GPU-Accelerated Generation of Correctly Rounded Elementary Functions

Abstract

Abstract-The IEEE 754-2008 standard recommends the correct rounding of some elementary functions. This requires to solve the Table Maker's Dilemma which implies a huge amount of CPU computation time. We consider in this paper accelerating such computations, namely Lefèvre algorithm on Graphics Processing Units (GPUs) which are massively parallel architectures with a partial SIMD execution (Single Instruction Multiple Data).We first propose an analysis of the Lefèvre hard-to-round argument search using the concept of continued fractions. We then propose a new parallel search algorithm much more efficient on GPU thanks to its more regular control flow. We also present an efficient hybrid CPU-GPU deployment of the generation of the polynomial approximations required in Lefèvre algorithm. In the end, we manage to obtain overall speedups up to 53.4x on one GPU over a sequential CPU execution, and up to 7.1x over a multi-core CPU, which enable a much faster solving of the Table Maker's Dilemma for the double precision format.

Solving the Table Maker's Dilemma on Current SIMD Architectures

“…These two steps are massively parallel over the domains D i since these numerous domains can all be processed independently. However, while the polynomial approximation generation has a regular control flow [9], the HR-case search presents divergence issues when executed on SIMD architectures. Moreover, the HR-case search is the most time consuming step when solving the TMD.…”

“…In [22], Lefèvre tested other variants of this strategy and concluded that in practice the most efficient strategy was this three phases refinement. This is mainly due to the fact that the number of arguments succeeding the existence test can be roughly predicted, and that the amount of time spent in the HR-case existence test has to be balanced with the amount of time spent in exhaustive search [9,22]. Algorithm 1: Lefèvre three phases isolation strategy for HR-case search.…”

“…Comparing this lower bound to ϵ ′ , we can then determine whether there is potentially a (p, ϵ ′ ) HR-case in the domain or not. Lefèvre existence test is presented in Algorithm 2 and is explained more thoroughly in [9,10].…”

“…We thus focus in this article on the double precision format (p = 53), for which the Lefèvre algorithm is as efficient as the SLZ algorithm in practice [25,6]. Moreover, the Lefèvre algorithm has already been used to generate all known hardness-to-round in double precision [25], and it offers fine-grained parallelism which is suitable for massively parallel architectures like GPUs [8,10]. We therefore study the Lefèvre algorithm here.…”

“…In this paper, we show that efficiently solving the TMD on various multi-core and many-core SIMD architectures (CPUs, GPUs, Intel Xeon Phi), and scaling performance with the number of SIMD lanes, requires to jointly handle this divergence at multiple levels: algorithm, programming and hardware. We start by describing a regular HR-case search algorithm, first presented in [9,10], which has been shown to drastically reduce divergence in the execution flow on NVIDIA GPUs [9,10]. We then extend the deployment of this HR-case search on other SIMD architectures: CPUs and the Intel Xeon Phi.…”