  /* softplus link function inverse to 'log1p_exp'
   * Args:
   *   x: a positive scalar
   * Returns:
   *   a scalar in (-Inf, Inf)
   */
   real log_expm1(real x) {
     return log(expm1(x));
   }
  /* softplus link function inverse to 'log1p_exp' (vectorized)
   * Args:
   *   x: a positive vector
   * Returns:
   *   a vector in (-Inf, Inf)
   */
   vector log_expm1(vector x) {
     return log(expm1(x));
   }
