1. A method of operating a computing system having a main storage and a plurality of different classes of registers to assign registers to requests serially referenced in an input instruction stream for quantities to be in a register, so as to improve utilization of the registers during execution of a compiled instruction stream, comprising the steps of: (1) during a serial scan of the input instruction stream, identifying register usage for quantities, including (a) quantities referenced in the input instruction stream which need to be assigned to a register and (b) quantities referenced in the input instruction stream which are either assigned to a register or logged to a set of register quantities awaiting assignment to a register; (2) upon identifying during step (1) a first quantity which needs to be assigned to a register, logging the first quantity to the set of register quantities awaiting assignment to a register with a control count indicative of the total number of registers for which the first quantity is not a candidate; (3) upon identifying during step (1) a second quantity either assigned to a register or logged to the set of register quantities awaiting assignment to a register, incrementing the control counts of quantities in the set of register quantities awaiting assignment to a register which are candidates for the same registers as the second quantity, and testing for complete sets; and (4) assigning a register to those quantities for which a complete set is detected.