You're right, any base-2 representation right of the decimal should be precise to represent in base-10. But it's the kind of thing where if you find yourself counting on it, you probably made a grave error earlier in your design :-).
-scott On Fri, Oct 23, 2015 at 10:40 AM, Alessandro Marzocchi < alessandro.marzocchi at gmail.com> wrote: > Sorry, i replied to wrong Scott Hess... mine was meant to be a reply to his > message... > " 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". > Sorry for my error. > Il 23/ott/2015 19:18, "Scott Robison" <scott at casaderobison.com> ha > scritto: > > On Fri, Oct 23, 2015 at 10:45 AM, Alessandro Marzocchi < > alessandro.marzocchi at gmail.com> wrote: > > > Scott.... actually all base2 fractions are rapresentable as base10 > > fractions... 10 is divisable by 2. > > As for Richard.... try to think about this... computer does 2 base > > calculations as we usually do calculation in base 10. But the same would > > happend to us when talking about different bases.. > > Let's say we want to add 3 times 20 minutes (expressed in decimal of > hours > > up to 6th digit). 0h20' is 0.333333 hours . If you multiply that for 3 > you > > get 0.999999.. not 1 as you expect. The > > > > Yes, they are. What did I write that leads you to believe I don't > understand that? Or are you addressing a different Scott (as there are > three in this conversation)? I'm not upset or angry or demanding > satisfaction, just trying to understand where my commentary went wrong and > if I need to correct anything for the record. > > When I say floating point calculations are "inexact" what I mean is > "potentially inexact" because conversion from decimal in source code to > binary floating point in memory is potentially inexact, the arithmetic > performed will potentially involve rounding of some form, and the final > conversion back from binary floating point to decimal representation for > humans can only work with what is left over after those previous potential > approximations. > > > > Il 23/ott/2015 18:31, "Scott Robison" <scott at casaderobison.com> ha > > scritto: > > > > > On Fri, Oct 23, 2015 at 9:34 AM, Rousselot, Richard A < > > > Richard.A.Rousselot at centurylink.com> wrote: > > > > > > > Scott, > > > > > > > > I agree with everything you said but... To me if a program/CPU > > evaluates > > > > something internally, then when it reports the result it should be > the > > > > result as it sees it. It shouldn't report something different. > > > > > > > > > > This is true to an extent, and there are ways to display something > "more > > > exact". But the library programmers wrote code to format floating point > > > numbers in a way that is appropriate for display to humans. Knowing > that > > > floating point calculations are inexact, they round values after a > > certain > > > number of decimal places, as most applications expect to see something > > like > > > "25" not "24.9999999999999995" (numbers made up). > > > > > > > > > > > > > > So using your analogy, I ask a English speaking person a two > > interrelated > > > > questions, they translate the questions to Japanese in their head, > then > > > > answers one question in Japanese and another in English. I say pick > a > > > > language and stick with it. Either answer my question all in English > > or > > > > all in Japanese don't mix it. > > > > > > > > I think we are getting to hung up on the details of what is going on > > > > internally. The real question is why don't the two results, which > are > > > > coming from the same program, agree? (i.e. return 22.99999999999999 > > not > > > > 23.0) > > > > > > > > Richard > > > > > > > > -----Original Message----- > > > > From: sqlite-users-bounces at mailinglists.sqlite.org [mailto: > > > > sqlite-users-bounces at mailinglists.sqlite.org] On Behalf Of Scott > Hess > > > > Sent: Friday, October 23, 2015 10:05 AM > > > > To: General Discussion of SQLite Database > > > > Subject: Re: [sqlite] Simple Math Question > > > > > > > > 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 > > > > _______________________________________________ > > > > sqlite-users mailing list > > > > sqlite-users at mailinglists.sqlite.org > > > > http://mailinglists.sqlite.org/cgi-bin/mailman/listinfo/sqlite-users > > > > This communication is the property of CenturyLink and may contain > > > > confidential or privileged information. Unauthorized use of this > > > > communication is strictly prohibited and may be unlawful. If you have > > > > received this communication in error, please immediately notify the > > > sender > > > > by reply e-mail and destroy all copies of the communication and any > > > > attachments. > > > > _______________________________________________ > > > > sqlite-users mailing list > > > > sqlite-users at mailinglists.sqlite.org > > > > http://mailinglists.sqlite.org/cgi-bin/mailman/listinfo/sqlite-users > > > > > > > > > > > > > > > > -- > > > Scott Robison > > > _______________________________________________ > > > sqlite-users mailing list > > > sqlite-users at mailinglists.sqlite.org > > > http://mailinglists.sqlite.org/cgi-bin/mailman/listinfo/sqlite-users > > > > > _______________________________________________ > > sqlite-users mailing list > > sqlite-users at mailinglists.sqlite.org > > http://mailinglists.sqlite.org/cgi-bin/mailman/listinfo/sqlite-users > > > > > > -- > Scott Robison > _______________________________________________ > sqlite-users mailing list > sqlite-users at mailinglists.sqlite.org > http://mailinglists.sqlite.org/cgi-bin/mailman/listinfo/sqlite-users > _______________________________________________ > sqlite-users mailing list > sqlite-users at mailinglists.sqlite.org > http://mailinglists.sqlite.org/cgi-bin/mailman/listinfo/sqlite-users >
