love this -- still recall how eye-opening my first course was on numerical solving and appreciating for the first time yet again the sheer potential of compute

Thank you! Yes it does feel a bit magical, both the mathematical aspects and the fact that it all boils down to a few lines of code in practice.

Sollya is probably the best modern tool for doing this. Under the hood, it does a Remez approximation followed by LLL to quantize to floating point. No use of Chebyshev directly.

The problem here likely is, that your first expression was not well defined for x=0 and seemingly the poor approximation code stumbled over it. Shame on it!

> Math.sin(x)/x (the sinc function) for 7 terms over [-3,3] gives coefficients c0...c6 that are all NaNs. Is this a bug?

That wouldn't exactly be a bug. The code is undoubtedly calculating the Chebyshev coefficients by evaluating the function on something like x_j = (xmin) + (xmax - xmin)/2(1 + cos(pi[0..j-1]/(j-1)). If one of those grid points happens to be exactly 0, it will try to evaluate Math.sin(0)/0, giving the NaN.

Another workaround is to have a slightly asymmetric range, such as [-3,+3.0000001]

Yes, that is a bug, good catch. The app should show an error if the function is not defined in all chebyshev nodes. Like you have already discovered, it's easy to work around this issue for now.

Thanks for the kind words. I also found it surprisingly hard to find working Chebyshev approximation code. Hopefully this project will change that :)

I'll second this. Their methods are very powerful and very fast. For those out of the loop, the Chebyshev (and ultra-spherical) machinery allows a very accurate (machine precision) approximation to most functions to be computed very quickly. Then, this representation can be manipulated more easily. This enables a variety of methods such as finding the solution to differential algebraic equations to machine precision or finding the global min/max of a 1-D function.

I believe they use a different algorithm now, but the basic methodology that used to be used by Chebfun can be found in the book Spectral Methods in Matlab by Trefethen. Look at chapter 6. The newer methodology with ultraspherical functions can be found in a SIAM review paper titled, "A Fast and Well-Conditioned Spectral Method," by Olver and Townsend.

Neural networks often have trigonometric functions internally, so it would be massively more computation than necessary.

If you have a few spare CPU cycles, a hybrid approximation could start with a sparse lookup table of values as the initial guess for a few rounds of a numerical approximation technique. Or you just store the first few coefficients of a polynomial approximation (as in the OP's work).

A neural network is essentially just a curve fitter, so yeah. You might find this[1] video illuminating.

The main strength of a neural network comes into play when there's a lot of different inputs, not just a handful. For the simpler cases like sin(x) we have other tools like the one posted here.

[1]: https://www.youtube.com/watch?v=FBpPjjhJGhk But what is a neural network REALLY?

Obviously you could train some kind of neural net to calculate any function, but this would never make sense for a well-known function like sine. Neural nets are a great solution when you need to evaluate something that isn't easy to analyze mathematically, but there are already many known techniques for calculating and approximating trigonometric functions.

Training a neural net to calculate sines is like the math equivalent of using an LLM to reverse a string. Sure, you *can*, but the idea only makes sense if you don't understand how fundamentally solvable the problem is with a more direct approach.

It's always worth looking if mathematicians already have a solution to a problem before reaching for AI/ML techniques. Unfortunately, a lot of effort is probably being spent these days programming some kind of AI/ML to solve problems that have a known, efficient, maybe even proven optimal solution that developers just don't know about.

The usual conservation trick is to have a table from 0 to PI/2 and use two additional index bits to generate the other three quadrants.

Take a look at CORDIC if you aren't familiar with it; that was a common trig hack back in the day, and still sees some use in the embedded space.

Neural nets can be useful when you have samples of a function but no idea how to approximate it, but that's not the case here.

You could also set the x_min to 0.001 or so.

You could look into using the ChebyshevExpansion class directly. It takes as one of its arguments a callback that returns f(x) for a given x. In your case, f(x) would be your sensor values with some suitable interpolation. A more ambitious route is to add support for somehow importing tabular data into the app.

With either of those, you're still representing your polynomial as a combination of powers: 1, x, x^2, x^3, x^4, etc.

For many purposes it's much better to represent a polynomial as a combination of Chebyshev polynomials: 1, x, 2x^2-1, 4x^3-3x, etc.

(Supposing you are primarily interested in values of x between -1 and +1. For other finite intervals, use Chebyshev polynomials but rescale x. If x can get unboundedly large, consider whether polynomials are really the best representation for the functions you're approximating.)

Handwavy account of why: Those powers of x are uncomfortably similar to one another; if you look at, say, x^4 and x^6, they are both rather close to 0 for smallish x and shoot up towards 1 once x gets close to +-1. So if you have a function whose behaviour is substantially unlike these and represent it as a polynomial, you're going to be relying on having those powers largely "cancel one another out", which means e.g. that when you evaluate your function you'll often be representing a smallish number as a combination of much larger numbers, which means you lose a lot of precision.

For instance, the function cos(10x) has 7 extrema between x=-1 and x=+1, so you should expect it to be reasonably well approximated by a polynomial of degree not too much bigger than 8. In fact you get a kinda-tolerable approximation with degree 12, and the coefficients of the best-fitting polynomial when represented as a combination of Chebyshev polynomials are all between -1 and +1. So far, so good.

If we represent the same function as a combination of powers, the odd-numbered coefficients are zero (as are those when we use the Chebyshev basis; in both cases this is because our function is an even function -- i.e., f(-x) = f(x)), but the even-numbered ones are now approximately 0.975, -4.733, 370.605, -1085.399, 1494.822, -994.178, 259.653. So we're representing this function that takes values between -1 and +1 as a sum of terms that take values in the thousands!

(Note: this isn't actually exactly the best-fitting function; I took a cheaty shortcut to produce something similar to not quite equal to the minimax fit. Also, I make a lot of mistakes and maybe there are some above. But the overall shape of the thing is definitely as I have described.)

Since our coefficients will be stored only to some finite precision, this means that when we compute the result we will be losing several digits of accuracy.

(In this particular case that's fairly meaningless, because when I said "kinda-tolerable" I meant it; the worst-case errors are on the order of 0.03, so losing a few places of accuracy in the calculation won't make much difference. But if we use higher-degree polynomials for better accuracy and work in single-precision floating point -- as e.g. we might do if we were doing our calculations on a GPU for speed -- then the difference may really bite us.)

It also means that if we want a lower-degree approximation we'll have to compute it from scratch, whereas if we take a high-degree Chebyshev-polynomial approximation and just truncate it by throwing out the highest-order terms it usually produces a result very similar to doing the lower-degree calculation from scratch.

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