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

TL;DR

This work addresses the limitation of existing EFT guarantees for FastTwoSum by deriving broader sufficient conditions that ensure the rounding error $\delta=a+b-\circ(a+b)$ lies in the floating-point set $\mathbb{F}$ under all faithful rounding modes, including round-to-odd (RO). The authors present general FR-based conditions that permit large exponent differences between inputs (up to nearly $2p-1$) and introduce RO-specific EFT criteria, leveraging RO's parity and saturation properties. They also adapt the ExtractScalar splitting to RO, providing an EFT for distributing a number across two FP components with configurable bit distribution, and extend these results to vector inputs. Collectively, these contributions widen the domain of inputs for which FastTwoSum acts as an EFT, enabling robust, faithful rounding-based algorithms and informing RO-enabled future standards and multi-precision arithmetic.

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).

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• Corollary 1
• Definition 4
• Lemma 2
• Corollary 2
• Lemma 3
• Theorem 1