The floating-point implementation of a function on an interval often reduces to polynomial approximation, the polynomial being typically provided by Remez algorithm. However, the floating-point evaluation of a Remez polynomial sometimes leads to catastrophic cancellations. This happens when some of the polynomial coefficients are very small in magnitude with respects to others. In this case, it is better to force these coefficients to zero, which also reduces the operation count. This technique, classically used for odd or even functions, may be generalized to a much larger class of functions. An algorithm is presented that forces to zero the smaller coefficients of the initial polynomial thanks to a modified Remez algorithm targeting an incomplete monomial basis. One advantage of this technique is that it is purely numerical, the function being used as a numerical black box. This algorithm is implemented within a larger polynomial implementation tool that is demonstrated on a range of examples, resulting in polynomials with less coefficients than those obtained the usual way.
On the influence of stochastic roundoff errors and their bias on the convergence of the gradient descent method with low-precision floating-point computation
Eindhoven University of Technology（埃因霍温理工大学）；Università di Pisa（比萨大学）
Verificarlo: checking floating point accuracy through Monte Carlo Arithmetic
CMLA - Centre de Mathématiques et de Leurs Applications, ENS Cachan Université Paris-Saclay et CNRS UMR 8536（CMLA - 数学及其应用中心，巴黎萨克雷大学卡尚分校及法国国家科学研究中心）；Université de Versailles Saint-Quentin-en-Yvelines Université Paris-Saclay（凡尔赛圣康坦昂伊夫林大学，巴黎萨克雷大学）
