Author here. Outward rounding to combat precision issues is what interval arithmetic is most known for (try 0.1+0.2 with "full precision mode" enabled), but that's really a shame in my opinion. Outward rounding is cool, but the "inclusion property", as it's known in research papers, works at every scale! This is what enables things like:
50 * (10 + [-1, 1])
[450, 550]
which is lovely, I think. Adding the union layer to it enables even cooler things, like the true inverse of the square function. Did you know it's not sqrt? Try 'sqinv(64)'.
I made interval calculator actually mostly as a way to test my implementation of interval union arithmetic [0], which I needed for another project: a backwards updating spreadsheet [1][2].
Nice! I am interested in how the arithmetic you implemented differs from the IEEE 1788 Standard for Interval Arithmetic (and how the two linked papers relate to it). To address the challenges you mention, did you have to start from scratch or was it something that can build on top of the IEEE standard?
fouronnes3 15 hours ago [-]
Interesting! I'm not familiar with IEEE 1788. The TypeScript library (not-so-float) that I wrote which powers the calculator uses the JS Number type which is double precision IEEE 754. Outward rounding is not supported by JS so I used a bit level manipulation hack by casting to TypedArray [0] to implement the equivalent of C's nextafter() function. Otherwise I mostly followed Hickey & van Emden paper which is really delightful [1]. The real hard work is actually generating all the test cases. Good luck getting 100% test coverage on interval division!
Very cool, I'll definitely be playing around with this some more! Two questions:
- How difficult would it be to add many-valued functions to this? It would be really nice to be able to get the full set of [pi/2, pi/2] + n[2pi, 2pi] from asin(1) without needing to break out Mathematica.
- And:
> Numbers input by the user are interpreted as the smallest interval that contains the IEEE 754 value closest to the input decimal representation but where neither bounds are equal to it
Am I missing something obvious, or should this be the other way round, i.e. the output bounds are the closest two IEEE 754 numbers that contain the input number?
The way it's written I'd interpret the smallest interval to be IEEE754(input)+[-epsilon, epsilon] for infinitesimally small epsilon.
iamwil 19 hours ago [-]
This is great. You might be interested in Matt Keeter's work on Implicit surfaces, and using interval math for its optimization:
This is based Raku’s built-in Junction and Range classes and was an interesting experiment.
_Microft 19 hours ago [-]
Very nice, thanks for sharing!
Maybe show which upper or lower values are included in the intervals?
A notation I am familiar with uses outward facing brackets if the value is not included in the interval. That always applies to infinity.
Applied to the cases here:
]-∞, -1] U [0.5, +∞[
The excluded interval in between becomes ]-1, 0.5[ then.
That’s how min (and analogously max) works, right?
min(A, B) = [lo(A,B), lo (hi(A), hi(B))].
Edit: idea: copy a formula from the results section to the input field if the user clicks/taps on it.
adito 18 hours ago [-]
From reading the linked paper[0], It explains closed interval only. "An interval union is a set of closed and disjoint intervals where the bounds of the extreme interval can be ±∞".
It's possible to support that but it makes the code very very much more complicated. I've decided early on to not support it. Would be a cool addition though!
globular-toast 18 hours ago [-]
I was also a bit confused by this. I thought the standard notation was round brackets, but maybe doesn't work well in ASCII?
qbit42 17 hours ago [-]
Round brackets are standard in the US but that notation is used in France and some other places.
meindnoch 16 hours ago [-]
(0, 1)
Is this an twice-open interval or a 2D vector?
See, this is why Bourbaki introduced the ]0,1[ notation.
streetfighter64 13 hours ago [-]
Is there any reasonable situation where you'd be confusing a vector with an interval? Having done mathematical writing and grading of tests using both styles of notation (simply adapting to what was used at the institution), I can't say that I ever noticed any practical difference between them.
akst 16 hours ago [-]
Very cool! I don't entirely understand some of the operations, but for what I do understand its pretty neat.
I wish in classes we were introduced to a notion of arithmetic on intervals as it comes up. Like in basic statistics with confidence intervals there's ±, as well as in the quadratic equation. It found some what dissatisfying we couldn't chain the resulting a series of operations and instead repeat the operations for the 2 seperate values of the ±. I get a teacher would rather not get hung up on this because they want to bring it back to the application generally, like solving a more complicated equation or hypothesis testing in basic stats. I just wish they hinted at the idea we can do arithmetic on these kinds of things more generally.
