The Ackermann function is a classic example of a recursive function, notable especially because it is not a primitive recursive function. It grows very quickly in value, as does the size of its call tree.



The Ackermann function is usually defined as follows:







A
(
m
,
n
)
=


{



n
+
1



if 

m
=
0




A
(
m
−
1
,
1
)



if 

m
>
0

 and 

n
=
0




A
(
m
−
1
,
A
(
m
,
n
−
1
)
)



if 

m
>
0

 and 

n
>
0.








{\displaystyle A(m,n)={\begin{cases}n+1&{\mbox{if }}m=0\\A(m-1,1)&{\mbox{if }}m>0{\mbox{ and }}n=0\\A(m-1,A(m,n-1))&{\mbox{if }}m>0{\mbox{ and }}n>0.\end{cases}}}






Its arguments are never negative and it always terminates.



Task

Write a function which returns the value of 



A
(
m
,
n
)


{\displaystyle A(m,n)}

. Arbitrary precision is preferred (since the function grows so quickly), but not required.



See also

 Conway chained arrow notation for the Ackermann function.


% Ackermann function
ack = proc (m, n: int) returns (int)
    if     m=0 then return(n+1)
    elseif n=0 then return(ack(m-1, 1))
    else            return(ack(m-1, ack(m, n-1)))
    end
end ack
 
% Print a table of ack( 0..3, 0..8 )
start_up = proc ()
    po: stream := stream$primary_output()
 
    for m: int in int$from_to(0, 3) do
        for n: int in int$from_to(0, 8) do
            stream$putright(po, int$unparse(ack(m,n)), 8)
        end
        stream$putl(po, "")
    end
end start_up 