Towards platform-independent verification of the standard mathematical functions: the square root function

TL;DR

This paper details a human-oriented specification and pen-and-paper verification of a platform-independent square root function, emphasizing properties of machine arithmetic to ensure correctness, with future work planned for computer-aided proof validation.

Contribution

It introduces a formal specification and verification approach for the square root function that is independent of specific hardware platforms, focusing on properties of machine arithmetic.

Findings

Explicit properties of machine arithmetic are sufficient for verification.

The approach uses Floyd-Hoare style verification with precise and machine arithmetic.

Future work includes computer-aided proof validation.

Abstract

The paper presents (human-oriented) specification and (pen-and-paper) verification of the square root function. The function implements Newton method and uses a look-up table for initial approximations. Specification is done in terms of total correctness assertions with use of precise arithmetic and the mathematical square root $\sqrt{\dots}$, algorithms are presented in pseudo-code with explicit distinction between precise and machine arithmetic, verification is done in Floyd-Hoare style and adjustment (matching) of runs of algorithms with precise arithmetic and with machine arithmetic. The primary purpose of the paper is to make explicit properties of the machine arithmetic that are sufficient to make verification presented in the paper. Computer-aided implementation and validation of the proofs (using some proof-assistant) is the topic for further studies.