Rounding Modes Research Articles

A comparison of three rounding algorithms for IEEE floating-point multiplication

A new IEEE compliant floating-point rounding algorithm for computing the rounded product from a carry-save representation of the product is presented. The new rounding algorithm is compared with the rounding algorithms of Yu and Zyner (1995) and of Quach et al. (1991). For each rounding algorithm, a logical description and a block diagram is given, the correctness is proven, and the latency is analyzed. We conclude that the new rounding algorithm is the fastest rounding algorithm, provided that an injection (which depends only on the rounding mode and the sign) can be added in during the reduction of the partial products into a carry-save encoded digit string. In double precision format, the latency of the new rounding algorithm is 12 logic levels compared to 14 logic levels in the algorithm of Quach et al. and 16 logic levels in the algorithm of Yu and Zyner.

Efficient and reliable methods for rounded-interval arithmetic

We present an efficient and reliable method for computing the unit-in-the-last-place ( ulp) of a double-precision floating-point number, taking advantage of the standard binary representation for floatingpoint numbers defined by IEEE Std 754-1985. The ulp is necessary to perform software rounding for robust rounded-interval arithmetic (RIA) operations. Hardware rounding, using two of the standard rounding modes defined by IEEE-754, may be more efficient. RIA has been used to produce robust software systems for the solution of systems of nonlinear equations, interrogation of geometric and differential properties of curves and surfaces, curve and surface intersections, and solid modeling.

Suppression of beam steering in an injection-locked laser diode array

Experimental measurements were made to demonstrate that the degree of beam steering in an injection-locked laser diode array is related to the divergence of the master laser beam. For a collimated master laser beam the beam steering was suppressed. The injection-locked laser diode array beam steers with the master laser wavelength because only a portion of the divergent master laser beam satisfies the round trip mode condition.

Computation of elementary functions on the IBM RISC System/6000 processor

The additional speed and precision of the IBM RISC System/6000 floating-point unit have motivated reexamination of algorithms to perform division, square root, and the elementary functions. New results are obtained which avoid the necessity of doing special testing to get the last bit rounded correctly in accordance with all of the IEEE rounding modes in the case of division and square root. For the elementary function library, a technique is described for always getting the last bit rounded correctly in the selected rounding mode.

Solution of linear systems with verified accuracy

In the past several years new methods have been derived for solving algebraic problems with high accuracy and with automatic verification of the correctness of the result. The introduction of the new methods consists of four parts: (1) TO obtain a result on a computer the correctness of which is verified to be correct, a precisely defined computer arithmetic is indispensable. Moreover, a computer arithmetic with maximum accurate results for an> single operation would be desirable. The definition of maximum accuracy for a computed result is easy, the theoretical and practical foundation is a new theory developed by Kulisch and Miranker [2] to be described in the following. A computer result is an approximation of the precisely defined real result by a floating-point number. Although the real (infinite precise) result is in general not computable, it may serve to define the term maximum accuracy. As long as no overflow or underflow occurs, there are essentially two cases. First, the precise result may be exactly a machine (floating-point) number. Second, there are two floating-point neighbours left and right to the precise real result. In the first case, the maximum accurate floating-point result is obviously the exact result which happens to be a machine number. In the second case, both floating-point neighbours of the precise result are of maximum accuracy and it depends on the rounding mode which result will be delivered. There are four essential rounding modes, namely rounding to nearest, rounding downwards, rounding upwards and rounding towards zero. For rounding to the nearest floating-point number there is the special case that the precise result is exactly the midpoint of the two floating-point neighbours. In this case the result is usually the floating-point number of larger magnitude. The definition of maximum accurate floating-point operations is clear, but the theoretical bases and implementation is not trivial. One reason for this is that the exact, infinite precise result is in general not known or difficult to compute. However, in numerical analysis not only bare numbers as approximations to real quantities occur but also complex numbers and, more general, vectors and matrices over real and complex numbers. It is desirable to provide the well-defined operations for vectors and matrices also with maximum accuracy. Obviously for that purpose a dot product with maximum accuracy is necessary. The implementation of a dot product with maximum accuracy is possible as has been shown by Bohlender [l]. It turns out that the maximum accurate dot product is essentially sufficient to provide all operations for real or complex scalars, vectors and matrices with maximum accuracy 121. (2) The ultimate goal is the implementation of algorithms delivering result with automatic verification of the correctness. A verification can be provided by delivering sets including the solution. The correctness, i.e. the fact that the delivered set actually contains the solution to the

A statistical study of the accuracy of floating point number systems

This paper presents the statistical results of tests of the accuracy of certain arithmetic systems in evaluating sums, products and inner products, and analytic error estimates for some of the computations. The arithmetic systems studied are 6-digit hexadecimal and 22-digit binary floating point number representations combined with the usual chop and round modes of arithmetic with various numbers of guard digits, and with a modified round mode with guard digits. In a certain sense, arithmetic systems differing only in their use of binary or hexadecimal number representations are shown to be approximately statistically equivalent in accuracy. Further, the usual round mode with guard digits is shown to be statistically superior in accuracy to the usual chop mode in all cases save one. The modified round mode is found to be superior to the chop mode in all cases.

A statistical study of the accuracy of floating point number systems

This paper presents the statistical results of tests of the accuracy of certain arithmetic systems in evaluating sums, products and inner products, and analytic error estimates for some of the computations. The arithmetic systems studied are 6-digit hexadecimal and 22-digit binary floating point number representations combined with the usual chop and round modes of arithmetic with various numbers of guard digits, and with a modified round mode with guard digits. In a certain sense, arithmetic systems differing only in their use of binary or hexadecimal number representations are shown to be approximately statistically equivalent in accuracy. Further, the usual round mode with guard digits is shown to be statistically superior in accuracy to the usual chop mode in all cases save one. The modified round mode is found to be superior to the chop mode in all cases.