Cheaptrick is "a [spectral-envelope] estimator for high-quality speech synthesis". It is used by the [world-vocoder].
Research paper: https://www.sciencedirect.com/science/article/pii/S0167639314000697
Judging by the research paper, it performs better than two other spectral envelope estimators that they compared it to.
Main source code: https://github.com/mmorise/World/blob/master/src/cheaptrick.cpp
Source seems pretty readable. I think it was ported over from Matlab, but doesn't seem to be an issue.
It looks to be doing both FFT and inverse FFT. It has mention of [cepstrum].