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| The '''Bailey–Borwein–Plouffe formula''' ('''BBP formula''') provides a [[spigot algorithm]] for the computation of the ''n''th [[binary digit]] of '''[[pi]]''' (symbol: {{pi}}) using [[Hexadecimal|{{nowrap|base 16}}]] math. The formula can directly calculate the value of any given digit of {{pi}} without the need to calculate the preceding digits. The BBP is a [[summation]]-style formula that was discovered in 1995 by [[Simon Plouffe]] and was named after the authors of the paper in which the formula was published, [[David H. Bailey]], [[Peter Borwein]], and [[Simon Plouffe]]. Before that paper, it had been published by Plouffe on his own site.<ref>{{cite journal
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| | author = Bailey, David H., Borwein, Peter B., and Plouffe, Simon
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| |date=April 1997
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| | title = On the Rapid Computation of Various Polylogarithmic Constants
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| | journal = Mathematics of Computation
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| | volume = 66 | issue = 218 | pages = 903–913 <!-- | url = http://crd.lbl.gov/~dhbailey/dhbpapers/digits.pdf PDF is free at doi link, and is better formatted. -->
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| | doi = 10.1090/S0025-5718-97-00856-9
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| }}</ref> The formula is
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| :<math> \pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \right]</math>.
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| The discovery of this formula came as a surprise. For centuries it had been assumed that there was no way to compute the ''n''th digit of {{pi}} without calculating all of the preceding ''n'' − 1 digits.
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| Since this discovery, many formulas for other irrational constants have been discovered of the general form
| | They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. It is very easy to customize plugins according to the needs of a particular business. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>Any business enterprise that is certainly worth its name should really shell out a good deal in making sure that they have the most effective website that provides related info to its prospect. Best of all, you can still have all the functionality that you desire when you use the Word - Press platform. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. From my very own experiences, I will let you know why you should choose WPZOOM Live journal templates. You can also get a free keyword tool that is to determine how strong other competing sites are and number of the searches on the most popular search sites. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. Are you considering getting your website redesigned. Our skilled expertise, skillfulness and excellence have been well known all across the world. Websites using this content based strategy are always given top scores by Google. <br><br>Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. One of the great features of Wordpress is its ability to integrate SEO into your site. If you beloved this article and you would like to acquire extra data pertaining to [http://ll.my/wordpressbackupplugin145414 wordpress backup] kindly pay a visit to our own web site. Contact Infertility Clinic Providing One stop Fertility Solutions at:. OSDI, a Wordpress Development Company based on ahmedabad, India. <br><br>Internet is not only the source for information, it is also one of the source for passive income. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities. |
| : <math>\alpha = \sum_{k = 0}^{\infty}\left[ \frac{1}{b^k} \frac{p(k)}{q(k)} \right]</math>
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| where α is the constant and ''p'' and ''q'' are [[polynomial]]s in integer coefficients and ''b'' ≥ 2 is an integer [[numerical base]].
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| Formulas in this form are known as '''BBP-type formulas'''.<ref name="Ref_a">{{MathWorld| title=BBP Formula| urlname=BBPFormula}}</ref> Certain combinations of specific ''p'', ''q'' and ''b'' result in well-known constants, but there is no systematic algorithm for finding the appropriate combinations; known formulas are discovered through [[experimental mathematics]].
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| == Specializations ==
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| A specialization of the general formula that has produced many results is
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| : <math> P(s,b,m,A) = \sum_{k=0}^{\infty}\left[ \frac{1}{b^k} \sum_{j=1}^{m}\frac{a_j}{(mk+j)^s} \right]</math>
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| where ''s'', ''b'' and ''m'' are integers and <math>A = (a_1, a_2, \dots , a_m)</math> is a vector of integers.
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| The P function leads to a compact notation for some solutions.
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| === Previously known BBP-type formulae ===
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| Some of the simplest formulae of this type that were well known before BBP, and that the P function leads to a compact notation, are
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| : <math>\begin{align} \ln\frac{10}{9} &
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| = \frac{1}{10} + \frac{1}{200} + \frac{1}{3\ 000} + \frac{1}{40\ 000} + \frac{1}{500\ 000} + \cdots \\ &
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| = \sum_{k=1}^{\infty} \frac{1}{10^k \cdot k} = \frac{1}{10} \sum_{k=0}^{\infty}\left[ \frac{1}{10^k} \left( \frac{1}{k+1} \right) \right] \\ &
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| = \frac{1}{10} P\left(1, 10, 1, (1) \right)
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| \end{align}</math>
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| : <math>\begin{align} \ln 2 &
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| = \frac{1}{2} + \frac{1}{2 \cdot 2^2} + \frac{1}{3 \cdot 2^3} + \frac{1}{4 \cdot 2^4} + \frac{1}{5 \cdot 2^5} + \cdots \\ &
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| = \sum_{k=1}^{\infty}\frac{1}{2^k \cdot k} = \frac{1}{2} \sum_{k=0}^{\infty}\left[ \frac{1}{2^k} \left( \frac{1}{k + 1} \right) \right] \\ &
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| = \frac{1}{2} P\left( 1, 2, 1, (1) \right).
