Terrell, George (2003), A Stabilized Lugannani-Rice Formula, Computing Science and Statistics, 35, I2003Proceedings/TerrellGeorge/TerrellGeorge.paper.pdf

The well-known approximate c.d.f formula proposed by Lugannani and Rice (1980) provides in many cases remarkably better accuracy than naive normal approximations. In extreme cases, however, it need not even be a probability. We propose a modification of the formula that is of the same asymptotic order of accuracy and similar typical performance, but is a stable approximation. That is, it is always a probability; and it is exact on an important family of arbitrarily skewed random variables.