[QUIZ] Digits of Pi (#202)

Discussion in 'Ruby' started by Daniel Moore, Apr 24, 2009.

  1. Daniel Moore

    Daniel Moore Guest

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    ## Digits of Pi (#202)

    Geia sas Rubyists,

    Pi or =F0 is a mathematical constant whose value is the ratio of any
    circle's circumference to its diameter in Euclidean space. Because =F0
    is an irrational number, its decimal expansion never ends and does not
    repeat. This infinite sequence of digits has fascinated mathematicians
    and laymen alike, and much effort over the last few centuries has been
    put into computing more digits and investigating the number's
    properties.[1]

    This week's quiz is to write a Ruby program that can compute the first
    100,000 digits of =F0.

    [1]: http://en.wikipedia.org/wiki/Pi

    Have Fun!
    --=20
    -Daniel
    http://rubyquiz.strd6.com
     
    Daniel Moore, Apr 24, 2009
    #1
    1. Advertising

  2. Daniel Moore wrote:
    > This week's quiz is to write a Ruby program that can compute the first
    > 100,000 digits of ð.


    Bonus points for determining the first non-trivial ruby program encoded
    in that sequence of digits.

    (Represent a program source string as a sequence of octal triplets that
    is, or is not, a subsequence of the octal representation of ð.)

    Non-trivial means not "", not just a literal, etc. Be imaginative.

    --
    vjoel : Joel VanderWerf : path berkeley edu : 510 665 3407
     
    Joel VanderWerf, Apr 24, 2009
    #2
    1. Advertising

  3. Daniel Moore

    Todd Benson Guest

    2009/4/24 Joel VanderWerf <>:
    > Daniel Moore wrote:
    >>
    >> This week's quiz is to write a Ruby program that can compute the first
    >> 100,000 digits of =F0.

    >
    > Bonus points for determining the first non-trivial ruby program encoded i=

    n
    > that sequence of digits.


    Do you mean non-trivial in that it will never fail with the correct
    initial conditions? Or simply that it's syntactically correct?
     
    Todd Benson, Apr 25, 2009
    #3
  4. Todd Benson wrote:
    > 2009/4/24 Joel VanderWerf <>:
    >> Daniel Moore wrote:
    >>> This week's quiz is to write a Ruby program that can compute the first
    >>> 100,000 digits of ð.

    >> Bonus points for determining the first non-trivial ruby program encoded in
    >> that sequence of digits.

    >
    > Do you mean non-trivial in that it will never fail with the correct
    > initial conditions? Or simply that it's syntactically correct?


    I dunno. That's why I said be imaginative :)

    --
    vjoel : Joel VanderWerf : path berkeley edu : 510 665 3407
     
    Joel VanderWerf, Apr 25, 2009
    #4
  5. Jeff Schwab wrote:
    > Joel VanderWerf wrote:
    >> Daniel Moore wrote:
    >>> This week's quiz is to write a Ruby program that can compute the first
    >>> 100,000 digits of ð.

    >>
    >> Bonus points for determining the first non-trivial ruby program
    >> encoded in that sequence of digits.

    >
    > Interesting idea!
    >
    > Is there any way of determining whether a given sequence of bytes is a
    > valid ruby program, other than running it? If this were my paternus
    > lingua, C++, I would start by chucking each candidate sequence at a
    > compiler.
    >
    >> (Represent a program source string as a sequence of octal triplets
    >> that is, or is not, a subsequence of the octal representation of ð.)

    >
    > What is an "octal triplet?" Do you mean that the integer value of each
    > byte in the source code should be represented by a sequence of three
    > octal digits (e.g. 077 for ?? and 141 for ?a)?
    >
    > I'm curious: Why octal? It seems an odd choice, given that an "octal
    > triplet" corresponds to 9 bits, rather than the 8 in a standard byte.
    > Why not "hex pairs?"


    You're right, that didn't make much sense. I was just thinking of "\nnn"
    and counting to three, but of course that's 9 bits. What I was trying
    to do was slice off just enough bits to make a printable char with high
    probability. Otherwise, the 128..255 will keep breaking up otherwise
    legal program strings.

