# Distant Decimals of $$\pi$$ π : Formal Proofs of Some Algorithms Computing Them and Guarantees of Exact Computation

Distant Decimals of $$\pi$$ π : Formal Proofs of Some Algorithms Computing Them and... We describe how to compute very far decimals of $$\pi$$ π and how to provide formal guarantees that the decimals we compute are correct. In particular, we report on an experiment where 1 million decimals of $$\pi$$ π and the billionth hexadecimal (without the preceding ones) have been computed in a formally verified way. Three methods have been studied, the first one relying on a spigot formula to obtain at a reasonable cost only one distant digit (more precisely a hexadecimal digit, because the numeration basis is 16) and the other two relying on arithmetic–geometric means. All proofs and computations can be made inside the Coq system. We detail the new formalized material that was necessary for this achievement and the techniques employed to guarantee the accuracy of the computed digits, in spite of the necessity to work with fixed precision numerical computation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Automated Reasoning Springer Journals

# Distant Decimals of $$\pi$$ π : Formal Proofs of Some Algorithms Computing Them and Guarantees of Exact Computation

, Volume 61 (4) – Dec 20, 2017
39 pages

/lp/springer_journal/distant-decimals-of-pi-formal-proofs-of-some-algorithms-computing-them-ovQRfH9lDh
Publisher
Springer Netherlands
Copyright © 2017 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Computer Science; Mathematical Logic and Formal Languages; Artificial Intelligence (incl. Robotics); Mathematical Logic and Foundations; Symbolic and Algebraic Manipulation
ISSN
0168-7433
eISSN
1573-0670
D.O.I.
10.1007/s10817-017-9444-2
Publisher site
See Article on Publisher Site

### Abstract

We describe how to compute very far decimals of $$\pi$$ π and how to provide formal guarantees that the decimals we compute are correct. In particular, we report on an experiment where 1 million decimals of $$\pi$$ π and the billionth hexadecimal (without the preceding ones) have been computed in a formally verified way. Three methods have been studied, the first one relying on a spigot formula to obtain at a reasonable cost only one distant digit (more precisely a hexadecimal digit, because the numeration basis is 16) and the other two relying on arithmetic–geometric means. All proofs and computations can be made inside the Coq system. We detail the new formalized material that was necessary for this achievement and the techniques employed to guarantee the accuracy of the computed digits, in spite of the necessity to work with fixed precision numerical computation.

### Journal

Journal of Automated ReasoningSpringer Journals

Published: Dec 20, 2017

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