Multi variable recursive Ackermann sequence

Multi-variable recursive functions often require more than one stopping condition or a mechanism for redirecting the recursion along with the primary recursive formula. The Ackermann function serves as a prime example.  This two-variable function is defined as follows:

1. Base Case (Stopping Condition): A function expression or formula without recursive call.  

2. Recursive Redirect: A formula that redirects the recursion when a second variable is zero. This acts as an intermediate step before reaching the base case.

3. Recursive Formula: The primary recursive formula that defines the function's behavior for other values of its variables.

1. A(0,1)=n+1

2. A(m,0)=A(m-1,1)

3. A(m,n)=A(m-1,Am,n-1))

The screenshot below shows calculation results for several parameters, raising the question of how the calculator app can handle this, especially with the constraint of a single function expression per user-defined function when using a map. The key is how the app cleverly stores and manages recursive function definitions and their arguments.

Here's a breakdown:

1. Function Storage (Map): A map is used to store user-defined functions. Because each function has only one expression, this map works well for basic functions.
   
   

2. Multi-Variable Recursion Trick: The challenge of multi-variable recursive functions (which often require multiple definitions with some parameters as constants) is overcome by storing the function and its arguments together as a single variable. This is the crucial innovation. Think of it as a specialized variable that encapsulates both the function and specific parameter values.
   
   

3. Function Evaluation Redirection: When a user-defined function is called, the app first searches the variable space. If a variable matching the function and its specific arguments is found, the app redirects the evaluation to that special variable expression rather than the main function.
   
   

4. True Function Calculation: If no matching variable (function with specific arguments) is found, the true function definition is used to perform the calculation.
   
   

5. Recursive Call Handling: Each recursive call within the function breaks down the problem into smaller parameter values. This process continues until it reaches a "stopping" or "redirecting" point (as described in step 3), or ultimately the true stopping condition of the function.
   
   

6. Memoization (Result Caching): Critically, after each calculation (whether it's a redirected lookup or a true function evaluation), the result is stored internally (likely in the variable map or a cache). This memoization dramatically speeds up subsequent calls because the app can avoid recalculating previously computed values.