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Unum Type 3
Unums (''universal numbers'') are a family of formats and arithmetic, similar to floating point, proposed by John L. Gustafson in 2015. They are designed as an alternative to the ubiquitous IEEE 754 floating-point standard. The latest version (known as posits) can be used as a drop-in replacement for programs that do not depend on specific features of IEEE 754. Type I Unum The first version of unums, formally known as Type I unum, was introduced in Gustafson's book ''The End of Error'' as a superset of the IEEE-754 floating-point format. The defining features of the Type I unum format are: * a variable-width storage format for both the significand and exponent, and * a ''u-bit'', which determines whether the unum corresponds to an exact number (''u'' = 0), or an interval between consecutive exact unums (''u'' = 1). In this way, the unums cover the entire extended real number line ˆ’∞,+∞ For computation with the format, Gustafson proposed using inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating-point Arithmetic
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be represented as a base-ten floating-point number: 12.345 = \underbrace_\text \times \underbrace_\text\!\!\!\!\!\!^ In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. The term ''floating point'' refers to the fact that the number's radix point can "float" anywhere to the left, right, or between the significant digits of the number. This position is indicated by the exponent, so floating point can be considered a form of scientific notation. A floating-point system can be used to represent, with a fixed number of digits, numbers of very different orders of magnitude — such as the number of meters between galaxies or between protons in an atom. For this reason, floating-poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Willard L
Willard may refer to: People * Willard (name) Geography Places in the United States * Willard, Colorado * Willard, Georgia * Willard, Kansas * Willard, Kentucky * Willard, Michigan, a small unincorporated community in Beaver Township, Bay County, Michigan * Willard, Missouri * Willard, New Mexico * Willard, New York * Willard, North Carolina * Willard, Ohio * Willard, Utah * Willard Bay, Utah, a reservoir * South Willard, Utah * Willard, Virginia * Willard, Washington * Willard, Rusk County, Wisconsin, a town * Willard, Clark County, Wisconsin, an unincorporated community * Willards, Maryland Places other than settlements * The Willard InterContinental Washington, a historic hotel in Washington, DC * Willard House (other), several houses * Willard Residential College, a Northwestern University residential hall * J. Willard Marriott Library, at the University of Utah * University of Illinois Willard Airport * Willard Drug Treatment Center, a specialized state p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chapman & Hall
Chapman & Hall is an imprint owned by CRC Press, originally founded as a British publishing house in London in the first half of the 19th century by Edward Chapman and William Hall. Chapman & Hall were publishers for Charles Dickens (from 1840 until 1844 and again from 1858 until 1870), Thomas Carlyle, William Thackeray, Elizabeth Barrett Browning, Anthony Trollope, Eadweard Muybridge and Evelyn Waugh. History Upon Hall's death in 1847, Chapman's cousin Frederic Chapman began his progress through the ranks of the company and eventually becoming a partner in 1858 and sole proprietor on Edward Chapman's retirement from Chapman & Hall in 1866. In 1868 author Anthony Trollope bought a third of the company for his son, Henry Merivale Trollope. From 1902 to 1930 the company's managing director was Arthur Waugh. In the 1930s the company merged with Methuen, a merger which, in 1955, participated in forming the Associated Book Publishers. The latter was acquired by The Thomson Corp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CRC Press
The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information technology. CRC Press is now a division of Taylor & Francis, itself a subsidiary of Informa. History The CRC Press was founded as the Chemical Rubber Company (CRC) in 1903 by brothers Arthur, Leo and Emanuel Friedman in Cleveland, Ohio, based on an earlier enterprise by Arthur, who had begun selling rubber laboratory aprons in 1900. The company gradually expanded to include sales of laboratory equipment to chemists. In 1913 the CRC offered a short (116-page) manual called the ''Rubber Handbook'' as an incentive for any purchase of a dozen aprons. Since then the ''Rubber Handbook'' has evolved into the CRC's flagship book, the '' CRC Handbook of Chemistry and Physics''. In 1964, Chemical Rubber decided to focus on its publishing ventures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tapered Floating Point
In computing, tapered floating point (TFP) is a format similar to Floating-point arithmetic, floating point, but with variable-sized entries for the significand and exponent instead of the fixed-length entries found in normal floating-point formats. In addition to this, tapered floating-point formats provide a fixed-size pointer entry indicating the number of digits in the exponent entry. The number of digits of the significand entry (including the sign) results from the difference of the fixed total length minus the length of the exponent and pointer entries. Thus numbers with a small exponent, i.e. whose order of magnitude is close to the one of 1, have a higher Precision (computer science), relative precision than those with a large exponent. History The tapered floating-point scheme was first proposed by Robert Morris (cryptographer), Robert Morris of Bell Laboratories in 1971, and refined with ''leveling'' by Masao Iri and Shouichi Matsui of University of Tokyo in 1981, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias Gamma Coding
Elias γ code or Elias gamma code is a universal code encoding positive integers developed by Peter Elias. It is used most commonly when coding integers whose upper-bound cannot be determined beforehand. Encoding To code a number ''x'' ≥ 1: # Let N = \lfloor \log_2 x \rfloor be the highest power of 2 it contains, so 2''N'' ≤ ''x'' < 2''N''+1. # Write out ''N'' zero bits, then # Append the binary form of ''x'', an ''N''+1-bit binary number. An equivalent way to express the same process: # Encode ''N'' in unary; that is, as ''N'' zeroes followed by a one. # Append the remaining ''N'' binary digits of ''x'' to this representation of ''N''. To represent a number x, Elias gamma (γ) uses 2 \lfloor \log_2(x) \rfloor + 1 bits. The code begins (the implied probability distribution for the code is added for clarity): Decoding To decode an Elias gamma-coded integer: #Read and count 0s from the stream until you reach the first 1. Call this count of zeroes ''N''. #Consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating-point Error Mitigation
