One simple example (from the above) is "sin (π − x) = sin x" for the implementation of the sine function not requiring the knowledge of its specific output values. Naturally, instead of the literal equality "=" one can use a more appropriate accuracy specification as in, say, relative ulp (https://en.wikipedia.org/wiki/Unit_in_the_last_place) error bound, cf. https://members.loria.fr/PZimmermann/papers/accuracy.pdf