code: def expand(input_shape: Shape, shape: Shape) -> DimMap:
      """Implement broadcast on multiple dimensions."""
      assert len(shape) >= len(input_shape)
  
      # 1. create padded input dimensions
      padded_input = dim_pad_left(len(input_shape), len(shape))
      # 2. check that input shapes are compatible
      mapping = []
      for p, desired_s in zip(padded_input, shape):
          if isinstance(p, Singleton):
              actual_s = 1
              assert desired_s >= 0
          else:
              assert isinstance(
                  p, InputDim
              ), f"DimSpec not supported in expand: {p}"
              actual_s = input_shape[p.input_dim]
              assert (
                  actual_s == 1
                  or desired_s == -1
                  or desired_s == actual_s
              )
          mapping.append(
              p
              if desired_s in (1, -1) or desired_s == actual_s
              else Broadcast.new(p, desired_s)
          )
      return tuple(mapping)
  
imports: from dataclasses import dataclass
  from typing import Callable, cast, Dict, Iterable, Optional, Sequence, Set, Tuple, Union
  import torch
  from torch import Tensor
  from torch._subclasses.fake_tensor import unset_fake_temporarily
  from torch.distributed._tensor._utils import compute_local_shape
  from torch.distributed._tensor.api import Shard
  from torch.distributed._tensor.op_schema import OpSchema, OutputSharding, RuntimeSchemaInfo
  from torch.distributed._tensor.ops.utils import normalize_dim, normalize_dims, prod, register_prop_rule
  from torch.distributed._tensor.placement_types import DTensorSpec, Placement, Replicate
  from torch.fx.experimental.proxy_tensor import disable_proxy_modes_tracing
label_desc: Use image node_id 3 for calling a global function with example usage: expand(input_shape, shape) and returns: tuple