code: def fatmoid(X: Tensor, tau: Union[float, Tensor] = 1.0) -> Tensor:
      """Computes a twice continuously differentiable approximation to the Heaviside
      step function with a fat tail, i.e. `O(1 / x^2)` as `x` goes to -inf.
  
      Args:
          X: A Tensor from which to compute the smoothed step function.
          tau: Temperature parameter controlling the smoothness of the approximation.
  
      Returns:
          A tensor of fat-tailed approximations to the Heaviside step function.
      """
      X = X / tau
      m = _inv_sqrt_3  # this defines the inflection point
      return torch.where(
          X < 0,
          2 / 3 * cauchy(X - m),
          1 - 2 / 3 * cauchy(X + m),
      )
  
imports: from __future__ import annotations
  import math
  from typing import Callable, Tuple, Union
  import torch
  from botorch.exceptions import UnsupportedError
  from botorch.utils.constants import get_constants_like
  from torch import finfo, Tensor
  from torch.nn.functional import softplus
label_desc: Use image node_id 27 for calling a global function with example usage: fatmoid(X, tau) and returns: torch