We present a pair arithmetic for the four basic operations and square root. It can be regarded as a simplified, more-efficient double-double arithmetic. The central assumption on the underlying arithmetic is the first standard model for error analysis for operations on a discrete set of real numbers. Neither do we require a floating-point grid nor a rounding to nearest property. Based on that, we define a relative rounding error unit u and prove rigorous error bounds for the computed result of an arbitrary arithmetic expression depending on u, the size of the expression, and possibly a condition measure. In the second part of this note, we extend the error analysis by examining requirements to ensure faithfully rounded outputs and apply our results to IEEE 754 standard conform floating-point systems. For a class of mathematical expressions, using an IEEE 754 standard conform arithmetic with base β, the result is proved to be faithfully rounded for up to 1 / √βu-2 operations. Our findings cover a number of previously published algorithms to compute faithfully rounded results, among them Horner's scheme, products, sums, dot products, or Euclidean norm. Beyond that, several other problems can be analyzed, such as polynomial interpolation, orientation problems, Householder transformations, or the smallest singular value of Hilbert matrices of large size.

We show how an IEEE-754 conformant precision-p base-β arithmetic can be implemented based on some binary floating-point and/or integer arithmetic. This includes the four basic operations and square root subject to the five IEEE-754 rounding modes, namely the nearest roundings with roundTiesToEven and roundTiesToAway, the directed roundings downwards and upwards, as well as rounding towards zero. Exceptional values like ∞ of NaN are covered according to the IEEE-754 arithmetic standard. The results of the precision-p base-β operations are computed using some underlying precision-q binary arithmetic. We distinguish two cases. When using a precision-q binary integer arithmetic, the base-β precision p is limited for all operations by β2p ≤ 2q, whereas using a precision-q binary floating-point arithmetic imposes stronger limits on the base-β precision, namely β2p ≤ 2q for addition and multiplication, β2p ≤ 2q-1 for division and β2p ≤ 2q-3 for the square root. Those limitations cannot be improved. The algorithms are implemented in a Matlab/Octave flbeta-toolbox with the choice of using uint64 or binary64 as underlying arithmetic. The former allows larger precisions, the latter is advantageous for the square root, whereas computing times are similar. The flbeta-toolbox offers precision-p base-β scalar, vector and matrix operations including sparse matrices as well as corresponding interval operations. The base β can be chosen in the range β [2,64]. The flbeta-toolbox will be part of Version 13 of INTLAB [18], the Matlab/Octave toolbox for reliable computing.

Suppose an m-digit floating-point arithmetic in base β ≥ 2 following the IEEE754 arithmetic standard is available. We show how a k-digit arithmetic with k < mcan be inherited solely using m-digit operations. This includes the rounding into kdigits, the four basic operations and the square root, all for even or odd base β. In particular, we characterize the relation between k and mso that no double rounding occurs when computing in mdigits and rounding the result into k digits. We discuss rounding to nearest as well as directed rounding, and our approach covers exceptional values including signed zero. For binary arithmetic, a Matlab toolbox based on binary64 including k-bit scalar, vector and matrix operations as well as k-bit interval arithmetic is part of Version 8 of INTLAB, the Matlab toolbox for reliable computing.