From: "Alexandre François Garreau" <galex-713@galex-713.eu>
To: help-gnu-emacs@gnu.org
Cc: Stefan Monnier <monnier@iro.umontreal.ca>
Subject: Re: (- 2.8 1.6) ; 1.1999999999999997
Date: Tue, 11 Feb 2020 02:50:42 +0100 [thread overview]
Message-ID: <4431552.mgxCkZHi0q@galex-713.eu> (raw)
In-Reply-To: <jwvftfhrk7b.fsf-monnier+emacs@gnu.org>
Le mardi 11 février 2020, 02:31:56 CET Stefan Monnier a écrit :
> > (- 2.8 1.6) ; 1.1999999999999997
>
> Emacs Lisp, like most common programming language doesn't support
> rational numbers (and even less real numbers, which can't be faithfully
> represented in a computer) but only floating point numbers. And like
> most programming languages it uses the standard IEEE representation
> supported natively by most CPUs. This representation can't represent
> 2.8 nor 1.6 (and nor 1.2) exactly.
>
> It so happens that "the representable number closest to 2.8" minus "the
> representable number closest to 1.6" is not equal to "the representable
> number closest to 1.2".
>
> > With the TI-84 Plus,
> >
> > 2.8-1.6=1.2
>
> IIUC those calculators used a decimal floating point representation,
> which suffers from the same kinds of problems of course, but works
> better in this specific case.
To add precision: IEEE (hence most computers) store floating point number
in base 2, so it’s like 2.8 = (2.5 = #b10.1) + (0.25 = #b0.01) - (0.05 =
…)… etc. while 1.6 = (1.5 = #b1.1) + (0.1 = ~#b0.0001…) etc. and 1.2 =
(1.25 = #b1.05) - 0.05, etc.
Base 2 is compatible with any base which is a power of 2 (2, 4, 16, etc.)
without rounding errors. We commonly day-to-day use base 10, which is
2×5. So some numbers convert smoothly without rounding error (“a half”
translates in 0.5 in base 10, and 0.1 in base 2), other won’t ever: “a
fivth” is tailor-made to be easily expressed in base 10: 0.2, but it
*can’t* be expressed in base two (without rational numbers (eg.
fractions)).
A good analogy to understand: in base 10, we can’t represent “a third” in
decimal notation (it gives some rounding like “0.33333333…” etc.), neither
it can in base 2 (because 3 is not a divisor of 2 nor of 10), but it *can*
in base 12! so in base twelve 1/3 would simply be 1/4 (exactely like 4
hours (or pennies or forgot how it worked) is already a third of a day).
So each base can turns out to be pretty arbitrary and incompatible with
some others. Some are more useful (base 12 have a lots of divisors),
others more common (base 10 since arabic numbers became widely used)… and
in case of computers, base 2 is ubiquitous. You can’t make electronics
efficiently with a base higher than 2 (afair CCCP made a base 3 computer
once but it was more complex and more expensive). So everything, even
TI84+ is base 2. The fact they store and treat base10 floating numbers
helps for education, because you won’t see rounding errors and won’t have
to learn about it. But with real efficient computers, that would be a waste
of both space and time. Space because to store a base 10 digit, in base
2, you anyway need 4 bits, that could store 16 different values (so you
waste 6 of them who won’t ever be used), slower because algorithm that
would naturally apply to base 2 won’t apply and you will have to do other
more complicated stuff (taking into account there are 6 possibilities every
4 bits that you are wasting and not taking into account).
next prev parent reply other threads:[~2020-02-11 1:50 UTC|newest]
Thread overview: 8+ messages / expand[flat|nested] mbox.gz Atom feed top
2020-02-10 21:18 (- 2.8 1.6) ; 1.1999999999999997 Emanuel Berg via Users list for the GNU Emacs text editor
2020-02-10 21:31 ` tomas
2020-02-10 21:33 ` Joost Kremers
2020-02-10 23:10 ` Emanuel Berg via Users list for the GNU Emacs text editor
2020-02-11 1:31 ` Stefan Monnier
2020-02-11 1:50 ` Alexandre François Garreau [this message]
2020-02-11 1:55 ` Emanuel Berg via Users list for the GNU Emacs text editor
2020-04-07 14:00 ` Bruno Félix Rezende Ribeiro
Reply instructions:
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
https://en.wikipedia.org/wiki/Posting_style#Interleaved_style
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
--in-reply-to=4431552.mgxCkZHi0q@galex-713.eu \
--to=galex-713@galex-713.eu \
--cc=help-gnu-emacs@gnu.org \
--cc=monnier@iro.umontreal.ca \
/path/to/YOUR_REPLY
https://kernel.org/pub/software/scm/git/docs/git-send-email.html
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
Code repositories for project(s) associated with this external index
https://git.savannah.gnu.org/cgit/emacs.git
https://git.savannah.gnu.org/cgit/emacs/org-mode.git
This is an external index of several public inboxes,
see mirroring instructions on how to clone and mirror
all data and code used by this external index.