    Base 128 (7 bits) would be better, but there are still some non-printing
    chars. Even better would be to skip ascii and use something else, but I
    didn't want to get too far into fantasy land...

    > Btw, isn't every finite sequence of digits a subsequence of Pi's
    > representation in that base? Or is that unknowable?


    IIRC that's true. But the _first_ ruby program... gee, that's got to
    mean something. :)

    --
    vjoel : Joel VanderWerf : path berkeley edu : 510 665 3407
     
    Joel VanderWerf, Apr 25, 2009
    #5
  6. >
    > This week's quiz is to write a Ruby program that can compute the first
    > 100,000 digits of =F0.
    >
    >


    # Is this cheating ? :)

    require 'bigdecimal'
    require 'bigdecimal/math'
    include BigMath

    puts PI(100_000)



    Harry


    --=20
    A Look into Japanese Ruby List in English
    http://www.kakueki.com/ruby/list.html
     
    Harry Kakueki, Apr 27, 2009
    #6
  7. Daniel Moore

    Robert Dober Guest

    2009/4/27 Harry Kakueki <>:
    >>
    >> This week's quiz is to write a Ruby program that can compute the first
    >> 100,000 digits of =F0.
    >>
    >>

    >
    > # Is this cheating ? =A0:)

    I would say no, but...
    I honestly could not come up with a solution, always getting confused
    how many significant digits I need in the involved constants. Thus I
    am really looking forward to the solutions and the summary. (Distant
    memories tell me I should do some range arithmetics, we will see)
    Sorry to say Harry, but your solution has not enlightened me on the topic ;=
    ).
    Cheers
    Robert
     
    Robert Dober, Apr 27, 2009
    #7
  8. Daniel Moore

    Todd Benson Guest

    2009/4/27 Robert Dober <>:
    > 2009/4/27 Harry Kakueki <>:
    >>>
    >>> This week's quiz is to write a Ruby program that can compute the first
    >>> 100,000 digits of =F0.
    >>>
    >>>

    >>
    >> # Is this cheating ? :)

    > I would say no, but...
    > I honestly could not come up with a solution, always getting confused
    > how many significant digits I need in the involved constants. Thus I
    > am really looking forward to the solutions and the summary. (Distant
    > memories tell me I should do some range arithmetics, we will see)
    > Sorry to say Harry, but your solution has not enlightened me on the topic=

    ;).
    > Cheers
    > Robert


    For what it's worth, I tried a Gauss-Legendre and it halted my PC at a
    delta of 1e-10 and smaller (only 10 good digits), and then started to
    diverge pretty rapidly, so much so that the program would halt (yes, I
    simply sat there and hit the gets over and over). I then tried
    Srinavasa's method, and it converged so quickly to 10 places, and
    never gave up after that, but it took about 3 hours to run k up to
    1024 (without my intervention), which amounts to around 12_000 correct
    places.

    There's Daniel Shanks, which I might try my hand at eventually, but
    the winner for this programming language might end up being
    brute-force by-digit deterministic approach mentioned by one of the
    first posters.

    I think, Robert, cheating would be writing a program that grabs the
    digits from the website :)

    Todd
     
    Todd Benson, Apr 27, 2009
    #8
  9. Daniel Moore

    Robert Dober Guest

    2009/4/25 Joel VanderWerf <>:
    >> Btw, isn't every finite sequence of digits a subsequence of Pi's
    >> representation in that base? =A0Or is that unknowable?

    >
    > IIRC that's true. But the _first_ ruby program... gee, that's got to mean
    > something. :)


    But that means that the representation of pi contains a ruby program
    that computes pi itself (although of course it never finishes), and
    that for any imaginable programming language capable of that task, eg
    a TuringMachine.

    I am confused now, but I bet the Ruby program starts exactly at the
    42**42 hexdigit - for a well chosen base of course ;)
    R.
     
