We present and analyze several implementations of the interval arithmetic narrowing function for multiplication. Starting from the Cleary algorithm for narrowing multiplication we describe two optimizations which produce code that is 10-15 times faster on the average. Finally, we propose a few simple RISC instructions which would allow eecient execution of interval arithmetic and narrowing rout...

Arithmetic coding is a data compression technique that encodes data (the data string) by creating a code string which represents a fractional value on the number line between 0 and 1. The coding algorithm is symbolwise recursive; i.e., it operates upon and encodes (decodes) one data symbol per iteration or recursion. On each recursion, the algorithm successively partitions an interval of the nu...

Program synthesis, or their fragments, is a way to write programs by providing only its meaning, without worrying about the implementation details. It avoids the drawback of writing sequential code, which might be di cult to check, error-prone or tedious. Our contribution is to provide complete program synthesis algorithms with unbounded data types in decidable theories. We present synthesis al...

We discuss brieey various phenomena to be tested when designing a robust code for delay diierential equations: choice of interpolant; tracking of discontinuities; vanishing delays; and problems arising from oating-point arithmetic.

The triple-code model, cognitive neuroimaging and developmental behavioral data suggest a specific association between phonological processing and arithmetic fact retrieval. Accordingly, individuals with deficits in phonological processing, such as individuals with developmental dyslexia, are expected to show difficulties in arithmetic fact retrieval. The present study tested this proposal in 2...

Binary arithmetic coding is, compression-wise, the most effective statistical coding method used in image and video compression. It is being used for compressing bi-level images/videos (JBIG, JBIG2, MPEG-4 shape coding) and is also optionally being utilized for coding of continuous-tone images/videos (JPEG, H.264). Despite its wide use, different arithmetic coders are generally incompatible wit...

At first sight, recursive saturation may seem an ‘unhappy marriage’ between model theory and recursion theory, as some authors put it. Nevertheless, it turns out to work wonderfully well with models of arithmetic. First and most important of all, since there are only countably many recursive sets, the number of recursive types over any infinite structure M is exactly card(M). So by an elementar...

The GNU Multi-Precision library is a widely used, safetycritical, library for arbitrary-precision arithmetic. Its source code is written in C and assembly, and includes intricate state-of-the-art algorithms for the sake of high performance. Formally verifying the functional behavior of such highly optimized code, not designed with verification in mind, is challenging. We present a fully verifie...