Estimated number (Heuristics) of operations between Two Integers

Question

I have two integers A and B and I want to get, WLOG say, from A to B with minimum numbers of the following operations:

• Add or subtract 1
• Multiple or divide by 2
• Square the number

Currently I am solving this problem using A* search where operation weights are all 1's with heuristics of 0 (Basically Dijkstra's Algorithm, I guess). But I want a better heuristics to explore less. So far, all my attempts have failed. For example, if both are positive and unequal, I tried to estimate by squaring and doubling as much as possible. I also tried to think in terms of bit strings but 111 + 1 = 1000 makes me feel like it will not be a good heuristics. I want some ideas, if not solutions, to come up with a good heuristics.

Answer 1

If A > B * 2^n , you know it must take at least n steps since you can only divide by 2. Solving for n we get ceil(log_2(A/B)) as a heuristic. This assumes A and B have the same sign.

If A < B then we're allowed to square, so unfortunately the equation to solve is A^(2^n) < B , or ceil(log_2(log_A(B))) as a lower-bound. This assumes A ≥ 2.