Backward Error Analysis for Floating-Point Circuit Design
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Solution Overview
Problem
Existing floating-point calculation circuits face challenges in accurately measuring performance due to unbounded relative forward errors, making maximum relative forward error an impractical indicator.
Innovation Solution
The use of maximum relative backward error as a design criterion, where parameters characterizing circuits for floating-point arithmetic operations are related to this error through an equation, allowing for the identification of circuits operable at desirable output accuracy.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If relative forward error is used as a performance indicator for floating-point calculation circuits, then it can measure the difference between true output and approximation, but it cannot provide bounded error metrics for operations like floating-point addition where maximum relative forward error is infinite
Solution Approach 1:
The patent inverts the traditional forward error approach by using backward error analysis. Instead of measuring the difference between true and approximate outputs directly, it determines the maximum relative backward error by analyzing how much the input would need to change to produce the approximate output. This inversion transforms an unbounded measurement problem into a bounded characterization problem that provides reliable performance metrics for floating-point circuits.
2Manufacturing precision
If maximum relative forward error is used to evaluate floating-point addition circuits, then it attempts to measure output accuracy, but the metric becomes infinite and thus impractical for performance indication
Solution Approach 1:
The patent introduces backward error as an intermediary metric that bridges the gap between theoretical output accuracy and practical performance evaluation. By characterizing the maximum relative backward error through circuit parameters rather than direct output comparison, it provides a practical, bounded metric that reflects output accuracy without encountering the infinity problem of forward error in floating-point addition.
3Manufacturing precision
If circuit parameters are varied to improve output accuracy, then desirable accuracy can be achieved, but the complexity of evaluating all parameter combinations increases
Solution Approach 1:
The patent performs preliminary characterization of the relationship between circuit parameters and maximum relative backward error. By establishing this characterization upfront, the patent enables direct evaluation of different parameter combinations using the backward error metric without requiring exhaustive simulation or measurement, thus reducing evaluation complexity while maintaining accuracy assessment capability.
Data Source
AI summary
Designing a circuit to perform a floating point arithmetic operation by identifying a multiple of parameters that characterize circuits for performing the floating point arithmetic operation and an equation relating the plurality of parameters to a maximum relative backward error parameter, the circuits respectively corresponding to combinations of values for the parameters; specifying a target maximum relative backward error for the floating point arithmetic operation; computing a maximum relative backward error for each of one or more of the combinations of values based on the equation; and when the maximum relative backward error for a respective combination of values is less than the target maximum relative backward error, identifying the circuit corresponding to the maximum relative backward error as a circuit operable to perform the floating point arithmetic operation at a desirable output accuracy.