Thanks for pointing that out. It should be fixed now. The shortening was done by the editor I was using (&quot;Buffer&quot;) to draft the tweets in - I wasn&#x27;t intending to track one but it probably does provide some means of seeing how many people clicked the link

The link at the end is both shortened (for tracking purposes?) and unclickable… so that’s unfortunate. Here is the real link to the paper, in a clickable format: <a href="https:&#x2F;&#x2F;dl.acm.org&#x2F;doi&#x2F;pdf&#x2F;10.1145&#x2F;3385412.3386037" rel="nofollow">https:&#x2F;&#x2F;dl.acm.org&#x2F;doi&#x2F;pdf&#x2F;10.1145&#x2F;3385412.3386037</a>

That does appear to equal exactly 5... would you care to show how it doesn&#x27;t?<pre><code> $ cat check_math.c
 #include &lt;stdio.h&gt;

 int main() {
 &#x2F;&#x2F; Define the values as float (32-bit floating point)
 float one_third = 1.0f &#x2F; 3.0f;
 float five = 5.0f;

 &#x2F;&#x2F; Compute the equation
 float result = one_third + five - one_third;

 &#x2F;&#x2F; Check for exact equality
 if (result == five) {
 printf(&quot;The equation evaluates EXACTLY to 5.0 (True)\n&quot;);
 } else {
 &#x2F;&#x2F; Print the actual result and the difference
 printf(&quot;The equation does NOT evaluate exactly to 5.0 (False)\n&quot;);
 printf(&quot;Computed result: %.10f\n&quot;, result);
 printf(&quot;Difference: %.10f\n&quot;, result - five);
 }

 return 0;
 }

 $ gcc -O0 check_math.c -o check_math; .&#x2F;check_math
 The equation evaluates EXACTLY to 5.0 (True)</code></pre>

not that simple, 1&#x2F;3 + 5 - 1&#x2F;3 should be 5. It doesn&#x27;t overflow in IEEE754

if &quot;answer&quot; overflows, switch to symbolic mode.

Uncomputable! That was the word I wanted. I only remembered constructability but that was something else. Thanks.

Wait. Could we in principle find more ways to express some of those uncomputable numbers, or have we conclusively proven we just can&#x27;t reach them - can&#x27;t identify any of them in any way we could express?EDIT: let me guess - there is a proof, and it&#x27;s probably a flavor of the diagonal argument, right?

SMBC manages to do it in a 4 panel comic: <a href="https:&#x2F;&#x2F;www.smbc-comics.com&#x2F;comic&#x2F;real-4" rel="nofollow">https:&#x2F;&#x2F;www.smbc-comics.com&#x2F;comic&#x2F;real-4</a>

&gt; existential angstThe best (and most educational) expression of that angst that I know: <a href="https:&#x2F;&#x2F;mathwithbaddrawings.com&#x2F;2016&#x2F;12&#x2F;28&#x2F;why-the-number-line-freaks-me-out&#x2F;" rel="nofollow">https:&#x2F;&#x2F;mathwithbaddrawings.com&#x2F;2016&#x2F;12&#x2F;28&#x2F;why-the-number-li...</a>.

This behaviour is what you get in say a cheap 1980s digital calculator, but it&#x27;s not what we actually want. We want correct answers and to do that you need to be smarter. Ideally impossibly smart, but if the calculator is smarter than the person operating it that&#x27;s a good start.You&#x27;re correct that the use of the calculator means we&#x27;re talking about computable numbers, so that&#x27;s nice - almost all Reals are non-computable but we ruled those out because we&#x27;re using a calculator. However just because our results are Computable doesn&#x27;t let us off the hook. There&#x27;s a difference between knowing the answer is exactly 40 and knowing only that you&#x27;ve computed a few hundred decimal places after 40 and so far they&#x27;re all zero, maybe the next one won&#x27;t be.

&gt; Almost all numbers cannot be practically expressedThat&#x27;s certainly true, but all numbers that can be entered on a calculator can be expressed (for example, by the button sequence entered in the calculator). The calculator app can&#x27;t help with the numbers that can&#x27;t be practically expressed, it just needs to accurately approximate the ones that can.

It doesn&#x27;t &quot;matter&quot;, but it&#x27;s interesting to probe the boundary between the easily accessible world and the probably inaccessible world.Non-discrete real-number-based Fractals are a beautiful visual version of this.

We know of at least one uncomputable number - Chaitin&#x27;s constant, the probability that any given Turing machine halts.Personally, I do wonder sometimes if real-world physical processes can involve uncomputable numbers. Can an object be placed X units away from some point, where X is an uncomputable number? The implications would be really interesting, no matter whether the answer is yes or no.

Well some classical techniques in standard undergraduate real analysis could lead to numbers outside the set of computable numbers, so if you don&#x27;t allow non-computable numbers you will need to be more careful in the theorems you derive in real analysis. I do not believe that is important however; it&#x27;s much simpler to just work with the set of real numbers rather than the set of computable numbers.

Does it matter that some numbers are inexpressible (i.e., cannot be computed)?I don&#x27;t think it matters on a practical level--it&#x27;s not like the cure for cancer is embedded in an inexpressible number (because the cure to cancer has to be a computable number, otherwise, we couldn&#x27;t actually cure cancer).But does it matter from a theoretical&#x2F;math perspective? Are there some theorems or proofs that we cannot access because of inexpressible numbers?[Forgive my ignorance--I&#x27;m just a dumb programmer.]

I worded that sentence carefully, when I said “almost all” :)

&gt; Almost all numbers cannot be practically expressed and it may even be that the probability of a random number being theoretically indescribable is about 100%. Depending on what a number is.A common rebuke is that the construction of the &#x27;real numbers&#x27; is so overwrought that most of them have no real claim to &#x27;existing&#x27; at all.

&gt; And almost all numbers cannot be expressed in IEEE floating points.It is a bit stronger than that. Almost all numbers cannot be practically expressed and it may even be that the probability of a random number being theoretically indescribable is about 100%. Depending on what a number is.&gt; Some problems can be avoided if you use bignums.Or that. My momentary existential angst has been assuaged. Thanks bignums.

It’s really not just for experts though. Even dealing with fractions is going to require more than a naive implementation.

For an everyday use calculator? Sure. It&#x27;s still fun and challenging to create a calculator that can handle &quot;as much&quot; math&#x2F;arithmetics as possible.

&gt; 1 + 10% is 0.11.I know adding % has multiple conventions, but this one seems odd, I&#x27;d interpret 1 + 10% as &quot;one plus 10 percent of one&quot; which is 1.1, or as 1 + 10 &#x2F; 100 which happens to be also 1.1 hereThe only interpretation that&#x27;d make it 0.11 is if it represents 1% + 10%, but then the question of 10% + 10% is answered: 0.2 or 20%. Or maybe there&#x27;s a typo and it was supposed to say &quot;0.1 + 10%&quot;

I’m confused. Why would 1 + 10% obviously be 0.11?I expected 1.1 (which is what my iOS calculator reported, when I got curious).I do understand the question of parsing. I just struggle to understand why the first one is confidently stated to correctly result in a particular answer. It feels like a perfect example itself of a problem with unclear parsing.

Maybe, though in the paper (not the article):&gt; We no longer receive bug reports about inaccurate results, as we occasionally did for the 2014 floating-point-based calculator(with a footnote: This excludes reports from one or two bugs that have now been fixed for many months. Unfortunately, we continue to receive complaints about
incorrect results, mostly for two reasons. Users often do not understand the
difference between degrees and radians. Second, there is no standard way
to parse calculator expressions. 1 + 10% is 0.11. 10% is 0.1. What’s 10% + 10%?)When you have 3 billion users, I can imagine that getting rid of bugs that only affect 0.001% of your userbase is still worthwhile and probably pays for itself in reduced support costs.

&gt; For the rest of your life there will be things to learn about them and their algebra.That’s just not true for the vast majority of people.

Totally agree. It bothered me when I was younger, though I had no idea how to explain why, but this should be deeply unsettling to everyone who encounters it:<pre><code> 1&#x2F;4 = 0.25 exact
 1&#x2F;3 = 0.333... infinitely repeating approximation</code></pre>

I think a big issue with how we teach math, is the casualness with which we introduce children to floating points.Its like: Hey little Bobby, now that you can count here are the ints and multiplication&#x2F;division. For the rest of your life there will be things to learn about them and their algebra.Tomorrow we&#x27;ll learn how to put a &quot;.25&quot; behind it. Nothing serious. Just adds multiple different types of infinities with profound impact on exactness and computability, which you have yet to learn about. But it lets you write 1&#x2F;4 without a fraction which means its simple!

