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25-459: BBP may refer to: Science [ edit ] Bailey–Borwein–Plouffe formula , a formula for computing the n th binary digit of pi Baseband processor , a device in a network interface that manages all the radio functions Benzyl butyl phthalate , a plasticizer Bloodborne pathogens , a virus that can be spread through contamination by blood and other body fluids Branchpoint Binding Protein ,

50-426: A spigot algorithm using this formula. We must first rewrite the formula as: Now, for a particular value of n and taking the first sum, we split the sum to infinity across the n th term: We now multiply by 16 , so that the hexadecimal point (the divide between fractional and integer parts of the number) shifts (or remains, if n = 0 ) to the left of the (n+1) -th fractional digit: Since we only care about

75-693: A benchmark Binibining Pilipinas , a Filipino beauty pageant BullionByPost , a British online bullion retailer Bobby's Burger Palace , a group of fast casual restaurants which opened in July 2008 in Lake Grove, New York " Boom Boom Pow ", a 2009 song by The Black Eyed Peas British Bangladeshi Power 100 , an annual publication listing the 100 leading British Bangladeshi figures Great Union Party (Turkish: Büyük Birlik Partisi), an extreme far-right nationalist political party in Turkey Topics referred to by

100-617: A park on the Brooklyn side of the East River in New York City Other [ edit ] Bayou Bridge Pipeline , an oil pipeline constructed from 2017 to 2019 in Louisiana that was the subject of protests due to its environmental impact Best Business Practice , a method or technique that has consistently shown results superior to those achieved with other means, and that is used as

125-566: A pre-mRNA processing factor involved in RNA splicing Blue Brain Project , a Swiss brain research initiative that aims to create a digital reconstruction of rodent and eventually human brains Places [ edit ] Bayport-Blue Point School District , a school district in Bayport and Blue Point, New York Bembridge Airport , Isle of Wight, England (IATA airport code: BBP) Brooklyn Bridge Park ,

150-415: A range of parameter values for s , b , and m , evaluating the sums out to many digits, and then using an integer relation-finding algorithm (typically Helaman Ferguson 's PSLQ algorithm ) to find a sequence A that adds up those intermediate sums to a well-known constant or perhaps to zero. The original BBP π summation formula was found in 1995 by Plouffe using PSLQ . It is also representable using

175-467: Is an integer base . Formulas of this form are known as BBP-type formulas . Given a number α {\displaystyle \alpha } , there is no known systematic algorithm for finding appropriate p ( k ) {\displaystyle p(k)} , q ( k ) {\displaystyle q(k)} , and b {\displaystyle b} ; such formulas are discovered experimentally . A specialization of

200-507: Is different from Wikidata All article disambiguation pages All disambiguation pages Bailey%E2%80%93Borwein%E2%80%93Plouffe formula The BBP formula gives rise to a spigot algorithm for computing the n th base-16 (hexadecimal) digit of π (and therefore also the 4n th binary digit of π ) without computing the preceding digits. This does not compute the n th decimal digit of π (i.e., in base 10). But another formula discovered by Plouffe in 2022 allows extracting

225-430: Is done at the same loop level, not nested . When its running 16 x product becomes greater than one, the modulus is taken, just as for the running total in each sum. Now to complete the calculation, this must be applied to each of the four sums in turn. Once this is done, the four summations are put back into the sum to π : Since only the fractional part is accurate, extracting the wanted digit requires that one removes

250-411: Is for s  = 1, but m  > 1. Many now-discovered formulae are known for b as an exponent of 2 or 3 and m as an exponent of 2 or it some other factor-rich value, but where several of the terms of sequence 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

275-555: Is the Riemann zeta function ), log 3 ⁡ 2 {\displaystyle \log ^{3}2} , log 4 ⁡ 2 {\displaystyle \log ^{4}2} , log 5 ⁡ 2 {\displaystyle \log ^{5}2} , and various products of powers of π {\displaystyle \pi } and log ⁡ 2 {\displaystyle \log 2} . These results are obtained primarily by

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300-426: The P function: which also reduces to this equivalent ratio of two polynomials: This formula has been shown through a fairly simple proof to equal π . We would like to define a formula that returns the ( n + 1 {\displaystyle n+1} )-th (with n ≥ 0 {\displaystyle n\geq 0} ) hexadecimal digit of π . A few manipulations are required to implement

