Circuit Transformations, Loop Fusion, and Inductive Proof

3 days ago

Carry-save addition is a hardware optimization that speeds up adding three bitvectors by using full-adders in parallel to reduce three addends to two, avoiding slow sequential carry propagation.
Loop fusion is a compiler optimization that merges multiple loops into one, reducing iteration overhead and enabling operation fusion for efficiency.
The blog explores whether loop fusion can be used to discover carry-save addition by analyzing a program that adds three bitvectors with two sequential loops and then fusing them.
A proof using bit-level analysis and induction shows that the fused loop program with carry-save addition is equivalent to the original sequential addition program, without relying on bitvector meaning.
The conclusion suggests that compiler optimizations like loop fusion could potentially discover other hardware-like tricks, bridging the gap between hardware and compiler optimization techniques.

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