A Grimoire of Functions | Hacker News Reader

A Grimoire of Functions

fredrikj.net

142 pointsakalin21 comments

Seconded! Fredrik is a hero.

Favorite line:

"Euler's formula... Discovered around 1740 by the Swiss Sorcerer Supreme, Leonhard Euler, this formula can transmute exponentials into trigonometric functions and vice versa. A magic formula indeed!"

Indeed! (And a mathematical IDENTITY... as are many others in this "Grimoire"...)

Great work, Fredrik!

> Discovered around 1740 by the Swiss Sorcerer Supreme

And earlier published in 1714 by the Englishman Roger Cotes.

See also Abramowitz and Stegun, of which the latest edition is https://dlmf.nist.gov/

They have a really nice changelog.

This is a delightful read. It has the neuroticism of someone whom I admire quite a lot and yet has the urge of curiosity. Thank you.

The "rewrite engine" ambitions for Fungrim seem potentially misplaced. It would probably be better to contribute to an existing open source state-of-the-art CAS like Fricas, than to make another one. The article tries to justify this in a weird way:

> Computer algebra systems can be viewed as having three components: a way to represent mathematical objects, algorithms for manipulating such objects, and user interfaces around this. The various components are usually entangled with each other without cleanly separating different levels of abstraction, so there is a lot of incidental complexity.

So? That is no reason to make another CAS, as opposed to helping refactor Fricas.

> The algorithms are incorrect by design, typically for efficiency reasons. A good example is ignoring special cases and assumptions on the variables when simplifying symbolic formulas (some examples given above). Another example is pretending that floating-point numbers are the same thing as real numbers. Calculating nonrigorously is not necessarily the wrong approach; it all depends on the application. It's sometimes better to do something fast and simple (but possibly incorrect), especially if the result can be checked afterwards. Sometimes you really want every step to be correct, even if it takes longer time and requires more complicated user input. For better or worse, current computer algebra systems are largely designed around the "fast and simple" principle, with optional correctness added more as an afterthought (and often not fully supported).

WTF! Only if one uses Wolfram Alpha exclusively!

Another comment [0] by fdej (the author) I just found by accident has some related misconceptions:

> However, nothing open source currently comes close to Mathematica for symbolic and symbolic-numerical calculus (expression simplification, symbolic integration, solving PDEs, etc.).

Try Fricas.

> Also, one of Mathematica's main strengths is providing a lot of functionality in a neat package where diverse components are very easy for the user to glue together, with automagical interfaces where the user doesn't need to know what happens behind the scenes

"Automagical" interfaces suck, they will inevitably break when you stray just a bit from the intended path. It is better to have a typed language like Fricas has.

[0] https://news.ycombinator.com/item?id=17694483

All good remarks.

> The "rewrite engine" ambitions for Fungrim seem potentially misplaced. It would probably be better to contribute to an existing open source state-of-the-art CAS like Fricas

For the record, I have contributed to several existing open source computer algebra systems (not Fricas though). I honestly don't think anyone has found the right foundations for computer algebra systems yet. Nor do I even think that there can necessarily be a "one size fits all solution" that works better than a multitude of specialized systems for different tasks. Fricas gets some things right where others don't, but it has its own weaknesses. I think it's valuable to explore different approaches. This possible future direction for Fungrim is more of a research project, and in any case, I think it's complementary to much of the work that is already being done on other open source computer algebra systems.

Sorry that this is still a bit vague. I have more specific issues in mind, but I don't have time to elaborate in this HN comment. I hope to do a better writeup in the future!

> WTF! Only if one uses Wolfram Alpha exclusively!

Or Maple, or Maxima, or SymPy, or Sage, or... even Fricas. Despite its type system, Fricas has the same problem as the rest with overzealous simplification. One of the very first examples in the Fricas book, "integrate (1/(x^2 + a),x)" gives a result that isn't valid when a = 0... And that's without even digging to look for something more subtle that will be more likely to catch the user off guard.

> Try Fricas.

