conj())
  batch2: maybe_multiply(batch1.transpose(1, 2).conj().bmm(grad), alpha.conj())
  result: maybe_multiply(self_t, beta) + maybe_multiply(batch1_t.bmm(batch2_p), alpha) + maybe_multiply(batch1_p.bmm(batch2_t), alpha)

- name: bernoulli(Tensor self, *, Generator? generator=None) -> Tensor
  self: zeros_like(grad)
  result: auto_element_wise

- name: bernoulli_.Tensor(Tensor(a!) self, Tensor p, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  p: zeros_like(p)
  result: self_t.zero_()

- name: bernoulli_.float(Tensor(a!) self, float p=0.5, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.zero_()

- name: bmm(Tensor self, Tensor mat2) -> Tensor
  self: grad.bmm(mat2.transpose(1, 2).conj())
  mat2: self.transpose(1, 2).conj().bmm(grad)
  result: self_t.bmm(mat2_p) + self_p.bmm(mat2_t)

- name: matmul(Tensor self, Tensor other) -> Tensor
  self, other: matmul_backward(grad, self, other, grad_input_mask)

- name: cat(Tensor[] tensors, int dim=0) -> Tensor
  tensors: cat_tensors_backward(grad, to_args_sizes_symint(tensors), to_args_scalartypes(tensors), dim)
  result: cat_jvp(tensors, dim)

- name: cauchy_(Tensor(a!) self, float median=0, float sigma=1, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.zero_()

- name: ceil(Tensor self) -> Tensor
  self: zeros_like(grad)
  result: auto_element_wise

- name: cholesky(Tensor self, bool upper=False) -> Tensor
  self: cholesky_backward(grad, upper, result)

- name: chunk(Tensor(a -> *) self, int chunks, int dim=0) -> Tensor(a)[]
  dispatch:
    Default:
      # the default case will use the CompositeImplicitAutograd
      self: not_implemented("chunk")
    AutogradNestedTensor:
      self: chunk_backward_nested(grads, self, chunks, dim)

- name: linalg_cholesky_ex(Tensor self, *, bool upper=False, bool check_errors=False) -> (Tensor L, Tensor info)
  self: cholesky_backward(grad, upper, L)
  L: cholesky_jvp(self_t, L, upper)

- name: cholesky_solve(Tensor self, Tensor input2, bool upper=False) -> Tensor
  self, input2: cholesky_solve_backward(grad, self, input2, result, upper, grad_input_mask)
  result: cholesky_solve_jvp(result, input2_p, input2_t, self_t, upper)

- name: cholesky_inverse(Tensor self, bool upper=False) -> Tensor
  self: cholesky_inverse_backward(grad, self, upper, result)
  result: cholesky_inverse_jvp(self_p, self_t, result, upper)

# For clamp, gradient is not defined at the boundaries. But empirically it's helpful
# to be able to get gradient on min and max, so we return the subgradient 1 for these cases.
- name: clamp.Tensor(Tensor self, Tensor? min=None, Tensor? max=None) -> Tensor
  self: clamp_backward(grad, self, min, max)
  min, max: clamp_backward_min_max(grad, self, min, max, grad_input_mask)
  result: clamp_jvp(self_p, self_t, min_p, min_t, max_p, max_t)

- name: clamp(Tensor self, Scalar? min=None, Scalar? max=None) -> Tensor
  self: clamp_backward(grad, self, min, max)
  result: auto_element_wise

- name: clamp_min(Tensor self, Scalar min) -> Tensor
  self: where(self >= min, grad, at::scalar_tensor(0., grad.options()))
  result: auto_element_wise

- name: clamp_min.Tensor(Tensor self, Tensor min) -> Tensor
  self: where(self >= min, grad, at::scalar_tensor(0., grad.options()))
  min: where(self < min, grad, at::scalar_tensor(0., grad.options()))
  result: where(self_p >= min_p, self_t, min_t)

- name: clamp_max(Tensor self, Scalar max) -> Tensor
  self: where(self <= max, grad, at::scalar_tensor(0., grad.options()))
  result: auto_element_wise

- name: clamp_max.Tensor(Tensor self, Tensor max) -> Tensor
  self: where(self <= max, grad, at::scalar_tensor(0., grad.options()))
  max: where(self > max, grad, at::scalar_tensor(0., grad.options()))
  result: where(self_p <= max_p, self_t, max_t)

- name: clone(Tensor self, *, MemoryFormat? memory_format=None) -> Tensor
  self: grad
  result: auto_linear

- name: _lazy_clone(Tensor self) -> Tensor
  self: grad
  result: auto_linear

- name: _to_copy(Tensor self, *, ScalarType? dtype=None, Layout? layout=None, Device? device=None, bool? pin_memory=None, bool non_blocking=False, MemoryFormat? memory_format=None) -> Tensor
  self: _to_copy_backward(grad, self.options())
  result: _to_copy(self_t, dtype, layout, device, pin_memory, non_blocking, memory_format)
  # The condition is: if dtype is not nullopt, then isDifferentiableType(*dtype)
  # (If dtype IS nullopt, we rely on the regular check that any input requires grad).
  output_differentiability: ["!dtype || isDifferentiableType(*dtype)"]

- name: _coalesce(Tensor self) -> Tensor
  self: grad

- name: complex(Tensor real, Tensor imag) -> Tensor
  real: at::real(grad)
  imag: at::imag(grad)
  result: at::complex(real_t, imag_t)

- name: polar(Tensor abs, Tensor angle) -> Tensor
  abs, angle: polar_backward(grad, result)
  result: at::complex(abs_t*angle_p.cos() - angle_t*abs_p*angle_p.sin(), abs_t*angle_p.sin() + angle_t*abs_p*angle_p.cos())

- name: _conj(Tensor(a) self) -> Tensor(a)
  self: grad.conj()
  result: self_t.conj()

- name: _neg_view(Tensor(a) self) -> Tensor(a)
  self: grad.neg()
  result: self_t._neg_view()

- name: _conj_physical(Tensor self) -> Tensor
  self: grad.conj_physical()
  result: self_t.conj_physical()

- name: conj_physical_(Tensor(a!) self) -> Tensor(a!)
  self: grad.conj_physical()
  result: self_t.conj_physical_()

- name: copysign.Tensor(Tensor self, Tensor other) -> Tensor
  self: copysign_tensor_self_backward(grad, self, result)
  other: zeros_like(other)
  result: copysign_tensor_self_backward(self_t, self_p, result)

- name: copysign.Scalar(Tensor self, Scalar other) -> Tensor
  self: copysign_tensor_self_backward(grad, self, result)
  result: auto_element_wise

- name: cos(Tensor self) -> Tensor
  self: grad * -self.sin().conj()
  result: auto_element_wise

- name: cosh(Tensor self) -> Tensor
  self: grad * self.sinh().conj()
  result: auto_element_wise

- name: count_nonzero.dim_IntList(Tensor self, int[] dim) -> Tensor
  output_differentiability: [False]

- name: count_nonzero(Tensor self, int? dim=None) -> Tensor
  output_differentiability: [False]

- name: linalg_cross(Tensor self, Tensor other, *, int dim=-1) -> Tensor
  self: at::linalg_cross(other.conj(), grad, dim)
  other: at::linalg_cross(grad, self.conj(), dim)
  result: "at::linalg_cross(self_t, other_p, dim) + at::linalg_cross(self_p, other_t, dim)"

- name: logcumsumexp(Tensor self, int dim) -> Tensor
  self: logcumsumexp_backward(grad, self, result, dim)
  result: logcumsumexp_jvp(self_p, self_t, dim)

- name: cumprod(Tensor self, int dim, *, ScalarType? dtype=None) -> Tensor
  self: cumprod_backward(grad.to(self.scalar_type()), self, dim, result)
  result: "cumprod_jvp(self_t, self_p, result, dim).to(dtype.has_value() ? *dtype : self_p.scalar_type())"

- name: cumsum(Tensor self, int dim, *, ScalarType? dtype=None) -> Tensor
  self: cumsum_backward(grad.to(self.scalar_type()), dim)
  result: auto_linear

- name: cummax(Tensor self, int dim) -> (Tensor values, Tensor indices)
  self: cummaxmin_backward(grad, self, indices, dim)
  values: self_t.gather(dim, indices)

- name: cummin(Tensor self, int dim) -> (Tensor values, Tensor indices)
  self: cummaxmin_backward(grad, self, indices, dim)
  values: self_t.gather(dim, indices)

- name: conv_tbc(Tensor self, Tensor weight, Tensor bias, int pad=0) -> Tensor
  self, weight, bias: "grad.defined() ? conv_tbc_backward(grad, self, weight, bias, pad) : std::tuple<Tensor, Tensor, Tensor>()"

- name: _ctc_loss(Tensor log_probs, Tensor targets, int[] input_lengths, int[] target_lengths, int blank=0, bool zero_infinity=False) -> (Tensor, Tensor)
  log_probs: _ctc_loss_backward(grad, log_probs, targets, input_lengths, target_lengths, result0, result1, blank, zero_infinity)

- name: _ctc_loss.Tensor(Tensor log_probs, Tensor targets, Tensor input_lengths, Tensor target_lengths, int blank=0, bool zero_infinity=False) -> (Tensor, Tensor)
  log_probs: _ctc_loss_backward(grad, log_probs, targets, input_lengths, target_lengths, result0, result1, blank, zero_infinity)

- name: deg2rad(Tensor self) -> Tensor
  self: deg2rad_backward(grad)
  result: auto_element_wise

- name: _linalg_det(Tensor A) -> (Tensor result, Tensor LU, Tensor pivots)
  A: linalg_det_backward(grad, result, A, LU, pivots)
  result: linalg_det_jvp(A_t, result, LU, pivots, A_p.is_contiguous() && !A_p.is_complex())
  output_differentiability: [True, False, False]

- name: _linalg_slogdet(Tensor A) -> (Tensor sign, Tensor logabsdet, Tensor LU, Tensor pivots)
  A: slogdet_backward(grad_sign, grad_logabsdet, A, sign, LU, pivots)
  sign, logabsdet: slogdet_jvp(LU, pivots, A_t, sign, A_p.is_contiguous() && !A_p.is_complex())
  output_differentiability: [True, True, False, False]

- name: block_diag(Tensor[] tensors) -> Tensor
  tensors: block_diag_backward(grad, to_args_sizes(tensors), to_args_scalartypes(tensors))
  result: block_diag_jvp(tensors)

- name: diag_embed(Tensor self, int offset=0, int dim1=-2, int dim2=-1) -> Tensor
  self: grad.diagonal(offset, dim1, dim2)
  result: auto_linear

- name: diagonal(Tensor(a) self, int offset=0, int dim1=0, int dim2=1) -> Tensor(a)
  self: diagonal_backward_symint(grad, self.sym_sizes(), offset, dim1, dim2)
  result: auto_linear

- name: diagonal_backward(Tensor grad_output, SymInt[] input_sizes, int offset, int dim1, int dim2) -> Tensor
  grad_output: grad.diagonal(offset, dim1, dim2)
  result: auto_linear

- name: dist(Tensor self, Tensor other, Scalar p=2) -> Tensor
  self: norm_backward(grad, self - other, p, result)
  other: -norm_backward(grad, self - other, p, result)
  result: norm_jvp(self_p - other_p, self_t - other_t, p, result, {}, false)

# The backward formula is done in this order to improve numerical stability
# of the higher order derivatives, see https://github.com/pytorch/pytorch/issues/43414
# Note that we don't use "result" because saving it would be BC-breaking when it is used in an inplace operation later
- name: div.Tensor(Tensor self, Tensor other) -> Tensor
  self: div_tensor_self_backward(grad, other, self.scalar_type())
  other: div_tensor_other_backward(grad, self, other)
  result: (self_t - other_t * result) / other_p

- name: div.Scalar(Tensor self, Scalar other) -> Tensor
  self: div_tensor_self_backward(grad, other, self.scalar_type())
  result: self_t / other

- name: div.Tensor_mode(Tensor self, Tensor other, *, str? rounding_mode) -> Tensor
  self: div_tensor_self_backward(grad, other, self.scalar_type(), rounding_mode)
  other: div_tensor_other_backward(grad, self, other, rounding_mode)
  result: "rounding_mode.has_value() ? result.new_zeros_symint(result.sym_sizes()) : self_t / other_p - other_t * (self_p / other_p) / other_p"

- name: div.Scalar_mode(Tensor self, Scalar other, *, str? rounding_mode) -> Tensor
  self: div_tensor_self_backward(grad, other, self.scalar_type(), rounding_mode)
  result: "rounding_mode.has_value() ? result.new_zeros_symint(result.sym_sizes()) : self_t / other"

- name: dot(Tensor self, Tensor tensor) -> Tensor
  self: grad * tensor.conj()
  tensor: grad * self.conj()
  result: at::dot(self_t, tensor_p) + at::dot(self_p, tensor_t)

- name: vdot(Tensor self, Tensor other) -> Tensor
  self: grad.conj() * other
  other: grad * self
  result: at::vdot(self_t, other_p) + at::vdot(self_p, other_t)

- name: _fused_dropout(Tensor self, float p, Generator? generator=None) -> (Tensor, Tensor)
  self: _fused_dropout_backward(grad, result1, p)

- name: native_dropout(Tensor input, float p, bool? train) -> (Tensor, Tensor)
  input: "GradMode::is_enabled() ? infinitely_differentiable_native_dropout_backward(grad, result1, (!train.has_value() || !train.value() ? 1 : (p == 1 ? 0.0 : 1.0 / (1.0 - p)))) : native_dropout_backward(grad, result1, (!train.has_value() || !train.value() ? 1 : (p == 1 ? 0.0 : 1.0 / (1.0 - p))))"
  result0: "(!train.has_value() || train.value()) ? (p == 1 ? 0.0 : 1.0 / (1.0 - p)) * input_t * result1 : input_t"

- name: native_dropout_backward(Tensor grad_output, Tensor mask, float scale) -> Tensor
  grad_output: "native_dropout_double_backward(grad, grad_output, mask, scale)"
  mask: 'not_implemented("native_dropout_backward: mask")'

- name: eq_.Scalar(Tensor(a!) self, Scalar other) -> Tensor(a!)
  self: zeros_like(self)
  result: self_t.zero_()

- name: eq_.Tensor(Tensor(a!) self, Tensor other) -> Tensor(a!)
  self: zeros_like(self)
  other: zeros_like(other)
  result: self_t.zero_()

- name: erf(Tensor self) -> Tensor
  self: 2.0 / sqrt(M_PI) * exp(-(self.pow(2))) * grad
  result: auto_element_wise

- name: erfc(Tensor self) -> Tensor
  self: -2.0 / sqrt(M_PI) * exp(-(self.pow(2))) * grad
  result: auto_element_wise

- name: special_erfcx(Tensor self) -> Tensor
  self: (2.0 * self * result - 2.0 / sqrt(M_PI)) * grad
  result: auto_element_wise

- name: erfinv(Tensor self) -> Tensor
  self: 0.5 * sqrt(M_PI) * exp(self.erfinv().pow(2)) * grad
  result: auto_element_wise

- name: exp(Tensor self) -> Tensor
  self: grad * result.conj()
  result: auto_element_wise

- name: exp2(Tensor self) -> Tensor
  self: grad * result.conj() * M_LN2
  result: auto_element_wise

- name: expm1(Tensor self) -> Tensor
  self: grad * (result.conj() + 1)
  result: auto_element_wise

# TODO: this derivative is not SymInt safe, need sum_to support
- name: expand(Tensor(a) self, SymInt[] size, *, bool implicit=False) -> Tensor(a)
  self: at::sum_to(grad, self.sym_sizes())
  result: auto_linear

- name: exponential_(Tensor(a!) self, float lambd=1, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.zero_()

- name: fake_quantize_per_tensor_affine_cachemask(Tensor self, float scale, int zero_point, int quant_min, int quant_max) -> (Tensor output, Tensor mask)
  self: fake_quantize_per_tensor_affine_cachemask_backward(grad, mask)

- name: _fake_quantize_per_tensor_affine_cachemask_tensor_qparams(Tensor self, Tensor scale, Tensor zero_point, Tensor fake_quant_enabled, int quant_min, int quant_max) -> (Tensor output, Tensor mask)
  self: fake_quantize_per_tensor_affine_cachemask_backward(grad, mask)

- name: _fake_quantize_learnable_per_tensor_affine(Tensor self, Tensor scale, Tensor zero_point, int quant_min, int quant_max, float grad_factor=1.0) -> Tensor
  self, scale, zero_point: "grad.defined() ? _fake_quantize_learnable_per_tensor_affine_backward(grad, self, scale, zero_point, quant_min, quant_max, grad_factor) : std::tuple<Tensor, Tensor, Tensor>()"

- name: fake_quantize_per_channel_affine_cachemask(Tensor self, Tensor scale, Tensor zero_point, int axis, int quant_min, int quant_max) -> (Tensor output, Tensor mask)
  self: fake_quantize_per_channel_affine_cachemask_backward(grad, mask)

- name: _fake_quantize_learnable_per_channel_affine(Tensor self, Tensor scale, Tensor zero_point, int axis, int quant_min, int quant_max, float grad_factor=1.0) -> Tensor
  self, scale, zero_point: "grad.defined() ? _fake_quantize_learnable_per_channel_affine_backward(grad, self, scale, zero_point, axis, quant_min, quant_max, grad_factor) : std::tuple<Tensor, Tensor, Tensor>()"

- name: _fused_moving_avg_obs_fq_helper(Tensor self, Tensor observer_on, Tensor fake_quant_on, Tensor(a!) running_min, Tensor(b!) running_max, Tensor(c!) scale, Tensor(d!) zero_point, float averaging_const, int quant_min, int quant_max, int ch_axis, bool per_row_fake_quant=False, bool symmetric_quant=False) -> (Tensor output, Tensor mask)
  self: fake_quantize_per_tensor_affine_cachemask_backward(grad, mask)

- name: fill.Scalar(Tensor self, Scalar value) -> Tensor
  self: zeros_like(grad)
  result: at::fill(self_t, 0)

- name: fill.Tensor(Tensor self, Tensor value) -> Tensor
  self: zeros_like(grad)
  value: grad.sum()
  result: at::fill(self_t, value_t)

- name: fill_.Scalar(Tensor(a!) self, Scalar value) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.fill_(0)

- name: fill_.Tensor(Tensor(a!) self, Tensor value) -> Tensor(a!)
  self: zeros_like(grad)
  value: grad.sum()
  result: self_t.fill_(value_t)

- name: floor(Tensor self) -> Tensor
  self: zeros_like(grad)
  result: auto_element_wise

- name: fmod.Scalar(Tensor self, Scalar other) -> Tensor
  self: grad
  result: auto_element_wise

- name: fmod.Tensor(Tensor self, Tensor other) -> Tensor
  self: grad
  other: -grad * self.div(other, /*rounding_mode=*/"trunc")
  result: self_t - other_t * self_p.div(other_p, /*rounding_mode=*/"trunc")

- name: frac(Tensor self) -> Tensor
  self: grad
  result: self_t

- name: frexp.Tensor(Tensor self) -> (Tensor mantissa, Tensor exponent)
  self: grad / exponent.exp2()
  mantissa: self_t / exponent.exp2()

- name: gather(Tensor self, int dim, Tensor index, *, bool sparse_grad=False) -> Tensor
  self: gather_backward(grad, self, dim, index, sparse_grad)
  index: non_differentiable
  result: auto_linear

- name: ge_.Scalar(Tensor(a!) self, Scalar other) -> Tensor(a!)
  self: zeros_like(self)
  result: self_t.zero_()

- name: ge_.Tensor(Tensor(a!) self, Tensor other) -> Tensor(a!)
  self: zeros_like(self)
  other: zeros_like(other)
  result: self_t.zero_()

- name: geometric_(Tensor(a!) self, float p, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.zero_()

- name: geqrf(Tensor self) -> (Tensor a, Tensor tau)
  self: not_implemented("geqrf")

- name: indices(Tensor(a) self) -> Tensor(a)
  output_differentiability: [False]

- name: _indices(Tensor(a) self) -> Tensor(a)
  output_differentiability: [False]

- name: crow_indices(Tensor(a) self) -> Tensor(a)
  output_differentiability: [False]

- name: col_indices(Tensor(a) self) -> Tensor(a)
  output_differentiability: [False]

- name: ccol_indices(Tensor(a) self) -> Tensor(a)
  output_differentiability: [False]

- name: row_indices(Tensor(a) self) -> Tensor(a)
  output_differentiability: [False]

- name: grid_sampler_2d(Tensor input, Tensor grid, int interpolation_mode, int padding_mode, bool align_corners) -> Tensor
  input, grid: "grad.defined() ? grid_sampler_2d_backward(grad, input, grid, interpolation_mode, padding_mode, align_corners, grad_input_mask) : std::tuple<Tensor, Tensor>()"

- name: grid_sampler_3d(Tensor input, Tensor grid, int interpolation_mode, int padding_mode, bool align_corners) -> Tensor
  input, grid: "grad.defined() ? grid_sampler_3d_backward(grad, input, grid, interpolation_mode, padding_mode, align_corners, grad_input_mask) : std::tuple<Tensor, Tensor>()"

# See NOTE [ grid_sample CPU fallback ]
- name: _grid_sampler_2d_cpu_fallback(Tensor input, Tensor grid, int interpolation_mode, int padding_mode, bool align_corners) -> Tensor
  input, grid: "grad.defined() ? _grid_sampler_2d_cpu_fallback_backward(grad, input, grid, interpolation_mode, padding_mode, align_corners) : std::tuple<Tensor, Tensor>()"

- name: gt_.Scalar(Tensor(a!) self, Scalar other) -> Tensor(a!)
  self: zeros_like(self)
  result: self_t.zero_()

- name: gt_.Tensor(Tensor(a!) self, Tensor other) -> Tensor(a!)
  self: zeros_like(self)
  other: zeros_like(other)
  result: self_t.zero_()

- name: hardsigmoid(Tensor self) -> Tensor
  self: hardsigmoid_backward(grad, self)
  result: auto_element_wise

