It is often said that it's only when you try to explain something to someone else that you come to realize that there are holes in your understanding of the topic in question. Such was the case when it came to presenting the concept of rounding in my most recent book, How Computers Do Math (John Wiley & Sons Inc.) featuring a virtual 8-bit computer-calculator called the DIY Calculator. As soon as I started my research into rounding, I was surprised to discover the wide range of algorithms in use. I was even more surprised to find that even the simplest technique--truncation, also known as chopping--has different results when working with signed vs. unsigned binary representations.
Let's start the ball rolling by considering the various rounding schemes in the context of the sign-magnitude decimal numbers we know and love so well. The most fundamental fact associated with rounding is that it involves transforming some quantity from a greater precision to a lesser precision; for example, suppose that we average out a range of prices and end up with a dollar value of $5.19286. In this case, rounding the more precise value of $5.19286 to the nearest cent would result in $5.19, which is less precise.
Given a choice, we would generally prefer to use a rounding algorithm that minimizes the effects of this loss of precision, especially in the case where multiple processing iterations--each involving rounding--can result in "creeping errors."
By this we mean that errors increase over time due to performing rounding operations on data that has previously been rounded. However, in the case of hardware implementations targeted toward tasks such as digital signal processing (DSP) algorithms, for example, we also have to be cognizant of the overheads associated with the various rounding techniques so as to make appropriate design trade-offs (see Fig. 1).
Round-toward-nearest: This is perhaps the most intuitive of the various rounding algorithms. In this case, values such as 3.1, 3.2, 3.3 and 3.4 would round down to 3, while values of 3.6, 3.7, 3.8 and 3.9 would round up to 4. The trick, of course, is to decide what to do in the case of the halfway value 3.5. In fact, round-toward-nearest may be considered to be a superset of two complementary options known as round-half-up and round-half-down, each of which treats the 3.5 value in a different manner as discussed below.
Round-half-up: This algorithm, which may also be referred to as arithmetic rounding, is the one that we typically associate with the rounding we learned at grade school. In this case, a halfway value such as 3.5 will round up to 4. One way to view this is that, at this level of precision and for this particular example, we can consider there to be 10 values that commence with a 3 in the most-significant place (3.0, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8 and 3.9). On this basis, it intuitively makes sense for five of the values to round down and for the other five to round up; that is, for the five values 3.0 through 3.4 to round down to 3, and for the remaining five values--3.5 through 3.9--to round up to 4.
