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

Laslo Hunhold, John Gustafson

TL;DR

This work addresses the memory bandwidth bottleneck in FFT-based spectral computations by evaluating emerging low- to high-precision number formats. It introduces NGAFFTB, a benchmark framework that systematically assesses round-trip FFTs on images and STFTs on audio, alongside spectral PDE solves such as the heat equation $u_t=\alpha\nabla^2 u$ and the Poisson problem $-\Delta u=f$ on $[0,1]^2$, with high-precision float128 references. The key finding is that OFP8 is generally unsuitable, while bfloat16 improves over float16 in some cases; posits and takums typically deliver better numerical stability, with takums often excelling at 16 bits and remaining competitive at higher precisions where discretization error dominates. These results suggest takum16 as a strong candidate to replace bf16 in HPC workloads and provide guidance for hardware design and numerical analysis of FFT-based spectral methods in emerging arithmetics.

Abstract

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.

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

TL;DR

This work addresses the memory bandwidth bottleneck in FFT-based spectral computations by evaluating emerging low- to high-precision number formats. It introduces NGAFFTB, a benchmark framework that systematically assesses round-trip FFTs on images and STFTs on audio, alongside spectral PDE solves such as the heat equation and the Poisson problem on , with high-precision float128 references. The key finding is that OFP8 is generally unsuitable, while bfloat16 improves over float16 in some cases; posits and takums typically deliver better numerical stability, with takums often excelling at 16 bits and remaining competitive at higher precisions where discretization error dominates. These results suggest takum16 as a strong candidate to replace bf16 in HPC workloads and provide guidance for hardware design and numerical analysis of FFT-based spectral methods in emerging arithmetics.

Abstract

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.
Paper Structure (12 sections, 5 equations, 4 figures)

This paper contains 12 sections, 5 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Experimental Methods
  3. Image and Audio Datasets
  4. Common Experiment Interface
  5. Spectral PDE Experiments
  6. FFT Experiments
  7. Results
  8. Heat Equation Experiments
  9. Poisson Equation Experiments
  10. Image FFT Experiments
  11. Audio STFT Experiments
  12. Conclusion

Figures (4)

  • Figure 1: Cumulative error distribution of the relative errors of the heat equation PDE test problem solutions computed using a range of machine number types, with the specified space discretization parameter $N_x$ and $N_t \in \{ 1,\dots,500 \}$. The symbol $\infty$ denotes an overflow in the underlying number type.
  • Figure 2: Cumulative error distribution of the relative errors of the Poisson equation PDE test problem solutions computed using a range of machine number types, with the specified right-hand side parameter $\sigma$ and $N_x \in \{ 2,\dots,100 \}$. The symbol $\infty$ denotes an overflow in the underlying number type.
  • Figure 3: Cumulative error distribution of the relative errors of the image FFT solutions computed using a range of machine number types. The symbol $\infty$ denotes an overflow in the underlying number type.
  • Figure 4: Cumulative error distribution of the relative errors of the audio STFT solutions computed using a range of machine number types. The symbol $\infty$ denotes an overflow in the underlying number type.