Computing FFTs at Target Precision Using Lower-Precision FFTs

2026-03-30

AI Summary

This work addresses the challenge of efficiently implementing high-precision fast Fourier transforms (FFTs) on modern processors that favor low-precision, high-throughput arithmetic. It introduces, for the first time, the Ozaki high-precision computation framework into the FFT domain, leveraging Bluestein’s algorithm to reformulate the FFT as a cyclic convolution. The method computes split components exactly using number-theoretic transforms (NTT) in low-precision arithmetic and reconstructs the high-precision result via the Chinese Remainder Theorem. A novel NTT-domain accumulation strategy is proposed to drastically reduce the number of required NTT calls. Experimental results demonstrate that double-precision FFTs can be achieved with only 64–96 invocations of 32-bit NTTs, yielding relative errors lower than those of both FFTW and Triple-Single approaches while maintaining consistent accuracy across varying transform lengths.

Technology Category

• Machine Learning: Hardware-aware ML
• Computer Vision: Other Foundations of Computer Vision
• Search and Optimization: Distributed Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Modern processors deliver higher throughput for lower-precision arithmetic than for higher-precision arithmetic. For matrix multiplication, the Ozaki scheme exploits this performance gap by splitting the inputs into lower-precision components and delegating the computation to optimized lower-precision routines. However, no similar approach exists for the fast Fourier transform (FFT). Here, we propose a method that computes target-precision FFTs using lower-precision FFTs by applying the Ozaki scheme to the cyclic convolution in the Bluestein FFT. The split component convolutions are computed exactly using the number theoretic transform (NTT), an FFT over a finite field, instead of floating-point FFTs, combined with the Chinese remainder theorem. We introduce an upper bound on the number of splits and an NTT-domain accumulation strategy to reduce the NTT call count. As a concrete implementation, we implement a double-precision FFT using 32-bit NTTs and confirm reduced relative error compared with those for FFTs based on FFTW and Triple-Single precision arithmetic, with stable error across FFT lengths, at most 96 NTT calls, or 64 NTT calls with NTT-domain accumulation. On an Intel Xeon Platinum 8468 for lengths $n=2^{10}$-$2^{18}$, the execution time is approximately 107-1315$\times$ that of FFTW's double-precision FFT, with NTTs accounting for approximately 80% of the total time.

Problem

• FFT
• lower-precision arithmetic
• target precision
• fast Fourier transform
• precision computation

Innovation

• Ozaki scheme
• Number Theoretic Transform
• Bluestein FFT
• Chinese Remainder Theorem
• low-precision computing