Towards Machine-Efficient Rational L∞-Approximations of Mathematical Functions

Software implementations of mathematical functions often use approximations that can be either polynomial or rational in nature. While polynomials are the preferred approximation in most cases, rational approximations are nevertheless an interesting alternative when dealing with functions that have a pronounced "nonpolynomial behavior" (such as poles close to the approximation domain, asymptotes or finite limits at ±∞).The major challenge is that of computing good rational approximations with machine number coefficients (e.g. floating-point or fixed-point) with respect to the supremum norm, a key step in most procedures for evaluating a mathematical function. This is made more complicated by the fact that even when dealing with real-valued coefficients, optimal supremum norm solutions are sometimes difficult to obtain. Here, we introduce flexible and fast algorithms for computing such rational approximations with both real and machine number coefficients. Their effectiveness is explored on several examples.