Recursive Question


I have a theoretical question. If I have a computer that has the
ability to double its speed every time a function is called
recursively. i.e. each time it computes an recursion it
does so in half the time it took on the previous recursion. how would I
find the complexity for an algorithm?

e.g. binary search is usually O(log n). I'm guessing in this case
binary search would be O(log(log n)) I right?

Thanks in advance