Widening integer arithmetic

Some codes require computations to use fewer bits of precision than are normal for the target machine. For example, Java requires 32-bit arithmetic even on a 64-bit target. To run narrow codes on a wide target machine, we present a widening transformation. Almost every narrow operation can be widened by sign- or zero-extending the operands and using a target-machine instruction at its natural width. But extensions can sometimes be avoided, and our transformation avoids as many as possible. The key idea is knowing what each operation can accept in the high bits of its arguments and what it can guarantee about the high bits of its result. This knowledge is formalized using fill types, which drive the widening transformation.