FFT butterfly ($(a,b)\mapsto (a+wb,a-wb)$)

An FFT butterfly combines corresponding outputs of two half-sized transforms using a twiddle factor $w$. Sharing the product $wb$ costs one complex multiplication and two additions. For the unnormalized positive-exponent inverse discrete Fourier transform of length $2m$, use $w=e^{2\pi i\ell /(2m)}$ at index $\ell$.