Optimal Piecewise Linear Approximations for Sigmoid, Tanh, Probability Density and Cumulative Distribution Functions: Tabular Form and R Package pwlapprox2d

We present information in tabular form about optimal piecewise linear (PWL) approximations
of the sigmoid and tanh activation functions as used in neural networks, as well as the probabil-
ity density and cumulative distribution functions of the Normal and log-Normal distributions.
The presented approximations minimise the maximum absolute difference between the PWL
function and the continuous function being modelled; we also provide information on opti-
mal over- and underestimators. We provide information on the optimal breakpoint locations
for different numbers of breakpoints in a series of tables, as well as the accuracy provided by
these approximations. This allows practitioners to utilise the PWL approximations whenever
needed; for example, to enable the use of mixed-integer linear programming techniques to
solve problems where these functions may appear, or to approximate integrals. The provided
optimal breakpoint locations lead to an improvement over a uniform breakpoint location
of the maximum absolute difference of up to 95% and an average of 84% among the six
functions.