- name: histc(Tensor self, int bins=100, Scalar min=0, Scalar max=0) -> Tensor
  output_differentiability: [False]

- name: hardswish(Tensor self) -> Tensor
  self: hardswish_backward(grad, self)
  result: auto_element_wise

- name: hardswish_backward(Tensor grad_output, Tensor self) -> Tensor
  grad_output: hardswish_backward(grad, self)
  self: at::where(at::logical_and(-3.0 < self, self < 3.0), grad * grad_output / 3.0, at::zeros({}, self.options()))
  result: "hardswish_backward(grad_output_t, self_p)
         + at::where(at::logical_and(-3.0 < self_p, self_p < 3.0), self_t * grad_output_p / 3.0, at::zeros({}, self_p.options()))"

- name: hypot(Tensor self, Tensor other) -> Tensor
  self: grad * self / result
  other: grad * other / result
  result: self_t * self_p / result + other_t * other_p / result

- name: i0(Tensor self) -> Tensor
  self: grad * at::special_i1(self)
  result: auto_element_wise

- name: special_i0e(Tensor self) -> Tensor
  self: grad * (at::special_i1e(self) - self.sgn() * result)
  result: auto_element_wise

- name: special_i1(Tensor self) -> Tensor
  self: i1_backward(grad, self, result)
  result: auto_element_wise

- name: special_i1e(Tensor self) -> Tensor
  self: i1e_backward(grad, self, result)
  result: auto_element_wise

- name: igamma(Tensor self, Tensor other) -> Tensor
  self: 'not_implemented("igamma: input")'
  other: grad * exp((self - 1) * log(other) - other - lgamma(self))

- name: igammac(Tensor self, Tensor other) -> Tensor
  self: 'not_implemented("igammac: input")'
  other: -grad * exp((self - 1) * log(other) - other - lgamma(self))

- name: index.Tensor(Tensor self, Tensor?[] indices) -> Tensor
  self: index_backward(grad.new_zeros_symint(self.sym_sizes(), self.options()), indices, grad)
  result: auto_linear

- name: _unsafe_index.Tensor(Tensor self, Tensor?[] indices) -> Tensor
  self: at::_unsafe_index_put(grad.new_zeros_symint(self.sym_sizes(), self.options()), indices, grad, true)
  result: auto_linear

- name: _unsafe_masked_index(Tensor self, Tensor mask, Tensor?[] indices, Scalar fill) -> Tensor
  self: at::_unsafe_masked_index_put_accumulate(grad.new_zeros_symint(self.sym_sizes(), self.options()), mask, indices, grad)
  mask: non_differentiable
  result: _unsafe_masked_index(self_t, mask, indices, 0)

- name: _unsafe_masked_index_put_accumulate(Tensor self, Tensor mask, Tensor?[] indices, Tensor values) -> Tensor
  self: grad
  mask: non_differentiable
  values: at::_unsafe_masked_index(grad, mask, indices, 0)
  result: at::_unsafe_masked_index_put_accumulate(self_t, mask, indices, values_t)

- name: index_add(Tensor self, int dim, Tensor index, Tensor source, *, Scalar alpha=1) -> Tensor
  self: grad
  # The case source.dim() == 0  is necessary to support scalar tensors of the form
  # source.dim() == 0 and index.dim() == 1 and index.size() == (1,),
  # This is because source is not broadcastable to index, as source.dim() < index.dim()
  source: "maybe_multiply(source.dim() > 0 ? grad.index_select(dim, index).expand_as(source) : grad.index_select(dim, index.squeeze(0)), alpha)"
  index: non_differentiable
  result: at::index_add(self_t, dim, index, maybe_multiply(source_t, alpha))

- name: index_reduce(Tensor self, int dim, Tensor index, Tensor source, str reduce, *, bool include_self=True) -> Tensor
  self, source: index_reduce_backward(grad, self, dim, index, source, reduce, include_self, result)
  index: non_differentiable

- name: index_copy(Tensor self, int dim, Tensor index, Tensor source) -> Tensor
  self: grad.index_fill(dim, index, 0)
  # The case source.dim() == 0 is necessary to support scalar tensors of the form
  # source.dim() == 0 and index.dim() == 1 and index.size() == (1,),
  # This is because source is not broadcastable to index, as source.dim() < index.dim()
  source: "source.dim() > 0 ? grad.index_select(dim, index).expand_as(source) : grad.index_select(dim, index.squeeze(0))"
  index: non_differentiable
  result: self_t.index_copy(dim, index, source_t)

- name: index_fill.int_Scalar(Tensor self, int dim, Tensor index, Scalar value) -> Tensor
  self: grad.index_fill(dim, index, 0)
  index: non_differentiable
  result: self_t.index_fill(dim, index, 0)

- name: index_fill.int_Tensor(Tensor self, int dim, Tensor index, Tensor value) -> Tensor
  self: grad.index_fill(dim, index, 0)
  value: grad.index_select(dim, std::get<0>(at::_unique(index, /*sorted=*/false))).sum()
  index: non_differentiable
  result: self_t.index_fill(dim, index, value_t)

- name: index_put(Tensor self, Tensor?[] indices, Tensor values, bool accumulate=False) -> Tensor
  self: "accumulate ? grad : grad.index_put(indices, zeros_like(values), false)"
  values: grad.index(indices)
  result: self_t.index_put(indices, values_t, accumulate)

- name: _unsafe_index_put(Tensor self, Tensor?[] indices, Tensor values, bool accumulate=False) -> Tensor
  self: "accumulate ? grad : at::_unsafe_index_put(grad, indices, zeros_like(values), false)"
  values: at::_unsafe_index(grad, indices)
  result: at::_unsafe_index_put(self_t, indices, values_t, accumulate)

- name: _index_put_impl_(Tensor(a!) self, Tensor?[] indices, Tensor values, bool accumulate=False, bool unsafe=False) -> Tensor(a!)
  self: "accumulate ? grad : grad.index_put(indices, zeros_like(values), false)"
  values: grad.index(indices)
  result: at::_index_put_impl_(self_t, indices, values_t, accumulate, unsafe)

- name: index_select(Tensor self, int dim, Tensor index) -> Tensor
  self: index_select_backward_symint(grad, self.sym_sizes(), dim, index)
  index: non_differentiable
  result: auto_linear

- name: linalg_inv_ex(Tensor A, *, bool check_errors=False) -> (Tensor inverse, Tensor info)
  A: -at::matmul(inverse.mH(), at::matmul(grad, inverse.mH()))
  inverse: -at::matmul(at::matmul(inverse, A_t), inverse)
  output_differentiability: [True, False]

- name: linalg_pinv.atol_rtol_tensor(Tensor self, *, Tensor? atol=None, Tensor? rtol=None, bool hermitian=False) -> Tensor
  self: pinv_backward(grad, result, self)
  result: pinv_jvp(self_p, result, self_t)

- name: isnan(Tensor self) -> Tensor
  self: non_differentiable

- name: kthvalue(Tensor self, int k, int dim=-1, bool keepdim=False) -> (Tensor values, Tensor indices)
  self: value_selecting_reduction_backward_symint(grad, dim, indices, self.sym_sizes(), keepdim)
  values: gather_with_keepdimed_indices(self_t, dim, indices, keepdim)

- name: le_.Scalar(Tensor(a!) self, Scalar other) -> Tensor(a!)
  self: zeros_like(self)
  result: self_t.zero_()

- name: le_.Tensor(Tensor(a!) self, Tensor other) -> Tensor(a!)
  self: zeros_like(self)
  other: zeros_like(other)
  result: self_t.zero_()

- name: lerp.Scalar(Tensor self, Tensor end, Scalar weight) -> Tensor
  self: "weight.isComplex() ? grad * (1 - weight.conj().toComplexDouble()) : grad * (1 - weight.toDouble())"
  end: grad * weight.conj()
  result: at::lerp(self_t, end_t, weight)

- name: lerp.Tensor(Tensor self, Tensor end, Tensor weight) -> Tensor
  self: grad * (1 - weight).conj()
  end: grad * weight.conj()
  weight: grad * (end - self).conj()
  result: at::lerp(self_t, end_t, weight_p) + weight_t * (end_p - self_p)

- name: lgamma(Tensor self) -> Tensor
  self: grad * digamma(self)
  result: auto_element_wise

- name: digamma(Tensor self) -> Tensor
  self: grad * polygamma(1, self)
  result: auto_element_wise

- name: polygamma(int n, Tensor self) -> Tensor
  self: grad * polygamma(n + 1, self)
  result: auto_element_wise

- name: polygamma_(Tensor(a!) self, int n) -> Tensor(a!)
  self: grad * polygamma(n + 1, self)
  result: self_t.mul_(polygamma(n + 1, original_self_p))

- name: log(Tensor self) -> Tensor
  self: grad.div(self.conj())
  result: auto_element_wise

- name: log10(Tensor self) -> Tensor
  self: grad / (self.conj() * 2.3025850929940456)
  result: auto_element_wise

- name: log1p(Tensor self) -> Tensor
  self: log1p_backward(grad, self)
  result: auto_element_wise

- name: log2(Tensor self) -> Tensor
  self: grad / (self.conj() * 0.6931471805599453)
  result: auto_element_wise

- name: logaddexp(Tensor self, Tensor other) -> Tensor
  self: grad / (1 + exp(other - self)).conj()
  other: grad / (1 + exp(self - other)).conj()
  result: self_t / (1 + exp(other_p - self_p)) + other_t / (1 + exp(self_p - other_p))

- name: logaddexp2(Tensor self, Tensor other) -> Tensor
  self: grad / (1 + pow(2, other - self))
  other: grad / (1 + pow(2, self - other))
  result: self_t / (1 + pow(2, other_p - self_p)) + other_t / (1 + pow(2, self_p - other_p))

# Note [Gradient formula for xlogy at x = 0, y <= 0]
# x * log(y) is not defined at y <= 0, so we cannot even talk about differentiability
# Now, xlogy(0, y) = 0 by definition.
# This does not make it differentiable as it's not defined in a neighbourhood of a point
# (0, y) when y <= 0.
# Now, when a function is non-differentiable, sometimes we return "a relatively sensible value"
# In this case, as per the discussion in https://github.com/pytorch/pytorch/issues/80770, we choose
# this value to be zero, which is the directional derivative along the line {x = 0}.
- name: xlogy.Tensor(Tensor self, Tensor other) -> Tensor
  self: at::xlogy(grad, other).masked_fill((self == 0.) & (other <= 0.), 0.)
  other: grad * self / other
  result: at::xlogy(self_t, other_p).masked_fill((self_p == 0.) & (other_p <= 0.), 0.) + other_t * self_p / other_p

- name: xlogy.Scalar_Self(Scalar self, Tensor other) -> Tensor
  other: grad * self / other
  result: auto_element_wise

- name: xlogy.Scalar_Other(Tensor self, Scalar other) -> Tensor
  self: "other.toDouble() > 0.
          ? at::xlogy(grad,  other)
          : at::xlogy(grad,  other).masked_fill(self == 0., 0.)"
  result: auto_element_wise

# See Note [Gradient formula for xlogy at x = 0, y <= 0]
# Same here but with y <= -1
- name: special_xlog1py(Tensor self, Tensor other) -> Tensor
  self: at::special_xlog1py(grad,  other).masked_fill((self == 0.) & (other <= -1.), 0.)
  other: grad * self / (other + 1)
  result: at::special_xlog1py(self_t,  other_p).masked_fill((self_p == 0.) & (other_p <= -1.), 0.) + other_t * self_p / (other_p + 1)

- name: special_xlog1py.self_scalar(Scalar self, Tensor other) -> Tensor
  other: grad * self / (other + 1)
  result: auto_element_wise

- name: special_xlog1py.other_scalar(Tensor self, Scalar other) -> Tensor
  self: "other.toDouble() > -1.
          ? at::special_xlog1py(grad,  other)
          : at::special_xlog1py(grad,  other).masked_fill(self == 0., 0.)"
  result: auto_element_wise

- name: special_zeta(Tensor self, Tensor other) -> Tensor
  self: not_implemented("zeta")
  other:  grad * -self * special_zeta(self + 1., other)

- name: special_zeta.self_scalar(Scalar self, Tensor other) -> Tensor
  other:  grad * -self * special_zeta(self.toDouble() + 1., other)

- name: special_zeta.other_scalar(Tensor self, Scalar other) -> Tensor
  self: not_implemented("zeta")

- name: log_normal_(Tensor(a!) self, float mean=1, float std=2, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.zero_()

- name: logsumexp(Tensor self, int[1] dim, bool keepdim=False) -> Tensor
  self: logsumexp_backward(grad, self, result, dim, keepdim)
  result: logsumexp_jvp(self_p, self_t, dim, keepdim)

- name: linalg_lstsq(Tensor self, Tensor b, float? rcond=None, *, str? driver=None) -> (Tensor solution, Tensor residuals, Tensor rank, Tensor singular_values)
  self, b: linalg_lstsq_backward(grad, self, b, grad_input_mask)
  solution: linalg_lstsq_jvp(self_p, b_p, self_t, b_t)
  output_differentiability: [True, False, False, False]

- name: lt_.Scalar(Tensor(a!) self, Scalar other) -> Tensor(a!)
  self: zeros_like(self)
  result: self_t.zero_()

- name: lt_.Tensor(Tensor(a!) self, Tensor other) -> Tensor(a!)
  self: zeros_like(self)
  other: zeros_like(other)
  result: self_t.zero_()

- name: linalg_lu_factor_ex(Tensor A, *, bool pivot=True, bool check_errors=False) -> (Tensor LU, Tensor pivots, Tensor info)
  A: lu_factor_ex_backward(grad, LU, pivots, pivot)
  LU: lu_factor_ex_jvp(A_t, LU, pivots, pivot)
  output_differentiability: [True, False, False]

- name: linalg_lu_factor(Tensor A, *, bool pivot=True) -> (Tensor LU, Tensor pivots)
  A: lu_factor_ex_backward(grad, LU, pivots, pivot)
  LU: lu_factor_ex_jvp(A_t, LU, pivots, pivot)
  output_differentiability: [True, False]

- name: linalg_lu(Tensor A, *, bool pivot=True) -> (Tensor P, Tensor L, Tensor U)
  A: linalg_lu_backward(grad_L, grad_U, P, L, U, pivot)
  L: std::get<0>(linalg_lu_jvp(A_t, P, L, U, pivot))
  U: std::get<1>(linalg_lu_jvp(A_t, P, L, U, pivot))
  output_differentiability: [False, True, True]

- name: linalg_lu_solve(Tensor LU, Tensor pivots, Tensor B, *, bool left=True, bool adjoint=False) -> Tensor
  LU: linalg_lu_solve_LU(grad, LU, pivots, result, left, adjoint)
  B: "at::linalg_lu_solve(LU, pivots, grad, left, !adjoint)"
  result: linalg_lu_solve_jvp(result, LU_p, pivots, LU_t, B_t, left, adjoint)

- name: lu_unpack(Tensor LU_data, Tensor LU_pivots, bool unpack_data=True, bool unpack_pivots=True) -> (Tensor P, Tensor L, Tensor U)
  LU_data: lu_unpack_backward(grad_L, grad_U, LU_data.sym_size(-2), LU_data.sym_size(-1))
  LU_pivots: non_differentiable
  L: "LU_data_t.sym_size(-2) >= LU_data_t.sym_size(-1) ? LU_data_t.tril(-1) : LU_data_t.narrow_symint(-1, 0, LU_data_t.sym_size(-2)).tril(-1)"
  U: "LU_data_t.sym_size(-1) >= LU_data_t.sym_size(-2) ? LU_data_t.triu() : LU_data_t.narrow_symint(-2, 0, LU_data_t.sym_size(-1)).triu()"
  output_differentiability: [False, True, True]

- name: masked_fill.Scalar(Tensor self, Tensor mask, Scalar value) -> Tensor
  self: grad.masked_fill(mask, 0)
  mask: non_differentiable
  result: self_t.masked_fill(mask, 0)

- name: masked_fill.Tensor(Tensor self, Tensor mask, Tensor value) -> Tensor
  self: grad.masked_fill(mask, 0)
  value: masked_fill_backward(grad, mask)
  mask: non_differentiable
  result: self_t.masked_fill(mask, value_t)

- name: masked_scatter(Tensor self, Tensor mask, Tensor source) -> Tensor
  self: grad.masked_fill(mask, 0)
  source: masked_scatter_backward_symint(grad, mask, source.sym_sizes())
  mask: non_differentiable
  result: self_t.masked_scatter(mask, source_t)

- name: masked_scatter_backward(Tensor grad_output, Tensor mask, SymInt[] sizes) -> Tensor
  grad_output: zeros_like(grad_output).masked_scatter(mask, grad)
  mask: non_differentiable
  result: masked_scatter_backward(grad_output_t, mask, grad_output_t.sizes())

- name: masked_select(Tensor self, Tensor mask) -> Tensor
  self: masked_select_backward(grad, self, mask)
  mask: non_differentiable
  result: auto_linear

- name: linalg_matrix_exp(Tensor self) -> Tensor
  self: linalg_matrix_exp_differential(self, grad, /*adjoint*/ true)
  result: linalg_matrix_exp_differential(self_p, self_t, /*adjoint*/ false)

- name: max.dim(Tensor self, int dim, bool keepdim=False) -> (Tensor values, Tensor indices)
  self: value_selecting_reduction_backward_symint(grad, dim, indices, self.sym_sizes(), keepdim)
  values: gather_with_keepdimed_indices(self_t, dim, indices, keepdim)

- name: max(Tensor self) -> Tensor
  self: evenly_distribute_backward(grad, self, result)
  result: evenly_read_jvp(self_t, self_p, result)

- name: maximum(Tensor self, Tensor other) -> Tensor
  self: at::where(self == other, grad / 2, grad).masked_fill_(self < other, 0)
  other: at::where(self == other, grad / 2, grad).masked_fill_(self > other, 0)
  result: other_t + at::where(self_p == other_p, at::scalar_tensor(0.5, result.options()), (self_p > other_p).to(result.scalar_type())) * (self_t - other_t)

- name: fmax(Tensor self, Tensor other) -> Tensor
  self: grad.masked_fill((self >= other).logical_or_(other.isnan()).logical_not_(), 0)
  other: grad.masked_fill((self >= other).logical_or_(other.isnan()), 0)
  result: other_t + (self_p > other_p).logical_or_(other_p.isnan()) * (self_t - other_t)

- name: mean(Tensor self, *, ScalarType? dtype=None) -> Tensor
  self: grad.expand_symint(self.sym_sizes()) / self.sym_numel()
  result: auto_linear

- name: mean.dim(Tensor self, int[1]? dim, bool keepdim=False, *, ScalarType? dtype=None) -> Tensor
  self: mean_backward(grad, self.sym_sizes(), dim, self.sym_numel(), keepdim)
  result: auto_linear

- name: median(Tensor self) -> Tensor
  self: evenly_distribute_backward(grad, self, result)
  result: evenly_read_jvp(self_t, self_p, result)

- name: nanmedian(Tensor self) -> Tensor
  self: evenly_distribute_backward(grad, self, result)
  result: evenly_read_jvp(self_t, self_p, result)

# This is in theory incorrect in the following case:
#   sorted list: [..., a, b, b, ..., b, b, c, ...] with median = b and the value
#                            |                     at middle position of the
#                            |                     list between two `b`s. E.g.,
#                            |
#                            ^the middle position
# The gradient exists and is essentially 0 in this case.
#
# In case where the middle position is at the boundary of `b` range, e.g.,
#   sorted list: [..., a, b, b, ..., b, b, c, ...]
#                                       |
#                                       ^the middle position
# The backward implementation is correct in the sense that it returns the
# subgradient on one side.
- name: median.dim(Tensor self, int dim, bool keepdim=False) -> (Tensor values, Tensor indices)
  self: value_selecting_reduction_backward_symint(grad, dim, indices, self.sym_sizes(), keepdim)
  values: gather_with_keepdimed_indices(self_t, dim, indices, keepdim)

- name: nanmedian.dim(Tensor self, int dim, bool keepdim=False) -> (Tensor values, Tensor indices)
  self: value_selecting_reduction_backward_symint(grad, dim, indices, self.sym_sizes(), keepdim)
  values: gather_with_keepdimed_indices(self_t, dim, indices, keepdim)

- name: min.dim(Tensor self, int dim, bool keepdim=False) -> (Tensor values, Tensor indices)
  self: value_selecting_reduction_backward_symint(grad, dim, indices, self.sym_sizes(), keepdim)
  values: gather_with_keepdimed_indices(self_t, dim, indices, keepdim)

- name: min(Tensor self) -> Tensor
  self: evenly_distribute_backward(grad, self, result)
  result: evenly_read_jvp(self_t, self_p, result)

- name: minimum(Tensor self, Tensor other) -> Tensor
  self: at::where(self == other, grad / 2, grad).masked_fill_(self > other, 0)
  other: at::where(self == other, grad / 2, grad).masked_fill_(self < other, 0)
  result: other_t + at::where(self_p == other_p, at::scalar_tensor(0.5, result.options()), (self_p < other_p).to(result.scalar_type())) * (self_t - other_t)

- name: fmin(Tensor self, Tensor other) -> Tensor
  self: grad.masked_fill((self <= other).logical_or_(other.isnan()).logical_not_(), 0)
  other: grad.masked_fill((self <= other).logical_or_(other.isnan()), 0)
  result: other_t + (self_p <= other_p).logical_or_(other_p.isnan()) * (self_t - other_t)

- name: amax(Tensor self, int[1] dim=[], bool keepdim=False) -> Tensor
  self: scale_grad_by_count(restore_reduced_dims(grad, dim, keepdim), restore_reduced_dims(result, dim, keepdim) == self, dim)
  result: amaxamin_jvp(self_p, self_t, result, dim, keepdim)

