This third version is extremely different from the first two: rather than trying to compute in-order FFTs without blowing caches, it generates code for bit-reversed FFTs. The idea came from Brad Lucier, who sent me a couple emails and showed how nicely his FFT scaled (it’s used in gambit’s bignum code). Bit-reversed FFTs don’t have to go through any special contortion to enjoy nice access patterns: everything is naturally sequential. The downside is that the output is in the wrong order (in bit-reversed order). However, it might still be an overall win over directly computing DFTs in order: we only need to execute one bit-reversal pass, and we can also provide FFT routines that work directly on bit-reversed inputs.
My hope when I started writing Napa-FFT3 was that I could get away with a single generator that’d work well at all sizes, and that bit-reversing would either not be too much of an issue, or usually not needed (e.g., for people who only want to perform convolutions or filtering).
Overview of the code
Generator for “flat” base cases output code for a specialised compiler geared toward large basic blocks. The specialised compiler takes potentially very long traces of simple operations on array elements, and performs two optimisations: array elements are cached in variables (registers), and variables are explicitly spilled back into arrays, following Belady’s algorithm. That allows us to easily exploit the register file, without taking its size directly into account in the domain-specific generators, and even when we have to cope with a relatively naïve machine code generator like SBCL’s.
Larger input sizes instead use a generator that outputs almost-normal recursive code; there’s one routine for each input size, which helps move as much address computation as possible to compile-time.
Even with code to handle scaling and convolution/filtering, I feel that the generators are easily understood and modified. They currently only support in-order input for the forward transform, and in-order output for the inverse, but the generators are simple enough that adding code for all four combinations (in-order input or output, forward or inverse transform) would be reasonable! I believe that’s a win.
Better: it seems my hope that we can execute bit reverses quickly was more than justified. I’m not quite sure how to describe it, but the code is based on recursing on the indices from the middle bits toward the least and most significant bits. The result is that the there’s exactly one swap at each leaf of the recursion, and that, when cache associativity is high enough (as is the case for the x86 chips I use), all the cache misses are mandatory. Better, the recursiveness ensures that the access patterns are also TLB optimal, when the TLB associativity is high enough (or infinite, as for my x86oids).
There’s one issue with that recursive scheme: it’s really heavy in integer arithmetic to compute indices. Again, I generate large basic blocks to work around that issue. The last couple levels (three, by default) of the recursion are unrolled and compiled into a long sequence of swaps. The rest of the recursion is executed by looping over a vector that contains indices that were computed at compile-time.
I have a hard time convincing myself that code generators are correct, especially without a nice static type system. Instead, I heavily tested the final generated code. I’m using Common Lisp, so array accesses were all checked automatically, which was very useful early in the development processes. Once I was convinced certain that all accesses were correct, I turned bound and type checking off. The first test file implements a set of randomised tests proposed by Funda Ergün. That was enough for me to assume that the FFTs themselves were correct. I then turned to a second set of tests to try and catch issues in the rest of the code that builds on straight FFTs.
The process did catch a couple bugs, and makes me feel confident enough to let other people use Napa-FFT3 in their programs.
Napa-FFT and Napa-FFT2 managed to come reasonably close to FFTW’s performance. When I started working on Napa-FFT3, I hoped that it could come as close, with much less complexity. In fact, it performs even better than expected: Napa-FFT3 is faster than Napa-FFT(2) at nearly all sizes, and outperforms FFTW’s default planner for out-of-cache transforms (even with the bit-reversal pass).