Real numbers are quite complex (no pun). Understanding the material well is a junior level math major course.If you really understand the existing math curriculum this should be high school level.

Not a calculator engineer but this seems hideously complex?

yours give 1 on (10^100)+1-(10^100), cool

Great resource! For my calculator I also wanted to tackle physics, which expands on the number definition with measurement error size.It is suprisingly hard problem.<a href="https:&#x2F;&#x2F;recomputer.github.io&#x2F;" rel="nofollow">https:&#x2F;&#x2F;recomputer.github.io&#x2F;</a>

Yep, I use that in the laptop. But the smartphone still needs a good one calculator App.

Proper ones are certainly usable for more than napkin math. I deal with fairly simple definite integrals and linear algebra occasionally. It&#x27;s easier for me to plug this into a programmable calculator than it is to scratch in the dirt on Maxima or Mathematica most of the time if I just need an answer.

I find that much of the time I want WolframAlpha for when basic arithmetic, because I like the way it tracks and converts units. It&#x27;s such a simple way to check that my calculation isn&#x27;t completely off base. If I forget to square something or I multiply when I meant to divide I get an obviously wrong answer.Plus of course not having to do even more arithmetic when one site gives me kilograms and another gives me ounces.

If you&#x27;re willing to learn to work with RPN calculators (which I think is a good idea), I can recommend RealCalc for Android. It has an RPN mode that is very economic in keypresses and it&#x27;s clear the developers understand how touchscreens work and how that ties into the kind of work pocket calculators are useful for.My only gripe with it is that it doesn&#x27;t solve compounding return equations, but for that one can use an emulated HP-12c.

I&#x27;m the developer of an Android calculator, called Algeo [1] and I wonder which part of it that makes it feel like slow&#x2F;not snappy? I&#x27;m trying to constantly improve it, though UX is a hard problem.[1] - <a href="https:&#x2F;&#x2F;play.google.com&#x2F;store&#x2F;apps&#x2F;details?id=com.algeo.algeo">https:&#x2F;&#x2F;play.google.com&#x2F;store&#x2F;apps&#x2F;details?id=com.algeo.alge...</a>

This relates to what I wrote in reply to the original tweet thread.Performing arithmetic on arbitrarily complex mathematical functions is an interesting area of research but not useful to 99% of calculator users. People who want that functionality with use Wolfram Alpha&#x2F;Mathematica, Matlab, some software library, or similar.Most people using calculators are probably using them for budgeting, tax returns, DIY projects (&quot;how much paint do I need?&quot;, etc), homework, calorie tracking, etc.If I was building a calculator app -- especially if I had the resources of Google -- I would start with trying to get inside the mind of the average calculator user and figuring out their actual problems. E.g., perhaps most people just use standard &#x27;napkin math&#x27;, but struggle a bit with multi-step calculations.&gt; But for some reason the authors of calculator apps never optimize them for the number of keypresses, unlike Casio&#x2F;TI&#x2F;HP. It&#x27;s a lost art. Even a simple operator repetition is a completely alien concept for new apps.Yes, there&#x27;s probably a lot of low-hanging fruit here.The Android calculator story sounded like many products that came out of Google -- brilliant technical work, but some sort of weird disconnect with the needs of actual users.(It&#x27;s not like the researchers ignored users -- they did discuss UI needs in the paper. But everything was distant and theoretical -- at no point did I see any mention of the actual workflow of calculator users, the problems they solve, or the particular UI snags they struggle with.)

It&#x27;s not even about features. Calculators are mostly useful for napkin math - if I can&#x27;t afford an error, I&#x27;ll take some full-fledged math software&#x2F;package and write a program that will be debuggable, testable, and have version control.But for some reason the authors of calculator apps never optimize them for the number of keypresses, unlike Casio&#x2F;TI&#x2F;HP. It&#x27;s a lost art. Even a simple operator repetition is a completely alien concept for new apps. Even the devs of the apps that are supposed to be snappy, like speedcrunch, seem to completely misunderstand the niche of a calculator, are they not using it themselves? Calculator is neither a CAS nor a REPL.For Android in particular, I&#x27;ve only found two non-emulated calculators worth using for that, HiPER Calc and 10BA by Segitiga.Pro. And I&#x27;m not sure I can trust the correctness.

Yes that. I use free42 on my phone and mac. And an actual real HP 42S.Edit: and Maxima as well on the mac (to back up another user&#x27;s comment)

&gt; Another app nobody has made is a simple random music player.Marvis on iOS is pretty good at this. I use it to shuffle music with some rules (&quot;low skip %, not added recently, not listened to recently&quot;)[0] and it always does a good job.[0] Because &quot;create playlist&quot; is still broken in iOS Shortcuts, incredibly.

I suspect nobody made xxx vlc plug-in because compiling it is far too hard.I tried to make a joystick controller for a particular use case on one platform (Linux) and I gave up.VLC solves a hard problem. Supporting lots of different libs, versions, platforms, hardware and on top of that licensing issues.

The Samsung music player can do that but you need a Samsung phone.

Vanilla Music can play and shuffle an entire folder

Foobar2000 has no problems in these areas

Anything that just shuffles on the filesystem&#x2F;folder level works for this. Even my Honda Civic&#x27;s stereo does it. Then you have iTunes, which uses playlists, and doesn&#x27;t work. It starts repeating songs before it exhausts the playlist.

I&#x27;d love to hear more about this. What was the other one you found? I wrote Tiny Player for iOS and another one for Mac and as more of an &quot;album listener&quot; myself I always struggled to keep the shuffle functionality up to other peoples expectations.

I haven&#x27;t used it in a while (now using streaming...), But Musicolet (<a href="https:&#x2F;&#x2F;krosbits.in&#x2F;musicolet&#x2F;" rel="nofollow">https:&#x2F;&#x2F;krosbits.in&#x2F;musicolet&#x2F;</a>) should be able to do this. Offline-only and lightweight.

Mediamonkey allows me to just go to tracks and hit shuffle and then it randomly adds all my tracks to a queue with no repeats. You can do it at any level of hierarchy, allmusic, playlist, album, artist, genre etc.Edit: I checked I can also shuffle a folder without adding it to the library.

Termux:<pre><code> pkg install mplayer
 cd &#x2F;sdcard&#x2F;Music
 find -type f | shuf | head -1 | xargs mplayer
</code></pre>
(Or whatever command-line player you already have installed. I just tested with espeak that audio in Termux works for me out of the box and saw someone else mentioning mplayer as working for them in Termux: <a href="https:&#x2F;&#x2F;android.stackexchange.com&#x2F;a&#x2F;258228" rel="nofollow">https:&#x2F;&#x2F;android.stackexchange.com&#x2F;a&#x2F;258228</a>)- It generates a list of all files in the current directory, one per line- Shuffles the list- Takes the top entry- Gives it to mplayer as an argument&#x2F;parameterRepeat the last command to play another random song. For infinite play:<pre><code> while true; do !!; done
</code></pre>
(Where !! substitutes the last command, so run this after the find...mplayer line)You can also stick these lines in a shell script, and I seem to remember you can have scripts as icons on your homescreen but I&#x27;m not super deep into Termux; it just seemed like a trivial problem to me, as in, small enough that piping like 3 commands does what you want for any size library with no specialised software needed

I&#x27;m pretty sure the paid version of PowerAmp for Android will do what you want, with or without explicitly creating a playlist.I have many thousands of mp3s on my phone in nested folders. PowerAmp has a &quot;shuffle all&quot; mode that handles them just fine, as well as other shuffle modes. I&#x27;ve never noticed it repeating a track before I do something to interrupt the shuffle.Earlier versions (&gt;~ 5 years ago) seemed to have trouble indexing over a few thousand tracks across the phone as a whole, but AFAIK that&#x27;s been fixed for awhile now.

Another app nobody has made is a simple random music player. Tried VLC on Android and adding 5000+ songs from SD card into a playlist for shuffling simply crashes the app. Why do we need a play list anyway, just play the folder! Is it trying to load the whole list at the same time into memory? VLC always works, but not on this task. Found another player that doesn&#x27;t require building a playlist but when the app is restarted it starts from the same song following the same random seed. Either save the last one or let me set the seed!

Well the 89 is a CAS in disguise most of the time which is mentioned in passing in the article.But, I agree I almost never want the full power of Mathematica&#x2F;sage initially but quickly become annoyed with calc apps. The 89 and hp prime&#x2F;&#x2F;50 have just enough to solve anything where I wouldn’t rather just use a full programming language.

