I've been studying interval arithmetic for the past few weeks and it's a really interesting field because while there is a ton of super interesting research published over the past decades, it has never really gotten the recognition that it deserves, IMO.

One reason for this is that standard interval arithmetic has really poor handling of division by intervals containing zero. If you compute 1 / [-1, 2] in regular interval arithmetic, you get either [-∞, +∞], or you have to say that the operation is undefined. Both solutions are virtually useless. The real answer of course is [-∞, -1] U [0.5, +∞]: i.e. a union of two disjoint intervals.

This is useful because you can confidently exclude a non empty set of the real numbers ([-1, 0.5]) from the set of possible values that you can get by dividing 1 by a number between -1 and 2.

But this definition of interval division yields a value that is not an interval. This is a problem if you want to define a closed arithmetic system, where you can build and evaluate arbitrary expression over interval values.

(This behavior extends to any non continuous function like tan() for example, which is implemented in my project - not without difficulties!)

Well the obvious solution is to define your arithmetic over disjoint unions of intervals. This is the subject of a 2017 paper called "Interval Unions" by by Schichl, H., Domes, F., Montanher, T. and Kofler, K..

This open-source project I made implements interval union arithmetic in TypeScript in the form of a simple interactive calculator, so you can try it out for yourself! The underlying TypeScript library is dependency free and implements interval union arithmetic over IEEE 754 double precision floats (JS native number type) with outward rounding. This guarantees accuracy of interval results in the presence of rounding issue inherent to floating point.

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.

https://github.com/juxt/tick

https://en.wikipedia.org/wiki/Allen's_interval_algebra

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]

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

This is great. You might be interested in Matt Keeter's work on Implicit surfaces, and using interval math for its optimization:

https://youtu.be/UxGxsGnbyJ4?si=Oo6Lmc4ACaSr5Dk6&t=1006

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.

You might be interested in this graphing calculator I made using interval arithmetic:

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

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.

This interval calculator is surprisingly robust. The way it handles boundary conditions and asymmetric intervals is clean and efficient.

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.

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.

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?

Why not use disks / exterior disks in the complex numbers plane instead of intervals ? It might make the mental model easier to reason about

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?

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.

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

Interals can be used to model errors and uncertainty and this lets you see how they conpound in calculations like speed = distance over time.

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.

Neat.

Very cool. This feels like one of those ideas that makes interval arithmetic go from “interesting but frustrating” to actually useful. I’d be curious how you handle the growth in the number of disjoint intervals over repeated operations, since that seems like the practical bottleneck.