Partial Square Root Calculation Using Exponent-Based Initialization
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Solution Overview
Problem
Calculating square roots using digit-recurrence methods is time-consuming due to the large number of iterations required, and existing methods increase circuit complexity and power consumption as radix increases.
Innovation Solution
A data processing apparatus and method that initializes the partial square root and remainder using predetermined values based on whether the exponent is odd or even, reducing the number of iterations by bypassing the initial iteration and utilizing redundant representation to efficiently determine the output values.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Loss of time
If the number of iterations is reduced to speed up calculation, then calculation time is improved, but accuracy may deteriorate
Solution Approach 1:
The patent applies preliminary action by pre-calculating and storing correction values in a lookup table before the main iterative process begins. The most significant bits of the square root are calculated in advance using a simplified method, and correction values are pre-computed based on these initial bits. This allows the iterative process to start from a closer approximation, reducing the number of iterations needed while maintaining accuracy.
Solution Approach 2:
The patent changes parameters by dynamically adjusting the precision and range of the iterative process based on the most significant bits determined in the preliminary stage. By using the pre-calculated most significant bits to select appropriate correction values from the lookup table, the system adapts the calculation parameters to match the actual input value, achieving high accuracy with fewer iterations.
2Productivity
If radix is increased to reduce iterations, then calculation speed is improved, but circuit complexity increases
Solution Approach 1:
The patent segments the square root calculation into two distinct parts: the most significant bits calculated using a simplified preliminary method, and the less significant bits calculated through iterative refinement. This segmentation allows each part to use optimization strategies suited to its requirements, avoiding the need for high radix throughout the entire calculation process.
Solution Approach 2:
The patent applies partial action by calculating only the most significant bits with high precision in the preliminary stage, then using these to guide a shorter iterative process for the remaining bits. The lookup table provides pre-computed correction values that partially complete the calculation, reducing the burden on the iterative circuit and allowing moderate radix values to achieve high overall productivity.
3Use of energy by moving object
If radix is increased to reduce iterations, then power consumption is reduced, but circuit complexity increases
Solution Approach 1:
By pre-calculating the most significant bits and generating correction values before the main iterative process, the patent reduces the total number of iterations required. This preliminary action decreases the overall energy consumption of the iterative circuit, as fewer iterations mean less energy used by the recurrent logic, while the lookup table can be implemented with energy-efficient memory structures.
Data Source
AI summary
A data processing apparatus is provided, to calculate an at least partial square root of a floating point number having an exponent and significand. Recurrence circuitry performs one or more iterations of an iterative square root operation, each of the one or more iterations receiving an input at least partial square root and an input remainder to produce the at least partial square root and a remainder of performing the iterative square root operation. The recurrence circuitry provides the at least partial square root and the remainder as the input at least partial square root and the input remainder for a subsequent iteration of the iterative square root operation. The recurrence circuitry includes initialization circuitry to provide the at least partial square root and the remainder after at least an initial iteration of the one or more iterations. The initialization produces the remainder by performing a selection of one of a plurality of predetermined values in dependence on whether the exponent is odd or even.