    Robert Dober, Apr 27, 2009
    #9
  10. On Mon, Apr 27, 2009 at 9:15 AM, Robert Dober <> wrot=
    e:
    > 2009/4/25 Joel VanderWerf <>:
    >>> Btw, isn't every finite sequence of digits a subsequence of Pi's
    >>> representation in that base? =A0Or is that unknowable?

    >>
    >> IIRC that's true. But the _first_ ruby program... gee, that's got to mea=

    n
    >> something. :)

    >
    > But that means that the representation of pi contains a ruby =A0program
    > that computes pi itself (although of course it never finishes), and
    > that for any imaginable programming language capable of that task, eg
    > a TuringMachine.



    Been reading Goedel, Escher, Bach again, have we Robert? <G>
    --=20
    Rick DeNatale

    Blog: http://talklikeaduck.denhaven2.com/
    Twitter: http://twitter.com/RickDeNatale
    WWR: http://www.workingwithrails.com/person/9021-rick-denatale
    LinkedIn: http://www.linkedin.com/in/rickdenatale
     
    Rick DeNatale, Apr 27, 2009
    #10
  11. Daniel Moore

    Andy Cooper Guest

    [Note: parts of this message were removed to make it a legal post.]

    On Mon, 2009-04-27 at 15:12 +0900, Todd Benson wrote:

    >
    > I think, Robert, cheating would be writing a program that grabs the
    > digits from the website :)
    >
    > Todd
    >



    I was unaware that one could cheat on a rubyquiz.
     
    Andy Cooper, Apr 27, 2009
    #11
  12. Daniel Moore

    Robert Dober Guest

    On Mon, Apr 27, 2009 at 3:27 PM, Rick DeNatale <> wr=
    ote:
    > On Mon, Apr 27, 2009 at 9:15 AM, Robert Dober <> wr=

    ote:
    >> 2009/4/25 Joel VanderWerf <>:
    >>>> Btw, isn't every finite sequence of digits a subsequence of Pi's
    >>>> representation in that base? =A0Or is that unknowable?
    >>>
    >>> IIRC that's true. But the _first_ ruby program... gee, that's got to me=

    an
    >>> something. :)

    >>
    >> But that means that the representation of pi contains a ruby =A0program
    >> that computes pi itself (although of course it never finishes), and
    >> that for any imaginable programming language capable of that task, eg
    >> a TuringMachine.

    >
    >
    > Been reading Goedel, Escher, Bach again, have we Robert? <G>
    > --
    > Rick DeNatale


    To my defense I can claim: "I did not understand a bit", but it is
    indeed a great read (actually never really finished it :-O )

    >
    > Blog: http://talklikeaduck.denhaven2.com/
    > Twitter: http://twitter.com/RickDeNatale
    > WWR: http://www.workingwithrails.com/person/9021-rick-denatale
    > LinkedIn: http://www.linkedin.com/in/rickdenatale
    >
    >




    --=20
    Si tu veux construire un bateau ...
    Ne rassemble pas des hommes pour aller chercher du bois, pr=E9parer des
    outils, r=E9partir les t=E2ches, all=E9ger le travail=85 mais enseigne aux
    gens la nostalgie de l=92infini de la mer.

    If you want to build a ship, don=92t herd people together to collect
    wood and don=92t assign them tasks and work, but rather teach them to
    long for the endless immensity of the sea.

    --
    Antoine de Saint-Exup=E9ry
     
    Robert Dober, Apr 27, 2009
    #12
  13. Daniel Moore

    Luke Cowell Guest

    Re: Digits of Pi (#202)

    Here's a solution using the "plus/minus fractiony solution for pi" from
    here:
    http://tinyurl.com/3ve7wy

    Note: this is also referred to as the Leibniz formula for pi


    My solution is definitely not a great solution as after about 5 minutes,
    I've got about 8 digits and it only gets slower. I doubt it will make it
    to 100,000 digits. I'm looking forward to seeing other solutions to this
    problem.

    require "rational"
    require 'enumerator'
    require 'bigdecimal'
    require 'bigdecimal/math'
    include BigMath

    iterations = 10000000000
    current = 1
    final = BigDecimal.new("4")
    other = false

    while(current < iterations) do
    #puts current
    current = current + 2
    if(other)
    final = final + Rational(4,current)
    else
    final = final - Rational(4,current)
    end

    other = !other
    print current.to_s + ":"
    puts final.to_f
    end
    --
    Posted via http://www.ruby-forum.com/.
     
