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Bellard's Formula
Bellard's formula is used to calculate the ''n''th digit of π in base 16. Bellard's formula was discovered by Fabrice Bellard in 1997. It is about 43% faster than the Bailey–Borwein–Plouffe formula (discovered in 1995). It has been used in PiHex, the now-completed distributed computing A distributed system is a system whose components are located on different computer network, networked computers, which communicate and coordinate their actions by message passing, passing messages to one another from any system. Distributed com ... project. One important application is verifying computations of all digits of pi performed by other means. Rather than having to compute all of the digits twice by two separate algorithms to ensure that a computation is correct, the final digits of a very long all-digits computation can be verified by the much faster Bellard's formula. Formula: : \begin \pi = \frac1 \sum_^\infty \frac \, \left(-\frac \right. & - \frac1 + \frac - \frac \left. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base 16
In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexadecimal uses 16 distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" (or alternatively "a"–"f") to represent values from 10 to 15. Software developers and system designers widely use hexadecimal numbers because they provide a human-friendly representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble). For example, an 8-bit byte can have values ranging from 00000000 to 11111111 in binary form, which can be conveniently represented as 00 to FF in hexadecimal. In mathematics, a subscript is typically used to specify the base. For example, the decimal value would be expressed in hexadecimal as . In programming, a number o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fabrice Bellard
Fabrice Bellard (; born 1972) is a French computer programmer known for writing FFmpeg, QEMU, and the Tiny C Compiler. He developed Bellard's formula for calculating single digits of pi. In 2012, Bellard co-founded Amarisoft, a telecommunications company, with Franck Spinelli. Life and career Bellard was born in 1972 in Grenoble, France and went to school in Lycée Joffre (Montpellier), where, at age 17, he created the executable compressor LZEXE. After studying at École Polytechnique, he went on to specialize at Télécom Paris in 1996. In 1997, he discovered a new, faster formula to calculate single digits of pi in hexadecimal representation, known as Bellard's formula. It is a variant of the Bailey–Borwein–Plouffe formula. Bellard's entries won the International Obfuscated C Code Contest three times. In 2000, he won in the category "Most Specific Output" for a program that implemented the modular Fast Fourier Transform and used it to compute the then biggest know ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bailey–Borwein–Plouffe Formula
The Bailey–Borwein–Plouffe formula (BBP formula) is a formula for . It was discovered in 1995 by Simon Plouffe and is named after the authors of the article in which it was published, David H. Bailey, Peter Borwein, and Plouffe. Before that, it had been published by Plouffe on his own site. The formula is : \pi = \sum_^\left frac \left(\frac-\frac-\frac-\frac\right)\right/math> The BBP formula gives rise to a spigot algorithm for computing the ''n''th base-16 (hexadecimal) digit of (and therefore also the ''4n''th binary digit of ) without computing the preceding digits. This does ''not'' compute the ''n''th decimal of (i.e., in base 10). But another formula discovered by Plouffe in 2022 allows extracting the ''n''th digit of in decimal. BBP and BBP-inspired algorithms have been used in projects such as PiHex for calculating many digits of using distributed computing. The existence of this formula came as a surprise. It had been widely believed that computing the ''n''th d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PiHex
PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of . 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. History To calculate the five trillionth digit (and the following seventy-six digits) took 13,500 CPU hours, using 25 computers from 6 different countries. The forty trillionth digit required 84,500 CPU hours and 126 computers from 18 different countries. The highest calculation, the one quadrillionth digit, took 1.2 million CPU hours and 1,734 computers from 56 different 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 the calculations alone. After setting three records, calculating the five trillionth bit, the forty trillionth bit, and the quadrillionth bit, the project ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributed Computing
A distributed system is a system whose components are located on different computer network, networked computers, which communicate and coordinate their actions by message passing, passing messages to one another from any system. Distributed computing is a field of computer science that studies distributed systems. The components of a distributed system interact with one another in order to achieve a common goal. Three significant challenges of distributed systems are: maintaining concurrency of components, overcoming the clock synchronization, lack of a global clock, and managing the independent failure of components. When a component of one system fails, the entire system does not fail. Examples of distributed systems vary from service-oriented architecture, SOA-based systems to massively multiplayer online games to peer-to-peer, peer-to-peer applications. A computer program that runs within a distributed system is called a distributed program, and ''distributed programming' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributed Computing Projects
Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a variable **Cumulative distribution function, in which the probability of being no greater than a particular value is a function of that value *Frequency distribution, a list of the values recorded in a sample *Inner distribution, and outer distribution, in coding theory *Distribution (differential geometry), a subset of the tangent bundle of a manifold *Distributed parameter system, systems that have an infinite-dimensional state-space *Distribution of terms, a situation in which all members of a category are accounted for *Distributivity, a property of binary operations that generalises the distributive law from elementary algebra *Distribution (number theory) *Distribution problems, a common type of problems in combinatorics where the goal is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pi Algorithms
The number (; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number appears in many formulas across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as \tfrac are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only sums, products, powers, and integers. The transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of appear to be randomly distributed, but no proof of this conjecture has been found. For thousands of years, mathematicians have attempted to extend their understanding of , sometimes by computing i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