I realise what you've got here is well beyond this, but seeing this was some level of validation that treating the interval as a piece of data with its own behaviour of certain operations does make some sense.
malcolmjuxt 13 hours ago [-]
I wish I had known about interval arithmetic when I first wrote tick, a time interval library in Clojure, which includes a. implementation of
Allen's Interval Algebra. It also embraces the notion of sets of discrete intervals which are useful for practical work calculations, like determining the set of intervals of your vacations that are in a particular year (for HR calculations). I accidentally stumbled on benefits of these sets without knowing much beyond Allen's work.
I can see valid uses of this but I also feel like a probabilistic calculator would be more useful.
e.g. the result for the 1 / [-1, 2] example doesn’t tell you how likely each value is and it clearly won’t be uniformly distributed (assuming the inputs are).
mike-the-mikado 14 hours ago [-]
I recently implemented something somewhat similar, but from the perspective of set membership.
I therefore needed to include a complement operation, so that I could do full Boolean analysis of interval membership.
Your intervals are all closed sets, consequently the complements are open intervals. I chose not to distinguish between open and closed intervals, since for my practical purposes whether the end points are members of the set is unimportant.
Of course, with inexact arithmetic, the question of whether the set is open of closed probably not well-defined.
SkiFire13 13 hours ago [-]
Expanding the logic to union of intervals looks cool, but what is the complexity of that? Since you introduce the the possibility of an operation on an interval producing two intervals I suspect executing N operations might have an exponential complexity, which unfortunately makes this unfeasible to use for some common intervals applciations like abstract interpretation, unless you start introducing approximations once you have enough intervals.
lou1306 12 hours ago [-]
Yes, this is well-known (eg. in abstract interpretation). As you said, usually you can set a "cap" to the size of these objects, and start merging intervals when you hit the cap. But at least in abstract interpretation it seems that they simply consider more sophisticated domains than intervals.
anematode 18 hours ago [-]
Excellent!! I love interval arithmetic and also wrote a TS implementation for a graphing calculator project. Agree that it's very underrated, and I wish that directed rounding was exposed in more languages.
fouronnes3 17 hours ago [-]
Yeah it's super interesting. Like you said, I learned that the IEEE 754 spec actually requires that complete implementations of floating point numbers expose a way to programmatically choose the rounding mode. As far as I know only C allows you to do that, and even then it depends on hardware support. For JS I had to use ugly typedarray casts. Which kinda only accidentally work due to endianess. But technically there should be an API for it!
There's other unused stuff in IEEE 754 like that: the inexact bit or signaling NaNs!
ted_dunning 7 minutes ago [-]
Julia supports full IEEE 754 rounding mode support.
JSR_FDED 17 hours ago [-]
I just read up on interval arithmetic. I understand its desirable properties. Where in practice have you applied it? What’s a real world application for interval arithmetic?
ngruhn 17 hours ago [-]
It can be used in static analysis or type checking. E.g.
if (x >= 0) {
x += 10
if (x =< 9) {
// unreachable
}
}
By maintaining an interval of possible values of x, you can detect the unreachable branch, because the interval becomes empty:
initial: [-oo, oo]
x >= 0 : [0, oo]
x += 10: [10, oo]
x =< 9 : [10, 9] (empty)
Oberdiah 15 hours ago [-]
I’m working on a static analyser at the moment that does this, and the inferences that can be made just from the information of intervals is quite impressive.
One thing you run into pretty quickly though in a lot of languages is integer overflow ruining your day - in your example above the commented section is reachable for signed ints that support overflow and that adds a whole other layer of complexity to things.
thekoma 15 hours ago [-]
We recently implemented this idea in an LLVM optimisation pass based on value-range information from sensor datasheets [1].
In physics, whenever you make a measurement it has a precision. Usually you represent this as a normal distribution, but for calculations it can be easier to represent this as an interval.
The police measure the distance my car travelled [ 99.9, 100.1 ] m and the time it took [ 3.3, 3.4 ] s - how fast was my car going? [29.38, 30.33] m/s according to the interval calculator.
Physics students learn exactly this method before they move on to more sophisticated analysis with error distributions.
ttoinou 15 hours ago [-]
And one might want to approximate error distribution calculus with a method that will look like the one here
nicolodev 17 hours ago [-]
It’s astonishing how nobody hasn’t mentioned abstract interpretation yet. Under classical static analysis, if you can “prove” that a variable does not have values in some unsound zones, you can e.g. “prove” soundness or apply further optimizations.
