Tak (function) explained

In computer science, the Tak function is a recursive function, named after Ikuo Takeuchi . It is defined as follows:

def tak(x, y, z): 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.^{[1] [2] [3] [4]}

tak vs. tarai

The original definition by Takeuchi was as follows:

def tarai(x, y, z): 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 たらい回し tarai mawashi, "to pass around" in Japanese.

John McCarthy named this function tak after Takeuchi.^[5]

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 -> Inttarai 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 mutually recursive helper function as follows.

def laziest_tarai(x, y, zx, zy, zz): 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, y, z): 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)Note the additional check for (x <= y) before z (the third argument) is evaluated, avoiding unnecessary recursive evaluation.

External links

• • TAK Function

Notes and References

1. Peter Coffee. Tak test stands the test of time . PC Week. 1996 . 13 . 39 .
2. http://buildingblocksjava.com/recursive-methods/ "Recursive Methods"
3. News: Six of the Best Against the Clock . Acorn User . June 1986 . 28 October 2020 . Johnson-Davies . David . 179, 181–182 .
4. News: Testing the Tak . Acorn User . November 1986 . 28 October 2020 . Johnson-Davies . David . 197, 199 .
5. John McCarthy . An Interesting LISP Function . ACM Lisp Bulletin . December 1979 . 3 . 6–8 . 10.1145/1411829.1411833. 31639459 .