Properties of the subtraction valid for any floating point system

In 1974, Sterbenz [20] presented a theorem about the exact subtraction of two floating point numbers x and y when they are very close one from another, that is y 2 ≤ x ≤ 2y. The theorem stating that x−y is exact under the preceding condition was presented for any radix provided the hardware was accurate enough. More recently, other authors [8,10] presented similar results with an emphasis on didactic aspects. We have recognized in [6] that Sterbenz’s theorem is not a property of the computing hardware but rather a property of the floating point number representation. Given x and y, the question is to know whether or not x− y can be represented in the working floating point system. This is clearly the key necessary condition for the implemented floating point subtraction x y to return the exact result x− y. With IEEE-like behavior, any floating point operation is cut down to two steps. An intermediate result is first computed to sufficient accuracy and then rounded. The designer must guarantee that the system always returns the result as if the infinitely precise mathematical operation were rounded. For example the subtraction is implemented as the composition of two mathematical functions, namely, the subtraction (−) and the user specified rounding function (◦) x y = ◦(x− y).