Least significant bit evaluation of arithmetic expressions in single-precision

Abstract

Single-precision floatingpoint computations may yield an arbitrary false result due to cancellation and rounding errors. This is true even for very simple, structured arithmetic expressions such as Horner's scheme for polynomial evaluation. A simple procedure will be presented for fast calculation of the value of an arithmetic expression to least significant bit accuracy in single precision computation. For this purpose in addition to the floating-point arithmetic only a precise scalar product (cf. [2]) is required. If the initial floatingpoint approximation is not too bad, the computing time of the new algorithm is approximately the same as for usual floating-point computation. If not, the essential progress of the presented algorithm is that the inaccurate approximation is recognized and corrected. The algorithm achieves high accuracy, i.e. between the left and the right bound of the result there is at most one more floating-point number. A rigorous estimation of all rounding errors introduced by floating-point arithmetic is given for general triangular linear systems. The theorem is applied to the evaluation of arithmetic expressions.