How can I write an algorithm that given a floating point number, and attempts to represent is as accurately as possible using a numerator and a denominator, both restricted to the range of a Java byte?
The reason for this is that an I2C device wants a numerator and denominator, while it would make sense to give it a float.
For example, 3.1415926535... would result in 245/78, rather than 314/100 or 22/7.
In terms of efficiency, this would be called around three times at the start of the program, but after that not at all. So a slow algorithm isn't too bad.
I've written some code (in Java, even) to do just the thing you're asking for. In my case, I needed to display a scaling factor as both a percentage and a ratio. The most familiar example of this is the zoom dialog you see in image editors, such as the GIMP.
You can find my code here, in the updateRatio() method starting at line 1161. You can simply use it, so long as the LGPL license works for you. What I did essentially follows what's done in the GIMP---this is one of those things where there's pretty much only one efficient, sensible way to do it.