def backward(self, y):
        r"""
        Inverse of the softplus transform:
        .. math::

           x = \log( \exp(y) - 1)

        The bound for the input y is [self._lower. inf[, self._lower is
        subtracted prior to any calculations. The implementation avoids overflow
        explicitly by applying the log sum exp trick:
        .. math::

           \log ( \exp(y) - \exp(0)) &= ys + \log( \exp(y-ys) - \exp(-ys)) \\
                                     &= ys + \log( 1 - \exp(-ys)

           ys = \max(0, y)

        As y can not be negative, ys could be replaced with y itself.
        However, in case :math:`y=0` this results in np.log(0). Hence the zero is
        replaced by a machine epsilon.
        .. math::

           ys = \max( \epsilon, y)


        """
        ys = np.maximum(y - self._lower, np.finfo(settings.float_type).eps)
        return ys + np.log(-np.expm1(-ys))