How to hunt wild constants

TL;DR

This paper reviews various computational tools that identify or approximate mathematical constants from floating-point numbers, highlighting their capabilities, differences, and practical uses in mathematical research and computation.

Contribution

It provides a comprehensive overview of existing programs for constant identification, explaining their mechanisms and how to effectively utilize them for mathematical discovery.

Findings

Multiple programs can identify or approximate constants from floats.

These tools can suggest exact limits, bounds, or insightful approximations.

Different tools have unique sets of constants they can recognize.

Abstract

There are now several comprehensive web applications, stand-alone computer programs and computer algebra functions that, given a floating point number such as 6.518670730718491, can return concise nonfloat constants such as 3 arctan 2 + ln 9 + 1, that closely approximate the float. Examples include AskConstants, Inverse Symbolic Calculator, the Maple identify function, MESearch, OEIS, RIES, and WolframAlpha. Usefully often such a result is the exact limit as the float is computed with increasing precision. Therefore these program results are candidates for proving an exact result that you could not otherwise compute or conjecture without the program. Moreover, candidates that are not the exact limit can be provable bounds, or convey qualitative insight, or suggest series that they truncate, or provide sufficiently close efficient approximations for subsequent computation. This article describes some of these programs, how they work, and how best to use each of them. Almost everyone who uses or should use mathematical software can benefit from acquaintance with several such programs, because these programs differ in the sets of constants that they can return.