Same here, except I use an HP48 emulator because TI sucks and HP rocks.

GeoGebra, in fact, embeds xcas and has a touchscreen friendly UI.

I just installed it, for about 2 Eur.Thanks for the heads up, I will be testing it for a few months, to see if it can replace the TI-89 emulator as my main calculator.Edit: that calculator gives a result of 0 on this test

HiPER Calc Pro looks like and works like a &quot;physical&quot; calculator, I&#x27;ve use it for years to get effect. I also have Wabbitemu but hardly ever use it, the former works fine for nearly everything.

Here I got 1 as the answer:<a href="https:&#x2F;&#x2F;imgur.com&#x2F;a&#x2F;TH14QZn" rel="nofollow">https:&#x2F;&#x2F;imgur.com&#x2F;a&#x2F;TH14QZn</a>

Just tried the TI-89 emulator on android and it says 1e100 + 1 - 1e100 is 0

I am seriously curious when it became not a violation of the principle of least surprise that a calculator app uses the network to communicate information from my device (which definitionally belongs to me) to the developer.Where I am standing, that never happened, but that would require that a simply staggering number of people be classified as unreasonable.

I&#x27;ve put a lot of effort into mine, TechniCalcI&#x27;ve been working on it for what will be a decade later this year. It tries to take all the features you had on these physical calculators, but present them in a modern way. It works on macOS, iOS, and iPad OSWith regards to the article, I wasn&#x27;t quite as sophisticated as that. I do track rationals, exponents, square roots, and multiples of pi; then fall back to decimal when needed. This part is open source, though!Marketing page - <a href="https:&#x2F;&#x2F;jacobdoescode.com&#x2F;technicalc" rel="nofollow">https:&#x2F;&#x2F;jacobdoescode.com&#x2F;technicalc</a>AppStore Link - <a href="https:&#x2F;&#x2F;apps.apple.com&#x2F;gb&#x2F;app&#x2F;technicalc-calculator&#x2F;id1504965415">https:&#x2F;&#x2F;apps.apple.com&#x2F;gb&#x2F;app&#x2F;technicalc-calculator&#x2F;id150496...</a>Open source components - <a href="https:&#x2F;&#x2F;github.com&#x2F;jacobp100&#x2F;technicalc-core">https:&#x2F;&#x2F;github.com&#x2F;jacobp100&#x2F;technicalc-core</a>

TI-89 doesn&#x27;t have infinite precision.Built-in Android calculator does.They are incomparable. TI-89 has tons of features, but can&#x27;t take a square foot to high accuracy.

<a href="https:&#x2F;&#x2F;f-droid.org&#x2F;en&#x2F;packages&#x2F;com.eanema.graph89&#x2F;" rel="nofollow">https:&#x2F;&#x2F;f-droid.org&#x2F;en&#x2F;packages&#x2F;com.eanema.graph89&#x2F;</a>

Can you tell me which emulator you&#x27;re using? I loved using the open source Wabbitemu on previous Android phones, but it seems to have been removed from the app store, so I can&#x27;t install it on newer devices :-&#x2F;

And the most usable calculator on iOS is easily GrafNCalc83 (a very close TI-83 homage), IMO, even for simple math.

That&#x27;s not surprising, considering the TI-89 was based on a full CAS, Derive. (A 3.3 MB program!)<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Derive_(computer_algebra_system)" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Derive_(computer_algebra_syste...</a>

Almost no one has done that calculator app, properly. When I mean properly, I mean a complete calculator, like the TI-89.I am using, in Android, and emulator for the TI-89 calculator.Because no Android app has half the features, and works as well.

Ahh, thank you. 
Yes indeed: bc says:<pre><code> &gt;&gt;&gt; sqrt(10) * sqrt(10)
 9.99999999999999999994
</code></pre>
Android calculator, on the other hand, gets this one right.

So, &#x27;bc&#x27; just has the (big) rationals. Rationals are the numbers you could make by taking one integer (say 5 or minus sixteen trillion and fifty-one) and dividing it by some positive integer (such as three or sixty-two thousand)If we have a &quot;Big Integer&quot; type which can represent arbitrarily huge integers, such 10 to the power 5000, we can use two of these to make a Big Rational, and so that&#x27;s what bc has.But the rationals aren&#x27;t enough for all the features on your calculator. What&#x27;s the square root of ten ? How about the square root of 40 ? Now, multiply those together. The correct answer is 20. Not 20.00000000000000001 but exactly 20.

Anyone know of a comparison of the linked paper&#x27;s algorithm to how Gavin Howard&#x27;s &#x27;bc&#x27; CLI calculator does it?

I noticed this too, but I was confused because the calculator article was informative and interesting. It&#x27;s entirely unlike the inept fluffy slop that gets posted to LinkedIn

I think it’s called broetry although perhaps the sentences are a little long for that.

Agreed.And furthermore the content isn&#x27;t as &quot;wow, who&#x27;d a thunk it&quot; as the author seems to think it is. I cannot be unusual in knowing that single and double precision floating point numbers just don&#x27;t cut it for a lot of arithmetic tasks. Surely this is taught in every comp-sci course? And doesn&#x27;t nearly everyone know the N ⊆ Z ⊆ Q ⊆ A ⊆ R ⊆ C hierarchy?, well nearly everyone doesn&#x27;t but surely everyone who either decides to write a calculator app or is tasked with writing one ought to know this? The claim &quot;A calculator app? Anyone could make that.&quot; is a patently ridiculous claim to me, anybody who would make such a claim is clearly ignorant of both software development and mathematics. Next article. &quot;A text editor? Anyone could make that.&quot;

&gt; (Also I decided to try writing this thread in the style of a linkedin influencer lol, sorry about that.)I hated reading this buzzfeedy style (or apparently LinkedIn-style?) moron-vomit.I shouldn&#x27;t complain, just ask my nearest LLM to rewrite this article^W scribbling to a less obnoxious form of writing..

Also the “highschool poem”-type writing style is quite jarring but forgiven when he acknowledged it at the end of the article.&gt; Also I decided to try writing this thread in the style of a linkedin influencer lol, sorry about that.

I&#x27;ve not intentionally implemented any scrolljacking (I&#x27;m using the default obsidian template), but I&#x27;m curious what you mean as I also don&#x27;t see where the scrolljacking would happen. Could you elaborate on the way in which the user experience is awful now, so I can improve it?

While we&#x27;re already breaking the HN guidelines—&quot;Please don&#x27;t complain about tangential annoyances—e.g. article or website formats&quot;—let me just say that the scrolljacking on this article is awful.

I&#x27;m happy to have spread the good word! Note that when you read the paper, some implementation details are slightly different than my description. For instance, they always store the recursive real form of the real part of each number, even when the symbolic part perfectly describes it. I removed this redundancy to try to simplify it for twitter, but I hope it doesn&#x27;t confuse those who go on to read the paper afterwards.

The tone of the article has given away the fact that the article is not serious. At least not the way it&#x27;s presented. You want something serious? Go read the pdf.And I don&#x27;t mind at all. Without this article, I probably will never know what&#x27;s in the paper and how they iterated. I&#x27;ll likely give up after reading the abstract -- &quot;oh, they solved a problem&quot;. But this article actually makes much more motivating to read the original paper, which I plan to do now.

Well the cats hate you right back, how dare you. The whole point of the internet is to post cats on it.

The original is a Twitter thread, not a serious article.

Bah, cats have a place in programming articles, regardless of seriousness

Cats rule our world. So nothing wrong about it.

This was not intended to be a serious article, like something you&#x27;d submit for publication in an ACM journal.The last sentence is: &quot;(Also I decided to try writing this thread in the style of a linkedin influencer lol, sorry about that.)&quot;

I really hate when people put cat images and memes in a serious article.Don&#x27;t get me wrong, the content is good and informative. But I just hate the format.That reminds me when SideFX started putting memes into their official tutorial youtube channel. At least this is just a webpage and we can scroll through them...

They fortunately fixed the repeating feature in iOS 18.3. Though it does seem a bit ridiculous that something like this is tied to the entire OS version.

I remember hearing stories that for a time there was no engineer inside Apple responsible for the iOS Calculator.Now it seems to be revived as there were some updates to it, but those also removed one of my favourite features -&gt; tapping equals button no longer repeats the last operation.

Oh no- I stand corrected. I tried it again and you are right. I had just woken up when I did my initial test and must have typoed something. I can no longer edit or delete my original comment :(

I&#x27;ve just tried the first example on iOS 18.3.1 and it absolutely reproduces perfectly for me.

