Fabrice Bellard's PI page

I have spent some time on the computation of Pi. This is not so uninteresting as it seems to be since it uses many beautiful algorithms.

Some new formulas and algorithms

  • (Jan 20 1997)
    While playing with the Taylor series of log, I found this new and faster formula to compute the n'th binary digit of Pi. The article (Jan 20 1997) in html or in PDF pi_bin.pdf (revised edition, Feb 2007). It is now used in almost every computation of the binary digits of pi.
  • (Jan 10 1997)
    Simon Plouffe has found an algorithm to compute the n'th digit of Pi in any base in O(n^3log(n)^3) with little memory. We give here an improvement of his algorithm to get a speed of O(n^2). The postscript file of the alpha version of the article: pi_n2.ps , the html version, and the corresponding implementation in C: pi.c . (Feb 27 1997) A faster implementation which uses the Gosper formula : pi1.c .

  • (Feb 4 1997)
    While testing some numerical relations with the PSLQ algorithm , I found this exotic formula for Pi:

    with

    (Aug 2003)Boris Gourevitch and Jesus Guillera show how to demonstrate similar formulas.

    • A world record !

    (Sept 22 1997)
    The 1000 billionth binary digit of Pi is '1' .

    See my previous computations of the binary digits of Pi.

    Now the PiHex project has computed binary digits of Pi much further.

    Classical Pi computation with the FFT multiplication

    (Jul 1995) I have implemented the Pollard's FFT fast multiplication method. I have used it to compute about 10 million digits of Pi on an Alpha DEC 3000 in about 1.5 day. I can send you the file containing them :-)
    A survey (in French) of the multiplication method: transparent.ps The complete sources of the program (in ANSI C for UNIX): pi_salamin.tar.gz

    Related information

  • The Pi page of the CECM. 
  • Favorite Mathematical Constants from Steven Finch.
  • The Pi entry in Eric's online encyclopedia.
  • The PiHex project by Colin Percival, a distributed effort to calculate Pi.
  • 3.14 et la suite, a French page written by Cadin Gildas.
  • L'univers de Pi, by Boris Gourevitch.
  • Jörg Arndt's home page .
  • Simon Plouffe's home page .
  • A French book about Pi: Le fascinant nombre Pi , Jean-Paul Delahaye, Editions Belin / Pour La Science, Paris, 1997, ISSN 0224-5159 ISBN 2-9029-1825-9. 
  • A German book about Pi by Jörg Arndt:  Pi - Algorithmen, Computer, Arithmetik, Springer-Verlag Heidelberg, ISBN 3-540-63419-3. English translation to come.
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    Fabrice Bellard - http://bellard.org/