The poster is talking about using symbolic math to obtain a closed-form expression for the Jacobian that is then numerically evaluated. This will often be faster and more accurate. For example, if your function is sin(x), then the symbolic math tells you your derivative is cos(x), so you put that expression in your code and compile it. When calling this on the angle x = 0.123, your code then just evaluates numerically cosf(0.123), rather than (sinf(0.124) - sinf(0.122)).

Well, the symbolic computation is only used to generate C code. The generated code then evaluates jacobians numerically. And we use several methods for numerical stability on top of that.

There are plenty of situations where the scenario goes like this: you first do symbolical calculations. then fill in some of the free variables, your resulting expression will simplify a lot. Then you compile that resulting expression to native code and evaluate the simplified specialized code for a zillion parameter vectors.

Anyway,if your day job needs something like this you're better off using a lisp than python.

I though finite difference differentiation was notoriously unstable.

Especially if you actually require vector-Jacobian or Jacobian-vector products instead of the full Jacobian.