What is the most efficient series to calculate sine?

Question

What formula is used to calculate sine in modern computers?

Is Taylor formula

the best? What formulas converge faster, especially out of $2\pi$ range?

Answer 1

There is not one formula. The best known method relies on the values of $\sin 1$, $\sin 0.1$, $\sin 10^{-2}$, $\dots,\sin 10^{-k}$, $\dots,\;$ for a small number of values of $k$, which can easily be calculated with Taylor's formula at any degree of accuracy.

Then the CORDIC algorithm can compute from these data the sine of any real number.

Answer 2

I don't think the chip-makers publish their microcode, but they almost certainly don't use Taylor series. It's more likely that they use either a CORDIC algorithm or an optimized polynomial approximation on some interval.