. Many modern algorithms for the transcendental functions rely on a large table of precomputed values together with a low-order polynomial to interpolate between them. In verifying such an algorithm, one is faced with the problem of bounding the error in this polynomial approximation. The most straightforward methods are based on numerical approximations, and are not prima facie reducible to a formal HOL proof. We discuss a technique for proving such results formally in HOL, via the formalization of a number of results in polynomial theory, e.g. squarefree decomposition and Sturm's theorem, and the use of a computer algebra system to compute results that are then checked in HOL. We demonstrate our method by tackling an example from the literature. 1 Introduction Many algorithms for the transcendental functions such as exp, sin and ln in floating point arithmetic are based on table lookup. Suppose that a transcendental function f(x) is to be calculated. Values of f(a i ) are...

Abstract. An inverse polynomial has a Chebyshev series expansion k∑ 1 / bjTj(x) = anTn(x) j=0 if the polynomial has no roots in [−1,1]. If the inverse polynomial is decomposed into partial fractions, the an are linear combinations of simple functions of the polynomial roots. Also, if the first k of the coefficients an are known, the others become linear combinations of these with expansion coefficients derived recursively from the bj’s. On a closely related theme, finding a polynomial with minimum relative error towards a given f(x) is approximately equivalent to finding the bj in f(x) / ∑k 0 bjTj(x) = 1 + ∑∞ k+1 anTn(x), and may be handled with a Newton method providing the Chebyshev expansion of f(x) is known. n=0