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Unum Type 3
Unums (''universal numbers'') are a family of number formats and arithmetic for implementing real numbers on a computer, proposed by John L. Gustafson in 2015. They are designed as an alternative to the ubiquitous IEEE 754 floating-point standard. The latest version is known as ''posits''. 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 interval arithmetic with a pair of unums, what he called a ''ubound'', provid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tapered Floating Point
In computing, tapered floating point (TFP) is a format similar to 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 relative precision than those with a large exponent. History The tapered floating-point scheme was first proposed by Robert Morris of Bell Laboratories in 1971, and refined with ''leveling'' by Masao Iri and Shouichi Matsui of University of Tokyo in 1981, and by Hozumi Hamada of Hitachi, Ltd. Alan Feldstein of Arizona State University and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias Gamma Coding
Elias \gamma 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''. #Considering the one that was r ... [...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, could not solve 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 Though not the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Significant Figures
Significant figures, also referred to as significant digits, are specific digits within a number that is written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presenting the outcome of a measurement (such as length, pressure, volume, or mass), if the number of digits exceeds what the measurement instrument can resolve, only the digits that are determined by the resolution are dependable and therefore considered significant. For instance, if a length measurement yields 114.8 mm, using a ruler with the smallest interval between marks at 1 mm, the first three digits (1, 1, and 4, representing 114 mm) are certain and constitute significant figures. Further, digits that are uncertain yet meaningful are also included in the significant figures. In this example, the last digit (8, contributing 0.8 mm) is likewise considered significant despite its uncertainty. Therefore, this measurement contains ... [...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. Specifically, how many bits are allocated for the integer portion, how many for the fractional portion, and whether there is a sign-bit. For example, in Q notation, Q7.8 means that the signed fixed point numbers in this format have 7 bits for the integer part and 8 bits for the fraction part. One extra bit is implicitly added for signed numbers. Therefore, Q7.8 is a 16-bit word, with the most significant bit representing the two's complement sign bit. There is an ARM variation of the Q notation that explicitly adds the sign bit to the integer part. In ARM Q notation, the above format would be called Q8.8. A number of other notations have been used for the same purpose. Definition General Format \underbrace_\mathrm\;\mathbf\;\underbrace_\mathrm\;\;\mathbf\;\underbrace_\mathrm Texas Instruments version The Q notation, as defined by Texas Instruments, consists of the letter follow ... [...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 The Karlsruhe Institute of Technology (KIT; ) is both a German public university, public research university in Karlsruhe, Baden-Württemberg, and a research center of the Helmholtz Association. KIT was created in 2009 when the University of Ka ... starting in the late 1960s. See also * Ulrich W. Kulisch * References Further reading * * * * * * * Computer arithmetic Numerical analysis {{Compu-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minifloat
In computing, minifloats are floating-point values represented with very few bits. This reduced precision makes them ill-suited for general-purpose numerical calculations, but they are useful for special purposes such as: * Computer graphics, where human perception of color and light levels has low precision. The 16-bit half-precision format is very popular. * Machine learning, which can be relatively insensitive to numeric precision. 16-bit, 8-bit, and even 4-bit floats are increasingly being used.https://developer.nvidia.com/blog/nvidia-arm-and-intel-publish-fp8-specification-for-standardization-as-an-interchange-format-for-ai/ (joint announcement by Intel, NVIDIA, Arm); https://arxiv.org/abs/2209.05433 (preprint paper jointly written by researchers from aforementioned 3 companies) Additionally, they are frequently encountered as a pedagogical tool in computer-science courses to demonstrate the properties and structures of floating-point arithmetic and IEEE 754 numbers. Depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IEEE 754-1985
IEEE 754-1985 is a historic 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, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LLVM
LLVM, also called LLVM Core, is a target-independent optimizer and code generator. It can be used to develop a Compiler#Front end, frontend for any programming language and a Compiler#Back end, backend for any instruction set architecture. LLVM is designed around a language-independent specification, language-independent intermediate representation (IR) that serves as a Software portability, portable, high-level assembly language that can be optimizing compiler, optimized with a variety of transformations over multiple passes. The name ''LLVM'' originally stood for ''Low Level Virtual Machine.'' However, the project has since expanded, and the name is no longer an acronym but an orphan initialism. LLVM is written in C++ and is designed for compile-time, Linker (computing), link-time, runtime (program lifecycle phase), runtime, and "idle-time" optimization. Originally implemented for C (programming language), C and C++, the language-agnostic design of LLVM has since spawned a wide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SUSE Linux
openSUSE () is a free and open-source Linux distribution developed by the openSUSE project. It is offered in two main variations: ''Tumbleweed'', an upstream rolling release distribution, and ''Leap'', a stable release distribution which is sourced from SUSE Linux Enterprise. The openSUSE project is sponsored by SUSE of Germany; the company released the first version as SUSE Linux in 1994. Its development was opened up to the community in 2005, which marked the creation of openSUSE. The focus of the developers is on creating a stable and user-friendly RPM-based operating system with a large target group for workstations and servers. Additionally, the project creates a variety of related tools, such as YaST, Open Build Service, openQA, Snapper, Portus, KIWI, and OSEM. Product history SUSE Linux In the past, the SUSE Linux company has focused on releasing the SUSE Linux Personal and SUSE Linux Professional box sets which included extensive printed documentation that w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