Floating-point error mitigation is the minimization of errors caused by the fact that real numbers cannot, in general, be accurately represented in a fixed space. By definition, floating-point error cannot be eliminated, and, at best, can only be managed. Huberto M. Sierra noted in his 1956 patent "Floating Decimal Point Arithmetic Control Means for Calculator": The Z1, developed by Konrad Zuse in 1936, was the first computer with floating-point arithmetic and was thus susceptible to floating-point error. Early computers, however, with operation times measured in milliseconds, were incapable of solving large, complex problems and thus were seldom plagued with floating-point error. Today, however, with supercomputer system performance measured in petaflops, floating-point error is a major concern for computational problem solvers. The following sections describe the strengths and weaknesses of various means of mitigating floating-point error. Numerical error analysis Thou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Significant Figures
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something. If a number expressing the result of a measurement (e.g., length, pressure, volume, or mass) has more digits than the number of digits allowed by the measurement resolution, then only as many digits as allowed by the measurement resolution are reliable, and so only these can be significant figures. For example, if a length measurement gives 114.8 mm while the smallest interval between marks on the ruler used in the measurement is 1 mm, then the first three digits (1, 1, and 4, showing 114 mm) are certain and so they are significant figures. Digits which are uncertain but ''reliable'' are also considered significant figures. In this example, the last digit (8, which adds 0.8 mm) is also considered a significant figure even though ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q (number Format)
The Q notation is a way to specify the parameters of a binary fixed point number format. For example, in Q notation, the number format denoted by Q8.8 means that the fixed point numbers in this format have 8 bits for the integer part and 8 bits for the fraction part. A number of other notations have been used for the same purpose. Definition Texas Instruments version The Q notation, as defined by Texas Instruments, consists of the letter followed by a pair of numbers ''m'n'', where ''m'' is the number of bits used for the integer part of the value, and ''n'' is the number of fraction bits. By default, the notation describes ''signed'' binary fixed point format, with the unscaled integer being stored in two's complement format, used in most binary processors. The first bit always gives the sign of the value(1 = negative, 0 = non-negative), and it is ''not'' counted in the ''m'' parameter. Thus the total number ''w'' of bits used is 1 + ''m'' + ''n''. For example, the spe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karlsruhe Accurate Arithmetic
Karlsruhe Accurate Arithmetic (KAA) or Karlsruhe Accurate Arithmetic Approach (KAAA), augments conventional floating-point arithmetic with good error behaviour with new operations to calculate scalar products with a single rounding error. The foundations for KAA were developed at the University of Karlsruhe starting in the late 1960s. See also * Ulrich W. Kulisch * * IBM 4361 * PCS Cadmus * FORTRAN-SC * PASCAL-SC * PASCAL-XSC * C-XSC * Extensions for Scientific Computation (XSC) * Triplex-ALGOL Karlsruhe * Interval arithmetic * Unum (number format), Unum * Catastrophic cancellation References Further reading * * * * * * {{cite journal , title=Numerische Mathematik: Rechnen mit garantierter Genauigkeit , trans-title=Numerical mathematics: Calculating with guaranteed accuracy , author-first=Christoph , author-last=Pöppe , journal=Spektrum der Wissenschaft , language=German , number=9 , volume=2000 , date=2000-09-01 , pages=54– , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IEEE 754-1985
IEEE 754-1985 was an industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by IEEE 754-2008, and then again in 2019 by minor revision IEEE 754-2019. During its 23 years, it was the most widely used format for floating-point computation. It was implemented in software, in the form of floating-point libraries, and in hardware, in the instructions of many CPUs and FPUs. The first integrated circuit to implement the draft of what was to become IEEE 754-1985 was the Intel 8087. IEEE 754-1985 represents numbers in binary, providing definitions for four levels of precision, of which the two most commonly used are: The standard also defines representations for positive and negative infinity, a "negative zero", five exceptions to handle invalid results like division by zero, special values called NaNs for representing those exceptions, denormal numbers to represent numbers smaller than shown above, and four roundi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LLVM
LLVM is a set of compiler and toolchain technologies that can be used to develop a front end for any programming language and a back end for any instruction set architecture. LLVM is designed around a language-independent intermediate representation (IR) that serves as a portable, high-level assembly language that can be optimized with a variety of transformations over multiple passes. LLVM is written in C++ and is designed for compile-time, link-time, run-time, and "idle-time" optimization. Originally implemented for C and C++, the language-agnostic design of LLVM has since spawned a wide variety of front ends: languages with compilers that use LLVM (or which do not directly use LLVM but can generate compiled programs as LLVM IR) include ActionScript, Ada, C#, Common Lisp, PicoLisp, Crystal, CUDA, D, Delphi, Dylan, Forth, Fortran, Free Basic, Free Pascal, Graphical G, Halide, Haskell, Java bytecode, Julia, Kotlin, Lua, Objective-C, OpenCL, PostgreSQL's SQL and PLpg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
