On Fri, Oct 23, 2015 at 7:39 AM, Dominique Devienne <ddevienne at gmail.com> wrote:
> On Fri, Oct 23, 2015 at 4:16 PM, Rousselot, Richard A < > Richard.A.Rousselot at centurylink.com> wrote: > > So I decided to output 1000 digits, because why not? So now I am more > > perplexed with all these digits showing it is working the opposite of > how I > > expected it. Why is the second set of equations evaluating to a "yes" > when > > it is the only one that is obviously NOT equal to the expression??? > > Indeed, that's puzzling :) Just to be clear, though, how floating-point numbers work is breaking your expectations because your expectations are wrong when applied to floating-point numbers. Internally, they are base-2 scientific notation, so asking for more significant digits in the base-10 representation won't help - base-10 fractional numbers cannot always be represented precisely in base-2, ALSO base-2 fractional numbers cannot always be represented precisely in base-10, so it's like a game of telephone where you can end up slightly removed from where you started out, even though it seems like it's a simple round trip. Since each individual digit cannot be represented perfectly, it doesn't matter how many digits of precision you ask for, you'll always be able to find cases where it doesn't line up like you expect. Think of it this way: Find an English sentence, and find an English to Japanese translator. Translate each individual word of the sentence from English to Japanese, then concatenate the results together. Then translate the entire original sentence to Japanese. The results will almost never be the same. Then do the same process translating the Japanese back to English. Again, the two routes will provide different results, _and_ both of those results will almost certainly not match the original English sentence. This isn't a reflection of the translator's abilities at all. I'm not saying the computer is always right, just that the computer is following a very strict recipe with reproducible results. I don't mean reproducible like your three examples make logical sense to you, the user, I mean reproducible like my Intel box gives the same results as my AMD box as my ARM box. If you want to be able to deal with fractional decimal values with high fidelity, you either need to arrange for base-10 representation (slow, because computers have to simulate it), or you have to do your math in shifted fashion (fast, but can be error prone). -scott