    Luke Cowell, Apr 27, 2009
    #13
  14. Daniel Moore

    Michael Kohl Guest

    2009/4/27 Harry Kakueki <>:
    > # Is this cheating ? =A0:)


    I guess the only thing I'd consider cheating would be something like this:

    require 'rubygems'
    require 'hpricot'
    require 'open-uri'

    doc =3D Hpricot(open('http://www.eveandersson.com/pi/digits/100000'))
    puts (doc/'pre').inner_html

    Michael
     
    Michael Kohl, Apr 29, 2009
    #14
  15. Daniel Moore wrote:
    > This week's quiz is to write a Ruby program that can compute the first
    > 100,000 digits of π.


    There's the impl on the benchmarks game (shootout):

    http://shootout.alioth.debian.org/u32q/benchmark.php?test=pidigits&lang=ruby&id=1

    But it doesn't seem fast enough to get to 100k digits in any amount of
    time I'm willing to wait for it. Here's timings for all the Ruby
    versions I have handy, calculating up to 10k digits:

    (jruby time is on Java 6 server VM)

    JRuby 1.3-dev: 0m47.903s
    Ruby 1.9.1: 1m27.527s
    Rubinius master: 2m50.545s
    MacRuby 0.4: 1m9.832s
    IKRuby*: 3m26.861s
    Ruby 1.8.7: 1m47.887s

    MacRuby 0.5-experimental crashed after only a few digits. IKRuby is
    JRuby on CLR (Mono, here) using IKVM.

    Seems like we need a faster algorithm!

    - Charlie
     
    Charles Oliver Nutter, Apr 29, 2009
    #15
  16. Daniel Moore

    Jay Anderson Guest

    Re: Digits of Pi (#202)

    > ## Digits of Pi (#202)
    >
    > This week's quiz is to write a Ruby program that can compute the first
    > 100,000 digits of pi.


    Attached is a solution using a machin formula
    (http://en.wikipedia.org/wiki/Machin-like_formula) translated into ruby
    from here:
    http://en.literateprograms.org/Category:Pi_with_Machin's_formula.

    $ time ruby pi.rb 100000 > pi.txt

    real 1m44.592s
    user 1m44.339s
    sys 0m0.212s

    No effort has been made to optimize, but it seems to have a reasonable
    running time.

    -----Jay

    Attachments:
    http://www.ruby-forum.com/attachment/3635/pi.rb

    --
    Posted via http://www.ruby-forum.com/.
     
    Jay Anderson, Apr 30, 2009
    #16
  17. Daniel Moore

    Robert Dober Guest

    Re: Digits of Pi (#202)

    On Thu, Apr 30, 2009 at 5:02 AM, Jay Anderson <> wrote:
    The good new is that I chose the correct algorithm from the beginning.
    The bad news is that I was waaaay to stupid to implement it, well
    done.
    BTW I tested your result, it seems to be correct :).
    Cheers
    Robert



    --=20
    Si tu veux construire un bateau ...
    Ne rassemble pas des hommes pour aller chercher du bois, pr=E9parer des
    outils, r=E9partir les t=E2ches, all=E9ger le travail=85 mais enseigne aux
    gens la nostalgie de l=92infini de la mer.

    If you want to build a ship, don=92t herd people together to collect
    wood and don=92t assign them tasks and work, but rather teach them to
    long for the endless immensity of the sea.

