Roundings in floating point arithmetic

Software for Interval Arithmetic: A Reasonably Portable Package

“…In addition, vital information concerning such phenomena as exponent range faults is generally not available in existing systems. Consequently, the package was designed to be based on a set of arithmetic primitives of the type described in [19]; these are implemented via software. Hollemth, X = interval 2 Variable names: The first letter (or pmr of letters) indicates the data type of the variable as above.…”

“…These algorithms, written in Algol, yield the smallest possible intervals; they utilize double precision arithmetic and various other manipulations to capture the information required for correct directed roundings. The present package employs simulated floating point operations using the algorithms in [19] to accomplish this.…”

Computer Arithmetic in Theory and Practice (Ulrich W. Kulisch and Willard L. Miranker)

“…Other examples of this type are known, even for floating-point software provided for popular microcomputers. This is in spite of the fact that algorithms for accurately rounded floating-point arithmetic have been known for some time [2]. Accuracy here refers not to the number of digits in the mantissas of the floating-point numbers used, but to the fact that results of operations should be rounded to an adjacent floating-point number, and not to some more distant value.…”