arXiv subjects

John Gustafson

Publications and source records attributed to John Gustafson.

6 recordsLinked to original sources

arXiv2026

An Energy-Efficient Approximate Posit Multiply-Divide Unit

In modern computing units, division operations are generally slower than other arithmetic operations and require more resources, such as area and power, than multiplication. To reduce the delay, fast division algorithms use an initial approximation of the reciprocal of the divisor and iteratively approach the correct value, followed by multiplication with the dividend. The hardware architecture and choice of algorithm can significantly alter the overall performance of the division unit. This paper proposes a reduced-accuracy division method for the posit number system, which is an alternative to the traditional floating-point system. The proposed design uses a Look-Up Table (LUT) and a single subtraction operation to perform approximate divisor reciprocation by leveragingthemathematicalsymmetriesofthepositnumbersystem.The paper also presents a hardware architecture that combines multiplication and division units. The reciprocal calculation has been incorporated into the posit Decoder, a common unit required to perform any hardware operation with posits. Compared to existing hardware implementations of division, the proposed method requires significantly fewer operations at the cost of perfect rounding for division. The proposed architecture was simulated using the Cadence RTL v7.1 E2 compiler at the TSMC 90 nm process node and achieves a Power Delay Product (PDP) reduction of 78.8% compared to an existing design that performs exact division, while only 46.33% of the area is required. The experimental results also demonstrate the effectiveness of the proposed system in improving the efficiency of multiplication in posit-based systems.

cs.AR↗

arXiv2025

Spectral Methods via FFTs in Emerging Machine Number Formats: OFP8, Bfloat16, Posit, and Takum Arithmetics

The Fast Fourier Transform (FFT) is one of the most widely used algorithms in high performance computing, with critical applications in spectral analysis for both signal processing and the numerical solution of partial differential equations (PDEs). These data-intensive workloads are primarily constrained by the memory wall, motivating the exploration of emerging number formats -- such as OFP8 (E4M3 and E5M2), bfloat16, and the tapered-precision posit and takum formats -- as potential alternatives to conventional IEEE 754 floating-point representations. This paper evaluates the accuracy and stability of FFT-based computations across a range of formats, from 8 to 64 bits. Round-trip FFT is applied to a diverse set of images, and short-time Fourier transform (STFT) to audio signals. The results confirm posit arithmetic's strong performance at low precision, with takum following closely behind. Posits show stability issues at higher precisions, while OFP8 formats are unsuitable and bfloat16 underperforms compared to float16 and takum.

math.NA↗

arXiv2024

Evaluation of Posits for Spectral Analysis Using a Software-Defined Dataflow Architecture

Spectral analysis plays an important role in detection of damage in structures and deep learning. The choice of a floating-point format plays a crucial role in determining the accuracy and performance of spectral analysis. The IEEE Std 754\textsuperscript{TM} floating-point format (IEEE~754 for short) is supported by most major hardware vendors for ``normal'' floats. However, it has several limitations. Previous work has attempted to evaluate posit format with respect to accuracy and performance. The accuracy of the posit has been established over IEEE~754 for a variety of applications. For example, our analysis of the Fast Fourier Transform shows 2x better accuracy when using a 32-bit posit vs. a 32-bit IEEE754 format. For spectral analysis, 32-bit posits are substantially more accurate than 32-bit IEEE~754 floats. Although posit has shown better accuracy than IEEE~754, a fair evaluation of posit with IEEE~754 format using a real hardware implementation has been lacking so far. A software simulation of posit format on an x86 CPU is about $\mathbf{69.3\times}$ slower than native IEEE~754 hardware for normal floats for a Fast Fourier Transform (FFT) of $\mathbf{2^{28}}$ points. We propose the use of a software-defined dataflow architecture to evaluate performance and accuracy of posits in spectral analysis. Our dataflow architecture uses reconfigurable logical elements that express algorithms using only integer operations. Our architecture does not have an FPU, and we express both IEEE~754 and posit arithmetic using the same integer operations within the hardware. On our dataflow architecture, the posit format is only $\mathbf{1.8\times}$ slower than IEEE~754 for a Fast Fourier Transform (FFT) of $\mathbf{2^{28}\approx 268}$ million points. With this implementation, we empirically propose a new lower bound for the performance of posit compared to IEEE~754 format.

cs.AR↗

arXiv2024

Algorithm-Hardware Co-Design of Distribution-Aware Logarithmic-Posit Encodings for Efficient DNN Inference

Traditional Deep Neural Network (DNN) quantization methods using integer, fixed-point, or floating-point data types struggle to capture diverse DNN parameter distributions at low precision, and often require large silicon overhead and intensive quantization-aware training. In this study, we introduce Logarithmic Posits (LP), an adaptive, hardware-friendly data type inspired by posits that dynamically adapts to DNN weight/activation distributions by parameterizing LP bit fields. We also develop a novel genetic-algorithm based framework, LP Quantization (LPQ), to find optimal layer-wise LP parameters while reducing representational divergence between quantized and full-precision models through a novel global-local contrastive objective. Additionally, we design a unified mixed-precision LP accelerator (LPA) architecture comprising of processing elements (PEs) incorporating LP in the computational datapath. Our algorithm-hardware co-design demonstrates on average <1% drop in top-1 accuracy across various CNN and ViT models. It also achieves ~ 2x improvements in performance per unit area and 2.2x gains in energy efficiency compared to state-of-the-art quantization accelerators using different data types.

cs.AR↗

arXiv2021

Brightening the Optical Flow through Posit Arithmetic

As new technologies are invented, their commercial viability needs to be carefully examined along with their technical merits and demerits. The posit data format, proposed as a drop-in replacement for IEEE 754 float format, is one such invention that requires extensive theoretical and experimental study to identify products that can benefit from the advantages of posits for specific market segments. In this paper, we present an extensive empirical study of posit-based arithmetic vis-à-vis IEEE 754 compliant arithmetic for the optical flow estimation method called Lucas-Kanade (LuKa). First, we use SoftPosit and SoftFloat format emulators to perform an empirical error analysis of the LuKa method. Our study shows that the average error in LuKa with SoftPosit is an order of magnitude lower than LuKa with SoftFloat. We then present the integration of the hardware implementation of a posit adder and multiplier in a RISC-V open-source platform. We make several recommendations, along with the analysis of LuKa in the RISC-V context, for future generation platforms incorporating posit arithmetic units.

cs.AR↗

arXiv2020

A Novel Approach to Generate Correctly Rounded Math Libraries for New Floating Point Representations

Given the importance of floating-point~(FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and Posits). These representations do not have correctly rounded math libraries. Further, the use of existing FP libraries for these new representations can produce incorrect results. This paper proposes a novel approach for generating polynomial approximations that can be used to implement correctly rounded math libraries. Existing methods generate polynomials that approximate the real value of an elementary function $f(x)$ and produce wrong results due to approximation errors and rounding errors in the implementation. In contrast, our approach generates polynomials that approximate the correctly rounded value of $f(x)$ (i.e., the value of $f(x)$ rounded to the target representation). It provides more margin to identify efficient polynomials that produce correctly rounded results for all inputs. We frame the problem of generating efficient polynomials that produce correctly rounded results as a linear programming problem. Our approach guarantees that we produce the correct result even with range reduction techniques. Using our approach, we have developed correctly rounded, yet faster, implementations of elementary functions for multiple target representations.

cs.MS↗