High-precision computations using residue-interval arithmetic on FPGAs

The problem of round-off errors arises in a large number of issues in various fields of knowledge, including computational mathematics, mathematical physics, biochemistry, quantum mechanics, mathematical programming. Today, experts place particular emphasis on accuracy, fault tolerance, stability, and reproducibility of computation results of numerical models when solving a wide range of industrial and scientific problems, such as: mathematical modeling and structural designs of aircrafts, cars, ships; process modeling and computations for solving large-scale problems in the field of nuclear physics, aerodynamics, gas, and hydrodynamics; problems on reliable predictive modeling of climatic processes and forecasting of global changes in the atmosphere and water environments; faithful modeling of chemical processes and synthesis of pharmaceuticals, etc.
Floating-point arithmetic is the dominant choice for most scientific applications. However, there are a lot of unsolvable with double-precision arithmetic problems. A vast number of floating-point arithmetic operations would be required to solve such problems. Each operation carries round-off errors leading to uncontrolled round-off errors and, consequently, incorrect results. Many modeling and simulation problems need to increase the accuracy of number representation to 100-1000 decimal digits or more to obtain reliable results. In this regard, arbitrary-precision arithmetic is becoming ever important. With this arithmetic, one can use numbers, whose arbitrary precision is many times greater than the word length of the conventional system.
The paper proposes a new way of representing integers and floats for computations in super-large ranges — hybrid residue-positional interval logarithmic number representation for performing high-precision and reliable calculations in super-large numerical ranges.