I've been implementing a function for calculating the nth Fibonacci number in F#. So far the best implementation I could come up with is this:

let fib n =
    let rec fib =
        | 0 -> 0I, 1I
        | n ->
            let f1, f2 = fib (n / 2)
            let f1' = f1 * (2I * f2 - f1)
            let f2' = f2 * f2 + f1 * f1
            if n % 2 = 0 then
                (f1', f2')
                (f2', f1' + f2')
    fib n |> fst

Can it be improved or written in a more F#-idiomatic way?

Also, as a separate question, can it be rewritten to be tail-recursive?

  • \$\begingroup\$ The question regarding tail-recursion is off-topic as we do not assist in adding additional implementation. As long as everything else works, it can still be reviewed. \$\endgroup\$
    – Jamal
    Commented Jul 5, 2014 at 17:12
  • 2
    \$\begingroup\$ @Jamal, I thought converting to tail-recursion can count as improvement? \$\endgroup\$
    – Regent
    Commented Jul 5, 2014 at 17:17
  • \$\begingroup\$ Only if it's already here in some form. \$\endgroup\$
    – Jamal
    Commented Jul 5, 2014 at 17:18
  • \$\begingroup\$ I don't like that the two functions have exactly the same name. Otherwise, this seems like a good way to write this. \$\endgroup\$
    – svick
    Commented Jul 7, 2014 at 18:09

1 Answer 1


A concise and idiomatic (I think) implementation (which happens to be tail recursive) but without your / 2 optimisation would be:

let fib n =
    let rec tail n1 n2 = function
    | 0 -> n1
    | n -> tail n2 (n2 + n1) (n - 1)
    tail 0I 1I n

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