I enjoyed the article, but it seems Apple has since improved their calculator app slightly. The first example is giving me the correct result today. However, the second example with the “Underflow” result is still occurring.

&gt; It&#x27;s too slow.&gt; 1 is not equal to 1 - e^(-e^1000). But for Richardson and Fitch&#x27;s algorithm to detect that, it would require more steps than there are atoms in the universe.&gt; They needed something faster.I&#x27;m disappointed after this paragraph I expected a better algorithm and instead they decided to give up. Fredrik Johansson in his paper &quot;Calcium: computing in exact real and complex fields&quot; gives a partial algorithm for the problem and writes &quot;Algorithm 2 is inspired by Richardson’s algorithm, but incomplete: it will
find logarithmic and exponential relations, but only if the extension tower is flattened (in
other words, we must avoid extensions such as e^log(z) or √z^2), and it does not handle all
algebraic functions.
Much like the Risch algorithm, Richardson’s algorithm has apparently never been implemented fully. We presume that Mathematica and Maple use similar heuristics to ours,
but the details are not documented [6], and we do not know to what extent True&#x2F;False
answers are backed up by a rigorous certification in those system&quot;.

People call that a JIT compiler nowadays (?)

That&#x27;s a really cool and simple solution to a difficult problem!! I love it!

I first learned to program on a Wang 2200 computer with 8KB of RAM, back in 1978. One of the math teachers stayed an hour late most days to allow us nerds to come in an use the two computers. There were more people than computers, so often you&#x27;d only get 10 or 15 minutes of time.Anyway, I wrote a program where you could enter an equation and it would draw an ASCII graph of the curve. I didn&#x27;t know how to parse expressions and even if I had I knew it would be slow. The machine had a cassette tape under computer control for storing and loading programs. What I did was to take the expression typed by the user and convert each one into its tokenized form and write it out to tape. The program would then load that just created overlay which contained something like &quot;1000 DEF FNY(X)=X^2-5&quot; and a FOR loop would sweep X over the designated range, and have &quot;LET Y=FNY(X)&quot; to evaluate the expression for me.As a result, after entering the equation, it would take about five seconds to write out the overlay, rewind a couple blocks, and load the overlay before it would start to plot. But once it started it went pretty fast.

years ago the daily wtf had a challenge for writing the worst calculator app.  my submission maintained calculation state by emitting it&#x27;s own source code, recompiling and running the new executable.

I just use the Win7 binary. Calc.exe is the definition of something that really doesn&#x27;t need to change.

Based on the thread, you can build it from the source code and telemetry won’t be enabled…

&gt; Won&#x27;t fix: <a href="https:&#x2F;&#x2F;github.com&#x2F;microsoft&#x2F;calculator&#x2F;issues&#x2F;148">https:&#x2F;&#x2F;github.com&#x2F;microsoft&#x2F;calculator&#x2F;issues&#x2F;148</a>I don&#x27;t see how one can expect them to take a report worded this way seriously. Perhaps if they actually reported the crash without the tantrum the team would fix it.

I did begin poking at the crash for security issues but gave up after 10 minutes

I don&#x27;t know if you&#x27;ve already heard of illegal numbers; otherwise, you&#x27;re one of today&#x27;s lucky ten thousand!<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Illegal_number" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Illegal_number</a>

So they send telemetry that shows what people are calculating.Does it mean that there are some &quot;dangerous&quot; numbers that can be used to flag someone?

I removed telemetry on my Win10 system and now calc.exe crashes on basic calculations. I&#x27;ve reported this but nobody cares because the next step in troubleshooting is to reinstall Windows. So if telemetry fails, calc.exe will silently explode. Therefore no, anyone cannot make it.Won&#x27;t fix: <a href="https:&#x2F;&#x2F;github.com&#x2F;microsoft&#x2F;calculator&#x2F;issues&#x2F;148">https:&#x2F;&#x2F;github.com&#x2F;microsoft&#x2F;calculator&#x2F;issues&#x2F;148</a>

Isn&#x27;t what you are asking for a CAS?

That&#x27;s a CAS, as mentioned. There are plenty of open source libraries available, but one that specifically implements the algorithms discussed in this article is flintlib. Here&#x27;s an example from their docs showing exactly what you want: 
<a href="https:&#x2F;&#x2F;flintlib.org&#x2F;doc&#x2F;examples_calcium.html#examples-calcium" rel="nofollow">https:&#x2F;&#x2F;flintlib.org&#x2F;doc&#x2F;examples_calcium.html#examples-calc...</a>

This is interesting.One of the first ideas I had for an app was a calculator that represented digits like shown in the article but allowed you to write them with variables and toggle between symbolic and actual responses.A use case would be: in a spreadsheet like interface you could verify if the operations produced the final equation you were modeling in order to help validate if the number was correct or not. I had a TI-89 that could do something close and even in 2006 that was not exactly brand new tech. I figured surely some open source library available on the desktop must get me close. I was wildly wrong. I stuck with programming but abandoned the calculator idea. Even nearly 20 years later, such a task doesn’t seem that much easier to me.

&quot;-2 ** 3 SyntaxError: unparenthesized unary expression can&#x27;t appear on the left-hand side of &#x27;*&#x27;
&quot;That&#x27;s actually a great error, I have made the mistake of expecting &quot;-2 ** 2&quot; would output 4 instead of -4 before.

Correct, but my goal was just to get the same result than JS `eval()`except for -n * m because in my opinion this shouln&#x27;t require parenthesis. It&#x27;s still a good learn to do this, I don&#x27;t want to deal with floating points things etc..

^ fyi, this comment reveals you didn&#x27;t RTFA

recursive descent parser, normally any decent developer can do that, my version: <a href="https:&#x2F;&#x2F;caub.github.io&#x2F;misc&#x2F;calculator" rel="nofollow">https:&#x2F;&#x2F;caub.github.io&#x2F;misc&#x2F;calculator</a>

The linked paper by Bill Gosper is about potentially infinite  continued fractions with potentially irrational symbolic terms.

&gt; No, all finite continued fractions express a rational numberAny real number x has an infinite continued fraction representation. By efficient I mean that the information of the continued fraction coefficients is an efficient way to compute rational upper and lower bounds that approximate x well (they are the best rational approximations to x).&gt; They are occasionally convenient for certain problem structures but, as I&#x27;m sure you&#x27;ve already discovered, somewhat less convenient for a wide range of common problems.I&#x27;m curious what you mean exactly. I&#x27;ve found them to be very convenient for evaluating arithmetic expressions (involving both rational and irrational numbers) to fairly high accuracy. They are not the most efficient solution for this, but their simplicity and not having to do error analysis is far better than any other purely numerical system.&gt; fundamentally equivalent to simple fractions.This feels like it is a bit too reductionist. I can come up with a lot of example, but it&#x27;s quite hard to find the best rational approximations of a number with just fractions, while it&#x27;s trivial with continued fractions. Likewise, a number like the golden ratio, e, or any algebraic number has a simple description in terms of continued fractions, while this is certainly not the case for normal fractions.That continued fractions can be easily converted to normal fractions and vice versa, is a strength of continued fractions, not a weakness.

&gt; finiteThat&#x27;s the issue, no? If you go infinite you can then express any real number. You can then actually represent all those whose sequence is equivalent to a computable function.

&gt; You can express both real and rational numbers efficiently using a continued fraction representation.No, all finite continued fractions express a rational number (for... obvious reasons), which is honestly kind of a disappointment, since arbitrary sequences of integers can, as a matter of principle, represent arbitrary computable numbers if you want them to. They&#x27;re powerful than finite positional representations, but fundamentally equivalent to simple fractions.They are occasionally convenient for certain problem structures but, as I&#x27;m sure you&#x27;ve already discovered, somewhat less convenient for a wide range of common problems.

In a purely numerical setting, you can only distinguish these two cases when you evaluate the expression with enough accuracy. This may feel like a weakness, but if you think about this it is a much more &quot;honest&quot; way of handling inaccuracy than just rounding like you would do with floating point arithmetic.A good way to think about the framework, is that for any expression you can compute a rational lower and upper bound for the &quot;true&quot; real solution. With enough computation you can get them arbitrarily close, but when an intermediate result is not rational, you will never be able to compute the true solution (even if it happens to be rational; a good example is that for sqrt(2) * sqrt(2) you will only be able to get a solution of the form 2 ± ϵ for some arbitrarily small ϵ).

How do you work out an answer for x - y when eg x = sqrt(2) and y = sqrt(2) - epsilon for arbitrarily small epsilon? How do you differentiate that from x - x?

