A Correctly Rounded Mixed-Radix Fused-Multiply-Add

Abstract

The IEEE 754–2008 Standard governs Floating-Point Arithmetic in all types of Computer Systems. The Standard provides for two radices, 2 and 10. It specifies conversion operations between these radices, but does not allow floating-point formats of different radices to be mixed in computational operations. In contrast, the Standard does provide for mixing formats of one radix in one operation. In order to enhance the Standard and make it closed under all basic computational operations, we propose an algorithm for a correctly rounded mixed-radix Fused-Multiply-and-Add (FMA). Our algorithm takes any combination of IEEE754 binary64 and decimal64 numbers in argument and provides a result in IEEE754 binary64 and decimal64, rounded according to any for the five IEEE754 rounding modes. Our implementation does not require any dynamic memory allocation; its runtime can be bounded statically. We compare our implementation to a basic mixed-radix FMA implementation based on the GMP Multiple Precision library.