An apparatus relating generally to accumulation is disclosed. In this apparatus, a first subtraction-bypass stage is coupled to receive an input operand and a modulus operand to provide a first difference and the input operand. An accumulation stage is coupled to the first subtraction-bypass stage to receive the first difference and the input operand. The accumulation stage is coupled to receive an offset operand for providing an offset-accumulated result. A second subtraction-bypass stage is coupled to receive the offset operand and the modulus operand to provide a second difference and the offset operand. A consolidation stage is coupled to receive the offset operand, the second difference and the offset-accumulated result to provide a consolidated accumulated result. The first subtraction-bypass stage, the accumulation stage, the second subtraction-bypass stage, and the consolidation stage are for a redundant number system.