New tightness lower and upper bounds for the standard normal distribution function and related functions

Most researches interested in finding the bounds of the cumulative standard normal distribution FoxTHORN are not tight for all positive values of the argument x. This paper mainly proposes new simple lower and upper bounds for FoxTHORN. Over the whole range of the positive argument x, the maximum absolute difference between the proposed lower bound and FoxTHORN is less than 3 x 10(-4), while it is less than 4:8 x 10(-4) between the proposed upper bound and FoxTHORN. Numerical comparisons have been made between the proposed bounds and some of the other existing bounds, which showed that the proposed bounds are more compact than most alternative bounds found in the literature.

Automated summary

This paper proposes new simple lower and upper bounds for the cumulative standard normal distribution FoxTHORN. The proposed bounds have smaller maximum absolute differences with FoxTHORN compared to existing bounds, making them more compact.