So far in this series, we’ve been looking at code piecemeal, writing little test modules to explore the algorithms we’ve been developing. Before we start trying to optimise this code, it makes sense to put it into a Cabal package with a more organised module structure, to provide a sensible API and to make a few other small changes.

From now on, the code we’re going to be talking about will be the `arb-fft`

package, which is hosted on GitHub. In this article, we’ll be talking about the version of the code tagged `pre-release-1`

. (We’ll be uploading a version of the code to Hackage once it’s ready, but there will be a few pre-release versions that we’ll look at before we get to that point.)

## “Cabalisation” and module structure

Setting up a new package with Cabal, Haskell’s package and build manager, is pretty easy: go to an empty directory, type `cabal init`

and answer some questions about your project – the project name, a one-line synopsis of what it does, author details, license details and a couple of other things. Cabal generates an initial Cabal package description file to which you add the details of libraries, executables, test suites and so on. Here’s the Cabal file for the `arb-fft`

package.

```
name: arb-fft
version: 0.1.0.0
synopsis: Pure Haskell arbitrary length FFT library
homepage: https://github.com/ian-ross/arb-fft
license: GPL-3
license-file: LICENSE
author: Ian Ross
maintainer: ian@skybluetrades.net
copyright: Copyright (2013) Ian Ross
category: Math
build-type: Simple
extra-source-files: README.md
cabal-version: >=1.10
Library
exposed-modules: Numeric.FFT
other-modules: Numeric.FFT.Plan
Numeric.FFT.Execute
Numeric.FFT.Special
Numeric.FFT.Types
Numeric.FFT.Utils
ghc-options: -O2
build-depends: base >= 4.6 && < 5,
containers >= 0.5.0.0 && < 0.6,
vector >= 0.10.9.1 && < 0.11
default-language: Haskell2010
Test-Suite basic-test
type: exitcode-stdio-1.0
hs-source-dirs: test
main-is: basic-test.hs
build-depends: arb-fft,
base >= 4.6 && < 5,
containers >= 0.5.0.0 && < 0.6,
vector >= 0.10.9.1 && < 0.11,
QuickCheck >= 2.6 && < 2.7,
tasty >= 0.3,
tasty-quickcheck >= 0.3
default-language: Haskell2010
```

We define a single library and one test suite. The modules in the library are:

`Numeric.FFT`

: The main “exposed” module implementing the public API. Exports functions for performing FFT and inverse FFT transforms, along with re-exporting some data types and additional API from a couple of the hidden modules.- `Numeric.FFT.Plan: Functions for pre-planning of FFT calculations.
`Numeric.FFT.Execute`

: Functions to execute pre-planned FFTs.`Numeric.FFT.Types`

: Type definitions, the most important of which are`Plan`

and`BaseTransform`

.`Numeric.FFT.Utils`

: Utility functions, including prime factorisation, primitive root calculation for $\mathbb{Z}_p^{\times}$ and some other small things.`Numeric.FFT.Special`

: Base transform implementations specialised to small fixed input lengths.

## Code changes for packaging

#### Pre-planning

The `plan`

function in the `Numeric.FFT.Plan`

module (also re-exported from `Numeric.FFT`

) takes a problem size as input and returns a value of the `Plan`

abstract data type that includes all of the information required to execute an FFT that can be calculated in advance: input vector permutation, size and factor information for the Danielson-Lanczos steps used to build up the final results, powers of $\omega_N$ for all the $N$ values required, and a “base transform” used at the “bottom” of the Cooley-Tukey decomposition, which is either a reference to a specialised hard-coded transform for small input lengths (defined in `Numeric.FFT.Special`

), or (for prime lengths) all the information needed to perform a prime-length transform using Rader’s algorithm, or (for any other lengths) an indicator that a simple DFT should be used for the base transform.

At the current stage of development of the code, this pre-planning is relatively lightweight, since we always use the same prime factor decomposition of the input size. Later on, we’ll add some benchmarking code to the planner to select the best decomposition of the input size from a list of likely good plans.

The plans generated by the `plan`

function are executed by code in the `Numeric.FFT.Execute`

module. This is where most of the work is done, and is where we’ll be concentrating our optimisation effort. As well as the main Cooley-Tukey driver function (`execute`

) and a function to perform a single Danielson-Lanczos step, this module also has code to apply the relevant “base transforms”.

#### Special base transforms

Up to now, we’ve been using a simple DFT for the small prime-length factors at the “bottom” of the FFT decomposition. For arbitrary prime lengths, we can use Rader’s algorithm, but this will be much less efficient than a “straight-line” hand-coded transform. The idea here is that for small input lengths that occur at the “bottom” of many transform decompositions, we can write optimal or near-optimal hand-coded transforms once and for all.

