Related papers: Arithmetic Operations Beyond Floating Point Number…
Simple floating point operations like addition or multiplication on normalized floating point values can be computed by current AMD and Intel processors in three to five cycles. This is different for denormalized numbers, which appear when…
Geometric predicates are at the core of many algorithms, such as the construction of Delaunay triangulations, mesh processing and spatial relation tests. These algorithms have applications in scientific computing, geographic information…
Level-index arithmetic appeared in the 1980s. One of its principal purposes is to abolish the issues caused by underflows and overflows in floating point. However, level-index arithmetic does not expand the set of numbers but spaces out the…
We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can…
We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the precision. We present a systematic recurrence for a martingale…
This is a draft of a book about algorithms for performing arithmetic, and their implementation on modern computers. We are concerned with software more than hardware - we do not cover computer architecture or the design of computer…
Support for arithmetic in multiple precisions and number formats is becoming increasingly common in emerging high-performance architectures. From a computational scientist's perspective, our goal is to determine how and where we can safely…
Scientific computing applications, such as computational fluid dynamics and climate modeling, typically rely on 64-bit double-precision floating-point operations, which are extremely costly in terms of computation, memory, and energy. While…
We consider the problem of solving floating-point constraints obtained from software verification. We present UppSAT --- a new implementation of a systematic approximation refinement framework [ZWR17] as an abstract SMT solver. Provided…
Floating-point arithmetic performance determines the overall performance of important applications, from graphics to AI. Meeting the IEEE-754 specification for floating-point requires that final results of addition, subtraction,…
Floating-point arithmetic plays a central role in science, engineering, and finance by enabling developers to approximate real arithmetic. To address numerical issues in large floating-point applications, developers must identify root…
In this paper, we propose a mixed-precision convolution unit architecture which supports different integer and floating point (FP) precisions. The proposed architecture is based on low-bit inner product units and realizes higher precision…
Debugging accumulation of floating-point errors is hard; ideally, computer should track it automatically. Here we consider twofold approximation of an exact real with value + error pair of floating-point numbers. Normally, value + error sum…
Floating-point computations are quickly finding their way in the design of safety- and mission-critical systems, despite the fact that designing floating-point algorithms is significantly more difficult than designing integer algorithms.…
Numerical underflow and overflow are major hurdles for rolling-out the modeling and simulation infrastructure for temperatures below about 50 K. Extending the numeric precision is computationally intensive and thus best avoided. The root…
Efficient number representation is essential for federated learning, natural language processing, and network measurement solutions. Due to timing, area, and power constraints, such applications use narrow bit-width (e.g., 8-bit) number…
Operator overloading algorithmic differentiation (AD) tools are usually only developed for floating-point values. Algorithmic optimization for, e.g., linear systems solvers or matrix-matrix multiplications are often introduced via external…
Computation, the use of a computer to solve, simulate, or visualize a physical problem, has revolutionized how physics research is done. Computation is used widely to model systems, to simulate experiments, and to analyze data. Yet, in most…
Within the past years, hardware vendors have started designing low precision special function units in response to the demand of the Machine Learning community and their demand for high compute power in low precision formats. Also the…
There is a growing interest in the use of reduced-precision arithmetic, exacerbated by the recent interest in artificial intelligence, especially with deep learning. Most architectures already provide reduced-precision capabilities (e.g.,…