Nice and clear way to think about it (and touching closer to the kind of numerical analysis I drilled most, too). I wonder, if having a good polynomial descript

Nice and clear way to think about it (and touching closer to the kind of numerical analysis I drilled most, too). I wonder, if having a good polynomial description of f then we can "just" use a Taylor integration method, approximating the inverses needed along the way with some high order approximation.