Related papers: On Approximate 8-bit Floating-Point Operations Usi…
A mixed precision Fast Fourier transform (FFT) implementation is presented. The procedure uses per-block microscaling (MX), a global power-of-two prescale, and prequantized low bit twiddles. We evaluate forward and round-trip FFT fidelity…
In computer science, transforming spherical coordinates into Cartesian coordinates is an important mathematical operation. The CORDIC (Coordinate Rotation Digital Computer) iterative algorithm can perform this operation, as well as…
We present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational…
We mechanize the fundamental properties of a rounding error model for floating-point arithmetic based on relative precision, a measure of error proposed as a substitute for relative error in rounding error analysis. A key property of…
Recently, the idea of using FP8 as a number format for neural network training has been floating around the deep learning world. Given that most training is currently conducted with entire networks in FP32, or sometimes FP16 with…
We introduce an approach to quantum information processing where the information is stored in the motional degrees of freedom of nanomechanical devices. The qubits of our approach are formed by the two lowest energy levels of mechanical…
In the work we discuss the benefit of using bitwise operations in programming. Some interesting examples in this respect have been shown. What is described in detail is an algorithm for sorting an integer array with the substantial use of…
This study aims to ensure consistency in accuracy throughout the entire design flow in the implementation of edge AI hardware for few-shot learning, by implementing fixed-point data processing in the pre-training and evaluation phases.…
We present Monte Carlo wavefunction simulations for quantum computations employing an exchange-coupled array of quantum dots. Employing a combination of experimentally and theoretically available parameters, we find that gate fidelities…
Finite-precision floating point arithmetic unavoidably introduces rounding errors which are traditionally bounded using a worst-case analysis. However, worst-case analysis might be overly conservative because worst-case errors can be…
Many algorithms feature an iterative loop that converges to the result of interest. The numerical operations in such algorithms are generally implemented using finite-precision arithmetic, either fixed- or floating-point, most of which…
Fluid dynamics simulations with the lattice Boltzmann method (LBM) are very memory-intensive. Alongside reduction in memory footprint, significant performance benefits can be achieved by using FP32 (single) precision compared to FP64…
We present an algorithm for the efficient numerical evaluation of integrals of the form \[ I(\omega) = \int_0^1 F( x,\mathrm e^{\mathrm i \omega x}; \omega) \, \mathrm d x \] for sufficiently smooth but otherwise arbitrary $F$ and $\omega…
Although not primarily designed for this purpose, floating-point numbers are often used to represent integral values, with some applications explicitly relying on this capability. However, the integral representation properties of IEEE 754…
The typical processors used for scientific computing have fixed-width data-paths. This implies that mathematical libraries were specifically developed to target each of these fixed precisions (binary16, binary32, binary64). However, to…
We show that quantum search can be used to compute the hardness to round an elementary function, that is, to determine the minimum working precision required to compute the values of an elementary function correctly rounded to a target…
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…
To demonstrate the ability in standard arithmetic operations to perform a variety of digit manipulation tasks, a closed-form representation of the Conway Base-13 Function over the integers is given.
This paper proposes a set of techniques to develop correctly rounded math libraries for 32-bit float and posit types. It enhances our RLibm approach that frames the problem of generating correctly rounded libraries as a linear programming…
Mainstream math libraries for floating point (FP) do not produce correctly rounded results for all inputs. In contrast, CR-LIBM and RLIBM provide correctly rounded implementations for a specific FP representation with one rounding mode.…