Here such a specialised base transform for input vectors of length 5:

```
-- | Length 5 hard-coded FFT.
kp951056516, kp559016994, kp250000000, kp618033988 :: Double
kp951056516 = 0.951056516295153572116439333379382143405698634
kp559016994 = 0.559016994374947424102293417182819058860154590
kp250000000 = 0.250000000000000000000000000000000000000000000
kp618033988 = 0.618033988749894848204586834365638117720309180
special5 :: Int -> VCD -> VCD
special5 sign xs =
let ar:+ai=xs!0 ; br:+bi=xs!1 ; cr:+ci=xs!2 ; dr:+di=xs!3 ; er:+ei=xs!4
ts = br - er ; t4 = br + er ; tt = cr - dr ; t7 = cr + dr
t8 = t4 + t7 ; ta = t4 - t7 ; te = bi - ei ; tm = bi + ei
tn = ci + di ; th = ci - di ; to = tm + tn ; tq = tm - tn
ti = te + kp618033988 * th ; tk = th - kp618033988 * te
t9 = ar - kp250000000 * t8 ; tu = ts + kp618033988 * tt
tw = tt - kp618033988 * ts ; tp = ai - kp250000000 * to
tb = t9 + kp559016994 * ta ; tj = t9 - kp559016994 * ta
tr = tp + kp559016994 * tq ; tv = tp - kp559016994 * tq
r4 = (tb + kp951056516 * ti) :+ (tr - kp951056516 * tu)
r3 = (tj - kp951056516 * tk) :+ (tv + kp951056516 * tw)
r2 = (tj + kp951056516 * tk) :+ (tv - kp951056516 * tw)
r1 = (tb - kp951056516 * ti) :+ (tr + kp951056516 * tu)
in generate 5 $ \i -> case i of
0 -> (ar + t8) :+ (ai + to)
1 -> if sign == 1 then r1 else r4
2 -> if sign == 1 then r2 else r3
3 -> if sign == 1 then r3 else r2
4 -> if sign == 1 then r4 else r1
```

This code (which is actually a bit of a cheat, as I’ll explain) demonstrates the principal problem with these hand-coded transforms: they are very tedious to write, and doing them properly is very error-prone. The code above is adapted from equivalent code in FFTW (translated from C to Haskell), and I still made a couple of confusing errors with the translation that took some time to track down.

The right way to do this (and the way that FFTW does it) is to *generate* the code for these small transforms. FTTW comes with an OCaml program called `genfft`

that generates C “codelets” for fixed transform sizes. It does this the way it should be done, which is to build a directed acyclic graph of the expressions relating the FFT output vector elements to the input vector elements, then recursively partitioning the expression DAG to find an optimal schedule of sub-expressions to calculate, exploiting sharing in the expression DAG. It performs a range of other optimisations and produces code with information about variable lifetimes to help a C compiler allocate variables to registers. It’s quite a complicated piece of code and while it would be quite possible to implement something similar here in Haskell, it’s not something I want to get into.

Instead, I’m just going to translate pre-computed codelets from FFTW into Haskell as needed. For the moment, there are just codelets for sizes 2, 3 and 5. I know that this is a bit of a cop-out!

## Benchmarking of full FFT with new prime length algorithm

We can benchmark the `pre-release-1`

version of our code as we did before. The only real difference in the benchmarking code is that we now split out the FFT pre-planning and execution steps, just as we do for the FFTW comparison:

```
doit :: Environment
-- ^ Criterion timing environment.
-> Int
-- ^ Problem size to benchmark.
-> Criterion (Int, Double, Double)
-- ^ (Problem size, DFT, my FFT, FFTW) result.
doit env sz = do
let v = tstvec sz
let myplan = FFT.plan sz
myffts <- runBenchmark env $ nf (FFT.fftWith myplan) v
let fftwPlan = FFTW.plan FFTW.dft sz
fftws <- runBenchmark env $ nf (FFTW.execute fftwPlan) v
let mean xs = VU.sum xs / fromIntegral (VU.length xs)
return (sz, mean myffts, mean fftws)
```

Here are some results (compare with the plot here):

Here, we show only results from our FFT code and FFTW, along with a couple of $O(N \log N)$ scaling lines. There are four things to observe in this plot:

The scaling behaviour of our FFT implementation has changed: instead of the upper bound to execution times scaling as $O(N^2)$ seen in the earlier benchmarking plot, we now have something like $O(N \log N)$ scaling for all input vector lengths. This is due to replacing simple DFTs for prime lengths with applications of Rader’s algorithm.

Because of the use of the next highest power of two size for the convolution in our implementation of Rader’s algorithm, the upper range of execution times display a series of plateaus – these all result from prime input sizes, where we need to perform a forward FFT and an inverse FFT both of a size given by the smallest power of two greater or equal than the input vector length (we only need to do one forward FFT because we pre-compute the FFT of the padded $b_q$ series in the Rader algorithm as part of the FFT pre-planning phase).

Although the overall scaling of the new code is $O(N \log N)$ as opposed to the $O(N^2)$ scaling of the original code without Rader’s algorithm, it is often the case that the new code is slightly

*slower*than the old code for a given input vector size. The main reason for this is that, for the base transforms for a given input vector length, we now use either a specialised hard-coded transform (which I’ve so far only implemented for vector lengths 2, 3 and 5), or we use Rader’s algorithm. The prior version of the code used a simple DFT for the base transforms in all cases. Although the asymptotic scaling of Rader’s algorithm is better than the simple DFT ($O(N \log N)$ compared to $O(N^2)$), Rader’s algorithm is more complex and has comparatively unfavourable constant factors compared to the simple DFT. The result is that the simple DFT can be faster for short vector lengths making the overall FFT execution times for the code using Rader’s algorithm slower. (Rader’s algorithm*is*good for larger prime input vector lengths.) There are two things we can do to fix this. First, we can implement more hard-coded straight-line base transforms. As described above, this is kind of a hassle, but we can cheat by translating FFTW’s codelets into Haskell. Second, we can use empirical measurements of FFT execution times to decide on whether to use Rader’s algorithm or the simple DFT for small factors. We’ll explore both of these issues in the next article when we look at various optimisation approaches.There’s still a lot of variability in the performance of our algorithm for different input vector sizes, particularly compared to the range of variability in performance for FFTW. Although both the bottom and top range of execution times appear to scale as $O(N \log N)$, there’s quite a wide band of variability between those limits. To some extent, this wide variability may just an artefact of the overall worse performance of our code, so we’ll come back and explore it later.

So, preamble over, we’ll get to optimisation in the next article!