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| \end{align}</math>
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| Plouffe was also inspired by the [[Inverse tangent#Infinite series|arctan power series]] of the form (the P notation can be also generalized to the case where ''b'' is not an integer) :
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| : <math>\begin{align} \arctan\frac{1}{b} &
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| = \frac{1}{b} - \frac{1}{b^3 3} + \frac{1}{b^5 5} - \frac{1}{b^7 7} + \frac{1}{b^9 9} + \cdots \\ &
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| = \sum_{k=1}^{\infty}\left[ \frac{1}{b^{k}} \frac{ \sin\frac{k\pi}{2} }{k} \right]
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| = \frac{1}{b} \sum_{k=0}^{\infty}\left[ \frac{1}{b^{4k}} \left( \frac{1}{4k+1} + \frac{-b^{-2}}{4k+3} \right) \right] \\ &
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| = \frac{1}{b} P\left( 1, b^4, 4, (1, 0, -b^{-2}, 0) \right).
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| \end{align}</math>
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| === The search for new equalities ===
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| Using the P function mentioned above, the simplest known formula for {{pi}} is for ''s = 1'' but ''m > 1''. Many now-discovered formulae are known for b as an exponent of 2 or 3 and m is an exponent of 2 or it is some other factor-rich value, but where several of the terms of vector A are zero. The discovery of these formulae involves a computer search for such linear combinations after computing the individual sums. The search procedure consists of choosing a range of parameter values for s, b, and m, evaluating the sums out to many digits, and then using an [[integer relation algorithm|integer relation finding algorithm]] (typically [[Helaman Ferguson]]'s PSLQ algorithm) to find a vector ''A'' that adds up those intermediate sums to a well-known constant or perhaps to zero.
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| === The BBP formula for {{pi}} ===
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| The original BBP {{pi}} summation formula was found in 1995 by Plouffe using [[integer relation algorithm|PSLQ]]. It is also representable using the ''P'' function above:
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| : <math>\begin{align}\pi &
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| = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \right] \\ &
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| = P\left( 1, 16, 8, (4, 0, 0, -2, -1, -1, 0, 0) \right)
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| \end{align}</math>
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| which also reduces to this equivalent ratio of two polynomials:
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| : <math>\pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15} \right) \right].</math>
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| This formula has been shown through a rigorous and fairly simple proof to equal {{pi}}.<ref name=QuestForPi>Bailey, Borwein, Plouffe, [http://crd.lbl.gov/~dhbailey/dhbpapers/pi-quest.pdf The Quest for Pi] (1997) ''Mathematical Intelligencer'', vol. 19, no. 1, 50–57.</ref>
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| ==== BBP digit-extraction algorithm for {{pi}} ====
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| The formula yields an algorithm for extracting [[hexadecimal]] digits of {{pi}}. In order to perform digit extraction, we must first rewrite the formula as
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| : <math>\pi = 4 \sum_{k = 0}^{\infty} \frac{1}{(16^k)(8k+1)} - 2 \sum_{k = 0}^{\infty} \frac{1}{(16^k)(8k+4)} - \sum_{k = 0}^{\infty} \frac{1}{(16^k)(8k+5)}
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| - \sum_{k = 0}^{\infty} \frac{1}{(16^k)(8k+6)}. \!</math>
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| A few manipulations are required to implement a [[spigot algorithm]] using this formula. We would like to find hexadecimal digit ''n'' of {{pi}}, so, taking the first sum, we split the [[series (mathematics)|sum]] to [[infinity]] across the ''n''th term
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| : <math>\sum_{k = 0}^{\infty} \frac{1}{(16^k)(8k+1)} = \sum_{k = 0}^{n} \frac{1}{(16^k)(8k+1)} + \sum_{k = n + 1}^{\infty} \frac{1}{(16^k)(8k+1)}. \!</math>
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| We now multiply by 16<sup> ''n''</sup> so that the hexadecimal point (the divide between fractional and integer parts of the number) is in the ''n''th place.
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| : <math>\sum_{k = 0}^{\infty} \frac{16^{n-k}}{8k+1} = \sum_{k = 0}^{n} \frac{16^{n-k}}{8k+1} + \sum_{k = n + 1}^{\infty} \frac{16^{n-k}}{8k+1}. \!</math>
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| Since we only care about the fractional part of the sum, we look at our two terms and realise that only the first sum is able to produce whole numbers; conversely, the second sum cannot produce whole numbers since the numerator can never be larger than the denominator for ''k'' > ''n''. Therefore, we need a trick to remove the whole numbers for the first sum. That trick is mod 8''k'' + 1. Our sum for the first fractional part then becomes:
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| : <math>\sum_{k = 0}^{n} \frac{16^{n-k} \bmod (8k+1)}{8k+1} + \sum_{k = n + 1}^{\infty} \frac{16^{n-k}}{8k+1}. \!</math>
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| Notice how the [[Modular arithmetic|modulo]] operator always guarantees that only the fractional sum will be kept. To calculate 16<sup> ''n'' − ''k''</sup> mod (8''k'' + 1) quickly and efficiently, use the [[modular exponentiation]] algorithm. When the running product becomes greater than one, take the modulo just as you do for the running total in each sum.
