@triton.autotune(configs=warps_kernel_configs(), key=['batch_dim', 'feat_dim'])
@triton.heuristics({'BLOCK_SIZE_BATCH': BLOCK_SIZE_BATCH_heuristic,
    'BLOCK_SIZE_FEAT': lambda args: next_power_of_2(args['feat_dim'])})
@triton.jit
def softmax_backward_kernel(output_grad_pointer, output_pointer,
    input_grad_pointer, batch_dim, feat_dim, output_grad_batch_stride,
    output_grad_feat_stride, output_batch_stride, output_feat_stride,
    input_grad_batch_stride, input_grad_feat_stride, log: tl.constexpr,
    BLOCK_SIZE_BATCH: tl.constexpr, BLOCK_SIZE_FEAT: tl.constexpr):
    """
    Calculates the input gradient of softmax.

    Args:
        output_grad_pointer: Pointer to softmax's output gradients.
            The output gradients must be of shape [batch_dim, feat_dim].
        output_pointer: Pointer to softmax's output.
            The output must be of shape [batch_dim, feat_dim].
        input_grad_pointer: Pointer to a container the input's gradients are written to.
            The container must be of shape [batch_dim, feat_dim].
        batch_dim: Batch dimension.
        feat_dim: Dimensionality of the features.
        output_grad_batch_stride: Stride necessary to jump one element along the
            output gradients' batch dimension.
        output_grad_feat_stride: Stride necessary to jump one element along the
            output gradients' feature dimension.
        output_batch_stride: Stride necessary to jump one element along the
            output's batch dimension.
        output_feat_stride: Stride necessary to jump one element along the
            output's feature dimension.
        input_grad_batch_stride: Stride necessary to jump one element along the
            input gradient container's batch dimension.
        input_grad_feat_stride: Stride necessary to jump one element along the
            input gradient container's feature dimension.
        log: Flag indicating if log of softmax was taken.
        BLOCK_SIZE_BATCH: Block size across the batch dimension.
        BLOCK_SIZE_FEAT: Block size across the feature dimension.
    """
    batch_pid = tl.program_id(axis=0)
    batch_offset = batch_pid * BLOCK_SIZE_BATCH + tl.arange(0, BLOCK_SIZE_BATCH
        )
    feat_offset = tl.arange(0, BLOCK_SIZE_FEAT)
    batch_mask = batch_offset < batch_dim
    feat_mask = feat_offset < feat_dim
    output_grad_pointer += output_grad_batch_stride * batch_offset[:, None
        ] + output_grad_feat_stride * feat_offset[None, :]
    output_pointer += output_batch_stride * batch_offset[:, None
        ] + output_feat_stride * feat_offset[None, :]
    input_grad_pointer += input_grad_batch_stride * batch_offset[:, None
        ] + input_grad_feat_stride * feat_offset[None, :]
    output_grad = tl.load(output_grad_pointer, mask=batch_mask[:, None] &
        feat_mask[None, :]).to(tl.float32)
    output = tl.load(output_pointer, mask=batch_mask[:, None] & feat_mask[
        None, :]).to(tl.float32)
    if log:
        input_grad = output_grad - tl.exp(output) * tl.sum(output_grad, axis=1
            )[:, None]
    else:
        input_grad = output * (output_grad - tl.sum(output_grad * output,
            axis=1)[:, None])
    tl.store(input_grad_pointer, input_grad, mask=batch_mask[:, None] &
        feat_mask[None, :])
