Polynomial Circuitry Using Fixed-Point Arithmetic for Resource Reduction
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Solution Overview
Problem
Computing floating-point polynomials in integrated circuit devices requires substantial resources and latency, especially when using double-precision arithmetic, which is inefficient for applications with small input ranges.
Innovation Solution
Implementing floating-point polynomial calculations using fixed-point resources by denormalizing coefficients and performing calculations as fixed-point operations, reducing the number of required resources and datapath length, and utilizing coefficient tables with shifted values based on exponents for efficient polynomial circuitry.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If double-precision floating-point arithmetic is used for polynomial calculations, then calculation precision is improved, but resource consumption increases substantially
Solution Approach 1:
The patent changes the parameter representation from double-precision floating-point to fixed-point arithmetic. By representing polynomial coefficients and input values in fixed-point format with appropriate bit widths, the system achieves sufficient precision for small input ranges while dramatically reducing the number of required adaptive look-up tables and multipliers in PLD hardware
Solution Approach 2:
The patent pre-calculates and stores polynomial coefficients in a lookup table during device configuration or initialization. These pre-computed coefficients are loaded into the PLD's memory resources, eliminating the need for complex real-time floating-point coefficient storage and reducing runtime computational complexity
2Measurement precision
If double-precision floating-point arithmetic is used for polynomial calculations, then calculation precision is improved, but datapath length increases and latency increases
Solution Approach 1:
The patent transitions from floating-point to fixed-point arithmetic, which uses simpler hardware circuits with shorter propagation delays. Fixed-point addition and multiplication operations complete in fewer clock cycles compared to floating-point operations, directly reducing calculation latency while maintaining adequate precision for the application's small input range
3Measurement precision
If double-precision floating-point arithmetic is used for polynomial calculations, then calculation precision is improved, but the number of adaptive look-up tables and multipliers increases
Solution Approach 1:
The patent changes the arithmetic parameter from double-precision floating-point to fixed-point representation. This parameter change allows the same polynomial calculation to be performed with significantly fewer hardware resources—specifically, fewer adaptive look-up tables and multipliers in the PLD—because fixed-point operations require smaller word widths and simpler control logic
Solution Approach 2:
The patent performs pre-computation of polynomial coefficients and stores them in a lookup table. This preliminary action allows the runtime calculation to simply retrieve pre-computed values and perform simple fixed-point arithmetic, eliminating the need for numerous multipliers that would be required for real-time floating-point polynomial evaluation
Data Source
AI summary
Polynomial circuitry for calculating a polynomial having terms including powers of an input variable, where the input variable has a mantissa and an exponent, and the circuitry has a number of bits of precision, includes multiplier circuitry that calculates a common power of the input variable factored out of terms of the polynomial having powers of the variable greater than 1. The polynomial circuitry further includes, for each respective remaining term of the polynomial that contributes to the number of bits of precision: (1) a coefficient memory loaded with a plurality of instances of a coefficient for the respective term, each instance being shifted by a different number of bits, (2) address circuitry for selecting one of the instances of the coefficient based on the exponent, and (3) circuitry for combining the selected instance of the coefficient with a corresponding power of the input variable to compute the respective term.