F# using sequence cache correctly

Question

I'm trying to use Seq.cache with a function that I made that returns a sequence of primes up to a number N excluding the number 1. I'm having trouble figuring out how to keep the cached sequence in scope but still use it in my definition.

let rec primesNot1 n = 
    {2 .. n} 
    |> Seq.filter (fun i -> 
        (primesNot1 (i / 2) |> Seq.for_all (fun o -> i % o <> 0)))
    |> Seq.append {2 .. 2}
    |> Seq.cache

Any ideas of how I could use Seq.cache to make this faster? Currently it keeps dropping from scope and is only slowing down performance.

Answer 1

Seq.cache caches an IEnumerable<T> instance so that each item in the sequence is only calculated once. In your case, though, you're caching the sequence returned by a function, and each time you call the function you get a new cached sequence, which doesn't do you any good. I don't think caching is really the right approach to your problem as you've outlined it; instead you should probably look into memoization.

If instead of defining a function giving the primes less than n you want to define an infinite enumerable sequence of primes, then caching makes more sense. That would look more like this:

let rec upFrom i =
  seq { 
    yield i
    yield! upFrom (i+1)
  }

let rec primes =
  seq { 
    yield 2
    yield!
      upFrom 3 |>
      Seq.filter (fun p -> primes |> Seq.takeWhile (fun j -> j*j <= p) |> Seq.forall (fun j -> p % j <> 0))
  }
  |> Seq.cache

I haven't compared the performance of this method compared to yours.

Answer 2

I figured out how to solve my problem with a fold but not my idea of using seq.cache.

let primesNot1 n = 
    {2 .. n}
    |> Seq.fold (fun primes i ->
        if primes |> Seq.for_all (fun o -> i % o <> 0) then
            List.append primes [i]
        else
            primes) [2]

Answer 3

Have you taken a look at LazyList? Seems like it's designed to solve the same problem. It's in PowerPack.