code: def fatplus(x: Tensor, tau: Union[float, Tensor] = TAU) -> Tensor:
      """Computes a fat-tailed approximation to `ReLU(x) = max(x, 0)` by linearly
      combining a regular softplus function and the density function of a Cauchy
      distribution. The coefficient `alpha` of the Cauchy density is chosen to guarantee
      monotonicity and convexity.
  
      Args:
          x: A Tensor on whose values to compute the smoothed function.
          tau: Temperature parameter controlling the smoothness of the approximation.
  
      Returns:
          A Tensor of values of the fat-tailed softplus.
      """
  
      def _fatplus(x: Tensor) -> Tensor:
          alpha = 1e-1  # guarantees monotonicity and convexity (TODO: ref + Lemma 4)
          return softplus(x) + alpha * cauchy(x)
  
      return tau * _fatplus(x / tau)
  
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 21 for calling a global function with example usage: fatplus(x, tau) and returns: unknown, unknown