Towards an API for the real numbers

If you don't like twitter, here is the paper, it discusses a hybrid approach for calculating mathematical expressions by combining symbolic and analytical methods: https://dl.acm.org/doi/pdf/10.1145/3385412.3386037

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: https://dl.acm.org/doi/pdf/10.1145/3385412.3386037

Haha, this is my thread. There’s a typo in the second tweet. Unfortunately there’s no way to edit threads on twitter. > (10^100) + 1 − (10^100) is 0, not 1. I should have said: (10^100) + 1 − (10^100) is 1, not 0. One other error is here: > I asked for 2π to within 0.01 decimals, so you need an approximation of π that's within 0.005 decimals. “0.01 decimals“ and “0.005 decimals“ should have simply been “0.01” and “0.005” respectively. I hope everyone enjoys! The original “API for the real numbers” paper is here: https://dl.acm.org/doi/pdf/10.1145/3385412.3386037 Someone is also implementing it in rust: https://crates.io/crates/realistic

> So you're saying my iPhone built-in calculator app is going to have problems....? Your Android phone's built-in calculator app, however, will not. :^) https://dl.acm.org/doi/pdf/10.1145/3385412.3386037

From the abstract, "simulations of such systems are usually done on digital computers, which are able to compute with finite sequences of rational numbers only." Not at all! Digital computers can use computable reals which are defined as any function from a rational value (the error bound) to a rational value which is within the supplied error-bounds from the true value. Do not mistake this for computing in the rationals, these functions which perform the described task are the computable real numbers. There are countable-infinity many of these functions, one for each computable real. For instance, note that you can always compare two rationals for equality, but you can't always compare two computable reals for equality, just like reals. Hans Boehm (of Boehm garbage collector fame) has been working on this for a long time, here is a recent paper on it: https://dl.acm.org/doi/pdf/10.1145/3385412.3386037

I believe you're basically referring to the computable reals; for the curious, here's the wikipedia page (kind of abstract): https://en.wikipedia.org/wiki/Computable_number and here's the paper where they implement them programmatically, including for the android calculator application (I believe this is how the android calculator works now): https://dl.acm.org/doi/abs/10.1145/3385412.3386037

If you want to move beyond standard floats, this paper could be good starting point: https://dl.acm.org/doi/pdf/10.1145/3385412.3386037 Right now 0.1+0.2-0.3 results = 2.775557562 \cdot 10^ {-17}, which depending on your viewpoint is either expected or unexpected. Also (0.1+0.2)-0.3 is simplified to 0.1+0.2-0.3 for display, but results a different value = 5.551115123 \cdot 10^ {-17}

Yeah there are a few number systems that can do this kind of thing, ranging from rational implementations where the numerator and denominator are stored, all the way to the computable reals (https://en.wikipedia.org/wiki/Computable_number), which can handle even trancendentals exactly! Fun fact, the built-in calculator on android actually uses the computable reals (https://dl.acm.org/doi/abs/10.1145/3385412.3386037), so you can do any calculation on it, and then ask for any number of digits (keep scrolling the result), and it will always be correct. The only downside is that they're a little too slow to use in a lot of performance-sensitive numerical applications.