I would be really surprised if any physical calculator used binary floating point numbers instead of binary-coded decimal. The appearance of repeating bits even for relatively simple decimal numbers like 0.1 is likely to confuse and annoy a calculator user expecting an exact answer.

CORE-MATH provides correctly rounded transcendental functions. (Though that doesn't mean that this calculation will return exactly 9. And I haven't tried it.)

https://core-math.gitlabpages.inria.fr/

chalk it up to floating point accuracy

I think it's forensic in the sense that, should you perform this operation on an unknown calculator, this chart could be used to identify it.

If they have, this site has their fingerprints.

Failure to identify an identity! Straight to calculator jail!

This is a good way to find out what fugitive chip an unknown calculator is harboring.

This is the result if you do it in radians, but the test actually requires that you set the calculator to degrees mode first.

Note that all of the inverse trig functions are multivalued because the trig functions are periodic. Here, Wolfram Alpha is giving you one of the possible answers. The entire family of answers should be +/-9 + n*pi for any integer n; the sign on the 9 is due to cos being an even function.

The neural networks we use today have really terrible accuracy, and we tend to make them worse, not better, as having more neurons is better than having more precision. Human brains are also a mess, but somehow, they work, and we are usually able to correct our own mistakes.

Since by AGI, we usually mean human-like, that system should be able to self correct the same way we do.

How do humans know? usually someone corrects someone else. we have repeatability in physics, or we wait 30 years and quash convictions etc. etc.

I'd presume it could reason around the wrong answer, at least to realize something was off. Current LLMs will sometimes hallucinate that this has happened when they're "thinking".

9 degrees. arcsin(arccos(arctan(tan(cos(sin(9)))))) basically makes a set of sin-cos-tan layers that arctan-arccos-arcsin unwrap one-by-one, which should result in nothing having changed, unless the functions used weren't accurate.

You can assume that sin(9) is within the range of all the functions that are post-composed w/ it so what you end up w/ in the end is arcsin(sin(9)). Naively you might think that's 9 but you have to be careful b/c the standard inverse branch of sin is defined to be [-1, 1] → [-π/2, π/2].

Edit: The assumption is that the calculators are using specific branches of the inverse functions but that's still a choice being made b/c the functions are periodic there are no unique choices of inverse functions. You have to pick a branch that is within the domain/range of periodicity.

arcsin(arccos(arctan(tan(cos(sin(9)))))) = 9 (in degrees mode - when regular trig functions output pure numbers, those numbers get interpreted as degrees for the next function and similar for inverses - calculator style), because each intermediate lands in the principal-value domain of the next inverse (e.g., arctan(tan(x)) = x when x \in (-90°, 90°) and the intermediates happen to be in those ranges). Specifically, sin(9°) ≈ 0.156434, cos(0.156434°) ≈ 0.999996, arctan(tan(0.999996°)) = 0.999996°, arccos(0.999996)≈0.156434°, arcsin(0.156434)≈9°.