Improved CMB (de-)lensing using general spherical harmonic transforms

Deep Cosmic Microwave Background polarization experiments allow in principle very precise internal reconstruction of the gravitational lensing signal. To this aim, likelihood-based or Bayesian methods are typically necessary, where performing a sometimes very large number of lensing and delensing remappings on the sphere is required before satisfactory convergence. We discuss here in some detail an optimized piece of numerical code able to perform both the lensing operation and its adjoint (closely related to delensing) efficiently, and to arbitrary accuracy, using non-uniform Fast Fourier Transform technology. Where applicable, we find the code outperforms by massive amounts current widespread software, being able to produce high-resolution maps accurate enough for next-generation CMB experiments on the timescale of seconds on a modern laptop. The adjoint operation performs similarly, and removes the need for computation of inverse deflection fields. This publicly available code enables de facto efficient spherical harmonic transforms on completely arbitrary grids, and could possibly find applications also in other areas.