Rounding Algorithms

dtmos writes "Clive Maxfield has an interesting article up on PL DesignLine cataloging most (all?) of the known rounding algorithms used in computer math. As he states, "...the mind soon boggles at the variety and intricacies of the rounding algorithms that may be used for different applications ... round-up, round-down, round-toward-nearest, arithmetic rounding, round-half-up, round-half-down, round-half-even, round-half-odd, round-toward-zero, round-away-from-zero, round-ceiling, round-floor, truncation (chopping), round-alternate, and round-random (stochastic rounding), to name but a few." It's a good read, especially if you *think* you know what your programs are doing."

Comments

Re:Most important...

[slothman32]

There's some straight line algorithm that uses a similar method.
It keeps adding the slope value for every x increment and when it overloads it also makes the y position go up one.
Or something like that. Bresenham's I believe.

To get on topic I would use the usual "(x).5 to (x+1).499~9 goes to (x+1)" way.
For negative, just ignore the sign when doing it, e.g. -1.6 -> -2

IEEE Standard

[Anonymous Coward]

And the IEEE standard for rounding is Banker's Rounding, or Even Rounding, plus whatever other names it goes by. When rounding to the nearest whole number, when the value is exactly halfway between, i.e. 2.5, the rounding algorithm chooses the nearest even number. This allows the distribution of rounding to happen in a more even distributed manner. Always rounding up, which is what US kids are taught in school, will eventually create a bias and throw the aggregates off.

2.5 = 2
3.5 = 4

Only with money in fractions

[MarkusQ]

"Bankers" rounding is only appropriate in a rather restricted range of problems; specifically, where you are more worried about "fairness" than about accuracy, and have a data set that is already biased towards containing exact halves (generally because you've already rounded it previously).

For pretty much all other cases it is broken, wrong, bad, very bad, and misguided. It is a kludge cut from the same cloth as using red and black ink, parenthesis, or location on the page (and all the permutations thereof) to indicate the sign of a number. Do not try to do any sort of scientific calculations, or engineering, or anything else that matters and round in this way.

Why? Because contrary to what some people think, there is no systematic bias in always rounding up. There are exactly as many values that will be rounded down as will be rounded up if you always round exact halves up. I think the trap that people fall into is forgetting that x.000... rounds down (they think of it as somehow "not rounding").

--MarkusQ

Re:Most important...

[Mr Z]

That's the basis behind delta-sigma modulation and Floyd-Steinberg dithering. You carry forward the cumulative error from previous quantization, adding it to the current term. Then you quantize as desired. Over multiple samples, the error gets spread out, such that the local average is very close to the original signal.