Yes, indeed. What I mean is, with Chebyshov polynomials, you will avoid Runge effect but it's not an issue for a small order polynomial. And to win against the hardware implementation, you have to keep your polynomial small.

I measured this for a contemporary Intel processor some maybe ten years ago and found out that 6th degree (actually, its 3 for x^2 since you can throw away all the even coefficients) makes sense performance-wise, yes the error is greater but that's the point of the trade-off. But even 8 or 10 is already doubtful.

Of course, there are architectures other than Intel and it might make sense to use higher order polynomials for better accuracy there.