> My original comment was about manned space flight in particular. If your application is relatively generic I think it is completely okay, if you are aware of it and mitigate the most pressing issues.
Everything we have been talking about relates to space flight. In fact, with humans on board, you can afford to be a lot less precise, because they can work around most numerical issues by hand. The Apollo guidance computers, for example, were prone to occasional instances of gimbal lock and numerical instability, and the astronauts just fixed it.
> You can use interval arithmetic, which guarantees that you at least know when cancellation has occurred. Interval arithmetic is fast enough for real time, although it has its own significant drawbacks.
Interval arithmetic does not prevent cancellation. It's just two floating point calculations, both of which are actually less precise than the one you would do otherwise (you don't use default rounding for interval arithmetic, you round the bottom down and the top up). You do know when things have been canceled, but you know that in a floating point calculation anyway if you have done the error analysis.
My overall point here is that NASA isn't missing anything by using floating point instead of using other weird or exotic arithmetic systems. Double-precision floating point combined with a rudimentary error analysis and some algebra is good enough for pretty much everything, and you may not be able to do better at all with fixed point. Designing fixed point algorithms also depends on a very careful analysis of interval ranges and precisions, and often gets you nothing over just using "double", where the error analysis is easier anyway.
If you need to do better than double, there's also double-double arithmetic for your hard parts, which is a similar speed to interval arithmetic and doubles the precision you get beyond double.
Everything we have been talking about relates to space flight. In fact, with humans on board, you can afford to be a lot less precise, because they can work around most numerical issues by hand. The Apollo guidance computers, for example, were prone to occasional instances of gimbal lock and numerical instability, and the astronauts just fixed it.
> You can use interval arithmetic, which guarantees that you at least know when cancellation has occurred. Interval arithmetic is fast enough for real time, although it has its own significant drawbacks.
Interval arithmetic does not prevent cancellation. It's just two floating point calculations, both of which are actually less precise than the one you would do otherwise (you don't use default rounding for interval arithmetic, you round the bottom down and the top up). You do know when things have been canceled, but you know that in a floating point calculation anyway if you have done the error analysis.
My overall point here is that NASA isn't missing anything by using floating point instead of using other weird or exotic arithmetic systems. Double-precision floating point combined with a rudimentary error analysis and some algebra is good enough for pretty much everything, and you may not be able to do better at all with fixed point. Designing fixed point algorithms also depends on a very careful analysis of interval ranges and precisions, and often gets you nothing over just using "double", where the error analysis is easier anyway.
If you need to do better than double, there's also double-double arithmetic for your hard parts, which is a similar speed to interval arithmetic and doubles the precision you get beyond double.