Are there functional alternatives to recursive calls that are more efficient or use less stack space?

Question

Are there functional alternatives to recursive calls that are more efficient or use less stack space?

Answer

One way to reduce the number of recursive calls is to use Dafny’s function-by-method construct to give a better performing implementation. For example

function Fib(n: nat): nat {
  if n < 2 then n else Fib(n-1) + Fib(n-2)
}

it a very natural but quite inefficient implementation of the Fibonacci function (it takes Fib(n) calls of Fib to compute Fib(n), though only to a stack depth of n).

With function-by-method we can still use the natural recursive expression as the definition but provide a better performing implementation:

function Fib(n: nat): nat
 {
  if n < 2 then n else Fib(n-1) + Fib(n-2)
} by method {
  var x := 0;
  var y := 1;
  for i := 0 to n 
    invariant x == Fib(i) && y == Fib(i+1)
  {
    x, y := y, x+y;
  }
  return x;
}

This implementation both combines the two computations of Fib and also, more importantly, turns a tail recursive recursion into a loop.