- name: amin(Tensor self, int[1] dim=[], bool keepdim=False) -> Tensor
  self: scale_grad_by_count(restore_reduced_dims(grad, dim, keepdim), restore_reduced_dims(result, dim, keepdim) == self, dim)
  result: amaxamin_jvp(self_p, self_t, result, dim, keepdim)

- name: mm(Tensor self, Tensor mat2) -> Tensor
  self: mm_mat1_backward(grad, mat2, self.sym_sizes(), self.sym_strides(), self.layout(), 1)
  mat2: mm_mat2_backward(grad, self, mat2.sym_sizes(), mat2.sym_strides(), mat2.layout(), 1)
  result: at::mm(self_t, mat2_p) + at::mm(self_p, mat2_t)

- name: mode(Tensor self, int dim=-1, bool keepdim=False) -> (Tensor values, Tensor indices)
  self: value_selecting_reduction_backward_symint(grad, dim, indices, self.sym_sizes(), keepdim)
  values: gather_with_keepdimed_indices(self_t, dim, indices, keepdim)

- name: mul.Tensor(Tensor self, Tensor other) -> Tensor
  self: mul_tensor_backward(grad, other, self.scalar_type())
  other: mul_tensor_backward(grad, self, other.scalar_type())
  result: other_t * self_p + self_t * other_p

- name: mul.Scalar(Tensor self, Scalar other) -> Tensor
  self: mul_tensor_backward(grad, other, self.scalar_type())
  result: self_t * other

- name: mv(Tensor self, Tensor vec) -> Tensor
  self: grad.ger(vec.conj())
  vec: self.conj().t().mv(grad)
  result: mv(self_t, vec_p) + mv(self_p, vec_t)

- name: mvlgamma(Tensor self, int p) -> Tensor
  self: mvlgamma_backward(grad, self, p)
  result: auto_element_wise

- name: nan_to_num(Tensor self, float? nan=None, float? posinf=None, float? neginf=None) -> Tensor
  self: grad * at::isfinite(self)
  result: auto_element_wise

- name: native_batch_norm(Tensor input, Tensor? weight, Tensor? bias, Tensor? running_mean, Tensor? running_var, bool training, float momentum, float eps) -> (Tensor, Tensor, Tensor)
  input, weight, bias: "grad.defined() ? native_batch_norm_backward(grad, input, weight, running_mean, running_var, result1, result2, training, eps, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"
  result0: batch_norm_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, running_mean, running_var, result1, result2, training, eps)

- name: _native_batch_norm_legit(Tensor input, Tensor? weight, Tensor? bias, Tensor(a!) running_mean, Tensor(b!) running_var, bool training, float momentum, float eps) -> (Tensor, Tensor, Tensor)
  input, weight, bias: "grad.defined() ? native_batch_norm_backward(grad, input, weight, running_mean, running_var, result1, result2, training, eps, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"
  result0: batch_norm_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, running_mean, running_var, result1, result2, training, eps)

- name: _native_batch_norm_legit_no_training(Tensor input, Tensor? weight, Tensor? bias, Tensor running_mean, Tensor running_var, float momentum, float eps) -> (Tensor, Tensor, Tensor)
  input, weight, bias: "grad.defined() ? native_batch_norm_backward(grad, input, weight, running_mean, running_var, result1, result2, /*training=*/false, eps, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"
  result0: batch_norm_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, running_mean, running_var, result1, result2, /*training=*/false, eps)

- name: _native_batch_norm_legit.no_stats(Tensor input, Tensor? weight, Tensor? bias, bool training, float momentum, float eps) -> (Tensor, Tensor, Tensor)
  input, weight, bias: "grad.defined() ? native_batch_norm_backward(grad, input, weight, Tensor(), Tensor(), result1, result2, training, eps, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"
  result0: batch_norm_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, Tensor(), Tensor(), result1, result2, training, eps)

- name: native_batch_norm_backward(Tensor grad_out, Tensor input, Tensor? weight, Tensor? running_mean, Tensor? running_var, Tensor? save_mean, Tensor? save_invstd, bool train, float eps, bool[3] output_mask) -> (Tensor, Tensor, Tensor)
  input, weight, grad_out: batchnorm_double_backward(input, weight, grads[0], grads[1], grads[2], grad_out, running_mean, running_var, train, eps, save_mean, save_invstd, grad_input_mask)
  save_mean: not_implemented("native_batch_norm_backward save_mean")
  save_invstd: not_implemented("native_batch_norm_backward save_invstd")

- name: native_layer_norm(Tensor input, SymInt[] normalized_shape, Tensor? weight, Tensor? bias, float eps) -> (Tensor, Tensor, Tensor)
  input, weight, bias: "grad.defined() ? native_layer_norm_backward_symint(grad, input, normalized_shape, result1, result2, weight, bias, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"
  result0: layer_norm_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, result1, result2, normalized_shape)

- name: native_layer_norm_backward(Tensor grad_out, Tensor input, SymInt[] normalized_shape, Tensor mean, Tensor rstd, Tensor? weight, Tensor? bias, bool[3] output_mask) -> (Tensor, Tensor, Tensor)
  input, weight, grad_out: layer_norm_double_backward(input, weight, grads[0], grads[1], grads[2], grad_out, mean, rstd, normalized_shape, grad_input_mask)
  bias: Tensor()
  mean: not_implemented("native_layer_norm_backward mean")
  rstd: not_implemented("native_layer_norm_backward rstd")

- name: native_group_norm(Tensor input, Tensor? weight, Tensor? bias, SymInt N, SymInt C, SymInt HxW, int group, float eps) -> (Tensor, Tensor, Tensor)
  input, weight, bias: "GradMode::is_enabled() || grads[1].defined() || grads[2].defined() ? infinitely_differentiable_native_group_norm_backward(grads[0], grads[1], grads[2], input, result1, result2, weight, N, C, HxW, group, eps, grad_input_mask) : (grads[0].defined() ? native_group_norm_backward_symint(grads[0].device().is_xpu() ? grads[0] : grads[0].contiguous(grads[0].device().is_cpu() ? input.suggest_memory_format() : c10::MemoryFormat::Contiguous), input.device().is_xpu() ? input : input.contiguous(input.device().is_cpu() ? input.suggest_memory_format() : c10::MemoryFormat::Contiguous), result1, result2, weight, N, C, HxW, group, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>())"
  result0: group_norm_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, result1, result2, group)
  result1: group_norm_mean_jvp(input_t, result1, group)
  result2: group_norm_invstd_jvp(input_p, input_t, result1, result2, group)

- name: ne_.Scalar(Tensor(a!) self, Scalar other) -> Tensor(a!)
  self: zeros_like(self)
  result: self_t.zero_()

- name: ne_.Tensor(Tensor(a!) self, Tensor other) -> Tensor(a!)
  self: zeros_like(self)
  other: zeros_like(other)
  result: self_t.zero_()

- name: neg(Tensor self) -> Tensor
  self: grad.neg()
  result: auto_element_wise

- name: _batch_norm_with_update(Tensor input, Tensor? weight, Tensor? bias, Tensor(a!) running_mean, Tensor(b!) running_var, float momentum, float eps) -> (Tensor, Tensor, Tensor, Tensor)
  input, weight, bias: "grad.defined() ? batch_norm_backward(grad, input, weight, running_mean, running_var, result1, result2, /*update*/true, eps, grad_input_mask, retain_variables ? result3.clone() : result3) : std::tuple<Tensor, Tensor, Tensor>()"
  result0: batch_norm_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, running_mean, running_var, result1, result2, true, eps)

- name: _batch_norm_no_update(Tensor input, Tensor? weight, Tensor? bias, Tensor? running_mean, Tensor? running_var, float momentum, float eps) -> (Tensor, Tensor, Tensor, Tensor)
  input, weight, bias: "grad.defined() ? batch_norm_backward(grad, input, weight, running_mean, running_var, result1, result2, /*update*/false, eps, grad_input_mask, retain_variables ? result3.clone() : result3) : std::tuple<Tensor, Tensor, Tensor>()"
  result0: batch_norm_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, running_mean, running_var, result1, result2, false, eps)

- name: batch_norm_backward(Tensor grad_out, Tensor input, Tensor weight, Tensor? running_mean, Tensor? running_var, Tensor? save_mean, Tensor? save_var, bool update, float eps, bool[3] output_mask, Tensor reserve) -> (Tensor, Tensor, Tensor)
  input, weight, grad_out: batchnorm_double_backward(input, weight, grads[0], grads[1], grads[2], grad_out, running_mean, running_var, update, eps, save_mean, save_var, grad_input_mask)
  save_mean: not_implemented("batch_norm_backward save_mean")
  save_var: not_implemented("batch_norm_backward save_var")
  reserve: not_implemented("batch_norm_backward reserve")

- name: nextafter(Tensor self, Tensor other) -> Tensor
  self: not_implemented("nextafter")
  other: not_implemented("nextafter")

- name: norm.Scalar(Tensor self, Scalar p=2) -> Tensor
  self: norm_backward(grad, self, p, result)
  result: norm_jvp(self_p, self_t, p, result)

- name: norm.ScalarOpt_dim(Tensor self, Scalar? p, int[1] dim, bool keepdim=False) -> Tensor
  self: norm_backward(grad, self, p, result, dim, keepdim)
  result: norm_jvp(self_p, self_t, p, result, dim, keepdim)

- name: norm.ScalarOpt_dtype(Tensor self, Scalar? p, *, ScalarType dtype) -> Tensor
  self: norm_backward(grad, self.to(grad.scalar_type()), p, result)
  result: norm_jvp(self_p, self_t, p, result)

- name: norm.ScalarOpt_dim_dtype(Tensor self, Scalar? p, int[1] dim, bool keepdim, *, ScalarType dtype) -> Tensor
  self: norm_backward(grad, self.to(grad.scalar_type()), p, result, dim, keepdim)
  result: norm_jvp(self_p, self_t, p, result, dim, keepdim)

- name: linalg_vector_norm(Tensor self, Scalar ord=2, int[1]? dim=None, bool keepdim=False, *, ScalarType? dtype=None) -> Tensor
  self: linalg_vector_norm_backward(grad, self, ord, result, dim, keepdim)
  result: linalg_vector_norm_jvp(self_p, self_t, ord, result, dim, keepdim)

- name: _pdist_forward(Tensor self, float p=2) -> Tensor
  self: _pdist_backward(grad, self, p, result)

- name: _pdist_backward(Tensor grad, Tensor self, float p, Tensor pdist) -> Tensor
  grad: not_implemented("_pdist_backward")
  self: not_implemented("_pdist_backward")
  pdist: not_implemented("_pdist_backward")

- name: _euclidean_dist(Tensor x1, Tensor x2) -> Tensor
  x1, x2: _euclidean_dist_backward(grad, x1, x2, result)

- name: _cdist_forward(Tensor x1, Tensor x2, float p, int? compute_mode) -> Tensor
  x1: _cdist_backward(grad.contiguous(), x1, x2, p, result)
  x2: _cdist_backward(grad.mT().contiguous(), x2, x1, p, result.mT().contiguous())

- name: _cdist_backward(Tensor grad, Tensor x1, Tensor x2, float p, Tensor cdist) -> Tensor
  grad: not_implemented("_cdist_backward")
  x1: not_implemented("_cdist_backward")
  x2: not_implemented("_cdist_backward")
  cdist: not_implemented("_cdist_backward")

- name: normal_(Tensor(a!) self, float mean=0, float std=1, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.zero_()

- name: normal.Tensor_float(Tensor mean, float std=1, *, Generator? generator=None) -> Tensor
  mean: at::zeros_symint(mean.sym_sizes(), grad.options())
  result: auto_element_wise

- name: normal.float_Tensor(float mean, Tensor std, *, Generator? generator=None) -> Tensor
  std: at::zeros_symint(std.sym_sizes(), grad.options())
  result: auto_element_wise

- name: normal.Tensor_Tensor(Tensor mean, Tensor std, *, Generator? generator=None) -> Tensor
  mean: at::zeros_symint(mean.sym_sizes(), grad.options())
  std: at::zeros_symint(std.sym_sizes(), grad.options())
  result: zeros_like(mean_t)

- name: linalg_householder_product(Tensor input, Tensor tau) -> Tensor
  input, tau: householder_product_backward(grad, result, input, tau)
  result: householder_product_jvp(input_t, tau_t, result, input_p, tau_p)

- name: ormqr(Tensor self, Tensor input2, Tensor input3, bool left=True, bool transpose=False) -> Tensor
  self, input2, input3: ormqr_backward(grad, result, self, input2, input3, left, transpose, grad_input_mask)

- name: permute(Tensor(a) self, int[] dims) -> Tensor(a)
  self: permute_backwards(grad, dims)
  result: auto_linear

- name: poisson(Tensor self, Generator? generator=None) -> Tensor
  self: zeros_like(self)
  result: auto_element_wise

- name: pow.Tensor_Scalar(Tensor self, Scalar exponent) -> Tensor
  self: pow_backward(grad, self, exponent)
  result: auto_element_wise

- name: pow.Tensor_Tensor(Tensor self, Tensor exponent) -> Tensor
  self: pow_backward_self(grad, self, exponent)
  exponent: pow_backward_exponent(grad, self, exponent, result)
  result: (pow_backward_self(self_t.conj(), self_p, exponent_p) + pow_backward_exponent(exponent_t.conj(), self_p, exponent_p, result)).conj()

- name: pow.Scalar(Scalar self, Tensor exponent) -> Tensor
  exponent: pow_backward_exponent(grad, self, exponent, result)
  result: auto_element_wise

- name: prod(Tensor self, *, ScalarType? dtype=None) -> Tensor
  self: prod_backward(grad, self.to(grad.scalar_type()), result)
  result: (prod_backward(at::ones({}, result.options()).expand_as(result), self_p.to(result.scalar_type()), result) * self_t.conj()).sum().conj()

- name: prod.dim_int(Tensor self, int dim, bool keepdim=False, *, ScalarType? dtype=None) -> Tensor
  self: prod_backward(grad, self.to(grad.scalar_type()), result, dim, keepdim)
  result: (prod_backward(at::ones({}, result.options()).expand_as(result), self_p.to(result.scalar_type()), result, dim, keepdim) * self_t.conj()).sum(dim, keepdim).conj()

- name: put(Tensor self, Tensor index, Tensor source, bool accumulate=False) -> Tensor
  self: "accumulate ? grad : grad.put(index, zeros_like(source), false)"
  index: non_differentiable
  source: grad.take(index).reshape_as(source)
  result: self_t.put(index, source_t, accumulate)

- name: linalg_qr(Tensor A, str mode='reduced') -> (Tensor Q, Tensor R)
  A: linalg_qr_backward(grad_Q, grad_R, Q, R, mode)
  Q, R: linalg_qr_jvp(A_t, Q, R, mode)

- name: rad2deg(Tensor self) -> Tensor
  self: rad2deg_backward(grad)
  result: auto_element_wise

- name: random_.from(Tensor(a!) self, int from, int? to, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.zero_()

- name: random_.to(Tensor(a!) self, int to, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.zero_()

- name: random_(Tensor(a!) self, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.zero_()

- name: reciprocal(Tensor self) -> Tensor
  self: -grad * (result * result).conj()
  result: auto_element_wise

- name: remainder.Scalar(Tensor self, Scalar other) -> Tensor
  self: grad
  result: auto_element_wise

- name: remainder.Tensor(Tensor self, Tensor other) -> Tensor
  self: grad
  other: -grad * self.div(other, /*rounding_mode=*/"floor")
  result: self_t - other_t * self_p.div(other_p, /*rounding_mode=*/"floor")

- name: renorm(Tensor self, Scalar p, int dim, Scalar maxnorm) -> Tensor
  self: renorm_backward(grad, self, p, dim, maxnorm)
  result: renorm_jvp(self_p, self_t, p, dim, maxnorm)

- name: repeat(Tensor self, SymInt[] repeats) -> Tensor
  self: repeat_backward(grad, repeats, self.sym_sizes())
  result: auto_linear

- name: special_entr(Tensor self) -> Tensor
  self: grad * (-(1 + self.log()))
  result: auto_element_wise

- name: special_ndtri(Tensor self) -> Tensor
  self: grad * std::sqrt(2 * M_PI) * (result.square() / 2).exp()
  result: auto_element_wise

- name: special_log_ndtr(Tensor self) -> Tensor
  self: grad / std::sqrt(2 * M_PI) * (result + self.pow(2) / 2).neg().exp()
  result: auto_element_wise

# [Note: Sometimes view derivatives]
# The following situation applies to other operations as well.
# TODO: This note is only referenced by to_dense and to_sparse*. Make
# this more generic if it's been referenced more than once.
#
# DO NOT define a backward for reshape!
# reshape is special in that it sometimes returns a view, and sometimes not.
# Defining a backward will make codegen spit out the forward call as
#     as_variable(baseType->reshape(self)),
# making it impossible (hard) to detect when it is actually a view.
# - name: reshape(Tensor self, IntArrayRef shape)

- name: _reshape_alias(Tensor(a) self, SymInt[] size, SymInt[] stride) -> Tensor(a)
  self: grad.reshape_symint(self.sym_sizes())
  result: auto_linear

- name: round(Tensor self) -> Tensor
  self: zeros_like(grad)
  result: auto_element_wise

- name: round.decimals(Tensor self, *, int decimals) -> Tensor
  self: zeros_like(grad)
  result: auto_element_wise

- name: rsqrt(Tensor self) -> Tensor
  self: -0.5 * grad * result.pow(3).conj()
  result: auto_element_wise

- name: scatter.src(Tensor self, int dim, Tensor index, Tensor src) -> Tensor
  self: grad.scatter(dim, index, 0)
  index: non_differentiable
  src: grad.gather(dim, index)
  result: self_t.scatter(dim, index, src_t)

- name: scatter.value(Tensor self, int dim, Tensor index, Scalar value) -> Tensor
  self: grad.scatter(dim, index, 0)
  index: non_differentiable
  result: self_t.scatter(dim, index, 0)

- name: scatter_add(Tensor self, int dim, Tensor index, Tensor src) -> Tensor
  self: grad
  index: non_differentiable
  src: grad.gather(dim, index)
  result: scatter_add(self_t, dim, index, src_t)

- name: select.int(Tensor(a) self, int dim, SymInt index) -> Tensor(a)
  dispatch:
    Default:
      self: select_backward_symint(grad, self.sym_sizes(), dim, index)
      result: auto_linear
    AutogradNestedTensor:
      self: _nested_select_backward_symint(grad, self, dim, index)

- name: select_backward(Tensor grad_output, SymInt[] input_sizes, int dim, SymInt index) -> Tensor
  grad_output: grad.select_symint(dim, index)
  result: auto_linear

- name: sigmoid(Tensor self) -> Tensor
  self: sigmoid_backward(grad, result)
  result: auto_element_wise

- name: logit(Tensor self, float? eps=None) -> Tensor
  self: "GradMode::is_enabled() ? infinitely_differentiable_logit_backward(grad, self, eps) : logit_backward(grad, self, eps)"
  result: auto_element_wise

- name: sign(Tensor self) -> Tensor
  self: zeros_like(grad)
  result: auto_element_wise

- name: sgn(Tensor self) -> Tensor
  self: sgn_backward(self, grad, result)
  # Cannot use auto_element_wise here because the Jacobian is *not* Hermitian (in fact, it is symmetric)
  # The function is not holomorphic, so there's no reason for its Jacobian to be Hermitian
  # auto_element_wise has a name that's a bit deceiving in the complex case
  result: sgn_backward(self_p, self_t, result)

- name: sin(Tensor self) -> Tensor
  self: grad * self.cos().conj()
  result: auto_element_wise

- name: sinc(Tensor self) -> Tensor
  self: sinc_backward(grad, self)
  result: auto_element_wise

- name: sinh(Tensor self) -> Tensor
  self: grad * self.cosh().conj()
  result: auto_element_wise

- name: slice.Tensor(Tensor(a) self, int dim=0, SymInt? start=None, SymInt? end=None, SymInt step=1) -> Tensor(a)
  self: slice_backward_wrapper(grad, self.sym_sizes(), dim, start, end, step)
  result: auto_linear

- name: slice_backward(Tensor grad_output, SymInt[] input_sizes, int dim, SymInt start, SymInt end, SymInt step) -> Tensor
  grad_output: grad.slice_symint(dim, start, end, step)
  result: auto_linear

- name: slice_inverse(Tensor(a) self, Tensor src, int dim=0, SymInt? start=None, SymInt? end=None, SymInt step=1) -> Tensor(a)
  self: grad.slice_symint(dim, start, end, step)
  src: slice_scatter_symint(grad, zeros_like(self), dim, start, end, step)
  result: auto_linear

- name: slice_scatter(Tensor self, Tensor src, int dim=0, SymInt? start=None, SymInt? end=None, SymInt step=1) -> Tensor
  self: slice_scatter_symint(grad, zeros_like(src), dim, start, end, step)
  src: grad.slice_symint(dim, start, end, step)
  result: auto_linear

- name: select_scatter(Tensor self, Tensor src, int dim, SymInt index) -> Tensor
  self: select_scatter_symint(grad, zeros_like(src), dim, index)
  src: grad.select_symint(dim, index)
  result: auto_linear

- name: diagonal_scatter(Tensor self, Tensor src, int offset=0, int dim1=0, int dim2=1) -> Tensor
  self: diagonal_scatter(grad, zeros_like(src), offset, dim1, dim2)
  src: grad.diagonal(offset, dim1, dim2)
  result: auto_linear

- name: as_strided_scatter(Tensor self, Tensor src, SymInt[] size, SymInt[] stride, SymInt? storage_offset=None) -> Tensor
  self: as_strided_scatter_backward(grad, TensorGeometry(self), TensorGeometry(src), size, stride, storage_offset)
  # See Note [as_strided_scatter backward support]
  src: grad.contiguous().as_strided_symint(size, stride, storage_offset)
  result: auto_linear

