When is double rounding innocuous?

Double rounding is the phenomenon that occurs when the result of an operation is rounded to fit some intermediate destination, and then again when delivered to its final destination. This can be a common occurrence when using some floating-point arithmetic engines which lack single precision registers: results of operations are typically rounded to fit in a register, whose width may be double precision or wider, before being stored in some memory location possibly in a format narrower than that of the registers. Examples of such floating-point arithmetic engines include Intel's x87 series and IBM's POWER architecture. (Implementations of the latter are found in some IBM workstations.)