    --
    Antoine de Saint-Exup=E9ry
     
    Robert Dober, Apr 30, 2009
    #17
  18. Daniel Moore

    Todd Benson Guest

    Re: Digits of Pi (#202)

    On Thu, Apr 30, 2009 at 7:21 AM, Robert Dober <> wrote:
    > On Thu, Apr 30, 2009 at 5:02 AM, Jay Anderson <> wrote:
    > The good new is that I chose the correct algorithm from the beginning.
    > The bad news is that I was waaaay to stupid to implement it, well
    > done.
    > BTW I tested your result, it seems to be correct :).
    > Cheers


    Jay's version seemed to work fine. It took about 5 minutes on my
    machine for 100_000.

    I haven't verified the digits yet, though.

    Todd
     
    Todd Benson, Apr 30, 2009
    #18
  19. Daniel Moore

    Robert Dober Guest

    Re: Digits of Pi (#202)

    On Thu, Apr 30, 2009 at 10:21 PM, Todd Benson <> wrote:
    > On Thu, Apr 30, 2009 at 7:21 AM, Robert Dober <> wr=

    ote:
    >> On Thu, Apr 30, 2009 at 5:02 AM, Jay Anderson <> wro=

    te:
    >> The good new is that I chose the correct algorithm from the beginning.
    >> The bad news is that I was waaaay to stupid to implement it, well
    >> done.
    >> BTW I tested your result, it seems to be correct =A0:).
    >> Cheers

    >
    > Jay's version seemed to work fine. =A0It took about 5 minutes on my
    > machine for 100_000.
    >
    > I haven't verified the digits yet, though.

    I have ;)

    Now that I learnt from Jay *not* to use BigDecimal :) I have
    implemented Chen-Lih's machin formula,
    last on this page: http://en.wikipedia.org/wiki/Machin-like_formula

    But the speed gain is minimal 20~30% so that is definitely not worth
    posting a *stolen* solution with this monster expression ;)
    It run 70s instead of 97s (yes I have top notch hardware ;) for 100_000 dig=
    its.

    Cheers
    Robert

    >
    > Todd
    >
    >




    --=20
    Si tu veux construire un bateau ...
    Ne rassemble pas des hommes pour aller chercher du bois, pr=E9parer des
    outils, r=E9partir les t=E2ches, all=E9ger le travail=85 mais enseigne aux
    gens la nostalgie de l=92infini de la mer.

    If you want to build a ship, don=92t herd people together to collect
    wood and don=92t assign them tasks and work, but rather teach them to
    long for the endless immensity of the sea.

    --
    Antoine de Saint-Exup=E9ry
     
    Robert Dober, Apr 30, 2009
    #19
  20. Daniel Moore

    Daniel Moore Guest

    [QUIZ][SUMMARY] Digits of Pi (#202)

    This week's quiz sparked quite a discussion!

    Dan Diebolt introduced the Bailey-Borwein-Plouffe formula[1][2], a
    formula for computing the nth binary digit of =F0 without computing the
    previous digits, though no solutions were provided that incorporated
    this formula.

    Additionally, there was some talk of finding the first non-trivial
    Ruby program encoded in the digits of =F0, but choosing an appropriate
    encoding also proved to be non-trivial.

    Harry Kakueki provided the first solution and made use of BigMath:

    require 'bigdecimal'
    require 'bigdecimal/math'
    include BigMath

    puts PI(100_000)

    Executing this solution took a little over five minutes on my machine.
    It is always good to know what libraries are already available.

    Luke Cowell's solution uses the Leibniz formula for pi[3].

    require "rational"
    require 'enumerator'
    require 'bigdecimal'
    require 'bigdecimal/math'
    include BigMath

    iterations =3D 10000000000
    current =3D 1
    final =3D BigDecimal.new("4")
    other =3D false

    while(current < iterations) do
    current =3D current + 2
    if(other)
    final =3D final + Rational(4,current)
    else
    final =3D final - Rational(4,current)
    end

    other =3D !other
    print current.to_s + ":"
    puts final.to_f
    end

    Unfortunately, Leibniz's formula is very inefficient for either
    mechanical or computer-assisted =F0 calculation. Calculating =F0 to 10
    correct decimal places using Leibniz' formula requires over
    10,000,000,000 mathematical operations[3]. Luke stated on the mailing
    list this algorithm was only able to generate about eight digits in
    five minutes.