Sounds a bit like <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Wiener%27s_attack" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Wiener%27s_attack</a>.(It&#x27;s not a good article when it comes to the attack details, unfortunately.)

What do you mean by &quot;[n]aturally&quot; here?

Why unnecessarily air this grievance in a public forum. If this person reads it they will be unhappy and I&#x27;m sure they have already suffered enough from this failure.

Continued fractions are very cool. I saw in a CTF competition once a question about breaking an RSA variant that relied on the fact that a certain ratio was a term in sequence of continued fraction convergents.Naturally the person pursing a PhD in number theory (whom I recruited to our team for specifically this reason) was unable to solve the problem and we finished in third place.

is that how that kid calculated super long pi?

That sounds kind of similar to Dedikin cuts but crossed with ordered sequence &lt; and &gt; the real? Cool web site.

I have been working on a new definition of real numbers which I think is a better foundation for real numbers and seems to be a theoretical version of what you are doing practically. I am currently calling them rational betweenness relations. Namely, it is the set of all rational intervals that contain the real number. Since this is circular, it is really about properties that a family of intervals must satisfy. Since real numbers are messy, this idealized form is supplemented with a fuzzy procedure for figuring out whether an interval contains the number or not. The work is hosted at (<a href="https:&#x2F;&#x2F;github.com&#x2F;jostylr&#x2F;Reals-as-Oracles">https:&#x2F;&#x2F;github.com&#x2F;jostylr&#x2F;Reals-as-Oracles</a>) with the first paper in the readme being the most recent version of this idea.The older and longer paper of Defining Real Numbers as Oracles contains some exploration of these ideas in terms of continued fractions. In section 6, I explore the use of mediants to compute continued fractions, as inspired by the old paper Continued Fractions without Tears ( <a href="https:&#x2F;&#x2F;www.jstor.org&#x2F;stable&#x2F;2689627" rel="nofollow">https:&#x2F;&#x2F;www.jstor.org&#x2F;stable&#x2F;2689627</a> ). I also explore a bit of Bill Gosper&#x27;s arithmetic in Section 7.9.2. In there, I square the square root of 2 and the procedure, as far as I can tell, never settles down to give a result as you seem to indicate in another comment.For fun, I am hoping to implement a version of some of these ideas in Julia at some point. I am glad to see a version in Python and I will no doubt draw inspiration from it and look forward to using it as a check on my work.

Nice story. An even more powerful way to express numbers is as a continued fraction (<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Continued_fraction" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Continued_fraction</a>). You can express both real and rational numbers efficiently using a continued fraction representation.As a fun fact, I have a not-that-old math textbook (from a famous number theorist) that says that it is most likely that algorithms for adding&#x2F;multiplying continued fractions do not exist. Then in 1972 Bill Gosper came along and proved that (in his own words) &quot;Continued fractions are not only perfectly amenable to arithmetic, they are amenable to perfect arithmetic.&quot;, see <a href="https:&#x2F;&#x2F;perl.plover.com&#x2F;yak&#x2F;cftalk&#x2F;INFO&#x2F;gosper.txt" rel="nofollow">https:&#x2F;&#x2F;perl.plover.com&#x2F;yak&#x2F;cftalk&#x2F;INFO&#x2F;gosper.txt</a>.I have been working on a Python library called reals (<a href="https:&#x2F;&#x2F;github.com&#x2F;rubenvannieuwpoort&#x2F;reals">https:&#x2F;&#x2F;github.com&#x2F;rubenvannieuwpoort&#x2F;reals</a>). The idea is that you should be able to use it as a drop-in replacement for the Decimal or Fraction type, and it should &quot;just work&quot; (it&#x27;s very much a work-in-progress, though). It works by using the techniques described by Bill Gosper to manipulate continued fractions. I ran into the problems described on this page, and a lot more. Fun times.

&gt; I think Google moved it from AOSP to the Google Play Services several years ago, but the old source is still available.For the curious, here is the source of ExactCalculator from the commit before all files were deleted: <a href="https:&#x2F;&#x2F;android.googlesource.com&#x2F;platform&#x2F;packages&#x2F;apps&#x2F;ExactCalculator&#x2F;+&#x2F;7265aa3a922a7e7c0389dae4b26cca562aa777a6" rel="nofollow">https:&#x2F;&#x2F;android.googlesource.com&#x2F;platform&#x2F;packages&#x2F;apps&#x2F;Exac...</a>, and here is the dependency CR.java <a href="https:&#x2F;&#x2F;android.googlesource.com&#x2F;platform&#x2F;external&#x2F;crcalc&#x2F;+&#x2F;refs&#x2F;heads&#x2F;main" rel="nofollow">https:&#x2F;&#x2F;android.googlesource.com&#x2F;platform&#x2F;external&#x2F;crcalc&#x2F;+&#x2F;...</a>

I played around with the calculator source code from the Android Open Source Project after a previous submission[1]. I think Google moved it from AOSP to the Google Play Services several years ago, but the old source is still available.It does solve some real problems that I&#x27;d love to have available in a library. The discussion on the previous article links to some libraries, but my recollection is that the calculator code is more accessible to an innumerate person like myself.Edit: the previous article under discussion doesn&#x27;t seem to be available, but it&#x27;s on archive.org[2].[1] <a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=24700705">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=24700705</a>[2] <a href="https:&#x2F;&#x2F;web.archive.org&#x2F;web&#x2F;20250126130328&#x2F;https:&#x2F;&#x2F;blog.acolyer.org&#x2F;2020&#x2F;10&#x2F;02&#x2F;toward-an-api-for-the-real-numbers&#x2F;" rel="nofollow">https:&#x2F;&#x2F;web.archive.org&#x2F;web&#x2F;20250126130328&#x2F;https:&#x2F;&#x2F;blog.acol...</a>

From the linked post: &#x27;A &quot;computer algebra system&quot; would have accomplished a similar goal, but been much slower and much more complicated&#x27;

I imagine you could do most or all of this with yacas, which is actually a computer algebra system (GPL).

He was not working on the iOS Calculator.

So did they fix the iOS Calculator bug?On another note. Since Calculator is so complex are there any open source cross platform library that makes it easier to implement?

One issue I have with Trello is that it has multiple types of items. And that it is not recursive.When I create an item &quot;Supermarket&quot; and then an item &quot;Bread&quot;, I cannot drag and drop the item &quot;Bread&quot; into &quot;Supermarket&quot;. But that is how I think. I have a lot of &quot;items&quot; and each item can contain other &quot;items&quot;. I don&#x27;t want any other type of object.Another problem is that I cannot customize the layout. I can&#x27;t remove every icon from the items in the list. I only want to see the item names, no other info like the icon that shows that there is a description or anything. But Trello seems to not support that.

I agree that Keep is pretty good. I switched over to Carnet when I setup my nextcloud. It&#x27;s OK, but not as good as Keep.

I use Google&#x27;s Keep but you may need to make your own

Why does a to-do list have to have any features by default? It could be a blank screen with a &quot;settings&quot; sign in the upper right, where you can enable just the features you need.If I don&#x27;t find such a software, I will write it myself. I actually already started:<a href="https:&#x2F;&#x2F;x.com&#x2F;marekgibney&#x2F;status&#x2F;1844077244903571549" rel="nofollow">https:&#x2F;&#x2F;x.com&#x2F;marekgibney&#x2F;status&#x2F;1844077244903571549</a>I am developing it on the side, while I try to get by with existing solutions.

To-Do List is an infinite product category.They are extremely personal and any unwanted features end up as friction.You&#x27;ll never find a perfect Todo app because it will have an audience of 1 so wouldn&#x27;t be made.Other examples of Todo apps:Things, 2Do, Todoist, OmniFocus, Due, Reminders (Apple), Clear, GoodTask, Notes, Google KeepThe list is literally neverending,

Just tried it. Way too much stuff on the screen for my liking. Plus it seems to be not recursive.Similar to my thoughts about Trello:<a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=43068867">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=43068867</a>

Any.do did me well until I adopted a new method of stacking tasks into a routine.The hard part is altering the routine.

I would love to have a To-Do app that is fluid for both one-off tasks and periodic checklists (daily&#x2F;weekly&#x2F;monthly&#x2F;etc.) Most importantly, I want it to yell at me to actually do it. I was pretty surprised that basically nothing seems to fit the bill and even what existing &quot;GTD&quot; type apps could do felt cumbersome and limited.

Just tried it, and this is very much the opposite of what I am looking for.What I would like is a very minimal layout. Basically with nothing on the screen. And I want to be able to organize my world by dragging, dropping, swiping recursive items.

