[mgaunard]

The canonical way to do these things is to reduce first to a strictly monotonic range. For cos that would be 0 to pi/2.

Surely you shouldn't get those effects then?

[gjm11]

The advantage of Chebyshev polynomials over ordinary power-basis polynomials will be much less in that situation. But sometimes you're approximating something that doesn't have convenient domain-reduction relations available.

[mgaunard]

Parallel execution and minimization of instructions makes a huge difference to performance.