Considering a "flop" as a real arithmetic operation and ignoring precision, and apropos of @skbmoore's r[eference](https://mathoverflow.net/questions/393388/minimal-number-of-operations-of-a-discrete-fourier-transform#comment1004235_393388), the paper [Johnson, S. G. and Frigo, M. "A Modified Split-Radix FFT With Fewer Arithmetic Operations." *IEEE Trans. Sig. Proc.* **55**, 111 (2007) ([doi][1])] by the authors of FFTW achieves a flop count of
$$\frac{43}{9} N \log_2 N - \frac{124}{27} N - 2 \log_2 N - \frac{2}{9}(-1)^{\log_2 N} \log_2 N + \frac{16}{27}(-1)^{\log_2 N} + 8.$$ Here $N$ is assumed a power of two. (See DSPSE answer along these lines [here][2].) In particular, $N = 1,2,4,8,16,32,64,128$ respectively gives flop counts of $4,4,16,56,168,456,1152,2792$. This paper also points out that "no tight lower bound on the flop count is known" in general.

In [Haynal, S. A. and Haynal, H. B. "Generating and searching families of FFT algorithms." J. Satisfiability, Boolean Modeling Comp. **7**, 145 (2011) ([doi][3])] the authors show that *certain types* of FFT algorithm cannot improve on this bound (or indeed in many cases a weaker one) for small $N$.

Finally, [Barseghyan and Sarukhanyan - New approach to FFT algorithms][4] claims improved flop counts for $N$.


  [1]: https://doi.org/10.1109/TSP.2006.882087
  [2]: https://dsp.stackexchange.com/a/52592
  [3]: https://doi.org/10.3233/SAT190084
  [4]: http://compsci.asj-oa.am/698/