- name: _linalg_solve_ex(Tensor A, Tensor B, *, bool left=True, bool check_errors=False) -> (Tensor result, Tensor LU, Tensor pivots, Tensor info)
  A, B: linalg_solve_backward(grad, result, A, LU, pivots, left, grad_input_mask[1])
  result: "linalg_solve_jvp(A_t, B_t, result, LU, pivots, left, A_p.is_contiguous() && !A_p.is_complex())"
  output_differentiability: [True, False, False, False]  # LU is an auxiliary tensor not exposed to the user

- name: sort(Tensor self, int dim=-1, bool descending=False) -> (Tensor values, Tensor indices)
  self: value_selecting_reduction_backward_symint(grad, dim, indices, self.sym_sizes(), true)
  output_differentiability: [True, False]
  values: gather_with_keepdimed_indices(self_t, dim, indices, true)

- name: sort.stable(Tensor self, *, bool? stable, int dim=-1, bool descending=False) -> (Tensor values, Tensor indices)
  self: value_selecting_reduction_backward_symint(grad, dim, indices, self.sym_sizes(), true)
  output_differentiability: [True, False]
  values: gather_with_keepdimed_indices(self_t, dim, indices, true)

- name: split.Tensor(Tensor(a -> *) self, SymInt split_size, int dim=0) -> Tensor(a)[]
  self: split_backward(grads, split_size, dim, self.sym_sizes(), self.options())
  result: auto_linear

- name: unsafe_split.Tensor(Tensor self, SymInt split_size, int dim=0) -> Tensor[]
  self: split_backward(grads, split_size, dim, self.sym_sizes(), self.options())
  result: auto_linear

- name: split_with_sizes(Tensor(a -> *) self, SymInt[] split_sizes, int dim=0) -> Tensor(a)[]
  dispatch:
    Default:
      self: split_with_sizes_backward(grads, split_sizes, dim, self.sym_sizes(), self.options())
      result: auto_linear
    AutogradNestedTensor:
      self: _nested_split_with_sizes_backward(grads, split_sizes, dim, at::native::get_nested_tensor_impl(self)->get_nested_sizes(), self.options())

- name: unsafe_split_with_sizes(Tensor self, SymInt[] split_sizes, int dim=0) -> Tensor[]
  self: split_with_sizes_backward(grads, split_sizes, dim, self.sym_sizes(), self.options())
  result: auto_linear

- name: sqrt(Tensor self) -> Tensor
  self: grad / (2 * result.conj())
  result: auto_element_wise

- name: squeeze(Tensor(a) self) -> Tensor(a)
  self: unsqueeze_to(grad, self.sym_sizes())
  result: auto_linear

- name: squeeze.dim(Tensor(a) self, int dim) -> Tensor(a)
  dispatch:
    Default:
      self: unsqueeze_to(grad, dim, self.sym_sizes())
      result: auto_linear
    AutogradNestedTensor:
      self: grad.unsqueeze(dim)

- name: squeeze.dims(Tensor(a) self, int[] dim) -> Tensor(a)
  dispatch:
    Default:
      self: unsqueeze_to(grad, dim, self.sym_sizes())
      result: auto_linear
    AutogradNestedTensor:
      self: unsqueeze_multiple(grad, dim, self.dim())

- name: squeeze_(Tensor(a!) self) -> Tensor(a!)
  self: unsqueeze_to(grad, self.sym_sizes())
  result: auto_linear

- name: squeeze_.dim(Tensor(a!) self, int dim) -> Tensor(a!)
  self: unsqueeze_to(grad, dim, self.sym_sizes())
  result: auto_linear

- name: squeeze_.dims(Tensor(a!) self, int[] dim) -> Tensor(a!)
  self: unsqueeze_to(grad, dim, self.sym_sizes())
  result: auto_linear

- name: std.correction(Tensor self, int[1]? dim=None, *, Scalar? correction=None, bool keepdim=False) -> Tensor
  self: std_backward(result, grad, self, dim, correction, keepdim)
  # pointwise (variance) + sum + sqrt
  result: (at::real(var_backward(self_t.conj(), self_p, dim, correction, true).sum(dim.value_or(IntArrayRef({})), keepdim)) / (2. * result)).masked_fill_(result == 0, 0)

- name: std_mean.correction(Tensor self, int[1]? dim=None, *, Scalar? correction=None, bool keepdim=False) -> (Tensor, Tensor)
  self: std_mean_backward(grads[0], grads[1], self, result0, dim, correction, keepdim)
  result0: (at::real(var_backward(self_t.conj(), self_p, dim, correction, true).sum(dim.value_or(IntArrayRef({})), keepdim)) / (2. * result0)).masked_fill_(result0 == 0, 0)
  # linear
  result1: mean(self_t, dim.value_or(IntArrayRef({})), keepdim)

- name: sub.Tensor(Tensor self, Tensor other, *, Scalar alpha=1) -> Tensor
  self: handle_r_to_c(self.scalar_type(), grad)
  other: handle_r_to_c(other.scalar_type(), maybe_multiply(-grad, alpha.conj()))
  result: self_t - maybe_multiply(other_t, alpha)

- name: sub.Scalar(Tensor self, Scalar other, Scalar alpha=1) -> Tensor
  self: handle_r_to_c(self.scalar_type(), grad)
  result: auto_element_wise

- name: rsub.Tensor(Tensor self, Tensor other, *, Scalar alpha=1) -> Tensor
  self: handle_r_to_c(self.scalar_type(), maybe_multiply(-grad, alpha.conj()))
  other: handle_r_to_c(other.scalar_type(), grad)
  result: -maybe_multiply(self_t, alpha) + other_t

- name: rsub.Scalar(Tensor self, Scalar other, Scalar alpha=1) -> Tensor
  self: handle_r_to_c(self.scalar_type(), maybe_multiply(-grad, alpha.conj()))
  result: auto_element_wise

- name: sum(Tensor self, *, ScalarType? dtype=None) -> Tensor
  self: grad.expand_symint(self.sym_sizes())
  result: auto_linear

- name: sum.dim_IntList(Tensor self, int[1]? dim, bool keepdim=False, *, ScalarType? dtype=None) -> Tensor
  dispatch:
    Default:
      self: sum_backward(grad, self.sym_sizes(), dim, keepdim)
      result: auto_linear
    AutogradNestedTensor:
      # TODO: replace this function once semantics for nested tensor expand have been settled on
      self: _nested_sum_backward(grad, self, dim, keepdim)

- name: nansum(Tensor self, int[1]? dim=None, bool keepdim=False, *, ScalarType? dtype=None) -> Tensor
  self: nansum_backward(grad.to(self.scalar_type()), self, dim, keepdim)
  result: at::where(self_p.isnan(), 0, self_t).sum(dim, keepdim, dtype)

# We never call _linalg_svd with compute_uv=False in an autograd context, so we don't even consider it here
- name: _linalg_svd(Tensor A, bool full_matrices=False, bool compute_uv=True, *, str? driver=None) -> (Tensor U, Tensor S, Tensor Vh)
  A: "svd_backward(full_matrices && grad_U.defined() ? grad_U.narrow_symint(-1, 0, S.sym_size(-1)) : grad_U,
                   grad_S,
                   full_matrices && grad_Vh.defined() ? grad_Vh.narrow_symint(-2, 0, S.sym_size(-1)) : grad_Vh,
                   full_matrices ? U.narrow_symint(-1, 0, S.sym_size(-1)) : U,
                   S,
                   full_matrices ? Vh.narrow_symint(-2, 0, S.sym_size(-1)) : Vh)"
  U, S, Vh: linalg_svd_jvp(A_t, U, S, Vh, full_matrices)

- name: _linalg_eigh(Tensor A, str UPLO="L", bool compute_v=True) -> (Tensor eigenvalues, Tensor eigenvectors)
  A: linalg_eig_backward(grads[0], grads[1], eigenvalues, eigenvectors, /*is_hermitian=*/true)
  eigenvalues, eigenvectors: linalg_eig_jvp(A_t, eigenvalues, eigenvectors, /*is_hermitian=*/true)

- name: linalg_eig(Tensor self) -> (Tensor eigenvalues, Tensor eigenvectors)
  self: handle_r_to_c(self.scalar_type(), linalg_eig_backward(grads[0], grads[1], eigenvalues, eigenvectors, /*is_hermitian=*/false))
  eigenvalues, eigenvectors: linalg_eig_jvp(self_t, eigenvalues, eigenvectors, /*is_hermitian=*/false)

- name: t(Tensor(a) self) -> Tensor(a)
  self: grad.t()
  result: auto_linear

- name: t_(Tensor(a!) self) -> Tensor(a!)
  self: grad.t()
  result: auto_linear

- name: one_hot(Tensor self, int num_classes=-1) -> Tensor
  self: non_differentiable

- name: flip(Tensor self, int[] dims) -> Tensor
  self: grad.flip(dims)
  result: auto_linear

- name: roll(Tensor self, SymInt[1] shifts, int[1] dims=[]) -> Tensor
  self: grad.roll_symint(fmap(reverse_list_symint(shifts), [](c10::SymInt i){return -i;}), reverse_list(dims))
  result: auto_linear

- name: rot90(Tensor self, int k=1, int[] dims=[0,1]) -> Tensor
  self: grad.rot90(-k, dims)
  result: auto_linear

- name: take(Tensor self, Tensor index) -> Tensor
  self: take_backward(grad, self, index)
  index: non_differentiable
  result: auto_linear

- name: tan(Tensor self) -> Tensor
  self: grad * (1 + result.pow(2)).conj()
  result: auto_element_wise

- name: tanh(Tensor self) -> Tensor
  self: tanh_backward(grad, result)
  result: auto_element_wise

- name: topk(Tensor self, SymInt k, int dim=-1, bool largest=True, bool sorted=True) -> (Tensor values, Tensor indices)
  self: value_selecting_reduction_backward_symint(grad, dim, indices, self.sym_sizes(), true)
  output_differentiability: [True, False]
  values: gather(self_t, dim, indices)

- name: trace(Tensor self) -> Tensor
  self: trace_backward_symint(grad, self.sym_sizes())
  result: auto_linear

- name: transpose.int(Tensor(a) self, int dim0, int dim1) -> Tensor(a)
  self: grad.transpose(dim0, dim1)
  result: auto_linear

- name: transpose_(Tensor(a!) self, int dim0, int dim1) -> Tensor(a!)
  self: grad.transpose(dim0, dim1)
  result: auto_linear

- name: triangular_solve(Tensor self, Tensor A, bool upper=True, bool transpose=False, bool unitriangular=False) -> (Tensor solution, Tensor cloned_coefficient)
  self, A: triangular_solve_backward(grad_solution, grad_cloned_coefficient, self, A, solution, upper, transpose, unitriangular, grad_input_mask)
  solution: triangular_solve_jvp(solution, A_p, A_t, self_t, upper, transpose, unitriangular)
  cloned_coefficient: A_t

- name: linalg_solve_triangular(Tensor self, Tensor B, *, bool upper, bool left=True, bool unitriangular=False) -> Tensor
  self, B: linalg_solve_triangular_backward(grad, self, result, upper, left, unitriangular, grad_input_mask)
  result: linalg_solve_triangular_forward_AD(self_t, B_t, self_p, result, upper, left, unitriangular)

- name: tril(Tensor self, int diagonal=0) -> Tensor
  self: grad.tril(diagonal)
  result: auto_linear

- name: triu(Tensor self, int diagonal=0) -> Tensor
  self: grad.triu(diagonal)
  result: auto_linear

- name: trunc(Tensor self) -> Tensor
  self: zeros_like(grad)
  result: auto_element_wise

# DO NOT define a backward for to_dense
# See [Note: Sometimes view derivatives]
# - name: to_dense(Tensor self, ScalarType? dtype=None, *, bool? masked_grad=None) -> Tensor
#
- name: _to_dense(Tensor self, ScalarType? dtype=None, bool? masked_grad=None) -> Tensor
  self: to_dense_backward(grad, self, masked_grad)

# DO NOT define a backward for to_sparse.sparse_dim
# See [Note: Sometimes view derivatives]
# - name: to_sparse.sparse_dim(Tensor self, int sparse_dim) -> Tensor
#
- name: _to_sparse.sparse_dim(Tensor self, int sparse_dim) -> Tensor
  self: to_sparse_backward(grad, self.layout(), self.sym_blocksize())

# DO NOT define a backward for to_sparse
# See [Note: Sometimes view derivatives]
# - name: to_sparse(Tensor self, *, Layout? layout=None, int[2]? blocksize=None, int? dense_dim=None) -> Tensor
#
- name: _to_sparse(Tensor self, *, Layout? layout=None, int[2]? blocksize=None, int? dense_dim=None) -> Tensor
  self: to_sparse_backward(grad, self.layout(), self.sym_blocksize())

# DO NOT define a backward for to_sparse_csr
# See [Note: Sometimes view derivatives]
# - name: to_sparse_csr(Tensor self, int? dense_dim=None) -> Tensor
#
- name: _to_sparse_csr(Tensor self, int? dense_dim=None) -> Tensor
  self: to_sparse_backward(grad, self.layout(), self.sym_blocksize())

# DO NOT define a backward for to_sparse_csc
# See [Note: Sometimes view derivatives]
# - name: to_sparse_csc(Tensor self, int? dense_dim=None) -> Tensor
#
- name: _to_sparse_csc(Tensor self, int? dense_dim=None) -> Tensor
  self: to_sparse_backward(grad, self.layout(), self.sym_blocksize())

# DO NOT define a backward for to_sparse_bsr
# See [Note: Sometimes view derivatives]
# - name: to_sparse_bsr(Tensor self, int[2] blocksize, int? dense_dim=None) -> Tensor
#
- name: _to_sparse_bsr(Tensor self, int[2] blocksize, int? dense_dim=None) -> Tensor
  self: to_sparse_backward(grad, self.layout(), self.sym_blocksize())

# DO NOT define a backward for to_sparse_bsc
# See [Note: Sometimes view derivatives]
# - name: to_sparse_bsc(Tensor self, int[2] blocksize, int? dense_dim=None) -> Tensor
#
- name: _to_sparse_bsc(Tensor self, int[2] blocksize, int? dense_dim=None) -> Tensor
  self: to_sparse_backward(grad, self.layout(), self.sym_blocksize())

- name: to_mkldnn(Tensor self, ScalarType? dtype=None) -> Tensor
  self: to_mkldnn_backward(grad, self)

- name: unfold(Tensor(a) self, int dimension, int size, int step) -> Tensor(a)
  self: unfold_backward_symint(grad, self.sym_sizes(), dimension, size, step)
  result: auto_linear

- name: unfold_backward(Tensor grad_in, SymInt[] input_sizes, int dim, int size, int step) -> Tensor
  grad_in: grad.unfold(dim, size, step)
  result: auto_linear

- name: uniform_(Tensor(a!) self, float from=0, float to=1, *, Generator? generator=None) -> Tensor(a!)
  self: zeros_like(grad)
  result: self_t.zero_()

- name: _unique(Tensor self, bool sorted=True, bool return_inverse=False) -> (Tensor, Tensor)
  output_differentiability: [True, False]
  self: not_implemented("_unique")

- name: unique_dim(Tensor self, int dim, bool sorted=True, bool return_inverse=False, bool return_counts=False) -> (Tensor, Tensor, Tensor)
  output_differentiability: [True, False, False]
  self: not_implemented("unique_dim")

- name: unique_consecutive(Tensor self, bool return_inverse=False, bool return_counts=False, int? dim=None) -> (Tensor, Tensor, Tensor)
  output_differentiability: [True, False, False]
  self: not_implemented("unique_consecutive")

- name: unique_dim_consecutive(Tensor self, int dim, bool return_inverse=False, bool return_counts=False) -> (Tensor, Tensor, Tensor)
  output_differentiability: [True, False, False]
  self: not_implemented("unique_dim_consecutive")

- name: _unique2(Tensor self, bool sorted=True, bool return_inverse=False, bool return_counts=False) -> (Tensor, Tensor, Tensor)
  output_differentiability: [True, False, False]
  self: not_implemented("_unique2")

- name: _unsafe_view(Tensor self, SymInt[] size) -> Tensor
  self: grad.reshape_symint(self.sym_sizes())
  result: auto_linear

- name: lift(Tensor self) -> Tensor
  self: grad
  result: auto_linear

- name: lift_fresh(Tensor(a) self) -> Tensor(a)
  self: grad
  result: auto_linear

- name: unsqueeze(Tensor(a) self, int dim) -> Tensor(a)
  self: grad.squeeze(dim)
  result: auto_linear

- name: unsqueeze_(Tensor(a!) self, int dim) -> Tensor(a!)
  self: grad.squeeze(dim)
  result: auto_linear

- name: var.correction(Tensor self, int[1]? dim=None, *, Scalar? correction=None, bool keepdim=False) -> Tensor
  self: var_backward(grad, self, dim, correction, keepdim)
  # pointwise + sum
  result: at::real(var_backward(self_t.conj(), self_p, dim, correction, true).sum(dim.value_or(IntArrayRef({})), keepdim))

- name: var_mean.correction(Tensor self, int[1]? dim=None, *, Scalar? correction=None, bool keepdim=False) -> (Tensor, Tensor)
  self: var_mean_backward(grads[0], grads[1], self, dim, correction, keepdim)
  result0: at::real(var_backward(self_t.conj(), self_p, dim, correction, true).sum(dim.value_or(IntArrayRef({})), keepdim))
  # linear
  result1: mean(self_t, dim.value_or(IntArrayRef({})), keepdim)

- name: view(Tensor(a) self, SymInt[] size) -> Tensor(a)
  dispatch:
    Default:
      self: grad.reshape_symint(self.sym_sizes())
      result: auto_linear
    AutogradNestedTensor:
      self: grad.reshape_as(self)
      result: auto_linear

- name: view.dtype(Tensor(a) self, ScalarType dtype) -> Tensor(a)
  output_differentiability: [False]

- name: view_as_real(Tensor(a) self) -> Tensor(a)
  self: at::view_as_complex(grad.contiguous()) # gx0 + 1j * gx1
  result: at::view_as_real(self_t)

- name: view_as_complex(Tensor(a) self) -> Tensor(a)
  self: at::view_as_real(grad.contiguous().resolve_conj()) # [gx, gy]
  result: at::view_as_complex(self_t)

- name: where.self(Tensor condition, Tensor self, Tensor other) -> Tensor
  condition: non_differentiable
  self: where(condition, grad, 0)
  other: where(condition, 0, grad)
  result: where(condition, self_t, other_t)

# weight_norm_cuda_interface_backward does not have an explicitly defined derivative, so if we do happen
# to be running backward with create_graph=True, fall back to a backward function that uses
# differentiable ops.
- name: _weight_norm_interface(Tensor v, Tensor g, int dim=0) -> (Tensor, Tensor)
  v, g: "grad.defined() ? (GradMode::is_enabled() ? _weight_norm_differentiable_backward(grad.contiguous(), v, g, result1, dim) : _weight_norm_interface_backward(grad.contiguous(), v, g, result1, dim)) : std::tuple<Tensor, Tensor>()"

- name: zero_(Tensor(a!) self) -> Tensor(a!)
  self: zeros_like(grad)
  result: auto_linear

- name: sparse_mask(Tensor self, Tensor mask) -> Tensor
  self: sparse_mask_backward(grad, mask, self.layout())
  mask: non_differentiable

- name: _sparse_coo_tensor_with_dims_and_tensors(int sparse_dim, int dense_dim, SymInt[] size, Tensor indices, Tensor values, *, ScalarType? dtype=None, Layout? layout=None, Device? device=None, bool? pin_memory=False, bool? is_coalesced=None) -> Tensor
  indices: non_differentiable
  values: grad.sparse_mask(result)._values()

- name: sparse_compressed_tensor.comp_plain_value_size(Tensor compressed_indices, Tensor plain_indices, Tensor values, SymInt[] size, *, ScalarType? dtype=None, Layout? layout=None, Device? device=None, bool? pin_memory=False) -> Tensor
  compressed_indices: non_differentiable
  plain_indices: non_differentiable
  # TODO: remove to_dense after gh-107381 is fixed
  values: grad.to_dense().sparse_mask(result).values()

- name: _sparse_sum.dim(Tensor self, int[1] dim) -> Tensor
  self: at::_sparse_sum_backward(grad, self, dim)

- name: _standard_gamma(Tensor self, Generator? generator=None) -> Tensor
  self: grad * _standard_gamma_grad(self, result)

- name: _standard_gamma_grad(Tensor self, Tensor output) -> Tensor
  self: not_implemented("_standard_gamma_grad")

- name: values(Tensor(a) self) -> Tensor(a)
  dispatch:
    Default:
      self: values_backward(grad, self)
    AutogradNestedTensor:
      self: at::_nested_view_from_buffer(grad.contiguous(), self._nested_tensor_size(), self._nested_tensor_strides(), self._nested_tensor_storage_offsets())

# Why is _values() not differentiable?
# See NOTE [ Sparse: autograd and API ]
- name: _values(Tensor(a) self) -> Tensor(a)
  output_differentiability: [False]

# NN
- name: _trilinear(Tensor i1, Tensor i2, Tensor i3, int[] expand1, int[] expand2, int[] expand3, int[] sumdim, int unroll_dim=1) -> Tensor
  i1, i2, i3: "_trilinear_backward(grad,
               wrap_opt_if(i1, grad_input_mask[1] || grad_input_mask[2]),
               wrap_opt_if(i2, grad_input_mask[0] || grad_input_mask[2]),
               wrap_opt_if(i3, grad_input_mask[0] || grad_input_mask[1]),
               expand1, expand2, expand3, sumdim, grad_input_mask)"
  result: "_trilinear(i1_t, i2_p, i3_p, expand1, expand2, expand3, sumdim, unroll_dim) +
           _trilinear(i1_p, i2_t, i3_p, expand1, expand2, expand3, sumdim, unroll_dim) +
           _trilinear(i1_p, i2_p, i3_t, expand1, expand2, expand3, sumdim, unroll_dim)"

- name: constant_pad_nd(Tensor self, SymInt[] pad, Scalar value=0) -> Tensor
  self: constant_pad_nd_backward(grad, pad)
  result: constant_pad_nd_symint(self_t, pad, 0)

