[eaglefield]

Theres no smoothing involved, it's three distinct domains with different function definitions. Top part is log(x) from [a,\infty), bottom part is -log(-x) from (-\infty,-a] and the middle part is cx from (-a,a). With c chosen to ensure continuity in the transform.

In the example from matplotlib I linked in the earlier comment they call out that the symlog transform has a discontinous gradient at the a's, and that the asinh transform can be used instead if that's a problem.

Edit: On reflection it's probably not entirely correct to talk about it as choosing an appropriate c. Since that transform seems to kinda break apart around a=1. Simpler to consider it a matter of plotting on a logarithmic scale down to some value. Then continuing the plot on a linear scale until you reach the negative value on the other side and then plotting on a negative logarithmic scale (-log(|x|)).