While it's cute use of mathematics, it's extremely inefficient in the real world because it introduces floating point multiplications and cos() which are very expensive. The only thing it lacks is branching which reduces the chances of a pipeline stall due to branch prediction miss.
This can be translated to the discrete domain pretty easily, just like the NTT. Pick a sufficiently large prime with order 15k, say, p = 2^61-1. 37 generates the whole multiplicative group, and 37^((2^61-2)/3) and 37^((2^61-2)/5) are appropriate roots of unity. Putting it all together yields
This involves 6 exponentiations by n with constant bases. Because in fizzbuzz the inputs are sequential, one can further precompute c^(2^i) and c^(-2^i) and, having c^n, one can go to c^(n+1) in average 2 modular multiplications by multiplying the appropriate powers c^(+-2^i) corresponding to the flipped bits.
(The divisions will get optimized away.)