In a previous paper, we’ve shown how the ideas of Ercegovac‘s “online arithmetic” – an arithmetic where computation proceeds from most significant digit to least significant digit, in contrast to the usual way we think about adding and multiplying – can be applied in the brave new world of clocking circuits faster than they “should” be able to run. The basic idea is simple: although sometimes beneficial, overclocking normal arithmetic – when it hurts – hurts bad, because errors tend to occur in the more significant digits. But if the flow of information were reversed, from more to less significant digits, then the error should hurt less too.

And so it did. But with one problem: to allow most significant digits to be generated first requires a redundant number system – a way of representing numbers where each number has more than one representation, for example, where there are two different ways of writing “10”. This redundancy cost a lot of silicon area.

This new paper shows that, in modern FPGA architectures, with careful design, the cost can be reduced significantly for adders. For multipliers, most significant digit first arithmetic has the important benefit that if you only want the most significant digits of the result, you don’t need to compute the least significant digits. In multiplication this is often the case, often in regular binary arithmetic we compute the 2n-bit result of an n by n-bit multiplication only to throw away the bottom n bits. We show in this paper that by judicious design, the area penalties of the redundant arithmetic can be eliminated in this case.

This work removes one of the last remaining hurdles that stops online arithmetic being industrially viable.