- name: binary_cross_entropy(Tensor self, Tensor target, Tensor? weight=None, int reduction=Mean) -> Tensor
  self: binary_cross_entropy_backward(grad, self, target, weight, reduction)
  target: binary_cross_entropy_target_backward(grad, self, target, weight, reduction)
  result: "apply_loss_reduction(
               binary_cross_entropy_backward(self_t, self_p, target_p, weight, at::Reduction::None)
             + binary_cross_entropy_target_backward(target_t, self_p, target_p, weight, at::Reduction::None),
           reduction)"

- name: binary_cross_entropy_backward(Tensor grad_output, Tensor self, Tensor target, Tensor? weight=None, int reduction=Mean) -> Tensor
  self: binary_cross_entropy_double_backward(grad_output, grad, self, target, weight, reduction)
  target: binary_cross_entropy_double_backward_target(grad, grad_output, self, target, weight, reduction)
  grad_output: binary_cross_entropy_double_backward_grad_output(grad, self, target, weight, reduction)
  result: " binary_cross_entropy_double_backward(grad_output_p, self_t, self_p, target_p, weight, reduction)
          + binary_cross_entropy_double_backward_target(target_t, grad_output_p, self_p, target_p, weight, reduction)
          + binary_cross_entropy_double_backward_grad_output(grad_output_t, self_p, target_p, weight, reduction)"

- name: binary_cross_entropy_with_logits(Tensor self, Tensor target, Tensor? weight=None, Tensor? pos_weight=None, int reduction=Mean) -> Tensor
  self: binary_cross_entropy_with_logits_backward(grad, self, target, weight, pos_weight, reduction)
  target: binary_cross_entropy_with_logits_target_backward(grad, self, target, weight, pos_weight, reduction)
  result: "apply_loss_reduction(
               binary_cross_entropy_with_logits_backward(self_t, self_p, target_p, weight, pos_weight, at::Reduction::None)
             + binary_cross_entropy_with_logits_target_backward(target_t, self_p, target_p, weight, pos_weight, at::Reduction::None),
           reduction)"

- name: embedding(Tensor weight, Tensor indices, SymInt padding_idx=-1, bool scale_grad_by_freq=False, bool sparse=False) -> Tensor
  indices: non_differentiable
  weight: embedding_backward_symint(grad, indices, weight.sym_size(0), padding_idx, scale_grad_by_freq, sparse)
  result: auto_linear

- name: embedding_dense_backward(Tensor grad_output, Tensor indices, SymInt num_weights, SymInt padding_idx, bool scale_grad_by_freq) -> Tensor
  grad_output: embedding_dense_double_backward_symint(grad, indices, padding_idx)
  indices: non_differentiable
  result: auto_linear

- name: _embedding_bag(Tensor weight, Tensor indices, Tensor offsets, bool scale_grad_by_freq=False, int mode=0, bool sparse=False, Tensor? per_sample_weights=None, bool include_last_offset=False, int padding_idx=-1) -> (Tensor, Tensor, Tensor, Tensor)
  indices: non_differentiable
  offsets: non_differentiable
  weight: _embedding_bag_backward_symint(grad, indices, offsets, result1, result2, result3, weight.sym_size(0), scale_grad_by_freq, mode, sparse, per_sample_weights, padding_idx)
  per_sample_weights: _embedding_bag_per_sample_weights_backward(grad, weight, indices, offsets, result1, mode, padding_idx)

- name: _embedding_bag_backward(Tensor grad, Tensor indices, Tensor offsets, Tensor offset2bag, Tensor bag_size, Tensor maximum_indices, SymInt num_weights, bool scale_grad_by_freq, int mode, bool sparse, Tensor? per_sample_weights, int padding_idx=-1) -> Tensor
  grad: not_implemented("_embedding_bag_backward")
  indices: non_differentiable
  offsets: non_differentiable
  offset2bag: non_differentiable
  bag_size: non_differentiable
  maximum_indices: non_differentiable
  per_sample_weights: not_implemented("_embedding_bag_backward")

- name: _embedding_bag_dense_backward(Tensor grad, Tensor indices, Tensor offset2bag, Tensor bag_size, Tensor maximum_indices, SymInt num_weights, bool scale_grad_by_freq, int mode, Tensor? per_sample_weights, int padding_idx=-1) -> Tensor
  grad: not_implemented("_embedding_bag_dense_backward")
  indices: non_differentiable
  offset2bag: non_differentiable
  bag_size: non_differentiable
  maximum_indices: non_differentiable
  per_sample_weights: not_implemented("_embedding_bag_dense_backward")

- name: embedding_renorm_(Tensor(a!) self, Tensor indices, float max_norm, float norm_type) -> Tensor(a!)
  indices: non_differentiable
  self: not_implemented("embedding_renorm")

- name: mse_loss(Tensor self, Tensor target, int reduction=Mean) -> Tensor
  self: mse_loss_backward(grad, self, target, reduction)
  target: mse_loss_backward(grad, target, self, reduction)
  result: apply_loss_reduction(mse_loss_backward(self_t.conj(), self_p, target_p, at::Reduction::None).conj() + mse_loss_backward(target_t.conj(), target_p, self_p, at::Reduction::None).conj(), reduction)

- name: multi_margin_loss(Tensor self, Tensor target, Scalar p=1, Scalar margin=1, Tensor? weight=None, int reduction=Mean) -> Tensor
  self: multi_margin_loss_backward(grad, self, target, p, margin, weight, reduction)
  target: non_differentiable

- name: multilabel_margin_loss_forward(Tensor self, Tensor target, int reduction) -> (Tensor output, Tensor is_target)
  self: multilabel_margin_loss_backward(grad, self, target, reduction, is_target)
  target: non_differentiable

- name: nll_loss_forward(Tensor self, Tensor target, Tensor? weight, int reduction, SymInt ignore_index) -> (Tensor output, Tensor total_weight)
  self: nll_loss_backward_symint(grad, self, target, weight, reduction, ignore_index, total_weight)
  target: non_differentiable
  output: std::get<0>(nll_loss_forward_symint(self_t, target, weight, reduction, ignore_index))

- name: nll_loss2d_forward(Tensor self, Tensor target, Tensor? weight, int reduction, SymInt ignore_index) -> (Tensor output, Tensor total_weight)
  self: nll_loss2d_backward_symint(grad, self, target, weight, reduction, ignore_index, total_weight)
  target: non_differentiable
  output: std::get<0>(nll_loss2d_forward_symint(self_t, target, weight, reduction, ignore_index))

- name: smooth_l1_loss(Tensor self, Tensor target, int reduction=Mean, float beta=1.0) -> Tensor
  self: smooth_l1_loss_backward(grad, self, target, reduction, beta)
  target: smooth_l1_loss_backward(grad, target, self, reduction, beta)
  result: apply_loss_reduction(smooth_l1_loss_backward(self_t.conj(), self_p, target_p, at::Reduction::None, beta).conj() + smooth_l1_loss_backward(target_t.conj(), target_p, self_p, at::Reduction::None, beta).conj(), reduction)

- name: huber_loss(Tensor self, Tensor target, int reduction=Mean, float delta=1.0) -> Tensor
  self: huber_loss_backward(grad, self, target, reduction, delta)
  target: huber_loss_backward(grad, target, self, reduction, delta)
  result: apply_loss_reduction(huber_loss_backward(self_t.conj(), self_p, target_p, at::Reduction::None, delta).conj() + huber_loss_backward(target_t.conj(), target_p, self_p, at::Reduction::None, delta).conj(), reduction)

- name: soft_margin_loss(Tensor self, Tensor target, int reduction=Mean) -> Tensor
  self: soft_margin_loss_backward(grad, self, target, reduction)
  result: apply_loss_reduction(soft_margin_loss_backward(self_t.conj(), self_p, target, at::Reduction::None).conj(), reduction)

- name: relu(Tensor self) -> Tensor
  self: threshold_backward(grad, result, 0)
  result: auto_element_wise

- name: silu(Tensor self) -> Tensor
  self: "GradMode::is_enabled() ? infinitely_differentiable_silu_backward(grad, self) : silu_backward(grad, self)"
  result: auto_element_wise

- name: mish(Tensor self) -> Tensor
  self: "GradMode::is_enabled() ? infinitely_differentiable_mish_backward(grad, self) : mish_backward(grad, self)"
  result: auto_element_wise

- name: elu(Tensor self, Scalar alpha=1, Scalar scale=1, Scalar input_scale=1) -> Tensor
  self: elu_backward(grad, alpha, scale, input_scale, /* is_result */ false, self)
  result: auto_element_wise

- name: elu_(Tensor(a!) self, Scalar alpha=1, Scalar scale=1, Scalar input_scale=1) -> Tensor(a!)
  self: elu_backward(grad, alpha, scale, input_scale, /* is_result */ true, result)
  result: self_t.copy_(elu_backward(original_self_t, alpha, scale, input_scale, /* is_result */ true, result))

- name: celu(Tensor self, Scalar alpha=1.0) -> Tensor
  self: elu_backward(grad, alpha, 1, 1.0/alpha.toFloat(), /* is_result */ false, self)
  result: auto_element_wise

- name: celu_(Tensor(a!) self, Scalar alpha=1.0) -> Tensor(a!)
  self: elu_backward(grad, alpha, 1, 1.0/alpha.toFloat(), /* is_result */ true, result)
  result: self_t.copy_(elu_backward(original_self_t, alpha, 1, 1.0/alpha.toFloat(), /* is_result */ true, result))

- name: gelu(Tensor self, *, str approximate='none') -> Tensor
  self: gelu_backward(grad, self, approximate)
  result: auto_element_wise

- name: gelu_backward(Tensor grad_output, Tensor self, *, str approximate='none') -> Tensor
  grad_output: gelu_backward(grad, self, approximate)
  self: gelu_double_backward(grad, grad_output, self, approximate)
  result: gelu_backward(grad_output_t, self_p, approximate) + gelu_double_backward(self_t, grad_output_p, self_p, approximate)

- name: glu(Tensor self, int dim=-1) -> Tensor
  # TODO: glu_backward can benefit from forward result,
  # and forward ad/forward over reverse ad for that matter
  self: glu_backward(grad, self, dim)
  result: glu_jvp(result, self_p, self_t, dim)

- name: hardshrink(Tensor self, Scalar lambd=0.5) -> Tensor
  self: hardshrink_backward(grad, self, lambd)
  result: auto_element_wise

- name: hardshrink_backward(Tensor grad_out, Tensor self, Scalar lambd) -> Tensor
  grad_out: hardshrink_backward(grad, self, lambd)
  self: zeros_like(grad)
  result: at::where((self_p > lambd).logical_or(self_p < -lambd), grad_out_t, at::zeros({}, result.options()).expand_as(result))

- name: hardtanh(Tensor self, Scalar min_val=-1, Scalar max_val=1) -> Tensor
  self: hardtanh_backward(grad, self, min_val, max_val)
  result: auto_element_wise

- name: leaky_relu(Tensor self, Scalar negative_slope=0.01) -> Tensor
  self: leaky_relu_backward(grad, self, negative_slope, false)
  result: auto_element_wise

- name: leaky_relu_(Tensor(a!) self, Scalar negative_slope=0.01) -> Tensor(a!)
  self: leaky_relu_backward(grad, result, negative_slope, true)
  result: self_t.copy_(leaky_relu_backward(original_self_t.conj(), result, negative_slope, true).conj())

- name: log_sigmoid_forward(Tensor self) -> (Tensor output, Tensor buffer)
  self: log_sigmoid_backward(grad, self, buffer)
  output: log_sigmoid_backward(self_t.conj(), self_p, buffer).conj()
  output_differentiability: [True, False]

- name: _log_softmax(Tensor self, int dim, bool half_to_float) -> Tensor
  self: _log_softmax_backward_data(grad, result, dim, self.scalar_type())
  result: self_t - logsumexp_jvp(self_p, self_t, {dim}, true)

- name: _sparse_log_softmax(Tensor self, int dim, bool half_to_float) -> Tensor
  self: _sparse_log_softmax_backward_data(grad, result, dim, self)

- name: _masked_softmax(Tensor self, Tensor mask, int? dim=None, int? mask_type=None) -> Tensor
  self: _masked_softmax_backward(grad, result, mask, dim)
  mask: non_differentiable

- name: _prelu_kernel(Tensor self, Tensor weight) -> Tensor
  self, weight: "grad.defined() ? _prelu_kernel_backward(grad, self, weight) : std::tuple<Tensor, Tensor>()"
  result: at::where(self_p >= 0, self_t, weight_p * self_t + weight_t * self_p)

- name: _prelu_kernel_backward(Tensor grad_output, Tensor self, Tensor weight) -> (Tensor, Tensor)
  grad_output: "grads[0].defined() ?
                (grads[1].defined() ? at::where(self >= 0, grads[0], grads[0] * weight + grads[1] * self)
                                    : at::where(self >= 0, grads[0], grads[0] * weight))
                                    : at::where(self >= 0, at::zeros({}, grad_output.options()), grads[1] * self)"
  self: "grads[1].defined() ? at::where(self >= 0, at::zeros({}, self.options()), grad_output * grads[1]) : zeros_like(self)"
  weight: "grads[0].defined() ? at::where(self >= 0, at::zeros({}, weight.options()), grad_output * grads[0]) : zeros_like(self)"
  result0: at::where(self_p >= 0, grad_output_t, grad_output_t * weight_p + grad_output_p * weight_t)
  result1: at::where(self_p >= 0, at::zeros({}, self_p.options()), grad_output_p * self_t + grad_output_t * self_p)

- name: rrelu_with_noise(Tensor self, Tensor noise, Scalar lower=0.125, Scalar upper=0.3333333333333333, bool training=False, Generator? generator=None) -> Tensor
  self: rrelu_with_noise_backward(grad, self, noise, lower, upper, training, false)
  result: auto_element_wise

- name: rrelu_with_noise_(Tensor(a!) self, Tensor noise, Scalar lower=0.125, Scalar upper=0.3333333333333333, bool training=False, Generator? generator=None) -> Tensor(a!)
  self: rrelu_with_noise_backward(grad, result, noise, lower, upper, training, true)

- name: _softmax(Tensor self, int dim, bool half_to_float) -> Tensor
  self: _softmax_backward_data(grad, result, dim, self.scalar_type())
  result: result * (self_t - logsumexp_jvp(self_p, self_t, {dim}, true))

- name: _sparse_softmax(Tensor self, int dim, bool half_to_float) -> Tensor
  self: _sparse_softmax_backward_data(grad, result, dim, self)

- name: _sparse_sparse_matmul(Tensor self, Tensor other) -> Tensor
  self: sparse_sparse_matmul_backward(grad, self, other, 0)
  other: sparse_sparse_matmul_backward(grad, self, other, 1)

- name: softplus(Tensor self, Scalar beta=1, Scalar threshold=20) -> Tensor
  self: softplus_backward(grad, self, beta, threshold)
  result: auto_element_wise

- name: softshrink(Tensor self, Scalar lambd=0.5) -> Tensor
  self: softshrink_backward(grad, self, lambd)
  result: auto_element_wise

- name: threshold(Tensor self, Scalar threshold, Scalar value) -> Tensor
  self: threshold_backward(grad, self, threshold)
  result: auto_element_wise

- name: threshold_(Tensor(a!) self, Scalar threshold, Scalar value) -> Tensor(a!)
  self: threshold_backward(grad, self, threshold)
  result: self_t.copy_(threshold_backward(self_t.conj(), original_self_p, threshold).conj())

- name: reflection_pad1d(Tensor self, SymInt[2] padding) -> Tensor
  self: reflection_pad1d_backward_symint(grad, self, padding)
  result: auto_linear

- name: reflection_pad2d(Tensor self, SymInt[4] padding) -> Tensor
  self: reflection_pad2d_backward_symint(grad, self, padding)
  result: auto_linear

- name: reflection_pad3d(Tensor self, SymInt[6] padding) -> Tensor
  self: reflection_pad3d_backward_symint(grad, self, padding)
  result: auto_linear

- name: replication_pad1d(Tensor self, SymInt[2] padding) -> Tensor
  self: replication_pad1d_backward_symint(grad, self, padding)
  result: auto_linear

- name: replication_pad2d(Tensor self, SymInt[4] padding) -> Tensor
  self: replication_pad2d_backward_symint(grad, self, padding)
  result: auto_linear

- name: replication_pad3d(Tensor self, SymInt[6] padding) -> Tensor
  self: replication_pad3d_backward_symint(grad, self, padding)
  result: auto_linear

- name: upsample_linear1d(Tensor self, SymInt[1] output_size, bool align_corners, float? scales=None) -> Tensor
  self: upsample_linear1d_backward_symint(grad, output_size, self.sym_sizes(), align_corners, scales)
  result: auto_linear

- name: upsample_bilinear2d(Tensor self, SymInt[2] output_size, bool align_corners, float? scales_h=None, float? scales_w=None) -> Tensor
  self: upsample_bilinear2d_backward_symint(grad, output_size, self.sym_sizes(), align_corners, scales_h, scales_w)
  result: auto_linear

- name: _upsample_bilinear2d_aa(Tensor self, SymInt[2] output_size, bool align_corners, float? scales_h=None, float? scales_w=None) -> Tensor
  self: _upsample_bilinear2d_aa_backward_symint(grad, output_size, self.sym_sizes(), align_corners, scales_h, scales_w)
  result: auto_linear

- name: upsample_bicubic2d(Tensor self, SymInt[2] output_size, bool align_corners, float? scales_h=None, float? scales_w=None) -> Tensor
  self: upsample_bicubic2d_backward_symint(grad, output_size, self.sym_sizes(), align_corners, scales_h, scales_w)
  result: auto_linear

- name: _upsample_bicubic2d_aa(Tensor self, SymInt[2] output_size, bool align_corners, float? scales_h=None, float? scales_w=None) -> Tensor
  self: _upsample_bicubic2d_aa_backward_symint(grad, output_size, self.sym_sizes(), align_corners, scales_h, scales_w)
  result: auto_linear

- name: upsample_trilinear3d(Tensor self, SymInt[3] output_size, bool align_corners, float? scales_d=None, float? scales_h=None, float? scales_w=None) -> Tensor
  self: upsample_trilinear3d_backward_symint(grad, output_size, self.sym_sizes(), align_corners, scales_d, scales_h, scales_w)
  result: auto_linear

- name: upsample_nearest1d(Tensor self, SymInt[1] output_size, float? scales=None) -> Tensor
  self: upsample_nearest1d_backward_symint(grad, output_size, self.sym_sizes(), scales)
  result: auto_linear

- name: _upsample_nearest_exact1d(Tensor self, SymInt[1] output_size, float? scales=None) -> Tensor
  self: _upsample_nearest_exact1d_backward_symint(grad, output_size, self.sym_sizes(), scales)
  result: auto_linear

- name: upsample_nearest2d(Tensor self, SymInt[2] output_size, float? scales_h=None, float? scales_w=None) -> Tensor
  self: upsample_nearest2d_backward_symint(grad, output_size, self.sym_sizes(), scales_h, scales_w)
  result: auto_linear

- name: _upsample_nearest_exact2d(Tensor self, SymInt[2] output_size, float? scales_h=None, float? scales_w=None) -> Tensor
  self: _upsample_nearest_exact2d_backward_symint(grad, output_size, self.sym_sizes(), scales_h, scales_w)
  result: auto_linear

- name: upsample_nearest3d(Tensor self, SymInt[3] output_size, float? scales_d=None, float? scales_h=None, float? scales_w=None) -> Tensor
  self: upsample_nearest3d_backward_symint(grad, output_size, self.sym_sizes(), scales_d, scales_h, scales_w)
  result: auto_linear

- name: _upsample_nearest_exact3d(Tensor self, SymInt[3] output_size, float? scales_d=None, float? scales_h=None, float? scales_w=None) -> Tensor
  self: _upsample_nearest_exact3d_backward_symint(grad, output_size, self.sym_sizes(), scales_d, scales_h, scales_w)
  result: auto_linear

- name: pixel_shuffle(Tensor self, int upscale_factor) -> Tensor
  self: pixel_unshuffle(grad, upscale_factor)
  result: auto_linear

- name: pixel_unshuffle(Tensor self, int downscale_factor) -> Tensor
  self: pixel_shuffle(grad, downscale_factor)
  result: auto_linear

- name: channel_shuffle(Tensor self, SymInt groups) -> Tensor
  self: channel_shuffle_symint(grad, grad.sym_size(1) / groups)
  result: auto_linear

- name: _adaptive_avg_pool2d(Tensor self, SymInt[2] output_size) -> Tensor
  self: _adaptive_avg_pool2d_backward(grad, self)
  result: auto_linear

- name: _adaptive_avg_pool3d(Tensor self, SymInt[3] output_size) -> Tensor
  self: _adaptive_avg_pool3d_backward(grad, self)
  result: auto_linear

- name: adaptive_max_pool2d(Tensor self, int[2] output_size) -> (Tensor, Tensor)
  self: adaptive_max_pool2d_backward(grad, self, result1)
  result0: gather(self_t.flatten(-2), -1, result1.flatten(-2)).view_as(result1)
  output_differentiability: [True, False]

- name: adaptive_max_pool3d(Tensor self, int[3] output_size) -> (Tensor, Tensor)
  self: adaptive_max_pool3d_backward(grad, self, result1)
  result0: gather(self_t.flatten(-3), -1, result1.flatten(-3)).view_as(result1)
  output_differentiability: [True, False]

- name: avg_pool2d(Tensor self, int[2] kernel_size, int[2] stride=[], int[2] padding=0, bool ceil_mode=False, bool count_include_pad=True, int? divisor_override=None) -> Tensor
  self: avg_pool2d_backward(grad, self, kernel_size, stride, padding, ceil_mode, count_include_pad, divisor_override)
  result: auto_linear

- name: avg_pool3d(Tensor self, int[3] kernel_size, int[3] stride=[], int[3] padding=0, bool ceil_mode=False, bool count_include_pad=True, int? divisor_override=None) -> Tensor
  self: avg_pool3d_backward(grad, self, kernel_size, stride, padding, ceil_mode, count_include_pad, divisor_override)
  result: auto_linear