The interval abstract domain works under interval analysis with an algebra that’s the same of this calculator. It’s funny to implement something like that on source/binary level :)
wvlia5 10 hours ago [-]
Hey, what about this idea?
redefine interval representation, such that [a, b] means the same if a<b; but if b<a, it means [-∞, b]U[a, +∞].
Then your example would become 1/[-1,2]=[0.5,-1]
The last point in your intro description can't be stressed enough: this allows for safe handling of rounding errors in floating point operations.
Though you are inherently losing precision: there are values in the output interval which don't have a corresponding input that causes this output.
Eric_Xua 10 hours ago [-]
Really cool project—turning interval unions into a usable, polished calculator makes the math feel practical.
petters 17 hours ago [-]
You could add a feature where it will compute the global optimum of any function of a small number of variables. Branch and bound with interval arithmetic works well for a small number of variables.
Disjoint unions of intervals seems like a nice thing to have
oncallthrow 4 hours ago [-]
have you considered implementing a +- operator?
For example a +- b would be [a - b, a + b]
adaptit 14 hours ago [-]
This interval calculator is surprisingly robust. The way it handles boundary conditions and asymmetric intervals is clean and efficient.
mbo 12 hours ago [-]
Could this support a native datetime type? I shipped a much worse of this for managing repeated events and schedules.
zamadatix 10 hours ago [-]
Naively, what breaks by doing datetime -> epoch -> interval -> datetime over some other form of implementation?
I tend to avoid datetimes as much as working with printers because the answer is always "more annoying than you'd first think" :D.
ttoinou 15 hours ago [-]
Why not use disks / exterior disks in the complex numbers plane instead of intervals ? It might make the mental model easier to reason about
dnnddidiej 16 hours ago [-]
Interals can be used to model errors and uncertainty and this lets you see how they conpound in calculations like speed = distance over time.
Falimonda 7 hours ago [-]
Random example generator would be nice
dfgtu 14 hours ago [-]
Very nice work. I was wondering if it might be useful to combine this with a library for arbitrary precision arithmetic. How difficult do you think that might be?
fouronnes3 14 hours ago [-]
Thanks! Arbitrary precision arithmetic is definitely something I'd like to learn more about, yeah. Haven't had time to study it so much yet unfortunately.
jakeinspace 9 hours ago [-]
This is great
boobsbr 17 hours ago [-]
Neat.
lou1306 12 hours ago [-]
Sorry to be a party pooper, the Web app is neat, but I have some reservations about the paper.
Namely, the "powerset of intervals" domain has been known since the '70s [1], and powerset domains have been generalised to arbitrary base domains decades ago [2]. A paper from the mid-2010s on these topics that lacks any engagement with the abstract interpretation literature is a bit disappointing.
As for the interpretation of division suggested here, it makes, say, 1 / S non-distinguishable from 1 / ([0, 0] U S) for any set of intervals S, which sounds suspicious.
[1] Patrick Cousot and Radhia Cousot. 1979. Systematic Design of Program Analysis Frameworks. In 6th ACM Symposium on Principles of Programming Languages (POPL), January 1979. ACM Press, San Antonio, TX, USA, 269–282. https://doi.org/10.1145/567752.567778
[2] Gilberto Filé and Francesco Ranzato. 1999. The Powerset Operator on Abstract Interpretations. Theor. Comput. Sci. 222, 1–2 (1999), 77–111. https://doi.org/10.1016/S0304-3975(98)00007-3
i_love_retros 13 hours ago [-]
Thank you for creating and sharing something that feels authentic and human made. No sign of AI vibe coded slop at all. It's beautiful.
mehmetkerem 11 hours ago [-]
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wallhz 15 hours ago [-]
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Sergey777 12 hours ago [-]
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LXforever 16 hours ago [-]
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fouronnes3 16 hours ago [-]
I don't handle it, ahah. You are right that if you take any classical numerical computing algorithm and replace the floating point reals by interval unions, most of the time the number of intervals in the unions in each of your variables will grow very fast. This is one of the problems of unions and as far as I'm aware it's a topic of active academic research.
Rendered at 00:24:41 GMT+0000 (Coordinated Universal Time) with Vercel.
I made interval calculator actually mostly as a way to test my implementation of interval union arithmetic [0], which I needed for another project: a backwards updating spreadsheet [1][2].
[0] https://github.com/victorpoughon/not-so-float
[1] https://victorpoughon.github.io/bidicalc/
[2] https://news.ycombinator.com/item?id=46234734
[0] https://github.com/victorpoughon/not-so-float/blob/main/src/...
[1] https://fab.cba.mit.edu/classes/S62.12/docs/Hickey_interval....