I&#x27;m one of the creators of Godspeed, which is a fast, 100% keyboard oriented to-do app (though we do support drag and drop as well!). And we&#x27;ve got a web app!<a href="https:&#x2F;&#x2F;godspeedapp.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;godspeedapp.com&#x2F;</a>

This is the sort of thing I like trying to make llms do, thanks for the idea. I have a discord bot set up already that sends notifications and accepts notes; I will try adding some endpoints and burning some credits I have to see how hard it is to make AI talk to alarm endpoints in a smart way, etc

It seems like you&#x27;re looking for an outliner? Workflowy might fit your needs: <a href="https:&#x2F;&#x2F;workflowy.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;workflowy.com&#x2F;</a>Like others have said, the perfect to-do list is impossible because each person wants wildly different functionality.My dream to-do list has minimal interaction, with the details handled like I have my own personal secretary. All I&#x27;d do is verbally say something like &quot;remind me to do laundry later&quot; and it would do the rest: Categorizing, organizing, prioritizing, scheduling and adding sub-tasks as needed.I love the idea of automatic sub-tasks created at level which helps with your particular procrastination level. For example &quot;do laundry&quot; would add in &quot;gather clothes, bring to laundry room, separate colors, add to washer, set timer, add to dryer, set timer, get clothes, fold clothes, put away, reschedule in a week (but hide until then). Maybe it&#x27;s even add in Pomodoro timers to help.LLMs with reasoning might get us there soon - we&#x27;ve been waiting for Knowledge Navigator like assistants for years.

To make any type of app really good is super hard.I have yet to see a good to-do list tool.I&#x27;m not kidding. I tried TickTick, Notion, Workflowy ... everything I tried so far feels cumbersome compared to how I would like to handle my To-Do list. The way you create, edit, browse, drag+drop items is not as all as fluid as I imagine it.So if anyone knows a good To-Do list software (must be web based, so I can use it anywhere without installing something) - let me know!

<a href="https:&#x2F;&#x2F;github.com&#x2F;based2&#x2F;KB&#x2F;blob&#x2F;main&#x2F;math&#x2F;aaa.md#calculator">https:&#x2F;&#x2F;github.com&#x2F;based2&#x2F;KB&#x2F;blob&#x2F;main&#x2F;math&#x2F;aaa.md#calculato...</a>

I&#x27;m on windows 11. I just did it and it replied &quot;7&quot;. I subtracted 7 to see if there was some epsilon error but it reported &quot;0&quot;. What do you experience?

Are you in standard or scientific? Each new operator (Not sure if thats the correct term) is calculated immediately. ie 1+2x3 is worked out as 1+2 (Stored into buffer as 3) x 3 = 9But scientific does it correctly where it just appends the new expression onto the buffer instead of applying it

There&#x27;s a point at which you&#x27;re really building a computer algebra system.Or you can do what the Windows 11 calculator does and not even get 1+2*3 right.

I solved this by making a calculator that rounds to the nearest 1&#x2F;16. Tiny app for Americans doing DIY work around the house:<a href="https:&#x2F;&#x2F;thomaspark.co&#x2F;projects&#x2F;calc-16&#x2F;" rel="nofollow">https:&#x2F;&#x2F;thomaspark.co&#x2F;projects&#x2F;calc-16&#x2F;</a>

lowkey this is why ieee 754 floating point is both a blessing and a curse, like yeah it’s fast n standardized but also introduces unavoidable precision loss, esp w iterative computations where rounding errors stack up in unpredictable ways. ppl act like increasing precision bits solves everything. but u just push the problem further down, still dealing w truncation, cancellation, etc. (and edge cases where numerical stability breaks down.)… and this is why interval arithmetic and arbitrary precision methods exist, so it gives guaranteed bounds on error instead of just hoping fp rounding doesn’t mess things up too bad. but obv those come w their own overhead: interval methods can be overly conservative, which leads to unnecessary precision loss, and arbitrary precision is computationally expensive, scaling non-linearly w operand size.wonder if hybrid approaches could be the move, like symbolic preprocessing to maintain exact forms where possible, then constrained numerical evaluation only when necessary. could optimize tradeoffs dynamically. so we’d keep things efficient while minimizing precision loss in critical operations. esp useful in contexts where precision requirements shift in real time. might even be interesting to explore adaptive precision techniques (where computations start at lower precision but refine iteratively based on error estimates).

The point underlying the problem is about the closure of operations [1]Typically one would like to be able to calculate things without making error, which accumulates.The symbolic representation you suggest use a growing memory to represent the point by all the operations which have been applied to it since the origin.What we would rather do is define a set of operation that are closed for a specific set of points, which allows to accumulate information by doing the computation rather than deferring the computation.One could for example think of using fixed point number to represent the coordinates, and define an extra point at the infinity to handle overflow. And then you have some property that you like and some that you like less. For example minimums distance which can define a point uniquely in continuous R^2, are no longer unique when you constrain yourself to integer grids by using fixed points.Or you could use some rational numbers to store the coordinates like in CGAL (which allows you to know on which sides of the planes you are without z-fighting), but they still require growing memory. You can maybe add some rule to handle the underflow and overflows.Or you can merge close points, but maybe you lose some information.Or you can define the operations on lattices, finite automaton, or do some error correcting codes, dynamic recombining graphs (aka the ruliad).It&#x27;s an open problem, see <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Robust_geometric_computation" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Robust_geometric_computation</a> for more.[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Closure_(mathematics)" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Closure_(mathematics)</a>

Just use Mathematica.<pre><code> ar = Sqrt[ax^2 + ay^2]; atheta = ArcTan[ax, ay];
 br = ar; btheta = atheta + theta;
 cr = ar; ctheta = atheta + Pi + theta + Pi;
 bx = br Sin[btheta]; by = br Cos[btheta];
 cx = cr Sin[ctheta]; cy = cr Cos[ctheta];

 In[2]:= bx == cx

 Out[2]= True

 In[3]:= by == cy

 Out[3]= True
</code></pre>
This seems so elementary that I think open source computer algebra systems can do it.

That&#x27;s just a single number to calculate.The real fun begins when you do geometry.Find a representation of finite memory to represent points, which allows exact addition, multiplication and rotation between them, (with all the nice standard math property like associativity and commutativity).For example your representation should be able to take a 2d point A, aka two coordinates, and rotate it around the origin by an angle theta to obtain the point B. Take the original point and rotate it by pi + theta, then reflect it around the origin to obtain the point C. Now answer the question whether B is coincident with C.

I use qalculate, it behaves well enough for my needs.<a href="https:&#x2F;&#x2F;qalculate.github.io&#x2F;" rel="nofollow">https:&#x2F;&#x2F;qalculate.github.io&#x2F;</a>

I just tried this in raku App::Crag...crag &#x27;say (10*100) + 1 − (10*100)&#x27; #1Raku uses Rats by default (Rational numbers) unless you ask for floating point.

Another way to see it is that floating point is the logical extension of fixed point math to log space to deal with numbers across a large orders of magnitude. I don&#x27;t know if &quot;beautiful&quot; is exactly the right word, but it&#x27;s an incredibly solid bit of engineering.

I feel like your description comes across as more negative on the design of IEEE-754 floats than you intend. Is there something else you think would have been better? Maybe I’m misreading it.Maybe the hardware focus can be blamed for the large exponents and small mantissas.The reasonable only non-IEEE things that comes to mind for me are:- bfloat16 which just works with the most significant half of a float32.- log8 which is almost all exponent.I guess in both cases they are about getting more out of available memory bandwidth and the main operation is f32 + x * y -&gt; f32 (ie multiply and accumulate into f32 result).Maybe they will be (or already are) incorporated into IEEE standards though

&gt; what you get if you started with the idea of sign, exponent, and fraction and made the most efficient hardware implementation of it possible. It&#x27;s not &quot;beautiful&quot;, but it falls out pretty straightforwardly from those starting assumptionsThis implies a strange way of defining what &quot;beautiful&quot; means in this context.

IEEE 754 is what you get if you started with the idea of sign, exponent, and fraction and made the most efficient hardware implementation of it possible. It&#x27;s not &quot;beautiful&quot;, but it falls out pretty straightforwardly from those starting assumptions, even the seemingly weirder parts like -0, subnormals and all the rounding modes. It was not really democratically designed, but done by numerical computing experts coupled with hardware design experts. Every &quot;simplified&quot; implementation of floating point that has appeared (e.g. auto-FTZ mode in vector units) has eventually been dragged kicking and screaming back to the IEEE standard.

