Relative error due to a single bit-flip in floating-point arithmetic

TL;DR

This paper analyzes the probability that a single random bit-flip in IEEE 754 double precision floating-point numbers causes a relative error below certain thresholds, providing probabilistic error bounds.

Contribution

It introduces a probabilistic framework for quantifying the impact of random single bit-flips on floating-point accuracy in double precision.

Findings

Over 25% chance of relative error below 10^-11

Over 50% chance of relative error below 10^-6

Provides probabilistic error bounds for random bit-flips

Abstract

We consider the error due to a single bit-flip in a floating point number. We assume IEEE 754 double precision arithmetic, which encodes binary floating point numbers in a 64-bit word. We assume that the bit-flip happens randomly so it has equi-probability (1/64) to hit any of the 64 bits. Since we want to mitigate the assumption on our initial floating-point number, we assume that it is uniformly picked among all normalized number. With this framework, we can summarize our findings as follows. The probability for a single bit flip to cause a relative error less than 10^-11 in a normalized floating-point number is above 25%; The probability for a single bit flip to cause a relative error less than 10^-6 in a normalized floating-point number is above 50%; Etc.