Approximation of Gaussian curvature of an SDF?

I want to try to approximate the gaussian curvature of a point on an SDF surface. https://en.wikipedia.org/wiki/Gaussian_curvature#Alternative_formulas

The analytic formula requires computing thedeterminant of a 4x4 matrix that also involves the hessian, this is not going to be super numerically stable nor fast.

There is this writeup that I found that discusses a simpler way of approximating the mean curvature:

https://rodolphe-vaillant.fr/entry/118/curvature-of-a-distance-field-implicit-surface

I am hoping that there is some other formulation for the gaussain curvature that may be less accurate but still sane for SDF's.

My only other approach is autodiff, which might have to be what I need to use.