    Jay Anderson provided a solution using a Machin-like formula[4] based
    on the implementation from the LiteratePrograms wiki[5]:

    def arccot(x, unity)
    xpow =3D unity / x
    n =3D 1
    sign =3D 1
    sum =3D 0
    loop do
    term =3D xpow / n
    break if term =3D=3D 0
    sum +=3D sign * (xpow/n)
    xpow /=3D x*x
    n +=3D 2
    sign =3D -sign
    end
    sum
    end

    def calc_pi(digits =3D 10000)
    fudge =3D 10
    unity =3D 10**(digits+fudge)
    pi =3D 4*(4*arccot(5, unity) - arccot(239, unity))
    pi / (10**fudge)
    end

    digits =3D (ARGV[0] || 10000).to_i
    p calc_pi(digits)

    This solution produces 100k digits in around 2 minutes on my machine.

    Robet Dober mentions a speed increase when using Hwang Chien-Lih's
    Machin-like formula with additional terms[6]:

    pi =3D 4*(183*arccot(239, unity) + 32*arccot(1023, unity) -
    68*arccot(5832, unity) + 12*arccot(110443, unity) - 12*arccot(4841182,
    unity) - 100*arccot(6826318, unity))

    This does indeed improve the efficiency of the algorithm, I achieved
    15-20% reduction in the time the algorithm took, down to about 100
    seconds.

    Charles Oliver Nutter pointed out the Ruby pidigits implementation on
    the Computer Language Benchmarks Game[7] and provided some benchmarks
    for various Ruby implementations:

    (jruby time is on Java 6 server VM)

    JRuby 1.3-dev: 0m47.903s
    Ruby 1.9.1: 1m27.527s
    Rubinius master: 2m50.545s
    MacRuby 0.4: 1m9.832s
    IKRuby*: 3m26.861s
    Ruby 1.8.7: 1m47.887s

    MacRuby 0.5-experimental crashed after only a few digits. IKRuby
    is JRuby on CLR (Mono, here) using IKVM.

    These benchmarks only produced 10k digits of =F0 (for expediency). One
    lesson to take away from this is that algorithm choice often has the
    biggest impact on performance.

    10,000 Digits (Ruby 1.8.6 on my machine)
    Machin-like formula : ~ 1s
    BigMath : ~ 2.5s
    pidigits : ~ 1m30s
    Leibniz formula : ~ ?

    And sometimes the fastest code is the code you don't write. Michael
    Kohl's solution:

    require 'rubygems'
    require 'hpricot'
    require 'open-uri'

    doc =3D Hpricot(open('http://www.eveandersson.com/pi/digits/100000'))
    puts (doc/'pre').inner_html

    It finishes in less than one second for the entire 100,000 digits!

    Thank you everyone for your solutions and discussion! =F0 is a timeless
    concept that has fascinated great minds for thousands of years and
    will continue to do so. If you are interested in learning more please
    follow the references as this summary barely scratches the surface.
    Thanks again to all who participated this week.

    [1]: http://everything2.com/title/Algorithm for calculating individua=
    l%20hexadecimal%20digits%20of%20pi
    [2]: http://en.wikipedia.org/wiki/Bailey–Borwein–Plouffe_fo=
    rmula
    [3]: http://en.wikipedia.org/wiki/Leibniz_formula_for_pi
    [4]: http://mathworld.wolfram.com/Machin-LikeFormulas.html
    [5]: http://en.literateprograms.org/Category:Pi_with_Machin's_formula
    [6]: http://en.wikipedia.org/wiki/Machin-like_formula#More_terms (It
    appears that this Wikipedia article is displaying arctan instead of
    arccot)
    [7]: http://shootout.alioth.debian.org/u32q/benchmark.php?test=3Dpidigits&l=
    ang=3Druby&id=3D1

    P.S. If you have any future quiz ideas be sure to submit them:
    http://rubyquiz.strd6.com/suggestions.

    --=20
    -Daniel
    http://rubyquiz.strd6.com
     
    Daniel Moore, May 3, 2009
    #20
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