An empirical study of error-free transformations for enhancing mathematical function precision
摘要
In numerical computation, the inherent rounding errors of floating-point operations often affect the precision of mathematical functions. The use of high-precision achieved through software-dependent simulation for precision compensation may result in significant performance overhead. Error-free transformations (EFT) technology, based on hardware-supported precision to approximate high-precision implementation, can effectively balance accuracy and performance. However, enhancing the precision of mathematical functions is a very complex and challenging issue. There is a lack of relevant research on when EFT technology can be used to improve the precision of mathematical functions, what effects can be achieved, and what impact it may have on program performance. In this work, we present an empirical study on the applicability and effectiveness of using error-free transformations (EFT) in floating-point computation to assess their potential and limitations in improving precision over mathematical functions. We select 42 mathematical functions from the GNU Scientific Library (GSL), known for significant rounding errors. We evaluate the EFT techniques from three aspects: the applicability of EFT for different mathematical functions (especially at the maximum error point and its vicinity), the precision improvement of EFT in input domains near the error-triggering input, and the performance of EFT compared with the high-precision versions. Experimental results show that EFT has advantages in reducing floating-point errors across 27 functions. Furthermore, while improving the accuracy of mathematical functions within specific input ranges near the maximum error input, EFT achieves a 10.92