Odd but Error-Free FastTwoSum: More General Conditions for FastTwoSum as an Error-Free Transformation for Faithful Rounding Modes

2026-01-23

AI Summary

This work addresses the limited applicability of the FastTwoSum algorithm as an error-free transformation (EFT), which traditionally fails to guarantee correctness across all operand ranges under arbitrary faithful rounding modes. We propose a more general sufficient condition that, for the first time, rigorously ensures the EFT property of FastTwoSum under any faithful rounding mode—including round-to-odd—thereby substantially expanding its domain of validity. To achieve this, we integrate floating-point arithmetic theory with the specific characteristics of round-to-odd to design ExtractScalar, a configurable floating-point splitting method that controls bit-level operand decomposition. Our contribution not only provides a theoretical foundation for FastTwoSum’s correctness under diverse faithful rounding modes but also delivers a practical tool for high-precision numerical computation.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

This paper proposes sufficient, yet more general conditions for applying FastTwoSum as an error-free transformation (EFT) under all faithful rounding modes. Additionally, it also identifies guarantees tailored to round-to-odd for establishing FastTwoSum as an EFT. This paper also describes a floating-point splitting tailored for round-to-odd that is an EFT where the distribution of bits is configurable (i.e., ExtractScalar for round-to-odd). Our sufficient conditions are much more general than those previously known in the literature (i.e., it applies to a wider operand domain).

Problem

• FastTwoSum
• error-free transformation
• faithful rounding
• round-to-odd
• floating-point arithmetic

Innovation

• FastTwoSum
• error-free transformation
• faithful rounding
• round-to-odd
• floating-point splitting