Newton-Raphson Divide and Square-Root Error Checking
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Solution Overview
Problem
Existing methods for performing Newton-Raphson divide and square-root operations are inefficient in performing error-check and exact-check computations, particularly when rounding results, as they often require multiple computations to determine the correctness and exactness of the final result.
Innovation Solution
A system that performs Newton-Raphson iterations followed by a multiply to produce a result with additional accuracy, then analyzes these additional bits to determine if the rounded result is correct and exact, potentially performing an additional error-check or exact-check computation only if necessary, thereby reducing the need for redundant computations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If multiple error-check and exact-check computations are performed to ensure result correctness, then reliability is improved, but computational overhead and processing time increase
Solution Approach 1:
The patent applies partial action by performing error-check and exact-check computations selectively rather than always. The system analyzes additional bits of accuracy from Newton-Raphson iterations to determine when full error-check/computation is necessary versus when the result can be accepted directly, thus reducing unnecessary computational overhead while maintaining reliability
Solution Approach 2:
The patent performs preliminary analysis of additional bits of accuracy from Newton-Raphson iterations before committing to full error-check computations. By examining these intermediate results first, the system can determine in advance whether further computations are needed, saving time when the preliminary analysis is sufficient to guarantee correctness
2Reliability
If multiple error-check and exact-check computations are performed to ensure result correctness, then reliability is improved, but device complexity increases
Solution Approach 1:
The patent reduces device complexity by implementing partial action - only performing full error-check computations when the preliminary analysis of additional bits indicates they are necessary. This conditional approach simplifies the overall computational structure compared to always performing complete verification sequences
Solution Approach 2:
The patent segments the verification process into two distinct phases: (1) preliminary analysis of additional bits from Newton-Raphson iterations, and (2) full error-check computations. This segmentation allows the system to handle simple cases efficiently while reserving complex computations for cases that truly require them
3Measurement precision
If additional bits of accuracy are retained beyond desired accuracy, then measurement precision is improved, but loss of information is reduced and processing efficiency improves
Solution Approach 1:
The patent performs preliminary action by retaining and analyzing additional bits of accuracy from Newton-Raphson iterations before final rounding. This preliminary retention of extra precision enables the system to make informed decisions about result correctness without requiring full error-check computations, thereby improving processing efficiency while maintaining necessary precision
Data Source
AI summary
One embodiment of the present invention provides a system that performs both error-check and exact-check operations for a Newton-Raphson divide or square-root computation. During operation, the system performs Newton-Raphson iterations followed by a multiply for a divide or a square-root operation to produce a result, which includes one or more additional bits of accuracy beyond a desired accuracy for the result. Next, the system rounds the result to the desired accuracy to produce a rounded result t. The system then analyzes the additional bits of accuracy to determine whether t is correct and whether t is exact.