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| Now to complete the calculation you must apply this to each of the four sums in turn. Once this is done, take the four summations and put them back into the sum to {{pi}}.
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| : <math>4 \Sigma_1 - 2 \Sigma_2 - \Sigma_3 - \Sigma_4. \,\!</math>
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| Since only the fractional part is accurate, extracting the wanted digit requires that one removes the integer part of the final sum and multiplies by 16 to "skim off" the hexadecimal digit at this position (in theory, the next few digits up to the accuracy of the calculations used would also be accurate).
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| This process is similar to performing [[long multiplication]], but only having to perform the summation of some middle columns. While there are some [[carry (arithmetic)|carries]] that are not counted, computers usually perform arithmetic for many bits (32 or 64) and they round and we are only interested in the most significant digit(s). There is a possibility that a particular computation will be akin to failing to add a small number (e.g. 1) to the number 999999999999999, and that the error will propagate to the most significant digit.
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| == BBP compared to other methods of computing {{pi}} ==
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| This algorithm computes {{pi}} without requiring custom data types having thousands or even millions of digits. The method calculates the ''n''th digit ''without'' calculating the first ''n'' − 1 digits, and can use small, efficient data types.
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| The algorithm is the fastest way to compute the ''n''th digit (or a few digits in a neighborhood of the ''n''th), but {{pi}}-computing algorithms using large data types remain faster when the goal is to compute all the digits from 1 to ''n''.{{citation needed|date=August 2013}}
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| Though the BBP formula can directly calculate the value of any given digit of {{pi}} with less computational effort than formulas that must calculate all intervening digits, BBP remains [[Time complexity#Linearithmic.2Fquasilinear time|linearithmic]] whereby successively larger values of ''n'' require increasingly more time to calculate; that is, the "further out" a digit is, the longer it takes BBP to calculate it, just like the standard {{pi}}-computing algorithms.<ref>{{cite web
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| | last = Bailey
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| | first = David H.
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| | title = The BBP Algorithm for Pi
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| | date = 8 September 2006
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| | quote = Run times for the BBP algorithm ... increase roughly linearly with the position ''d''.
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| | url = http://www.experimentalmath.info/bbp-codes/bbp-alg.pdf
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| | format = PDF
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| | accessdate = 17 January 2013}}</ref>
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| == Generalizations ==
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| D.J. Broadhurst provides a generalization of the BBP algorithm that may be used to compute a number of other constants in nearly linear time and logarithmic space.<ref name="Ref_f">D.J. Broadhurst, "[http://arxiv.org/abs/math.CA/9803067 Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)]", (1998) ''arXiv'' math.CA/9803067</ref> Explicit results are given for [[Catalan's constant]], <math>\pi^3</math>, <math>\log^32</math>, [[Apéry's constant]] <math>\zeta(3)</math> (where <math>\zeta(x)</math> is the [[Riemann zeta function]]), <math>\pi^4</math>, <math>\log^42</math>, <math>\log^52</math>, <math>\zeta(5)</math>, and various products of powers of <math>\pi</math> and <math>\log2</math>. These results are obtained primarily by the use of [[polylogarithm ladder]]s.
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| == See also ==
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| * [[Computing π|Computing {{pi}}]]
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| * [[Experimental mathematics]]
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| * [[Bellard's formula]]
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| * [[Feynman point]]
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| == References ==
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| {{reflist}}
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| ==Further reading==
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| * D.J. Broadhurst, "[http://arxiv.org/abs/math.CA/9803067 Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)]", (1998) ''[[arXiv]]'' math.CA/9803067
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| == External links ==
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| * [[Richard J. Lipton]], "[http://rjlipton.wordpress.com/2010/07/14/making-an-algorithm-an-algorithm-bbp/ Making An Algorithm An Algorithm — BBP]", weblog post, July 14, 2010.
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| * [[Richard J. Lipton]], "[http://rjlipton.wordpress.com/2009/03/15/cooks-class-contains-pi/ Cook’s Class Contains Pi]", weblog post, March 15, 2009.
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| * {{cite web
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| |url=http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf
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| |first1=David H.
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| |last1=Bailey
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| |title=A compendium of BBP-type formulas for mathematical constants
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| |accessdate=2010-04-30}}
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| {{DEFAULTSORT:Bailey-Borwein-Plouffe formula}}
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| [[Category:Pi algorithms]]
| |
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