def METHOD_NAME(a_inv: types.Float, x: types.Float) -> types.Float:
  """Updates the inverse using the Woodbury matrix identity.
  Given a matrix `A` of size d-by-d and a matrix `X` of size k-by-d, this
  function computes the inverse of B = A + X^T X, assuming that the inverse of
  A is available.
  Reference:
  https://en.wikipedia.org/wiki/Woodbury_matrix_identity
  Args:
    a_inv: a `Tensor` of shape [`d`, `d`]. This is the current inverse of `A`.
    x: a `Tensor` of shape [`k`, `d`].
  Returns:
    The update that needs to be added to 'a_inv' to compute the inverse.
    If `x` is empty, a zero matrix is returned.
  """
  batch_size = tf.shape(x)[0]
  def true_fn():
    return tf.zeros_like(a_inv)
  def false_fn():
    return simplified_woodbury_update(a_inv, x)
  a_inv_update = tf.cond(tf.equal(batch_size, 0), true_fn, false_fn)
  return a_inv_update