Memoize a compiled function

Question

Let f be some function, I know that i can memoize a function so that it does not compute again previously calculated values, I can also compile the function to speed up the calculations. But how can I memoize a compiled function ? Ideally I would want to do something like this

Clear[f]
f[x_] := Total[Table[i*x + i*x^2 + i*x^3, {i, 1000000}]];
f[12.] // RepeatedTiming (* timing for the non memoized non compile function*)
Clear[f]
f = Compile[{{x, _Real}}, Total[Table[i*x + i*x^2 + i*x^3, {i, 1000000}]]];
f[12.] // RepeatedTiming  (*timing for the non memoized compiled function,this ideally should be faster than the non compiled function  *)
f[-12.] // RepeatedTiming (*timing for the non memoized compiled function,this ideally should be faster than the non compiled function   *)
Clear[f]
f[x_] := 
  f[x] = f[-x] = 
    Compile[{{x, _Real}}, 
     Total[Table[
       i*x + i*x^2 + i*x^3, {i, 
        1000000}]]]; (* This syntax is incorrect, what I am trying to say is f is a compiled function, and the result for x and -x are the same anfd should be stored*)
f[12.] // RepeatedTiming(*timing for the memoized compiled function,this ideally should be faster than the non compiled function*)
f[-12.] // RepeatedTiming(*timing for the memoized compiled function, this ideally should be "instantaneous" *)

Answer 1

You can use a separate CompiledFunction cf as a fallback when the argument does not appear in the DownValues of f.

cf = Compile[{{x, _Real}},
   Sum[i (x + x^2 + x^3), {i, 1, 1000000}],
   CompilationTarget -> "C"
   ];

ClearAll[f];
f[x_] := f[x] = cf[x]; 

I would not use f[x_] := f[x] = f[-x] = cf[x]; as the result is not independent of the sign of the argument.