By viewing the nonuniform fast Fourier transform (NUFFT) as a perturbed version of an FFT, we propose a fast and simple algorithm for the NUFFT that adapts to any working precision. Our accuracy and computational times are competitive with the state-of-the-art algorithm by Greengard and Lee. Motivated by the Kadec-1/4 theorem, we have also been able to fully understand the impact of nonuniformity on the complexity and stability of NUFFTs.