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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:
Double-precision floating-point format is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point.
In computing, a vector processor or array processor is a central processing unit (CPU) that implements an instruction set where its instructions are designed to operate efficiently and effectively on large one-dimensional arrays of data called vectors. This is in contrast to scalar processors, whose instructions operate on single data items only, and in contrast to some of those same scalar processors having additional single instruction, multiple data (SIMD) or SWAR Arithmetic Units. Vector processors can greatly improve performance on certain workloads, notably numerical simulation and similar tasks. Vector processing techniques also operate in video-game console hardware and in graphics accelerators.
In computing, floating point operations per second is a measure of computer performance, useful in fields of scientific computations that require floating-point calculations. For such cases, it is a more accurate measure than measuring instructions per second.
Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $23.4476 with $23.45, the fraction 312/937 with 1/3, or the expression √2 with 1.414.
In computing, especially digital signal processing, the multiply–accumulate (MAC) or multiply-add (MAD) operation is a common step that computes the product of two numbers and adds that product to an accumulator. The hardware unit that performs the operation is known as a multiplier–accumulator ; the operation itself is also often called a MAC or a MAD operation. The MAC operation modifies an accumulator a:
The IEEE Standard for Floating-Point Arithmetic is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portably. Many hardware floating-point units use the IEEE 754 standard.
In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most arithmetic logic unit (ALU) hardware, which typically offers between 8 and 64 bits of precision.
In computer architecture, 128-bit integers, memory addresses, or other data units are those that are 128 bits wide. Also, 128-bit central processing unit (CPU) and arithmetic logic unit (ALU) architectures are those that are based on registers, address buses, or data buses of that size.
Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root of a real number. Arithmetically, it means given , a procedure for finding a number which when multiplied by itself, yields ; algebraically, it means a procedure for finding the non-negative root of the equation ; geometrically, it means given two line segments, a procedure for constructing their geometric mean.
A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software.
In computing, minifloats are floating-point values represented with very few bits. Predictably, they are not well suited for general-purpose numerical calculations. They are used for special purposes, most often in computer graphics, where iterations are small and precision has aesthetic effects. Machine learning also uses similar formats like bfloat16. 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.
Extended precision refers to floating-point number formats that provide greater precision than the basic floating-point formats. Extended precision formats support a basic format by minimizing roundoff and overflow errors in intermediate values of expressions on the base format. In contrast to extended precision, arbitrary-precision arithmetic refers to implementations of much larger numeric types using special software.
Fast inverse square root, sometimes referred to as Fast InvSqrt or by the hexadecimal constant 0x5F3759DF, is an algorithm that estimates , the reciprocal of the square root of a 32-bit floating-point number in IEEE 754 floating-point format. This operation is used in digital signal processing to normalize a vector, such as scaling it to length 1. For example, computer graphics programs use inverse square roots to compute angles of incidence and reflection for lighting and shading. Predated by similar video game algorithms, this one is best known for its implementation in 1999 in Quake III Arena, a first-person shooter video game heavily based on 3D graphics. The algorithm only started appearing on public forums between 2002 and 2003. Computation of square roots usually depends upon many division operations, which for floating point numbers are computationally expensive. The fast inverse square generates a good approximation with only one division step.
In computing, half precision is a binary floating-point computer number format that occupies 16 bits in computer memory. It is intended for storage of floating-point values in applications where higher precision is not essential, in particular image processing and neural networks.
In computing, quadruple precision is a binary floating point–based computer number format that occupies 16 bytes with precision at least twice the 53-bit double precision.
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem
Fermi is the codename for a graphics processing unit (GPU) microarchitecture developed by Nvidia, first released to retail in April 2010, as the successor to the Tesla microarchitecture. It was the primary microarchitecture used in the GeForce 400 series and GeForce 500 series. It was followed by Kepler, and used alongside Kepler in the GeForce 600 series, GeForce 700 series, and GeForce 800 series, in the latter two only in mobile GPUs. In the workstation market, Fermi found use in the Quadro x000 series, Quadro NVS models, as well as in Nvidia Tesla computing modules. All desktop Fermi GPUs were manufactured in 40nm, mobile Fermi GPUs in 40nm and 28nm. Fermi is the oldest microarchitecture from NVIDIA that received support for the Microsoft's rendering API Direct3D 12 feature_level 11.
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.
The bfloat16 floating-point format is a computer number format occupying 16 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. This format is a truncated (16-bit) version of the 32-bit IEEE 754 single-precision floating-point format (binary32) with the intent of accelerating machine learning and near-sensor computing. It preserves the approximate dynamic range of 32-bit floating-point numbers by retaining 8 exponent bits, but supports only an 8-bit precision rather than the 24-bit significand of the binary32 format. More so than single-precision 32-bit floating-point numbers, bfloat16 numbers are unsuitable for integer calculations, but this is not their intended use. Bfloat16 is used to reduce the storage requirements and increase the calculation speed of machine learning algorithms.
References
- 1 2 3 "Difference Between Single-, Double-, Multi-, Mixed-Precision". NVIDIA Blog. Retrieved 30 December 2020.
- 1 2 Abdelfattah, Ahmad; Anzt, Hartwig; Boman, Erik G.; Carson, Erin; Cojean, Terry; Dongarra, Jack; Gates, Mark; Grützmacher, Thomas; Higham, Nicholas J.; Li, Sherry; Lindquist, Neil; Liu, Yang; Loe, Jennifer; Luszczek, Piotr; Nayak, Pratik; Pranesh, Sri; Rajamanickam, Siva; Ribizel, Tobias; Smith, Barry; Swirydowicz, Kasia; Thomas, Stephen; Tomov, Stanimire; Tsai, Yaohung M.; Yamazaki, Ichitaro; Urike Meier Yang (2020). "A Survey of Numerical Methods Utilizing Mixed Precision Arithmetic". arXiv: 2007.06674 [cs.MS].
- ↑ Holt, Kris. "The US again has the world's most powerful supercomputer". Engadget. Retrieved 20 July 2018.