The RISC-V Vector instruction set extension (RVV) provides scalable data-parallel instructions suitable for accurate and performant implementations of numerical algorithms across many application domains [1]. The primary objective of this paper is to share our experience implementing vector C99(libm) functions using RVV. Our contributions are threefold: First, we contributed an RVV port of SLEEF, a multi-platform open-source vector libm. Second, we show that while SLEEF simplifies porting efforts, it also precludes some RVV-specific optimization opportunities. With SiFive’s X280 vector processor micro-architecture as a case-study, we highlight RVV features that optimized code can use. We also expand the discussion to how these features might be used differently when optimizing for other cores. Third, we compare the performance of our SLEEF RVV port to our own RVV-native routines. We present results from 1-ulp accurate implementations of Libm functions in a cycle-accurate simulation of the X280 pipeline to show the impact of RVV-enabled optimizations.

Besides scalar instructions, modern micro-architectures also provide support for vector instructions. They enable to treat packed inputs (typically 4 or 8) in a single instruction. The challenge is now to write vector programs to support mathematical functions like sin, cos, exp, log, ... which efficiently exploit those vector instructions. This article focuses on the design of vectorized implementation of log(x) function, and more particularly on its automation for different formats and micro-architectures. First it rewrites a classic range reduction in a branchless fashion so as to use at best recent micro-architecture features, like rcp (reciprocal) instruction, and to treat all inputs in the same flow. Second it details rigorously how to achieve “faithfully rounded” implementations. Third it shows how to automate this implementation process using the MetaLibm framework, on SSE/AVX and AVX2 supporting micro-architectures. Finally we illustrate that this process enables to achieve high throughput implementations for the binary32 and binary64 formats in a fully automated way.

Elementary mathematical functions are pervasively used in many applications such as electronic calculators, computer simulations, or critical embedded systems. Their evaluation is always an approximation, which usually makes use of mathematical properties, precomputed tabulated values, and polynomial approximations. Each step generally combines error of approximation and error of evaluation on finite-precision arithmetic. When they are used, tabulated values generally embed rounding error inherent to the transcendence of elementary functions. In this article, we propose a general method to use error-free values that is worthy when two or more terms have to be tabulated in each table row. For the trigonometric and hyperbolic functions, we show that Pythagorean triples can lead to such tables in little time and memory usage. When targeting correct rounding in double precision for the same functions, we also show that this method saves memory and floating-point operations by up to 29 and 42 percent, respectively.

Software evaluation of elementary functions usually requires three steps: a range reduction, a polynomial evaluation, and a reconstruction step. These evaluation schemes are designed to give the best performance for a given accuracy, which requires a fine control of errors. One of the main issues is to minimize the number of sources of error and/or their influence on the final result. The work presented in this article addresses this problem as it removes one source of error for the evaluation of trigonometric functions. We propose a method that eliminates rounding errors from tabulated values used in the second range reduction for the sine and cosine evaluation. When targeting correct rounding, we show that such tables are smaller and make the reconstruction step less expensive than existing methods. This approach relies on Pythagorean triples generators. Finally, we show how to generate tables indexed by up to 10 bits in a reasonable time and with little memory consumption.