Sometimes you need a number system which is 1. approximate 2. compact and fast 3. high dynamic rangeYou’re going to have a hard time doing better than floats with those constraints.

There&#x27;s no &quot;accurate results&quot; most of the timeYou can only have a result that&#x27;s exact enough in your desired precision

&gt; so long as it produces accurate resultsThat&#x27;s doing a lot of work. IEE-754 does very in terms of error vs representation size.What system has accurate results? I don&#x27;t know any number system at all in usage that 1) represents numbers with a fixed size 2) Can represent 1&#x2F;n accurately for reasonable integers 3) can do exponents accurately

It&#x27;s ideal for engineering calculations which is a common use of computers. There, nobody cares if 1-1=0 exactly or not because you could never have measured those values exactly in the first place. Single precision is good enough for just about any real-world measurement or result while double precision is good for intermediate results without losing accuracy that&#x27;s visible in the single precision input&#x2F;output as long as you&#x27;re not using a numerically unstable algorithm.

Electronic computers were created to be faster and cheaper than a pool of human computers (who may have had slide rules or mechanical adding machines). Human computers were basically doing decimal floating point with limited precision.

Given a computer have finite memory but there are infinitely many real numbers in any range, any system using real numbers will have to use rounding.

&gt; IEEE 754 is like democracy in a sense that it is the worst, except for all the others.I can&#x27;t see what would be worse. The entire raison d&#x27;etre for computers is to give accurate results. Introducing a math system which is inherently inaccurate to computers cuts against the whole reason they exist! Literally any other math solution seems like it would be better, so long as it produces accurate results.

William Kahan worked on both IEEE 754 and HP calculators.  The speed gap between something like an 8087 and a calculator was not that big back then, either.

&gt; equal precision across the rangeWhat? Pretty sure there&#x27;s more precision in [0-1] than there is in really big numbers.

Billions or trillions of ops per second and 1987 don&#x27;t really go together.

IEEE 754 is what you get when you want numbers to have huge dynamic range, equal precision across the range, and fixed bit width. It balances speed and accuracy, and produces a result that is very close to the expected result 99.9999999% of the time. A competent numerical analyst can take something you want to do on paper and build a sequence of operations in floating point that compute that result almost exactly.I don&#x27;t think anyone who worked on IEEE 754 (and certainly nobody who currently works on it) contemplated calculators as an application, because a calculator is solving a fundamentally different problem. In a calculator, you can spend 10-100 ms doing one operation and people won&#x27;t mind. In the applications for which IEEE 754 is made, you are expecting to do billions or trillions of operations per second.

&gt; IEEE754 is not great for pure maths, however, it is fine for real life.Partially. It can be fine for pretty much any real-life use case. But many naive implementations of formulae involve some gnarly intermediates despite having fairly mundane inputs and outputs.

It becomes a problem when precision errors accumulate in a system, right?The issue isn&#x27;t so much that a single calculation is slightly off, it&#x27;s that many calculations together will be off by a lot at the end.Is this stupid or..?

IEEE754 is not great for pure maths, however, it is fine for real life.In real life, no instrument is going to give you a measurement with the 52 bits of precision a double can offer, and you are probably never going to get quantities are in the 10^1000 range. No actuator is precise enough either. Even single precision is usually above what physical devices can work with. When drawing a pixel on screen, you don&#x27;t need to know its position down to the subatomic level.For these real life situations, improving on the usual IEEE 754 arithmetic would probably be better served with interval arithmetic. It would fail at maths, but in exchange you get support for measurement errors.Of course, in a calculator, precision is important because you don&#x27;t know if the user is working with real life quantities or is doing abstract maths.

As soon as I read the title, I chuckled, because coming from the computational mathematics background I already knew what it roughly is going to be about. IEEE 754 is like democracy in a sense that it is the worst, except for all the others. Immediately when I saw the example I thought: it is either going to be either a Kahan summation or full-scale computer algebra system. It turned out to be some subset of the latter and I have to admit I have never heard of Recursive Real Arithmetic (I knew of Real Analysis though).If anything that was a great insight about one of my early C++ heroes, and what they did in their professional life outside of the things they are known for. But most importantly it was a reminder how deep seemingly simple things can be.

Thanks, that &quot;responsive pixel art&quot; is very cool!

Thanks, that&#x27;s a lovely analogy and I&#x27;m excited to listen to that podcast.I think the general idea of converting things from discrete and implementation-motivated representations to higher-level abstract descriptions (bitmaps to vectors, in your example) is great. It&#x27;s actually something I&#x27;m very interested in, since the higher-level representations are usually much easier to do interesting transformations to. (Another example is going from meshes to SDFs for 3D models.)You might get a kick out of the &quot;responsive pixel art&quot; HN post from 2015 which implements this idea in a unique way: <a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=11253649">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=11253649</a>

The way this article talks about using &quot;recursive real arithmetic&quot; (RRA) reminds me of an excellent discussion with Conal Elliot on the Type Theory For All podcast. He talked about moving from representing things discretely to representing things continually (and therefore more accurately). For instance, before, people represented fonts as blocks of pixels, (discrete.) They were rough approximations of what the font really was. But then they started to be recognized as lines&#x2F;vectors (continual), no matter the size, they represented exactly what a font was.Conal gave a beautiful case for how comp sci should be about pursuing truth like that, and not just learning the latest commercial tool. I see the same dogged pursuit of true, accurate representation in this beatiful story.- <a href="https:&#x2F;&#x2F;www.typetheoryforall.com&#x2F;episodes&#x2F;the-lost-elegance-of-computation" rel="nofollow">https:&#x2F;&#x2F;www.typetheoryforall.com&#x2F;episodes&#x2F;the-lost-elegance-...</a>- <a href="https:&#x2F;&#x2F;www.typetheoryforall.com&#x2F;episodes&#x2F;denotational-design" rel="nofollow">https:&#x2F;&#x2F;www.typetheoryforall.com&#x2F;episodes&#x2F;denotational-desig...</a>

Everything in life is estimation. A calculator that tells you the perfect answer in some highly unusual situation probably isn&#x27;t fixing the most common source of error.e.g I measure an angle and I am not sure about whether it&#x27;s 45 degrees or 46 then an answer like this is pointless:
0.7071067811865476cos of 46 (if I&#x27;ve converted properly to radians) is 
0.6946583704589973so my error is about 0.01 and those long lists of digits imply a precision I don&#x27;t have.I think it would be more useful for most people to tell them how much error there is in their results after guessing or letting them assign the estimated error in their inputs.

This is a crazy take to me. Most people don’t care that a calculator gives them a correct answer? It’s just an estimator?Why would we not expect it to work when we know how to build ones that do, and people use it as a replacement for math on paper?

I doubt that most people using the calc app expect it to handle such situations. It&#x27;s nice that it does of course but IMO it misses the point that the inputs to a lot of real world calculations are inaccurate to start with.i.e it&#x27;s more likely that I&#x27;ve made a few mm mistake when measuring the radius of my table than that I&#x27;m not using an precise enough version of Pi. The area of the table will have more error because one is squaring the radius, obviously.It would be interesting to have a calculator that let you add in your estimated measurement error (or made a few reasonable guesses about it for you) and told you the error in your result e.g. the standard deviation.I sometimes want to buy stuff at a hardware shop and I think : &quot;how much paint do I need to buy?&quot; I haven&#x27;t planned so I&#x27;m thinking &quot;it&#x27;s about 4m by 5m...I think?&quot; I try to do a couple of calculations with worst case numbers so I at least get enough paint and save another trip to the shop but not comically too much so that I have a tin of it for the next 5 years.I remember having to estimate error in results that were calculated from measured values for physics 101 and it was a pain.

&gt; the dividing line seems to be at 1e33.. Not sure what to make of thatThat’s not too bad. They are probably using hand-rolled FP128 format for their numbers. If they were using hardware-provided FP64 arithmetic, the threshold would have been 2^53 ≈ 9E+15: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Double-precision_floating-point_format#IEEE_754_double-precision_binary_floating-point_format:_binary64" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Double-precision_floating-poin...</a>

Tried with the HP Prime and it gave the precise 1 for the test. One need to put it in the CAS mode and use the exact form of 10^100 instead of 1E100. You shall get the right answer if the calculator is instructed to use its very powerful CAS engine.

