In computer science, the Tak function is a recursive function, named after . It is defined as follows:
def tak(x: int, y: int, z: int) -> int: if y < x: return tak( tak(x - 1, y, z), tak(y - 1, z, x), tak(z - 1, x, y) ) else: return z
This function is often used as a benchmark for languages with optimization for recursion. "Recursive Methods" by Elliotte Rusty Harold
tak() vs. tarai()
The original definition by Takeuchi was as follows:
def tarai(x: int, y: int, z: int) -> int: if y < x: return tarai( tarai(x - 1, y, z), tarai(y - 1, z, x), tarai(z - 1, x, y) ) else: return y # not z!
tarai is short for in Japanese.
John McCarthy named this function tak() after Takeuchi.
However, in certain later references, the y somehow got turned into the z. This is a small, but significant difference because the original version benefits significantly from lazy evaluation.
Though written in exactly the same manner as others, the Haskell code below runs much faster.
tarai :: Int -> Int -> Int -> Int tarai x y z | x <= y = y | otherwise = tarai (tarai (x-1) y z) (tarai (y-1) z x) (tarai (z-1) x y)
One can easily accelerate this function via memoization yet lazy evaluation still wins.
The best known way to optimize tarai is to use a mutually recursive helper function as follows.
def laziest_tarai(x: int, y: int, zx: int, zy: int, zz: int) -> int: if not y < x: return y else: return laziest_tarai( tarai(x-1, y, z), tarai(y-1, z, x), tarai(zx, zy, zz)-1, x, y)
def tarai(x: int, y: int, z: int) -> int: if not y < x: return y else: return laziest_tarai( tarai(x-1, y, z), tarai(y-1, z, x), z-1, x, y)
Here is an efficient implementation of tarai() in C: int tarai(int x, int y, int z) { while (x > y) { int oldx = x, oldy = y; x = tarai(x - 1, y, z); y = tarai(y - 1, z, oldx); if (x <= y) break; z = tarai(z - 1, oldx, oldy);
return y; } Note the additional check for (x <= y) before z (the third argument) is evaluated, avoiding unnecessary recursive evaluation.
References
External links
TAK Function