Improved computing processor. In an embodiment, one or more roots of a perturbed polynomial equation, comprising a plurality of terms, are computed, assuming a non-zero coefficient for a highest-order one of the plurality of terms. For at least a highest-order term, an error upper bound of an unperturbed coefficient of the term is computed, it is determined whether a perturbed coefficient of the term is less than or equal to the error upper bound, and, when the perturbed coefficient of the term is less than or equal to the error upper bound, one or more roots of the perturbed polynomial equation are computed, assuming a zero coefficient for the term. Each computed root is added to a root set.

Apparatuses and methods of manufacturing same, systems, and methods for performing recursive operations using a partial remainder-divisor (PD) table are described. In one aspect, it is determined whether a current cell in the PD table indicated by a current partial remainder/radicand row value and a current divisor/root column value is outside a primary region of the PD table. If the current cell is outside the primary region of the PD table, at least one of the current partial remainder/radicand row value and the current divisor/root column value are adjusted so that the indicated current cell falls within the primary region of the PD table.

Apparatuses and methods of manufacturing same, systems, and methods for generating a starting estimate for radix-16 square root iterative calculation using hardware, including a radix-4 partial remainder-divisor (PD) table, which is used for both division and square root operations, are described. In one aspect, a part of a radicand for a radix-16 square root iterative operation is used to determine column/root and row/partial radicand values, which are then used to determine a starting estimate from a radix-4 PD table for the radix-16 square root iterative operation.

Apparatuses and methods of manufacturing same, systems, and methods for performing recursive operations using a partial remainder-divisor (PD) table are described. In one aspect, it is determined whether a current cell in the PD table indicated by a current partial remainder/radicand row value and a current divisor/root column value is outside a primary region of the PD table. If the current cell is outside the primary region of the PD table, at least one of the current partial remainder/radicand row value and the current divisor/root column value are adjusted so that the indicated current cell falls within the primary region of the PD table.

Apparatuses and methods of manufacturing same, systems, and methods for generating a starting estimate for radix-16 square root iterative calculation using hardware, including a radix-4 partial remainder-divisor (PD) table, which is used for both division and square root operations, are described. In one aspect, a part of a radicand for a radix-16 square root iterative operation is used to determine column/root and row/partial radicand values, which are then used to determine a starting estimate from a radix-4 PD table for the radix-16 square root iterative operation.