Based on the Tonemap function originally posted by Daniele Siragusano here on acescentral: https://community.acescentral.com/t/output-transform-tone-scale/3498/14
Based on the Michaelis-Menten model of enzyme kinetics. https://en.wikipedia.org/wiki/Michaelis-Menten_kinetics
First here is a simple form solved for mid-grey and peak white intersections, which does not include a toe or surround compensation.
https://www.desmos.com/calculator/3xbbc6alcm
Second, here is a more complex form which includes a display-referred toe adjustment for flare compensation. In this form, the mid-grey intersection is maintained by adjusting the power function.
As far as I can tell there is no closed-form solution for $s_0$ in terms of the mid-grey intersection. So I have resorted to this as one of many workarounds that I have tried.
This also includes an unconstrained power function for a simple surround compensation model.
https://www.desmos.com/calculator/d6eknotb9c
There is also this alternative model which I posted a while ago on acescentral. This form does not maintain the middle-grey intersection point when $t_0$ is adjusted, but does maintain the peak white intersection. This solution may actually be preferable because when the mid-grey intersection is constrained, contrast changes when the toe is adjusted.