325-764: The n th digit of π in decimal. BBP and BBP-inspired algorithms have been used in projects such as PiHex for calculating many digits of π using distributed computing. The existence of this formula came as a surprise. It had been widely believed that computing the n th digit of π is just as hard as computing the first n digits. Since its discovery, formulas of the general form: have been discovered for many other irrational numbers α {\displaystyle \alpha } , where p ( k ) {\displaystyle p(k)} and q ( k ) {\displaystyle q(k)} are polynomials with integer coefficients and b ≥ 2 {\displaystyle b\geq 2}

350-790: The "further out" a digit is, the longer it takes BBP to calculate it, just like the standard π -computing algorithms. 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. Explicit results are given for Catalan's constant , π 3 {\displaystyle \pi ^{3}} , π 4 {\displaystyle \pi ^{4}} , Apéry's constant ζ ( 3 ) {\displaystyle \zeta (3)} , ζ ( 5 ) {\displaystyle \zeta (5)} , (where ζ ( x ) {\displaystyle \zeta (x)}

375-482: The calculations alone. After setting three records, calculating the five trillionth bit, the forty trillionth bit, and the quadrillionth bit, the project ended on September 11, 2000. While the PiHex project calculated the least significant digits of π ever attempted at the time in any base, the second place is held by Peter Trueb who computed some 22+ trillion digits in 2016 and third place by houkouonchi who derived

400-549: The first n  − 1 digits and can use small, efficient data types. Fabrice Bellard found a variant of BBP, Bellard's formula , which is faster. Though the BBP formula can directly calculate the value of any given digit of π with less computational effort than formulas that must calculate all intervening digits, BBP remains linearithmic ( O ( n log ⁡ n ) {\displaystyle O(n\log n)} ), whereby successively larger values of n require increasingly more time to calculate; that is,

425-454: The following seventy-six digits) took 13,500 CPU hours, using 25 computers from 6 countries. The forty trillionth digit required 84,500 CPU hours and 126 computers from 18 countries. The highest calculation, the one quadrillionth digit, took 1.2 million CPU hours and 1,734 computers from 56 countries. Total resources: 1,885 computers donated 1.3 million CPU hours. The average computer that was used to calculate would have taken 148 years to complete

450-432: The fractional part of the sum, we look at our two terms and realise that only the first sum contains terms with an integer part; conversely, the second sum doesn't contain terms with an integer part, since the numerator can never be larger than the denominator for k  >  n . Therefore, we need a trick to remove the integer parts, that we don't need, from the terms of the first sum, in order to speed up and increase

475-433: The general formula that has produced many results is: where s , b , and m are integers, and A = ( a 1 , a 2 , … , a m ) {\displaystyle A=(a_{1},a_{2},\dots ,a_{m})} is a sequence of integers. The P function leads to a compact notation for some solutions. For example, the original BBP formula: can be written as: Some of

500-526: The integer part of the final sum, multiplies it by 16 and keeps the integer part to "skim off" the hexadecimal digit at the desired position (in theory, the next few digits up to the accuracy of the calculations used would also be accurate). This process is similar to performing long multiplication , but only having to perform the summation of some middle columns. While there are some carries that are not counted, computers usually perform arithmetic for many bits (32 or 64) and round, and we are only interested in

525-405: 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. This algorithm computes π without requiring custom data types having thousands or even millions of digits. The method calculates the n th digit without calculating

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550-410: The precision of the calculations. That trick is to reduce modulo  8 k  + 1. Our first sum (out of four) to compute the fractional part then becomes: Notice how the modulus operator always guarantees that only the fractional parts of the terms of the first sum will be kept. To calculate 16  mod (8 k  + 1) quickly and efficiently, the modular exponentiation algorithm

575-403: The same term [REDACTED] This disambiguation page lists articles associated with the title BBP . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=BBP&oldid=1206917578 " Category : Disambiguation pages Hidden categories: Short description

600-413: The simplest formulae of this type that were well known before BBP and for which the P function leads to a compact notation, are: (In fact, this identity holds true for a > 1: Plouffe was also inspired by the arctan power series of the form (the P notation can be also generalized to the case where b is not an integer): Using the P function mentioned above, the simplest known formula for π

625-521: The use of polylogarithm ladders . PiHex PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of π . 1,246 contributors used idle time slices on almost two thousand computers to make its calculations. The software used for the project made use of Bellard's formula , a faster version of the BBP formula . To calculate the five trillionth digit (and

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