- name: fractional_max_pool2d(Tensor self, int[2] kernel_size, int[2] output_size, Tensor random_samples) -> (Tensor, Tensor)
  self: fractional_max_pool2d_backward(grad, self, kernel_size, output_size, result1)
  result0: gather(self_t.flatten(-2), -1, result1.flatten(-2)).view_as(result1)
  output_differentiability: [True, False]

- name: fractional_max_pool3d(Tensor self, int[3] kernel_size, int[3] output_size, Tensor random_samples) -> (Tensor, Tensor)
  self: fractional_max_pool3d_backward(grad, self, kernel_size, output_size, result1)
  result0: gather(self_t.flatten(-3), -1, result1.flatten(-3)).view_as(result1)
  output_differentiability: [True, False]

- name: linear(Tensor input, Tensor weight, Tensor? bias=None) -> Tensor
  input, weight, bias: "grad.defined() ? linear_backward(input, grad, weight, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: linear_backward(Tensor self, Tensor grad_output, Tensor weight, bool[3] output_mask) -> (Tensor, Tensor, Tensor)
  self, grad_output, weight: linear_double_backward(grads, self, grad_output, weight)

#mps
- name: max_pool2d(Tensor self, int[2] kernel_size, int[2] stride=[], int[2] padding=0, int[2] dilation=1, bool ceil_mode=False) -> Tensor
  self: max_pool2d_backward(grad, self, kernel_size, stride, padding, dilation, ceil_mode)

- name: _mps_convolution(Tensor self, Tensor weight, Tensor? bias, SymInt[] padding, SymInt[] stride, SymInt[] dilation, SymInt groups) -> Tensor
  self, weight, bias: "grad.defined() ? mps_convolution_backward_symint(self, grad, weight, padding, stride, dilation, groups, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: mps_convolution_backward(Tensor self, Tensor grad_output, Tensor weight, SymInt[] padding, SymInt[] stride, SymInt[] dilation, SymInt groups, bool[3] output_mask) -> (Tensor, Tensor, Tensor)
  grad_output, self, weight: _convolution_double_backward_symint(grads[0], grads[1], grads[2], grad_output, weight, self, stride, padding, dilation, false, std::vector<c10::SymInt>(padding.size(), 0), groups, grad_input_mask)

- name: max_pool2d_with_indices(Tensor self, int[2] kernel_size, int[2] stride=[], int[2] padding=0, int[2] dilation=1, bool ceil_mode=False) -> (Tensor, Tensor)
  self: max_pool2d_with_indices_backward(grad, self, kernel_size, stride, padding, dilation, ceil_mode, result1)
  result0: gather(self_t.flatten(-2), -1, result1.flatten(-2)).view_as(result1)
  output_differentiability: [True, False]

- name: max_pool3d_with_indices(Tensor self, int[3] kernel_size, int[3] stride=[], int[3] padding=0, int[3] dilation=1, bool ceil_mode=False) -> (Tensor, Tensor)
  self: max_pool3d_with_indices_backward(grad, self, kernel_size, stride, padding, dilation, ceil_mode, result1)
  result0: gather(self_t.flatten(-3), -1, result1.flatten(-3)).view_as(result1)
  output_differentiability: [True, False]

- name: max_unpool2d(Tensor self, Tensor indices, SymInt[2] output_size) -> Tensor
  self: max_pool_double_backward(grad, indices, 2)
  indices: non_differentiable
  result: auto_linear

- name: max_unpool3d(Tensor self, Tensor indices, SymInt[3] output_size, int[3] stride, int[3] padding) -> Tensor
  self: max_pool_double_backward(grad, indices, 3)
  indices: non_differentiable
  result: auto_linear

- name: convolution(Tensor input, Tensor weight, Tensor? bias, SymInt[] stride, SymInt[] padding, SymInt[] dilation, bool transposed, SymInt[] output_padding, SymInt groups) -> Tensor
  input, weight, bias: "grad.defined() ? convolution_backward_symint(grad, input, weight, bias->sym_sizes(), stride, padding, dilation, transposed, output_padding, groups, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"
  result: convolution_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, stride, padding, dilation, transposed, output_padding, groups)

# TorchScript serializes calls to _convolution so this entry is present until that is changed to use convolution.
# Note that the benchmark, deterministic, cudnn_enabled, and allow_tf32 flags are queried from the global context
# by convolution_backward instead of being passed along from the forward pass.
- name: _convolution(Tensor input, Tensor weight, Tensor? bias, SymInt[] stride, SymInt[] padding, SymInt[] dilation, bool transposed, SymInt[] output_padding, SymInt groups, bool benchmark, bool deterministic, bool cudnn_enabled, bool allow_tf32) -> Tensor
  input, weight, bias: "grad.defined() ? convolution_backward_symint(grad, input, weight, bias->sym_sizes(), stride, padding, dilation, transposed, output_padding, groups, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"
  result: _convolution_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, stride, padding, dilation, transposed, output_padding, groups, benchmark, deterministic, cudnn_enabled, allow_tf32)

- name: convolution_backward(Tensor grad_output, Tensor input, Tensor weight, SymInt[]? bias_sizes, SymInt[] stride, SymInt[] padding, SymInt[] dilation, bool transposed, SymInt[] output_padding, SymInt groups, bool[3] output_mask) -> (Tensor, Tensor, Tensor)
  grad_output, input, weight: _convolution_double_backward_symint(grads[0], grads[1], grads[2], grad_output, weight, input, stride, padding, dilation, transposed, output_padding, groups, grad_input_mask)
  result0: std::get<0>(convolution_backward_symint(grad_output_p, input_p, weight_t, bias_sizes, stride, padding, dilation, transposed, output_padding, groups, {true, false, false})) + std::get<0>(convolution_backward_symint(grad_output_t, input_p, weight_p, bias_sizes, stride, padding, dilation, transposed, output_padding, groups, {true, false, false}))
  result1: std::get<1>(convolution_backward_symint(grad_output_p, input_t, weight_p, bias_sizes, stride, padding, dilation, transposed, output_padding, groups, {false, true, false})) + std::get<1>(convolution_backward_symint(grad_output_t, input_p, weight_p, bias_sizes, stride, padding, dilation, transposed, output_padding, groups, {false, true, false}))
  result2: convolution_backward_jvp_grad_bias(grad_output_t, result2)

- name: convolution_overrideable(Tensor input, Tensor weight, Tensor? bias, SymInt[] stride, SymInt[] padding, SymInt[] dilation, bool transposed, SymInt[] output_padding, SymInt groups) -> Tensor
  input, weight, bias: "grad.defined() ? convolution_backward_overrideable_symint(grad, input, weight, stride, padding, dilation, transposed, output_padding, groups, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: convolution_backward_overrideable(Tensor grad_output, Tensor input, Tensor weight, SymInt[] stride, SymInt[] padding, SymInt[] dilation, bool transposed, SymInt[] output_padding, SymInt groups, bool[3] output_mask) -> (Tensor grad_input, Tensor grad_weight, Tensor grad_bias)
  grad_output, input, weight: _convolution_double_backward_symint(grads[0], grads[1], grads[2], grad_output, weight, input, stride, padding, dilation, transposed, output_padding, groups, grad_input_mask)

- name: slow_conv_transpose2d(Tensor self, Tensor weight, SymInt[2] kernel_size, Tensor? bias=None, SymInt[2] stride=1, SymInt[2] padding=0, SymInt[2] output_padding=0, SymInt[2] dilation=1) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad, self, weight, bias->sym_sizes(), stride, padding, dilation, true, output_padding, 1, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: slow_conv_transpose3d(Tensor self, Tensor weight, SymInt[3] kernel_size, Tensor? bias=None, SymInt[3] stride=1, SymInt[3] padding=0, SymInt[3] output_padding=0, SymInt[3] dilation=1) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad, self, weight, bias->sym_sizes(), stride, padding, dilation, true, output_padding, 1, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: _slow_conv2d_forward(Tensor self, Tensor weight, SymInt[2] kernel_size, Tensor? bias, SymInt[2] stride, SymInt[2] padding) -> Tensor
  self, weight, bias: "grad.defined() ? _slow_conv2d_backward_symint(grad, self, weight, kernel_size, stride, padding, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: _slow_conv2d_backward.output_mask(Tensor grad_output, Tensor self, Tensor weight, SymInt[2] kernel_size, SymInt[2] stride, SymInt[2] padding, bool[3] output_mask) -> (Tensor grad_input, Tensor grad_weight, Tensor grad_bias)
  grad_output, self, weight: _convolution_double_backward_symint(grads[0], grads[1], grads[2], grad_output, weight, self, stride, padding, {{1, 1}}, false, {{0, 0}}, 1, grad_input_mask)

- name: _conv_depthwise2d(Tensor self, Tensor weight, SymInt[2] kernel_size, Tensor? bias, SymInt[2] stride, SymInt[2] padding, SymInt[2] dilation) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad.contiguous(), self, weight, bias->sym_sizes(), stride, padding, dilation, /*transposed=*/ false, /*output_padding=*/ {{0, 0}}, /*groups=*/ 1, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: conv_depthwise3d(Tensor self, Tensor weight, SymInt[3] kernel_size, Tensor? bias, SymInt[3] stride, SymInt[3] padding, SymInt[3] dilation) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad.contiguous(), self, weight, bias->sym_sizes(), stride, padding, dilation, /*transposed=*/ false, /*output_padding=*/ {{0, 0, 0}}, /*groups=*/ 1, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: slow_conv3d_forward(Tensor self, Tensor weight, SymInt[3] kernel_size, Tensor? bias, SymInt[3] stride, SymInt[3] padding) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad, self, weight, bias->sym_sizes(), stride, padding, /*dilation=*/ {{1, 1, 1}}, false, /*output_padding=*/ {{0, 0, 0}}, 1, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: slow_conv_dilated2d(Tensor self, Tensor weight, SymInt[2] kernel_size, Tensor? bias=None, SymInt[2] stride=1, SymInt[2] padding=0, SymInt[2] dilation=1) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad, self, weight, bias->sym_sizes(), stride, padding, dilation, false, std::vector<c10::SymInt>(padding.size(), 0), 1, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: slow_conv_dilated3d(Tensor self, Tensor weight, SymInt[3] kernel_size, Tensor? bias=None, SymInt[3] stride=1, SymInt[3] padding=0, SymInt[3] dilation=1) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad, self, weight, bias->sym_sizes(), stride, padding, dilation, false, std::vector<c10::SymInt>(padding.size(), 0), 1, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: col2im(Tensor self, SymInt[2] output_size, int[2] kernel_size, int[2] dilation, int[2] padding, int[2] stride) -> Tensor
  self: im2col(grad, kernel_size, dilation, padding, stride)
  result: auto_linear

- name: im2col(Tensor self, int[2] kernel_size, int[2] dilation, int[2] padding, int[2] stride) -> Tensor
  self: col2im_symint(grad, {self.sym_size(-2), self.sym_size(-1)}, kernel_size, dilation, padding, stride)
  result: auto_linear

- name: _adaptive_avg_pool2d_backward(Tensor grad_output, Tensor self) -> Tensor
  grad_output: _adaptive_avg_pool2d_symint(grad, {grad_output.sym_size(-2), grad_output.sym_size(-1)})
  self: zeros_like(self)
  result: _adaptive_avg_pool2d_backward(grad_output_t, self_p)

- name: _adaptive_avg_pool3d_backward(Tensor grad_output, Tensor self) -> Tensor
  grad_output: _adaptive_avg_pool3d_symint(grad, { grad_output.sym_size(-3), grad_output.sym_size(-2), grad_output.sym_size(-1) })
  self: zeros_like(self)
  result: _adaptive_avg_pool3d_backward(grad_output_t, self_p)

- name: adaptive_max_pool2d_backward(Tensor grad_output, Tensor self, Tensor indices) -> Tensor
  grad_output: max_pool_double_backward(grad, indices, 2)
  self: zeros_like(self)
  result: auto_linear

- name: adaptive_max_pool3d_backward(Tensor grad_output, Tensor self, Tensor indices) -> Tensor
  grad_output: max_pool_double_backward(grad, indices, 3)
  self: zeros_like(self)
  result: auto_linear

- name: avg_pool2d_backward(Tensor grad_output, Tensor self, int[2] kernel_size, int[2] stride, int[2] padding, bool ceil_mode, bool count_include_pad, int? divisor_override) -> Tensor
  grad_output: avg_pool2d(grad, kernel_size, stride, padding, ceil_mode, count_include_pad, divisor_override)
  self: zeros_like(self)
  result: avg_pool2d_backward(grad_output_t, self_p, kernel_size, stride, padding, ceil_mode, count_include_pad, divisor_override)

- name: avg_pool3d_backward(Tensor grad_output, Tensor self, int[3] kernel_size, int[3] stride, int[3] padding, bool ceil_mode, bool count_include_pad, int? divisor_override) -> Tensor
  grad_output: avg_pool3d(grad, kernel_size, stride, padding, ceil_mode, count_include_pad, divisor_override)
  self: zeros_like(self)
  result: avg_pool3d_backward(grad_output_t, self_p, kernel_size, stride, padding, ceil_mode, count_include_pad, divisor_override)

- name: elu_backward(Tensor grad_output, Scalar alpha, Scalar scale, Scalar input_scale, bool is_result, Tensor self_or_result) -> Tensor
  grad_output: elu_backward(grad, alpha, scale, input_scale, is_result, self_or_result)
  self_or_result: elu_double_backward(grad, grad_output, alpha, scale, input_scale, is_result, self_or_result)
  result: elu_backward(grad_output_t, alpha, scale, input_scale, is_result, self_or_result_p) + elu_double_backward(self_or_result_t, grad_output_p, alpha, scale, input_scale, is_result, self_or_result_p)

- name: fractional_max_pool2d_backward(Tensor grad_output, Tensor self, int[2] kernel_size, int[2] output_size, Tensor indices) -> Tensor
  grad_output: max_pool_double_backward(grad, indices, 2)
  self: zeros_like(self)
  result: auto_linear

- name: fractional_max_pool3d_backward(Tensor grad_output, Tensor self, int[3] kernel_size, int[3] output_size, Tensor indices) -> Tensor
  grad_output: max_pool_double_backward(grad, indices, 3)
  self: zeros_like(self)
  result: auto_linear

- name: glu_backward(Tensor grad_output, Tensor self, int dim) -> Tensor
  grad_output: glu_double_backward_grad_output(grad, self, dim)
  self: glu_double_backward(grad, grad_output, self, dim)
  result: glu_backward_jvp(result, grad_output_p, self_p, grad_output_t, self_t, dim)

- name: hardtanh_backward(Tensor grad_output, Tensor self, Scalar min_val, Scalar max_val) -> Tensor
  grad_output: hardtanh_backward(grad, self, min_val, max_val)
  self: zeros_like(grad)
  result: at::where((self_p > min_val).logical_and(self_p < max_val), grad_output_t, at::zeros({}, result.options()).expand_as(result))

- name: log_sigmoid_backward(Tensor grad_output, Tensor self, Tensor buffer) -> Tensor
  grad_output: log_sigmoid_backward(grad, self, buffer)
  self: log_sigmoid_double_backward(grad * grad_output, self)
  result: log_sigmoid_backward(grad_output_t, self_p, buffer) + log_sigmoid_double_backward(self_t * grad_output_p, self_p)

- name: _log_softmax_backward_data(Tensor grad_output, Tensor output, int dim, ScalarType input_dtype) -> Tensor
  grad_output: grad.to(output.dtype()) - (grad.to(output.dtype()) * output.exp()).sum(dim, true)
  output: (-grad_output.sum(dim, true) * output.exp() * grad.to(output.dtype())).to(output.dtype())

- name: leaky_relu_backward(Tensor grad_output, Tensor self, Scalar negative_slope, bool self_is_result) -> Tensor
  # self_is_result is always false here since double backward call is an out-of-place call, self is input itself
  grad_output: leaky_relu_backward(grad, self, negative_slope, false)
  self: zeros_like(grad)
  # leaky_relu_backward(grad_output, self, negative_slope, false)
  # computes grad_output * at::where(self_p > 0, 1, negative_slope)
  # so the jvp formula is the following:
  # grad_output_t * at::where(self_p > 0, self_p.new_ones([]), negative_slope);
  #
  # leaky_relu_backward(grad_output, result, negative_slope, true)
  # computes grad_output * at::where(result > 0, 1, negative_slope)
  # under the assumption that `negative_slope` is positive (otherwise,
  # it is not possible to compute the gradient).
  #
  # so the jvp formula is the following:
  # grad_output_t * at::where(result_p > 0, result_p.new_ones([]), negative_slope);
  # with the assumption that negative_slope is positive.
  #
  # Combined together that results in the following optimized kernel which
  # also checks the assumption that negative_slope is positive when self_is_result
  # is True:
  result: leaky_relu_backward(grad_output_t, self_p, negative_slope, self_is_result)

# This derivative is mps-only, and `error_for_max_pool2d_double_backward` just raises an error.
- name: max_pool2d_backward(Tensor grad_output, Tensor self, int[2] kernel_size, int[2] stride=[], int[2] padding=0, int[2] dilation=1, bool ceil_mode=False) -> Tensor
  grad_output: error_for_max_pool2d_double_backward()
  self: zeros_like(self)
  result: auto_linear

- name: max_pool2d_with_indices_backward(Tensor grad_output, Tensor self, int[2] kernel_size, int[2] stride, int[2] padding, int[2] dilation, bool ceil_mode, Tensor indices) -> Tensor
  grad_output: max_pool_double_backward(grad, indices, 2)
  self: zeros_like(self)
  indices: non_differentiable
  result: auto_linear

- name: max_pool3d_with_indices_backward(Tensor grad_output, Tensor self, int[3] kernel_size, int[3] stride, int[3] padding, int[3] dilation, bool ceil_mode, Tensor indices) -> Tensor
  grad_output: max_pool_double_backward(grad, indices, 3)
  self: zeros_like(self)
  indices: non_differentiable
  result: auto_linear

- name: mse_loss_backward(Tensor grad_output, Tensor self, Tensor target, int reduction) -> Tensor
  grad_output: mse_loss_backward(grad, self, target, reduction)
  self: mse_loss_double_backward(grad * grad_output, self, reduction)
  target: -mse_loss_double_backward(grad * grad_output, target, reduction)
  result: "  mse_loss_double_backward(self_t * grad_output_p, self_p, reduction)
           - mse_loss_double_backward(target_t * grad_output_p, target_p, reduction)
           + mse_loss_backward(grad_output_t, self_p, target_p, reduction)
          "

- name: nll_loss_backward(Tensor grad_output, Tensor self, Tensor target, Tensor? weight, int reduction, SymInt ignore_index, Tensor total_weight) -> Tensor
  grad_output: nll_loss_symint(grad, target, weight, reduction, ignore_index)
  self: zeros_like(grad)
  target: non_differentiable

- name: nll_loss2d_backward(Tensor grad_output, Tensor self, Tensor target, Tensor? weight, int reduction, SymInt ignore_index, Tensor total_weight) -> Tensor
  grad_output: nll_loss2d_symint(grad, target, weight, reduction, ignore_index)
  self: zeros_like(grad)
  target: non_differentiable

- name: rrelu_with_noise_backward(Tensor grad_output, Tensor self, Tensor noise, Scalar lower, Scalar upper, bool training, bool self_is_result) -> Tensor
  # self_is_result is always false here since double backward call is an out-of-place call, self is input itself
  grad_output: rrelu_with_noise_backward(grad, self, noise, lower, upper, training, false)
  self: zeros_like(grad)
  result: rrelu_with_noise_backward(grad_output_t, self_p, noise, lower, upper, training, false)

- name: reflection_pad1d_backward(Tensor grad_output, Tensor self, SymInt[2] padding) -> Tensor
  grad_output: reflection_pad1d_symint(grad, padding)
  self: zeros_like(self)
  result: reflection_pad1d_backward_symint(grad_output_t, self_p, padding)

- name: reflection_pad2d_backward(Tensor grad_output, Tensor self, SymInt[4] padding) -> Tensor
  grad_output: reflection_pad2d_symint(grad, padding)
  self: zeros_like(self)
  result: reflection_pad2d_backward_symint(grad_output_t, self_p, padding)

- name: reflection_pad3d_backward(Tensor grad_output, Tensor self, SymInt[6] padding) -> Tensor
  grad_output: reflection_pad3d_symint(grad, padding)
  self: zeros_like(self)
  result: reflection_pad3d_backward_symint(grad_output_t, self_p, padding)

- name: replication_pad1d_backward(Tensor grad_output, Tensor self, SymInt[2] padding) -> Tensor
  grad_output: replication_pad1d_symint(grad, padding)
  self: zeros_like(self)
  result: replication_pad1d_backward_symint(grad_output_t, self_p, padding)

- name: replication_pad2d_backward(Tensor grad_output, Tensor self, SymInt[4] padding) -> Tensor
  grad_output: replication_pad2d_symint(grad, padding)
  self: zeros_like(self)
  result: replication_pad2d_backward_symint(grad_output_t, self_p, padding)

- name: replication_pad3d_backward(Tensor grad_output, Tensor self, SymInt[6] padding) -> Tensor
  grad_output: replication_pad3d_symint(grad, padding)
  self: zeros_like(self)
  result: replication_pad3d_backward_symint(grad_output_t, self_p, padding)

- name: sparse_sampled_addmm(Tensor self, Tensor mat1, Tensor mat2, *, Scalar beta=1, Scalar alpha=1) -> Tensor
  self, mat1, mat2: "sparse_sampled_addmm_backward(grad,
                                                   self,
                                                   wrap_opt_if(mat1, grad_input_mask[2]),
                                                   wrap_opt_if(mat2, grad_input_mask[1]),
                                                   alpha, beta, grad_input_mask)"

