Adrian Cochrane
Fascinating, Julia implemented parts of its Jitter in Julia... Which would not be usable until a basic Jitter is bootstrapped in C++ & a custom Lisp is bootstrapped with which to interpret the full Jitter!
The standard library is otherwise not shy about language-binding existing libraries like LibPCRE, LibUV, LibGMP & Ryu. Whereas most language devs seem to enjoy reimplementing these things...
I guess if the Julia devs are writing the code they'd prefer to use Julia... That much seems typical.
Adrian Cochrane
Ooooo! They implement trigonometry themselves, I was wondering how you do that!
https://github.com/JuliaLang/julia/blob/master/base/special/trig.jl#L69
Now can any mathematicians explain why this works?
Urusan
@alcinnz Reading through this code, I'm simultaneously impressed and disgusted by the use of y² and y³ as Unicode variable names to represent y squared and y cubed. Not sure whether that's a good idea or not.
Adrian Cochrane
@wizzwizz4 Here's where it is defined: https://github.com/JuliaLang/julia/blob/52af407745f23512d48f489658bbe07cb21b5808/base/math.jl#L252
The "@" sign is how you call macros, which are evaluated at JIT-time.
wizzwizz4
@alcinnz I'm not sure where @horner comes from, but it looks like it's probably Horner's root-finding method, rather than Horner's evaluation method.
(Without knowing for certain what the definitions mean, I'm not certain how it works – but it doesn't seem incomprehensible.)
Adrian Cochrane
@wizzwizz4 O.K., I'll put some time aside to do it!
And I would be keen to fold this topic into a bonus season for my GPU videotutorial series...
Currently I'm reading: https://gowers.wordpress.com/2014/03/02/how-do-the-power-series-definitions-of-sin-and-cos-relate-to-their-geometrical-interpretations/
wizzwizz4
@alcinnz This normally takes between 10 min and an hour of staring at the page, thinking (perhaps with doodling, or fiddly algebra).
wizzwizz4
@alcinnz Okay, looking at it more closely? That's definitely Horner's evaluation method, implementing a Taylor series expansion: https://en.wikipedia.org/wiki/Taylor_series
I recommend trying to solve this problem yourself, though, rather than looking at Wikipedia. It's a simple, fun piece of mathematics – and as with all mathematics, you can only discover it once:
Using algebra, find the polynomials f(x) where f(0)=sin(0). Then, add the condition f'(0)=sin'(0), and f''(0)=sin''(0), and the third derivative….
Silmathoron
@alcinnz
Honestly, if you figure it out, I'm interested: I understand the maths but the actual implementation is a bit strange ^^"
They approximate sin by its Taylor expansion up to the 13th order (x^13) and then use Horner's method to evaluate the polygon (it's an optimal algorithm for polynomial evaluation).
They use the fact that all exponents are separated by 2. But I'm not sure why they separate the call (faster?) or why they define y3 after using it in the previous expression
@wizzwizz4
Adrian Cochrane
@silmathoron I can tell you right now that the `horner` macro is inlined as it JITs this `sin` function. Beyond that, I'll put some time aside this weekend to explore!
Until then, any infodumping on sin() would be appreciated!
Silmathoron
@alcinnz @wizzwizz4 well, honestly, you don't need anything beyond its Taylor expansion and you already have the approximation written above the DS variables so I think you're good.
Maybe just: it's anti-symmetric (sin(-x) = -sin(x)) and that's why the exponents are all odd.
Eamon
@urusan @alcinnz I'm learning Julia now and abusing Unicode in variable names (and for operators!) is part of the culture. But I like it when languages are weird (e.g., I love that R uses `<-` as its assignment operator, or that `.` can be placed used in names without meaning anything special).
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