All the calculators that I just tried for the article&#x27;s expression give the wrong answer (HP Prime, TI-36X Pro, some casio thing). Even google&#x27;s own online calculator gives the wrong answer, which is mildly ironic. [<a href="https:&#x2F;&#x2F;www.google.com&#x2F;search?q=1e101%2B1-1e101&amp;oq=1e101%2B1" rel="nofollow">https:&#x2F;&#x2F;www.google.com&#x2F;search?q=1e101%2B1-1e101&amp;oq=1e101%2B1</a>]I played around with the macOS calculator and discovered that the dividing line seems to be at 1e33. I.e. 1e33+1-1e33 gives the correct answer of 1 but 1e34+1-1e34 gives 0. Not sure what to make of that.

Due to backwards compatibility modern PC CPUs have some mathematical constants in hardware, one of them Pi <a href="https:&#x2F;&#x2F;www.felixcloutier.com&#x2F;x86&#x2F;fld1:fldl2t:fldl2e:fldpi:fldlg2:fldln2:fldz" rel="nofollow">https:&#x2F;&#x2F;www.felixcloutier.com&#x2F;x86&#x2F;fld1:fldl2t:fldl2e:fldpi:f...</a> Moreover, that FLDPI instruction delivers 80 bits of precision, i.e. more precise than FP64.That’s pretty much useless in modern world because the whole x87 FPU is deprecated. Modern compilers are generating SSE1 and SSE2 instructions for floating-point arithmetic, instead of x87.

No? They made a goal to show 0.0000 in as few places as possible, and they got as close to it as they could without compromising their other requirements.

&gt; Showing 0.0000000000000 on the screen, when the answer is exactly 0, would be a horrible user experience.&gt; They realized that it&#x27;s not the end of the world if they show &quot;0.000000...&quot; in a case where the answer is exactly 0so... devs self-made a requirement, got into trouble (complexity) - removed the requirement, trouble didn&#x27;t go anywherejust keep saying &quot;it&#x27;s a win&quot; and you&#x27;ll be winning, I guess

I commented about this on X.As a developer, &quot;infinite scroll to get more digits&quot; sounds really cool. It sounds conceptually similar to lazily-evaluated sequences in languages like Clojure and Haskell (where you can have a &#x27;virtually-infinite&#x27; list or array -- basically a function -- and can access arbitrarily large indices).As a user, it sounds like an annoying interface. On the rare case I want to compute e^(-10000), I do not want to scroll for 3 minutes through screens filled with 0s to find the significant digits.Furthermore, it&#x27;s not very usable. A key question in this scenario would be: how many zeroes were there?It&#x27;s basically impossible to tell with this UI. A better approach is simply to switch to scientific notation for very large or very small numbers, and leave decimal expansion as an extra option for users who need it. (Roughly similar to what Wolfram Alpha gives you for certain expressions.)

The thing about this calculator app is that it can display any number of digits just by scrolling the display field. The UX is &quot;any number of digits the user wants&quot; not some predetermined fixed number of digits.

Interesting article, and kudos to Boehm for going the extra mile(s), but it seems like overkill to me.I wouldn&#x27;t expect, or use, a calculator for any calculation requiring more accuracy than the number of digits it can display. I&#x27;m OK with with iPhone&#x27;s 10^100 + 1 = 1e100.If I really needed something better, I&#x27;d try Wolfram Alpha.

Sounds like he&#x27;s just using stock math functions. Both Javascript and Python act the same way when you save the result immediately after subtracting two numbers multiple times, rather than just 8.7 - (2.9*3).

Holy shit, I thought you just made mindblowing mechanical devices.I can see why you wouldn&#x27;t necessarily just want to use it, but I thought the RIM pager had a JVM with floating point?I mostly just used mine for email.

&gt; the &quot;exponent&quot; part wasn&#x27;t how many bits to shift, but what power of two to divide by, so that 0.1, 0.001 etc can be represented exactlyYou mean what power of ten to divide by?

I wrote the calculator for the original blackberry.  Floating point won&#x27;t do.  I implemented decimal based floating point functions to avoid these rounding problems.  This sounds harder than it was, basically, the &quot;exponent&quot; part wasn&#x27;t how many bits to shift, but what power of two to divide by, so that 0.1, 0.001 etc can be represented exactly.  Not sure if I had two or three digits of precision beyond whats on the display.  1 digit is pretty standard for 5 function calculators, scientific ones typically have two.
It was only a 5 function calculator, so not that hard, plus there was no floating point library by default so doing any floating point really ballooned the size of an app with the floating point library.

The NYC subway fare is $2.90. I was using PCalc on iOS to step through remaining MetroCard values per swipe and discovered that AC, 8.7, m+, 2.9, m-, m-, m- evaluates to -8.881784197E-16 instead of zero. This doesn&#x27;t happen when using Apple&#x27;s calculator. I wrote to the developer and he replied, &quot;Apple has now got their own private maths library which isn&#x27;t available to developers, which they&#x27;re using in their own calculator. What I need to do is replace the Apple libraries with something else - that&#x27;s on my list!&quot;

Was given the task to build a simple calculator app as a project for a Java class I took in college.No parens or anything like that, nothing nearly so fancy. Classic desk calculator where you set the infix operation to apply to the previous value, followed by the second value of the operation.It was frankly an unexpected challenge. There&#x27;s a lot more to it than meets the eye.I only got as far as rational numbers though. PI accurate to the 8 digit display was good enough for me.Honestly though, I think it was a great exercise for students, showing how seemingly simple tasks can actually be more complex than they seem. I&#x27;m still here thinking about it some twenty years later.

I wrote an OCaml implementation of this paper a few years ago, which I&#x27;ve now extracted into its own [repo](<a href="https:&#x2F;&#x2F;github.com&#x2F;joelburget&#x2F;constructive-reals&#x2F;blob&#x2F;main&#x2F;Constructive_reals.ml">https:&#x2F;&#x2F;github.com&#x2F;joelburget&#x2F;constructive-reals&#x2F;blob&#x2F;main&#x2F;C...</a>)The link in the paper to their Java implementation is now broken: does anyone have a current link?

At some point, when I get a spare 5 years (and&#x2F;or if people start paying for software again), I will start to work on a calculator application. Number system wrangling is quite fun and challenging, and I am hoping to incorporate units as a first-class citizen.

Saw the thread on Twitter. Kudos to the author for going in so much detail!

There’s a pleasantly elegant “hey, we’ve solved the practical functional complement to this category of problems over here, so let’s just split the general actual user problem structurally” vibe to this journey.It often pays off to revisit what the actual “why” is behind the work that you’re doing, and this story is a delightful example.I wrote an arbitrary precision arithmetic C++ library back in the 90’s. We used it to compute key pairs for our then new elliptic-curve based software authentication&#x2F;authorization system. I think the full cracks of the software were available in less than two weeks, but it was definitely a fun aside and waaaay too strong of a solution to a specific problem. I was young and stupid… now I’m old and stupid, so I’d just find an existing tool chain to solve the problem.

The point of the article is to show building a calculator requires a CAS, which should have been obvious to anyone with a basic understanding of how a calculator works.The premise of the article is itself somewhat bogus, but I suppose there are programmers today who never had to work with a graphing calculator.While RRA is an interesting approach, ultimately it wasn&#x27;t sufficient.Re-using an off-the-shelf CAS would have been the more practical solution, avoiding all the extra R&amp;D on a novel number representation that wasn&#x27;t quite sufficient to do the job.

The point of the article is to teach you how calculators work. Not find a piece of software to unblock you.You may well find yourself in the field of computing having to compute something!

Could have just used an off-the-shelf CAS.

I thought this was going to be about how Apple completely destroyed the calculator on iOS with the latest update.Now it does the running ticker tape thing, which means you can&#x27;t use the AC button to quickly start over, because there is no AC button anymore!I know it&#x27;s supposed to be easier&#x2F;better for the user, but they didn&#x27;t even give me a way to go back to the old behavior.

A what?I&#x27;m also reading this on an sPhone but don&#x27;t remember seeing anything that looked like, well... what you said

There should have been an &quot;x&quot; on the right of the &quot;contact me&quot; portion that you could click to make it go away. Sounds like it didn&#x27;t show up for you, so sorry about that. Unfortunately I don&#x27;t have an iPhone SE to test against and the &quot;x&quot; does seem to show up on the iPhone SE screen-size simulator in Chrome. This means I don&#x27;t know how to reproduce the issue and probably won&#x27;t be able to resolve it without removing the &quot;contact me&quot; page entirely, which I&#x27;m not willing to do right now.

Why does the author insist on using a dick bar? With the &quot;contact me&quot; portion, it takes up 30% of my screen on an iPhone SE.

“A calculator app? Anyone could make that”