Floating-Point Accumulation Using Exponent-Based Partial Sums
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Solution Overview
Problem
Floating-point addition is non-associative due to the large range of floating-point number encoding, leading to loss of precision and non-deterministic outcomes, which are undesirable in data processing applications.
Innovation Solution
An apparatus and method that maintain multiple partial sum floating-point values and select a partial sum based on the least significant portion of the exponent of an input value, performing addition by replacing mantissa values or selecting new partial sums based on exponent equality or inequality, ensuring associative accumulation across the range of representable floating-point values.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If floating-point addition is performed using standard IEEE 754 format, then the computation speed is high, but the precision is lost and the result becomes non-deterministic due to non-associative property
Solution Approach 1:
The patent divides the accumulation process into multiple segments by maintaining multiple partial sums (e.g., 4 partial sums) instead of a single accumulator. Each partial sum accumulates a subset of input values, and the final result is obtained by summing these partial sums. This segmentation allows the system to handle the full range of floating-point values while maintaining precision, as each partial sum operates within a narrower range where precision is preserved.
Solution Approach 2:
The patent introduces an additional dimension to the accumulation process by using multiple partial sums indexed by exponent ranges. Instead of a single one-dimensional accumulator, the system uses a multi-dimensional structure where partial sums are selected based on the exponent of input values. This dimensional expansion enables the system to maintain precision across the full floating-point range by routing values to appropriate partial sums based on their magnitude.
2Reliability
If the precision of the output sum is limited to hide variations, then the final sum becomes exact, but the measurement precision is reduced
Solution Approach 1:
By segmenting the accumulation into multiple partial sums, each operating within specific exponent ranges, the system maintains full precision for each segment. The deterministic outcome is achieved by consistently routing values to the same partial sum based on their exponent, eliminating non-associativity while preserving measurement precision through the segmented structure.
3Reliability
If constraints are imposed on the order of summation to achieve deterministic results, then the outcome becomes deterministic, but the ease of operation is reduced
Solution Approach 1:
The patent changes the parameter used for selecting the accumulator from a fixed order constraint to an exponent-based selection criterion. Instead of requiring inputs to be processed in a specific sequence, the system selects the appropriate partial sum based on the exponent of each input value. This parameter change maintains deterministic outcomes while preserving operational flexibility, as values can be accumulated in any order and the system will automatically route them to the correct partial sum.
Data Source
AI summary
An apparatus and method for generating a sum of floating-point input values are provided. To sum the values multiple partial sum floating-point values are maintained and the partial sum to which an input value may be added is selected by a least significant portion of the exponent of the input value. If the exponent of the input value is equal to the exponent of the value stored in the selected partial sum a mantissa sum of the input value and stored partial sum value replaces the mantissa value of the selected partial sum value. If the exponent of the input value is larger than the exponent of the value stored in the selected partial sum the selected partial sum value is replaced with the input value. An associative and deterministic summation is thus provided.