- name: _sparse_mm_reduce_impl(Tensor self, Tensor other, str reduce) -> (Tensor, Tensor)
  output_differentiability: [True, False]
  self, other: "grad.defined() ? _sparse_mm_reduce_impl_backward(self, grad, other, reduce, result1, grad_input_mask) :  std::tuple<Tensor, Tensor>()"

- name: smooth_l1_loss_backward(Tensor grad_output, Tensor self, Tensor target, int reduction, float beta) -> Tensor
  grad_output: smooth_l1_loss_backward(grad, self, target, reduction, beta)
  self: smooth_l1_loss_double_backward(grad * grad_output, self, target, reduction, beta)
  target: -smooth_l1_loss_double_backward(grad * grad_output, self, target, reduction, beta)
  result: "  smooth_l1_loss_double_backward(self_t * grad_output_p, self_p, target_p, reduction, beta)
           - smooth_l1_loss_double_backward(target_t * grad_output_p, self_p, target_p, reduction, beta)
           + smooth_l1_loss_backward(grad_output_t, self_p, target_p, reduction, beta)
          "

- name: huber_loss_backward(Tensor grad_output, Tensor self, Tensor target, int reduction, float delta) -> Tensor
  grad_output: huber_loss_double_backward_grad_output(grad, grad_output, self, target, reduction, delta)
  self: huber_loss_double_backward(grad * grad_output, self, target, reduction, delta)
  target: -huber_loss_double_backward(grad * grad_output, self, target, reduction, delta)

- name: softplus_backward(Tensor grad_output, Tensor self, Scalar beta, Scalar threshold) -> Tensor
  grad_output: softplus_backward(grad, self, beta, threshold)
  self: softplus_double_backward(grad * grad_output, self, beta, threshold)
  result: "softplus_backward(grad_output_t, self_p, beta, threshold)
         + softplus_double_backward(self_t * grad_output_p, self_p, beta, threshold)"

- name: _softmax_backward_data(Tensor grad_output, Tensor output, int dim, ScalarType input_dtype) -> Tensor
  grad_output: _softmax_backward_data(grad.to(output.dtype()), output, dim, input_dtype)
  output: softmax_double_backward(grad.to(output.dtype()), grad_output, dim, output).to(output.dtype())

- name: soft_margin_loss_backward(Tensor grad_output, Tensor self, Tensor target, int reduction) -> Tensor
  grad_output: soft_margin_loss_double_backward_grad_output(grad, grad_output, self, target, reduction)
  self: soft_margin_loss_double_backward(grad * grad_output, self, target, reduction)

- name: softshrink_backward(Tensor grad_output, Tensor self, Scalar lambd) -> Tensor
  grad_output: softshrink_backward(grad, self, lambd)
  self: zeros_like(grad)
  result: at::where((self_p > lambd).logical_or(self_p < -lambd), grad_output_t, at::zeros({}, result.options()).expand_as(result))

- name: threshold_backward(Tensor grad_output, Tensor self, Scalar threshold) -> Tensor
  grad_output: threshold_backward(grad, self, threshold)
  self: zeros_like(grad)
  result: zeros_like(self_t) + threshold_backward(grad_output_t, self_p, threshold)

- name: upsample_linear1d_backward(Tensor grad_output, SymInt[1] output_size, SymInt[3] input_size, bool align_corners, float? scales=None) -> Tensor
  grad_output: upsample_linear1d_symint(grad, output_size, align_corners, scales)
  result: auto_linear

- name: upsample_bilinear2d_backward(Tensor grad_output, SymInt[2] output_size, SymInt[4] input_size, bool align_corners, float? scales_h=None, float? scales_w=None) -> Tensor
  grad_output: upsample_bilinear2d_symint(grad, output_size, align_corners, scales_h, scales_w)
  result: auto_linear

- name: _upsample_bilinear2d_aa_backward(Tensor grad_output, SymInt[2] output_size, SymInt[4] input_size, bool align_corners, float? scales_h=None, float? scales_w=None) -> Tensor
  grad_output: _upsample_bilinear2d_aa_symint(grad, output_size, align_corners, scales_h, scales_w)
  result: auto_linear

- name: upsample_bicubic2d_backward(Tensor grad_output, SymInt[2] output_size, SymInt[4] input_size, bool align_corners, float? scales_h=None, float? scales_w=None) -> Tensor
  grad_output: upsample_bicubic2d_symint(grad, output_size, align_corners, scales_h, scales_w)
  result: auto_linear

- name: _upsample_bicubic2d_aa_backward(Tensor grad_output, SymInt[2] output_size, SymInt[4] input_size, bool align_corners, float? scales_h=None, float? scales_w=None) -> Tensor
  grad_output: _upsample_bicubic2d_aa_symint(grad, output_size, align_corners, scales_h, scales_w)
  result: auto_linear

- name: upsample_trilinear3d_backward(Tensor grad_output, SymInt[3] output_size, SymInt[5] input_size, bool align_corners, float? scales_d=None, float? scales_h=None, float? scales_w=None) -> Tensor
  grad_output: upsample_trilinear3d_symint(grad, output_size, align_corners, scales_d, scales_h, scales_w)
  result: auto_linear

- name: upsample_nearest1d_backward(Tensor grad_output, SymInt[1] output_size, SymInt[3] input_size, float? scales=None) -> Tensor
  grad_output: upsample_nearest1d_symint(grad, output_size, scales)
  result: auto_linear

- name: _upsample_nearest_exact1d_backward(Tensor grad_output, SymInt[1] output_size, SymInt[3] input_size, float? scales=None) -> Tensor
  grad_output: _upsample_nearest_exact1d_symint(grad, output_size, scales)
  result: auto_linear

- name: upsample_nearest2d_backward(Tensor grad_output, SymInt[2] output_size, SymInt[4] input_size, float? scales_h=None, float? scales_w=None) -> Tensor
  grad_output: upsample_nearest2d_symint(grad, output_size, scales_h, scales_w)
  result: auto_linear

- name: _upsample_nearest_exact2d_backward(Tensor grad_output, SymInt[2] output_size, SymInt[4] input_size, float? scales_h=None, float? scales_w=None) -> Tensor
  grad_output: _upsample_nearest_exact2d_symint(grad, output_size, scales_h, scales_w)
  result: auto_linear

- name: upsample_nearest3d_backward(Tensor grad_output, SymInt[3] output_size, SymInt[5] input_size, float? scales_d=None, float? scales_h=None, float? scales_w=None) -> Tensor
  grad_output: upsample_nearest3d_symint(grad, output_size, scales_d, scales_h, scales_w)
  result: auto_linear

- name: _upsample_nearest_exact3d_backward(Tensor grad_output, SymInt[3] output_size, SymInt[5] input_size, float? scales_d=None, float? scales_h=None, float? scales_w=None) -> Tensor
  grad_output: _upsample_nearest_exact3d_symint(grad, output_size, scales_d, scales_h, scales_w)
  result: auto_linear

- name: sigmoid_backward(Tensor grad_output, Tensor output) -> Tensor
  grad_output: sigmoid_backward(grad, output.conj())
  output: grad.conj() * grad_output * (-2 * output.conj() + 1)
  result: sigmoid_backward(grad_output_t, output_p) + output_t.conj() * grad_output_p * (-2 * output_p.conj() + 1)

- name: tanh_backward(Tensor grad_output, Tensor output) -> Tensor
  grad_output: tanh_backward(grad, output.conj())
  output: grad.conj() * (-2 * output.conj() * grad_output)
  result: tanh_backward(grad_output_t, output_p) + output_t.conj() * (-2 * output_p.conj() * grad_output_p)

# cudnn
- name: _cudnn_ctc_loss(Tensor log_probs, Tensor targets, int[] input_lengths, int[] target_lengths, int blank, bool deterministic, bool zero_infinity) -> (Tensor, Tensor)
  log_probs: _cudnn_ctc_loss_backward(grad, result0, result1, zero_infinity)

- name: _cudnn_ctc_loss.Tensor(Tensor log_probs, Tensor targets, Tensor input_lengths, Tensor target_lengths, int blank, bool deterministic, bool zero_infinity) -> (Tensor, Tensor)
  log_probs: _cudnn_ctc_loss_backward(grad, result0, result1, zero_infinity)

- name: cudnn_convolution_transpose(Tensor self, Tensor weight, SymInt[] padding, SymInt[] output_padding, SymInt[] stride, SymInt[] dilation, SymInt groups, bool benchmark, bool deterministic, bool allow_tf32) -> Tensor
  self, weight: "_cudnn_convolution_backward(self, grad, weight, padding, output_padding, stride, dilation, true, groups, {grad_input_mask[0], grad_input_mask[1]})"

- name: _mps_convolution_transpose(Tensor self, Tensor weight, SymInt[] padding, SymInt[] output_padding, SymInt[] stride, SymInt[] dilation, SymInt groups) -> Tensor
  self, weight: "grad.defined() ? mps_convolution_transpose_backward_symint(self, grad, weight, padding, output_padding, stride, dilation, groups, grad_input_mask) : std::tuple<Tensor, Tensor>()"

- name: cudnn_convolution(Tensor self, Tensor weight, SymInt[] padding, SymInt[] stride, SymInt[] dilation, SymInt groups, bool benchmark, bool deterministic, bool allow_tf32) -> Tensor
  self, weight: "_cudnn_convolution_backward(self, grad, weight, padding, std::vector<c10::SymInt>(padding.size(), 0), stride, dilation, false, groups, {grad_input_mask[0], grad_input_mask[1]})"

- name: cudnn_grid_sampler(Tensor self, Tensor grid) -> Tensor output
  self, grid: "grad.defined() ? cudnn_grid_sampler_backward(self, grid, grad) : std::tuple<Tensor, Tensor>()"

- name: cudnn_affine_grid_generator(Tensor theta, int N, int C, int H, int W) -> Tensor grid
  theta: cudnn_affine_grid_generator_backward(grad, N, C, H, W)

# NB: Why is the backwards here so complicated?  CuDNN cannot be used to compute
# backward in evaluation mode, because the math for backward in evaluation mode
# is different (since the forward math is different), and CuDNN does not support
# it.  And in any case, you shouldn't be using this bn in evaluation mode,
# because it should be merged into the previous convolution (left for future
# work.)
# NB2: The quotes around the gradient are needed to appease YAML parsing rules.
- name: cudnn_batch_norm(Tensor input, Tensor weight, Tensor? bias, Tensor? running_mean, Tensor? running_var, bool training, float exponential_average_factor, float epsilon) -> (Tensor, Tensor, Tensor, Tensor)
  input, weight, bias: "grad.defined() ? (training ? cudnn_batch_norm_backward(input, grad.contiguous(input.suggest_memory_format()), weight, running_mean, running_var, result1, result2, epsilon, retain_variables ? result3.clone() : result3) : native_batch_norm_backward(grad, input, weight, running_mean, running_var, result1, result2, training, epsilon, grad_input_mask)) : std::tuple<Tensor, Tensor, Tensor>()"
  result0: batch_norm_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, running_mean, running_var, result1, result2, training, epsilon)

# HACK: save_mean and save_var are going to be passed in as
# requires_grad variables (even though we'll never backprop through
# them) so we need to prevent the unpacking from triggering an error.
- name: cudnn_batch_norm_backward(Tensor input, Tensor grad_output, Tensor weight, Tensor? running_mean, Tensor? running_var, Tensor? save_mean, Tensor? save_var, float epsilon, Tensor reserveSpace) -> (Tensor, Tensor, Tensor)
  save_mean: not_implemented("cudnn_batch_norm_backward save_mean")
  save_var: not_implemented("cudnn_batch_norm_backward save_var")
  reserveSpace: not_implemented("cudnn_batch_norm_backward reserveSpace")
  input, weight, grad_output: batchnorm_double_backward(input, weight, grads[0], grads[1], grads[2], grad_output, running_mean, running_var, true, epsilon, save_mean, save_var, grad_input_mask)

# nnpack

- name: _nnpack_spatial_convolution(Tensor input, Tensor weight, Tensor? bias, SymInt[2] padding, SymInt[2] stride=1) -> Tensor
  # NNPACK does not support strided convolutions in the backwards path, which is the reason why we are using the closest available function that does here.
  input, weight, bias: "grad.defined() ? convolution_backward_symint(grad, input, weight, bias->sym_sizes(), stride, padding, std::vector<c10::SymInt>(padding.size(), 1), false, std::vector<c10::SymInt>(padding.size(), 0), 1, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

#LSTM MPS
- name: _lstm_mps(Tensor input, Tensor[] hx, Tensor[] params, bool has_biases, int num_layers, float dropout, bool train, bool bidirectional, bool batch_first) -> (Tensor, Tensor, Tensor, Tensor, Tensor, Tensor)
  output_differentiability: [True, True, True, False, False, False]
  input, hx, params: "lstm_mps_backward(grads[0], grads[1], grads[2], result3, result4, input, result5, hx, params, has_biases, num_layers, dropout, train, bidirectional, batch_first)"

- name: lstm_mps_backward(Tensor? grad_y, Tensor? grad_hy, Tensor? grad_cy, Tensor z_state, Tensor cell_state_fwd, Tensor input, Tensor layersOutputs, Tensor[] hx, Tensor[] params, bool has_biases, int num_layers, float dropout, bool train, bool bidirectional, bool batch_first) -> (Tensor, Tensor[], Tensor[])



# Only frst three of _cudnn_rnn outputs can have gradients.
# _cudnn_rnn outputs: (output, hy, cy, reserve, weight_buf)
- name: _cudnn_rnn(Tensor input, Tensor[] weight, int weight_stride0, Tensor? weight_buf, Tensor hx, Tensor? cx, int mode, SymInt hidden_size, SymInt proj_size, int num_layers, bool batch_first, float dropout, bool train, bool bidirectional, SymInt[] batch_sizes, Tensor? dropout_state) -> (Tensor, Tensor, Tensor, Tensor, Tensor)
  dropout_state: non_differentiable
  output_differentiability: [True, True, True, False, False]
  input, hx, cx, weight: "_cudnn_rnn_backward_symint(input, weight, weight_stride0, result4, hx, cx, result0, grads[0], grads[1], grads[2], mode, hidden_size, proj_size, num_layers, batch_first, dropout, train, bidirectional, batch_sizes, dropout_state, retain_variables ? result3.clone() : result3, grad_input_mask)"

- name: _cudnn_rnn_backward(Tensor input, Tensor[] weight, int weight_stride0, Tensor weight_buf, Tensor hx, Tensor? cx, Tensor output, Tensor? grad_output, Tensor? grad_hy, Tensor? grad_cy, int mode, SymInt hidden_size, SymInt proj_size, int num_layers, bool batch_first, float dropout, bool train, bool bidirectional, SymInt[] batch_sizes, Tensor? dropout_state, Tensor reserve, bool[4] output_mask) -> (Tensor, Tensor, Tensor, Tensor[])
  dropout_state: non_differentiable
  input: not_implemented("_cudnn_rnn_backward", kCudnnDoubleBackwardMsg)
  weight: not_implemented_list("_cudnn_rnn_backward", kCudnnDoubleBackwardMsg)
  hx: not_implemented("_cudnn_rnn_backward", kCudnnDoubleBackwardMsg)
  cx: not_implemented("_cudnn_rnn_backward", kCudnnDoubleBackwardMsg)
  output: not_implemented("_cudnn_rnn_backward", kCudnnDoubleBackwardMsg)
  grad_output: not_implemented("_cudnn_rnn_backward", kCudnnDoubleBackwardMsg)
  grad_hy: not_implemented("_cudnn_rnn_backward", kCudnnDoubleBackwardMsg)
  grad_cy: not_implemented("_cudnn_rnn_backward", kCudnnDoubleBackwardMsg)

# miopen

- name: miopen_convolution_transpose(Tensor self, Tensor weight, Tensor? bias, SymInt[] padding, SymInt[] output_padding, SymInt[] stride, SymInt[] dilation, SymInt groups, bool benchmark, bool deterministic) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad, self, weight, bias->sym_sizes(), stride, padding, dilation, true, output_padding, groups, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: miopen_convolution(Tensor self, Tensor weight, Tensor? bias, SymInt[] padding, SymInt[] stride, SymInt[] dilation, SymInt groups, bool benchmark, bool deterministic) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad, self, weight, bias->sym_sizes(), stride, padding, dilation, false, std::vector<c10::SymInt>(padding.size(), 0), groups, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: miopen_depthwise_convolution(Tensor self, Tensor weight, Tensor? bias, SymInt[] padding, SymInt[] stride, SymInt[] dilation, SymInt groups, bool benchmark, bool deterministic) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad, self, weight, bias->sym_sizes(), stride, padding, dilation, false, std::vector<c10::SymInt>(padding.size(), 0), groups, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: miopen_batch_norm(Tensor input, Tensor weight, Tensor? bias, Tensor? running_mean, Tensor? running_var, bool training, float exponential_average_factor, float epsilon) -> (Tensor, Tensor, Tensor)
  input, weight, bias: "grad.defined() ? (training ? miopen_batch_norm_backward(input, grad.contiguous(), weight, running_mean, running_var, result1, result2, epsilon) : native_batch_norm_backward(grad, input, weight, running_mean, running_var, result1, result2, training, epsilon, grad_input_mask)) : std::tuple<Tensor, Tensor, Tensor>()"
  result0: batch_norm_jvp(input_p, input_t, weight_p, weight_t, bias_p, bias_t, running_mean, running_var, result1, result2, training, epsilon)

- name: miopen_batch_norm_backward(Tensor input, Tensor grad_output, Tensor weight, Tensor? running_mean, Tensor? running_var, Tensor? save_mean, Tensor? save_var, float epsilon) -> (Tensor, Tensor, Tensor)
  save_mean: not_implemented("miopen_batch_norm_backward save_mean")
  save_var: not_implemented("miopen_batch_norm_backward save_var")
  input, weight, grad_output: batchnorm_double_backward(input, weight, grads[0], grads[1], grads[2], grad_output, running_mean, running_var, true, epsilon, save_mean, save_var, grad_input_mask)

- name: miopen_rnn(Tensor input, Tensor[] weight, int weight_stride0, Tensor hx, Tensor? cx, int mode, int hidden_size, int num_layers, bool batch_first, float dropout, bool train, bool bidirectional, int[] batch_sizes, Tensor? dropout_state) -> (Tensor, Tensor, Tensor, Tensor, Tensor)
  dropout_state: non_differentiable
  output_differentiability: [True, True, True, False, False]
  input, hx, cx, weight: "miopen_rnn_backward(input, weight, weight_stride0, result4, hx, cx, result0, grads[0], grads[1], grads[2], mode, hidden_size, num_layers, batch_first, dropout, train, bidirectional, batch_sizes, dropout_state, retain_variables ? result3.clone() : result3, grad_input_mask)"

- name: miopen_rnn_backward(Tensor input, Tensor[] weight, int weight_stride0, Tensor weight_buf, Tensor hx, Tensor? cx, Tensor output, Tensor? grad_output, Tensor? grad_hy, Tensor? grad_cy, int mode, int hidden_size, int num_layers, bool batch_first, float dropout, bool train, bool bidirectional, int[] batch_sizes, Tensor? dropout_state, Tensor reserve, bool[4] output_mask) -> (Tensor, Tensor, Tensor, Tensor[])
  dropout_state: non_differentiable

- name: mkldnn_rnn_layer(Tensor input, Tensor weight0, Tensor weight1, Tensor weight2, Tensor weight3, Tensor hx_, Tensor cx_, bool reverse, int[] batch_sizes, int mode, int hidden_size, int num_layers, bool has_biases, bool bidirectional, bool batch_first, bool train) -> (Tensor, Tensor, Tensor, Tensor)
  output_differentiability: [True, True, True, False]
  input, weight0, weight1, weight2, weight3, hx_, cx_: "GradMode::is_enabled() ? mkldnn_rnn_layer_differentiable_backward(input, weight0, weight1, weight2, weight3, hx_, cx_, result0, result1, result2, grads[0], grads[1], grads[2], reverse, mode, hidden_size, num_layers, has_biases, train, bidirectional, batch_sizes, batch_first, result3) : mkldnn_rnn_layer_backward(input, weight0, weight1, weight2, weight3, hx_, cx_, result0, result1, result2, grads[0], grads[1], grads[2], reverse, mode, hidden_size, num_layers, has_biases, train, bidirectional, batch_sizes, batch_first, result3)"

- name: mkldnn_rnn_layer_backward(Tensor input, Tensor weight1, Tensor weight2, Tensor weight3, Tensor weight4, Tensor hx_, Tensor cx_tmp, Tensor output, Tensor hy_, Tensor cy_, Tensor? grad_output, Tensor? grad_hy, Tensor? grad_cy, bool reverse, int mode, int hidden_size, int num_layers, bool has_biases, bool train, bool bidirectional, int[] batch_sizes, bool batch_first, Tensor workspace) -> (Tensor, Tensor, Tensor, Tensor, Tensor, Tensor, Tensor)

# mkldnn
- name: mkldnn_convolution(Tensor self, Tensor weight, Tensor? bias, SymInt[] padding, SymInt[] stride, SymInt[] dilation, SymInt groups) -> Tensor
  self, weight, bias: "grad.defined() ? convolution_backward_symint(grad, self, weight, bias->sym_sizes(), stride, padding, dilation, /*transposed=*/ false, /*output_padding=*/ std::vector<c10::SymInt>(padding.size(), 0), groups, grad_input_mask) : std::tuple<Tensor, Tensor, Tensor>()"

- name: mkldnn_linear(Tensor self, Tensor weight, Tensor? bias=None) -> Tensor
  self, weight, bias: mkldnn_linear_backward(self, grad, weight, grad_input_mask)

- name: mkldnn_max_pool2d(Tensor self, int[2] kernel_size, int[2] stride=[], int[2] padding=0, int[2] dilation=1, bool ceil_mode=False) -> Tensor
  self: mkldnn_max_pool2d_backward(grad, result, self, kernel_size, stride, padding, dilation, ceil_mode)

- name: mkldnn_max_pool3d(Tensor self, int[3] kernel_size, int[3] stride=[], int[3] padding=0, int[3] dilation=1, bool ceil_mode=False) -> Tensor
  self: mkldnn_max_pool3d_backward(grad, result, self, kernel_size, stride, padding, dilation, ceil_mode)

