All good stuff, I’d add though that for many numerical routines you end up testing on simple well known systems that have well defined analytic answers.

So for e.g. if you’re writing solvers for ODE systems, then you tend to test it on things like simple harmonic oscillators.

If you’re very lucky, some algorithms have bounded errors. For e.g. the fast multipole method for evaluating long range forces does, so you can check this for very simple systems.