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Comment by the-Black-Dread | Hacker News Reader

Then what's the best way to handle these cases? Are there any set of rules we should use while implementing the mathematical equations dealing with limits in floating points.

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Floating point numbers have the highest precision when they're near 0. So if you know a number is going to be near 1, like exp(x) when x is small, then it's better to calculate the offset from 1 rather than the number itself. Programming languages make 'expm1' available for this purpose, which computes exp(x)-1.

Basically

- don't subtract almost equal numbers

- don't add numbers of vastly different magnitudes

Useful list of tricks: http://www.plunk.org/~hatch/rightway.html

I love that that page is nearly the same as it when I saw it ~20 years ago. Things things are still the right way.

Other commenters have pointed to tricks that are undoubtedly useful, but the real answer imo is to understand where rounding error will be introduced in your problem, and where it can become catastrophic so that you can use careful alternatives in those cases.

I'll second this. Implementing tricks can improve some use cases but make others worse.

For a summation, add the smaller numbers first. Smaller as in 0.0000000000053, not like -5172365126.

For summation just go with Kahan summation or some improved variant of it https://en.wikipedia.org/wiki/Kahan_summation_algorithm

And if you can't remember that, you can use a pairwise summation like (((a+b)+(c+d))+((e+f)+(g+h)))+... which gives a decent error reduction in return.

how about small as in ²³⁷

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