- How difficult would it be to add many-valued functions to this? It would be really nice to be able to get the full set of [pi/2, pi/2] + n[2pi, 2pi] from asin(1) without needing to break out Mathematica.
- And:
> Numbers input by the user are interpreted as the smallest interval that contains the IEEE 754 value closest to the input decimal representation but where neither bounds are equal to it
Am I missing something obvious, or should this be the other way round, i.e. the output bounds are the closest two IEEE 754 numbers that contain the input number?
The way it's written I'd interpret the smallest interval to be IEEE754(input)+[-epsilon, epsilon] for infinitesimally small epsilon.
https://youtu.be/UxGxsGnbyJ4?si=Oo6Lmc4ACaSr5Dk6&t=1006
https://memalign.github.io/m/formulagraph/index.html
Some detail on how this works, including links to the relevant interval math code:
https://memalign.github.io/p/formulagraph.html
http://www.peda.com/grafeq/
This is based Raku’s built-in Junction and Range classes and was an interesting experiment.
Applied to the cases here:
]-∞, -1] U [0.5, +∞[
The excluded interval in between becomes ]-1, 0.5[ then.
That’s how min (and analogously max) works, right? min(A, B) = [lo(A,B), lo (hi(A), hi(B))].
Edit: idea: copy a formula from the results section to the input field if the user clicks/taps on it.
[0]: https://www.ime.usp.br/~montanhe/unions.pdf
See, this is why Bourbaki introduced the ]0,1[ notation.
I wish in classes we were introduced to a notion of arithmetic on intervals as it comes up. Like in basic statistics with confidence intervals there's ±, as well as in the quadratic equation. It found some what dissatisfying we couldn't chain the resulting a series of operations and instead repeat the operations for the 2 seperate values of the ±. I get a teacher would rather not get hung up on this because they want to bring it back to the application generally, like solving a more complicated equation or hypothesis testing in basic stats. I just wish they hinted at the idea we can do arithmetic on these kinds of things more generally.
I realise what you've got here is well beyond this, but seeing this was some level of validation that treating the interval as a piece of data with its own behaviour of certain operations does make some sense.
https://github.com/juxt/tick
https://en.wikipedia.org/wiki/Allen's_interval_algebra
e.g. the result for the 1 / [-1, 2] example doesn’t tell you how likely each value is and it clearly won’t be uniformly distributed (assuming the inputs are).
I therefore needed to include a complement operation, so that I could do full Boolean analysis of interval membership.
Your intervals are all closed sets, consequently the complements are open intervals. I chose not to distinguish between open and closed intervals, since for my practical purposes whether the end points are members of the set is unimportant.
Of course, with inexact arithmetic, the question of whether the set is open of closed probably not well-defined.
There's other unused stuff in IEEE 754 like that: the inexact bit or signaling NaNs!
[1]: https://dl.acm.org/doi/pdf/10.1145/3640537.3641576
The police measure the distance my car travelled [ 99.9, 100.1 ] m and the time it took [ 3.3, 3.4 ] s - how fast was my car going? [29.38, 30.33] m/s according to the interval calculator.
Physics students learn exactly this method before they move on to more sophisticated analysis with error distributions.
The interval abstract domain works under interval analysis with an algebra that’s the same of this calculator. It’s funny to implement something like that on source/binary level :)
Though you are inherently losing precision: there are values in the output interval which don't have a corresponding input that causes this output.
Disjoint unions of intervals seems like a nice thing to have
For example a +- b would be [a - b, a + b]
I tend to avoid datetimes as much as working with printers because the answer is always "more annoying than you'd first think" :D.
Namely, the "powerset of intervals" domain has been known since the '70s [1], and powerset domains have been generalised to arbitrary base domains decades ago [2]. A paper from the mid-2010s on these topics that lacks any engagement with the abstract interpretation literature is a bit disappointing.
As for the interpretation of division suggested here, it makes, say, 1 / S non-distinguishable from 1 / ([0, 0] U S) for any set of intervals S, which sounds suspicious.
[1] Patrick Cousot and Radhia Cousot. 1979. Systematic Design of Program Analysis Frameworks. In 6th ACM Symposium on Principles of Programming Languages (POPL), January 1979. ACM Press, San Antonio, TX, USA, 269–282. https://doi.org/10.1145/567752.567778
[2] Gilberto Filé and Francesco Ranzato. 1999. The Powerset Operator on Abstract Interpretations. Theor. Comput. Sci. 222, 1–2 (1999), 77–111. https://doi.org/10.1016/S0304-3975(98)00007-3