I admit that I haven't tried Fricas in a long time, but I'm also confident that I wouldn't need to look hard to find examples. Here is just one quote from the Fricas book: "FriCAS can compute some forward Laplace transforms, mostly of elementary functions not involving logarithms, although some cases of special functions are handled." Mathematica, on the other hand, handles a huge number of Laplace transforms involving logarithms and special functions (largely using Meijer G-function methods in combination with internal lookup tables and many different heuristics). How well does Fricas support definite integration with piecewise functions or functions involving branch cuts? Mathematica has improved a lot in this area in the last few years (even if it isn't perfect). Of course, I'm sure Fricas has its strong areas too, and Mathematica is terrible in some areas.

> "Automagical" interfaces suck, they will inevitably break when you stray just a bit from the intended path. It is better to have a typed language like Fricas has.

Oh, I do agree completely about the problem with automagical interfaces that work until they don't :) Nevertheless, even "automagic" that breaks down at some point is great for discovery and prototyping. I think in many cases, we can do a better job providing interfaces that scale smoothly between automatic and user-controlled.

This is awesome. I have just started https://nixpulvis.com/math/10-overview as a way to index notation and LaTeX syntax for myself. I'll be sure to read though this from time to time. I was going to start adding D3 and all that, but now I'm wondering if I can just incorporate this. I too am using MathJax.

Great idea. It might be more useful if it were compatible to and integrated with jupyter notebooks which has become widely used. See https://www.jupyter.org for more information.

> [Regarding CAS-es:] Another example is pretending that floating-point numbers are the same thing as real numbers.

What kind of lousy CAS does not support arbitrarily precise numbers? For example Fricas/Axiom/Scratchpad gives arbitrarily precise results.

Arbitrarily precise is not the same as exact. There are many possible examples to illustrate why this matters. Here's one: let's say that you compute floor(f(x)) where f(x) falls right on an integer n. If you evaluate f(x) in floating-point arithmetic, you probably get n +/- epsilon and not n exactly because of rounding errors. This makes the floor function go to either n or n - 1, so you sometimes end up with an error of 1 no matter how precise the floating-point numbers.

Right, and with a little extra effort, you can make an arbitrarily small error in one of your terms give a result that can be anywhere from negative infinity to positive infinity.

Very nice! But is Python the right choice for this? It seems like the macro rewriting abilities of Lisp (especially Racket) would be a perfect fit for this project. And he's already using S-exps, so why reinvent the wheel?

That's a good question. The S-expressions are actually a trivial part of the project right now, because there is no symbolic computation going on, just composing S-expressions without any form of rewriting and then generating the website from the final S-expressions. So there is wheel reinvention, but it's just a few minutes of work :) Python happens to be convenient for the website generation, plus I use it because I have other tools in Python. The code is a bit of a mess right now, but I will clean it up gradually.

Later on I plan to make the formula database itself available in some language-agnostic form, with the idea that you could interface it from any programming language or computer algebra system you like. Both Python and Julia are on my radar. And Racket would be quite nice too, since I agree that it's a very nice language for this kind of thing!

Lisp macros are meant to rewrite S-expressions that represent source code in controlled, predictable and well-behaved ways, whereas S-expressions in Fungrim represent data and they can be transformed in generic and aggressive ways to deduce something useful.

Different purposes, with different needs and only a limited technology overlap (the theory of how to maintain program correctness through transformation is related to the theory of how to squash hidden assumptions and subtle type mismatches in algebraic formulas).

> Lisp macros are meant to rewrite S-expressions that represent source code

'source code' is slightly misleading, since the s-expressions don't need to be valid source code. What's valid is only limited by what the macro wants do. If a macro accepts the s-expression form, then it is valid.

For example that means that the macro form itself does not need to be parseable to an AST by an existing language grammar. The macro itself can parse the s-expression in arbitrary ways. If data is source for the macro, then it may do something with it...

I think you misunderstand the genericicity of macros in Lisp. Macros take one or more S-exps and return and S-exp. nothing more, nothing less.

There’s no reason why macros need to generate valid code. Though in Lisp, it’s pretty easy to make any code you want valid.

Not a bad TeX reference either, great stuff!

Awesome! This guy's a hero.

Cheers :)

a grimoire never sounded so cool

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