- name: mkldnn_adaptive_avg_pool2d(Tensor self, int[2] output_size) -> Tensor
  self: mkldnn_adaptive_avg_pool2d_backward(grad, self)

- name: _mkldnn_reshape(Tensor self, int[] shape) -> Tensor
  self: grad.reshape_symint(self.sym_sizes())

# NestedTensor
- name: _nested_tensor_from_tensor_list(Tensor[] list, ScalarType? dtype=None, Layout? layout=None, Device? device=None, bool? pin_memory=None) -> Tensor
  list: "grad.defined()? at::unbind(grad) : std::vector<Tensor>(list.size())"

- name: _nested_tensor_from_mask(Tensor t, Tensor mask, bool mask_check=True) -> Tensor
  t: grad.to_padded_tensor_symint(0, t.sym_sizes())
  mask: non_differentiable

- name: _nested_from_padded(Tensor padded, Tensor cpu_nested_shape_example, bool fuse_transform_0213=False) -> Tensor
  padded: _nested_from_padded_backward(grad, padded, fuse_transform_0213)
  cpu_nested_shape_example: non_differentiable

- name: to_padded_tensor(Tensor self, float padding, SymInt[]? output_size=None) -> Tensor
  self: "self.layout() == c10::kJagged ? at::_nested_from_padded_tensor_symint(grad, at::_nested_get_offsets(self), at::_nested_get_jagged_dummy(self), at::_nested_get_ragged_idx(self), at::_nested_get_min_seqlen(self).defined() ? std::optional<Tensor>(at::_nested_get_min_seqlen(self)) : ::std::nullopt, at::_nested_get_max_seqlen(self).defined() ? std::optional<Tensor>(at::_nested_get_max_seqlen(self)) : ::std::nullopt, std::optional<c10::SymInt>(at::_nested_get_values(self).sym_size(0))) : at::_nested_from_padded(grad, self._nested_tensor_size())"
  padding: non_differentiable

- name: _nested_from_padded_tensor(Tensor padded, Tensor offsets, Tensor dummy, int ragged_idx=1, Tensor? min_seqlen=None, Tensor? max_seqlen=None, SymInt? sum_S=None) -> Tensor
  padded: grad.to_padded_tensor_symint(0.0, at::OptionalArrayRef<c10::SymInt>(padded.sym_sizes()))
  offsets: non_differentiable
  dummy: non_differentiable

- name:  _nested_view_from_buffer(Tensor(a) self, Tensor nested_size, Tensor nested_strides, Tensor offsets) -> Tensor(a)
  self: grad.values()
  nested_size: non_differentiable
  nested_strides: non_differentiable

- name: _nested_view_from_jagged(Tensor(a) self, Tensor offsets, Tensor dummy, Tensor? lengths=None, int ragged_idx=1, Tensor? min_seqlen=None, Tensor? max_seqlen=None) -> Tensor(a)
  self: grad.values()
  offsets: non_differentiable
  lengths: non_differentiable
  dummy: non_differentiable

- name: _nested_get_values(Tensor(a) self) -> Tensor(a)
  self: "_nested_view_from_jagged(grad, at::_nested_get_offsets(self), at::_nested_get_jagged_dummy(self), at::_nested_get_lengths(self), at::_nested_get_ragged_idx(self), at::_nested_get_min_seqlen(self).defined() ? std::optional<Tensor>(at::_nested_get_min_seqlen(self)) : ::std::nullopt, at::_nested_get_max_seqlen(self).defined() ? std::optional<Tensor>(at::_nested_get_max_seqlen(self)) : ::std::nullopt)"

# Transformer
- name:  _safe_softmax(Tensor self, int dim, ScalarType? dtype=None) -> Tensor
  self: _softmax_backward_data(grad, result, dim, self.scalar_type())
  result: result * (self_t - safe_logsumexp_jvp(self_p, self_t, {dim}, true))

- name: _scaled_dot_product_efficient_attention(Tensor query, Tensor key, Tensor value, Tensor? attn_bias, bool compute_log_sumexp, float dropout_p=0.0, bool is_causal=False, *, float? scale=None) -> (Tensor output, Tensor log_sumexp, Tensor philox_seed, Tensor philox_offset)
  output_differentiability: [True, False, False, False]
  query, key, value, attn_bias: _scaled_dot_product_efficient_attention_backward(grad, query, key, value, attn_bias, output, log_sumexp, philox_seed, philox_offset, dropout_p, grad_input_mask, is_causal, scale)

- name: _scaled_dot_product_flash_attention(Tensor query, Tensor key, Tensor value, float dropout_p=0.0, bool is_causal=False, bool return_debug_mask=False, *, float? scale=None) -> (Tensor output, Tensor logsumexp, Tensor cum_seq_q, Tensor cum_seq_k, SymInt max_q, SymInt max_k, Tensor philox_seed, Tensor philox_offset, Tensor debug_attn_mask)
  output_differentiability: [True, False, False, False, False, False, False, False, False]
  query, key, value: _scaled_dot_product_flash_attention_backward_symint(grad, query, key, value, output, logsumexp, cum_seq_q, cum_seq_k, max_q, max_k, dropout_p, is_causal, philox_seed, philox_offset, scale)

- name: _scaled_dot_product_flash_attention_for_cpu(Tensor query, Tensor key, Tensor value, float dropout_p=0.0, bool is_causal=False, *, Tensor? attn_mask=None, float? scale=None) -> (Tensor output, Tensor logsumexp)
  output_differentiability: [True, False]
  query, key, value: _scaled_dot_product_flash_attention_for_cpu_backward(grad, query, key, value, output, logsumexp, dropout_p, is_causal, attn_mask, scale)

- name: _flash_attention_forward(Tensor query, Tensor key, Tensor value, Tensor? cum_seq_q, Tensor? cum_seq_k, SymInt max_q, SymInt max_k, float dropout_p, bool is_causal, bool return_debug_mask, *, float? scale=None, SymInt? window_size_left=None, SymInt? window_size_right=None, Tensor? seqused_k=None, Tensor? alibi_slopes=None) -> (Tensor output, Tensor softmax_logsumexp, Tensor philox_seed, Tensor philox_offset, Tensor debug_attn_mask)
  output_differentiability: [True, False, False, False, False]
  query, key, value: _flash_attention_backward_symint(grad, query, key, value, output, softmax_logsumexp, cum_seq_q, cum_seq_k, max_q, max_k, dropout_p, is_causal, philox_seed, philox_offset, scale, window_size_left, window_size_right)

- name: _efficient_attention_forward(Tensor query, Tensor key, Tensor value, Tensor? bias, Tensor? cu_seqlens_q, Tensor? cu_seqlens_k, SymInt? max_seqlen_q, SymInt? max_seqlen_k, float dropout_p, int custom_mask_type, bool compute_log_sumexp=False, *, float? scale=None, Tensor? seqlen_k=None, int? window_size=None) -> (Tensor output, Tensor logsumexp, Tensor philox_seed, Tensor philox_offset, SymInt max_seqlen_batch_q, SymInt max_seqlen_batch_k)
  output_differentiability: [True, False, False, False, False, False]
  query, key, value, bias: _efficient_attention_backward_symint(grad, query, key, value, bias, output, cu_seqlens_q, cu_seqlens_k, max_seqlen_batch_q, max_seqlen_batch_k, logsumexp, dropout_p, philox_seed, philox_offset, custom_mask_type, bias.requires_grad(), scale)

- name: _scaled_dot_product_cudnn_attention(Tensor query, Tensor key, Tensor value, Tensor? attn_bias, bool compute_log_sumexp, float dropout_p=0.0, bool is_causal=False, bool return_debug_mask=False, *, float? scale=None) -> (Tensor output, Tensor logsumexp, Tensor cum_seq_q, Tensor cum_seq_k, SymInt max_q, SymInt max_k, Tensor philox_seed, Tensor philox_offset, Tensor debug_attn_mask)
  output_differentiability: [True, False, False, False, False, False, False, False, False]
  query, key, value: _scaled_dot_product_cudnn_attention_backward_symint(grad, query, key, value, output, logsumexp, philox_seed, philox_offset, attn_bias, cum_seq_q, cum_seq_k, max_q, max_k, dropout_p, is_causal, scale)

- name: _scaled_dot_product_fused_attention_overrideable(Tensor query, Tensor key, Tensor value, Tensor? attn_bias=None, float dropout_p=0.0, bool is_causal=False, bool return_debug_mask=False, *, float? scale=None) -> (Tensor output, Tensor logsumexp, Tensor cum_seq_q, Tensor cum_seq_k, SymInt max_q, SymInt max_k, Tensor philox_seed, Tensor philox_offset, Tensor debug_attn_mask)
  output_differentiability: [True, False, False, False, False, False, False, False, False]
  query, key, value, attn_bias: _scaled_dot_product_fused_attention_overrideable_backward_symint(grad, query, key, value, attn_bias, grad_input_mask, output, logsumexp, cum_seq_q, cum_seq_k, max_q, max_k, dropout_p, is_causal, philox_seed, philox_offset, scale)

# fft
- name: _fft_r2c(Tensor self, int[] dim, int normalization, bool onesided) -> Tensor
  self: fft_r2c_backward(grad, dim, normalization, onesided, self.sym_size(dim.back()))
  result: auto_linear

- name: _fft_c2r(Tensor self, int[] dim, int normalization, SymInt last_dim_size) -> Tensor
  self: fft_c2r_backward(grad, dim, normalization)
  result: auto_linear

- name: _fft_c2c(Tensor self, SymInt[] dim, int normalization, bool forward) -> Tensor
  self: _fft_c2c_symint(grad, dim, normalization, !forward)
  result: auto_linear

- name: unbind.int(Tensor(a -> *) self, int dim=0) -> Tensor(a)[]
  dispatch:
    Default:
      self: unbind_backward(grads, dim)
      result: auto_linear
    AutogradNestedTensor:
      self: "self.layout() == c10::kJagged ? unbind_backward_nested_jagged(grads, self, dim) : unbind_backward_nested(grads, at::native::get_nested_tensor_impl(self)->get_nested_sizes(), dim, self.options())"
      result: auto_linear

- name: stack(Tensor[] tensors, int dim=0) -> Tensor
  tensors: stack_tensors_backward(grad, dim, to_args_scalartypes(tensors))
  result: stack_jvp(tensors, dim)

# fused RNN kernels

# Only frst two of _thnn_fused_lstm_cell outputs can have gradients.
# _thnn_fused_lstm_cell outputs: (hy, cy, workspace)
- name: _thnn_fused_lstm_cell(Tensor input_gates, Tensor hidden_gates, Tensor cx, Tensor? input_bias=None, Tensor? hidden_bias=None) -> (Tensor, Tensor, Tensor)
  output_differentiability: [True, True, False]
  input_gates, hidden_gates, cx, input_bias, hidden_bias: "GradMode::is_enabled() ? _thnn_differentiable_lstm_cell_backward(grads[0], grads[1], input_gates, hidden_gates, input_bias, hidden_bias, cx, result1) : _thnn_fused_lstm_cell_backward(grads[0], grads[1], cx, result1, result2, input_bias.defined())"

- name: _thnn_fused_gru_cell(Tensor input_gates, Tensor hidden_gates, Tensor hx, Tensor? input_bias=None, Tensor? hidden_bias=None) -> (Tensor, Tensor)
  input_gates, hidden_gates, hx, input_bias, hidden_bias: "grad.defined() ? (GradMode::is_enabled() ? _thnn_differentiable_gru_cell_backward(grad, input_gates, hidden_gates, hx, input_bias, hidden_bias) : _thnn_fused_gru_cell_backward(grad, result1, input_bias.defined())) : std::tuple<Tensor, Tensor, Tensor, Tensor, Tensor>()"

# PackedSequence helpers
- name: _pack_padded_sequence(Tensor input, Tensor lengths, bool batch_first) -> (Tensor, Tensor)
  input: _pack_padded_sequence_backward_symint(grad, input.sym_sizes(), result1, batch_first)

# TH wrappers
- name: eq.Scalar(Tensor self, Scalar other) -> Tensor
  output_differentiability: [False]

- name: eq.Tensor(Tensor self, Tensor other) -> Tensor
  output_differentiability: [False]

- name: ge.Scalar(Tensor self, Scalar other) -> Tensor
  output_differentiability: [False]

- name: ge.Tensor(Tensor self, Tensor other) -> Tensor
  output_differentiability: [False]

- name: gt.Scalar(Tensor self, Scalar other) -> Tensor
  output_differentiability: [False]

- name: gt.Tensor(Tensor self, Tensor other) -> Tensor
  output_differentiability: [False]

- name: le.Scalar(Tensor self, Scalar other) -> Tensor
  output_differentiability: [False]

- name: le.Tensor(Tensor self, Tensor other) -> Tensor
  output_differentiability: [False]

- name: lt.Scalar(Tensor self, Scalar other) -> Tensor
  output_differentiability: [False]

- name: lt.Tensor(Tensor self, Tensor other) -> Tensor
  output_differentiability: [False]

- name: ne.Scalar(Tensor self, Scalar other) -> Tensor
  output_differentiability: [False]

- name: ne.Tensor(Tensor self, Tensor other) -> Tensor
  output_differentiability: [False]

- name: multinomial(Tensor self, int num_samples, bool replacement=False, *, Generator? generator=None) -> Tensor
  output_differentiability: [False]

- name: nonzero(Tensor self) -> Tensor
  output_differentiability: [False]

- name: segment_reduce(Tensor data, str reduce, *, Tensor? lengths=None, Tensor? indices=None, Tensor? offsets=None, int axis=0, bool unsafe=False, Scalar? initial=None) -> Tensor
  data: _segment_reduce_backward(grad, result, data, reduce, lengths, offsets, axis, initial)

- name: _pin_memory(Tensor self, Device? device=None) -> Tensor
  self: grad

- name: _new_zeros_with_same_feature_meta(Tensor self, Tensor other, *, int self_num_batch_dims=0) -> Tensor
  self: non_differentiable
  other: non_differentiable
  output_differentiability: [False]

- name: _test_warn_in_autograd(Tensor self) -> Tensor
  self: warn_backwards(grad)

- name: _test_autograd_multiple_dispatch.fullcoverage(Tensor self) -> Tensor
  dispatch:
    Default:
      self: grad.expand_symint(self.sym_sizes()) + 1
      result: auto_linear
    AutogradNestedTensor:
      self: grad.mul(grad)
    AutogradCUDA:
      self: grad.expand_symint(self.sym_sizes()) * 2

- name: _test_autograd_multiple_dispatch.ntonly(Tensor self, bool b) -> Tensor
  dispatch:
    AutogradNestedTensor:
      self: grad.mul(grad).add(grad)

- name: _test_autograd_multiple_dispatch_view(Tensor(a) self) -> Tensor(a)
  dispatch:
    Default:
      self: grad.reshape_as(self)
    AutogradCUDA:
      self: grad.reshape_as(self) + 1

- name: _efficientzerotensor(SymInt[] size, *, ScalarType? dtype=None, Layout? layout=None, Device? device=None, bool? pin_memory=None) -> Tensor
  output_differentiability: [False]

- name: scatter_reduce.two(Tensor self, int dim, Tensor index, Tensor src, str reduce, *, bool include_self=True) -> Tensor
  self, src: scatter_reduce_backward(grad, self, dim, index, src, reduce, include_self, result)
  index: non_differentiable
  result: scatter_reduce_jvp(self_p, self_t, dim, index, src_p, src_t, reduce, include_self, result)

- name: special_airy_ai(Tensor x) -> Tensor
  x: non_differentiable

- name: special_bessel_j0(Tensor self) -> Tensor
  self: non_differentiable

- name: special_bessel_j1(Tensor self) -> Tensor
  self: non_differentiable

- name: special_bessel_y0(Tensor self) -> Tensor
  self: non_differentiable

- name: special_bessel_y1(Tensor self) -> Tensor
  self: non_differentiable

- name: special_chebyshev_polynomial_t(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_chebyshev_polynomial_t.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_chebyshev_polynomial_t.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_chebyshev_polynomial_u(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_chebyshev_polynomial_u.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_chebyshev_polynomial_u.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_chebyshev_polynomial_v(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_chebyshev_polynomial_v.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_chebyshev_polynomial_v.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_chebyshev_polynomial_w(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_chebyshev_polynomial_w.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_chebyshev_polynomial_w.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_hermite_polynomial_h(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_hermite_polynomial_h.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_hermite_polynomial_h.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_hermite_polynomial_he(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_hermite_polynomial_he.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_hermite_polynomial_he.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_laguerre_polynomial_l(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_laguerre_polynomial_l.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_laguerre_polynomial_l.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_legendre_polynomial_p(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_legendre_polynomial_p.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_legendre_polynomial_p.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_modified_bessel_i0(Tensor self) -> Tensor
  self: non_differentiable

- name: special_modified_bessel_i1(Tensor self) -> Tensor
  self: non_differentiable

- name: special_modified_bessel_k0(Tensor self) -> Tensor
  self: non_differentiable

- name: special_modified_bessel_k1(Tensor self) -> Tensor
  self: non_differentiable

- name: special_scaled_modified_bessel_k0(Tensor x) -> Tensor
  x: non_differentiable

- name: special_scaled_modified_bessel_k1(Tensor x) -> Tensor
  x: non_differentiable

- name: special_shifted_chebyshev_polynomial_t(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_shifted_chebyshev_polynomial_t.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_shifted_chebyshev_polynomial_t.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_shifted_chebyshev_polynomial_u(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_shifted_chebyshev_polynomial_u.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_shifted_chebyshev_polynomial_u.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_shifted_chebyshev_polynomial_v(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_shifted_chebyshev_polynomial_v.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_shifted_chebyshev_polynomial_v.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_shifted_chebyshev_polynomial_w(Tensor x, Tensor n) -> Tensor
  x: non_differentiable
  n: non_differentiable

- name: special_shifted_chebyshev_polynomial_w.x_scalar(Scalar x, Tensor n) -> Tensor
  n: non_differentiable

- name: special_shifted_chebyshev_polynomial_w.n_scalar(Tensor x, Scalar n) -> Tensor
  x: non_differentiable

- name: special_spherical_bessel_j0(Tensor x) -> Tensor
  x: non_differentiable

- name: _reshape_copy(Tensor self, SymInt[] size) -> Tensor
  self: grad.reshape_symint(self.sym_sizes())
  result: auto_linear

# note(crcrpar): `torchgen/api/autograd` logic would unwantedly replace substrings of `self` and `other` of function names.
- name: _foreach_div.List(Tensor[] self, Tensor[] other) -> Tensor[]
  self: div_tensor_self_backward(grads[i], other[i], self[i].scalar_type())
  other: div_tensor_other_backward(grads[i], self[i], other[i])
  result: (self_t - other_t * result[i]) / other_p

- name: _foreach_pow.List(Tensor[] self, Tensor[] exponent) -> Tensor[]
  self: pow_backward_self(grads[i], self[i], exponent[i])
  exponent: pow_backward_exponent(grads[i], self[i], exponent[i], result[i])
  result: (pow_backward_self(self_t.conj(), self_p, exponent_p) + pow_backward_exponent(exponent_t.conj(), self_p, exponent_p, result[i])).conj()

- name: _foreach_pow.ScalarList(Tensor[] self, Scalar[] exponent) -> Tensor[]
  self: pow_backward(grads[i], self[i], exponent[i])
  result: pow_backward(self_t.conj(), self_p, exponent[i]).conj()

- name: _foreach_pow.ScalarAndTensor(Scalar self, Tensor[] exponent) -> Tensor[]
  exponent: pow_backward_exponent(grads[i], self, exponent[i], result[i])

# note(crcrpar): following definitions seem necessary because the reference native functions
# of `maximum` and `minimum` don't have the overload def with Scalar as their second argument.
- name: _foreach_minimum.Scalar(Tensor[] self, Scalar scalar) -> Tensor[]
  self: at::where(self[i] == scalar, grads[i] / 2, grads[i]).masked_fill_(self[i] > scalar, 0)
  result: scalar + at::where(self_p == scalar, at::scalar_tensor(0.5, result[i].options()), (self_p < scalar).to(result[i].scalar_type())) * (self_t - scalar)

- name: _foreach_minimum.ScalarList(Tensor[] self, Scalar[] scalars) -> Tensor[]
  self: at::where(self[i] == scalars[i], grads[i] / 2, grads[i]).masked_fill_(self[i] > scalars[i], 0)
  result: scalars[i] + at::where(self_p == scalars[i], at::scalar_tensor(0.5, result[i].options()), (self_p < scalars[i]).to(result[i].scalar_type())) * (self_t - scalars[i])

- name: _foreach_maximum.Scalar(Tensor[] self, Scalar scalar) -> Tensor[]
  self: at::where(self[i] == scalar, grads[i] / 2, grads[i]).masked_fill_(self[i] < scalar, 0)
  result: scalar + at::where(self_p == scalar, at::scalar_tensor(0.5, result[i].options()), (self_p > scalar).to(result[i].scalar_type())) * (self_t - scalar)

- name: _foreach_maximum.ScalarList(Tensor[] self, Scalar[] scalars) -> Tensor[]
  self: at::where(self[i] == scalars[i], grads[i] / 2, grads[i]).masked_fill_(self[i] < scalars[i], 0)
  result: scalars[i] + at::where(self_p == scalars[i], at::scalar_tensor(0.5, result[i].options()), (self_p > scalars[i]).to(result[i].scalar_type())) * (self_t - scalars[i])

# note(crcrpar): forward-mode AD is tricky for a simple string replace to handle:
#   formula.replace("p", "ord") produces `norm_jvord(self_ord, self_t, ord, result)`
- name: _foreach_norm.Scalar(Tensor[] self, Scalar ord=2, ScalarType? dtype=None) -> Tensor[]
  self: norm_backward(grads[i], self[i], ord, result[i])
  result: norm_jvp(self_p, self_t, ord, result[i])