Dual-Select FMA Butterfly for FFT: Eliminating Twiddle Factor Singularities with Bounded Precomputed Ratios

The fused multiply-add (FMA) instruction enables the radix-2 FFT butterfly to be computed in 6~FMA operations – the proven minimum. The classical factorization by Linzer and Feig~\cite{linzer1993} precomputes the ratio $\cot \theta =\cos \theta \mathrm{/}\sin \theta$, which is singular when the twiddle factor is $W^0=1$ (i.e., $\sin \theta =0$). Standard practice clamps $\sin \theta$ to a small epsilon, degrading numerical precision. We observe that an alternative factorization using $\cos \theta$ as the outer multiplier (precomputing $\tan \theta$) avoids this particular singularity but introduces a new one at $W^{N\mathrm{/}4}$. We then propose a \emph{dual-select} strategy that chooses, per twiddle factor, whichever factorization yields $\mathrm{\mid }\text{ratio}\mathrm{\mid }\le 1$. This eliminates all singularities, requires no epsilon clamping, and bounds the precomputed ratio to unity for all twiddle factors. For $N=1024$, the worst-case ratio drops from 163 (Linzer-Feig) to exactly~1.0 (dual-select), yielding a $235\times$ tighter error bound in FP16 arithmetic over 10~FFT passes. The strategy adds zero computational overhead – only the precomputed twiddle table changes.