mathematics geometry path-finding
This is a fairly simple question in concept. Shouldn't be too hard to make a simple program for - the tricky part is finding how to shave off those extra characters. Good luck!

## Task

Input is a string of indeterminate length, composed of any number of eight characters; [ ⭢⭠⭡⭣⭧⭨⭦⭩ ]

For a pointer starting at (0,0) on a Cartesian plane, of the form (X, Y), below is the list of what each of the arrows will do to the pointer.

|Input | Hex Value | Action | Cartesian|
|:-:|:-:| :-- |:-:|
|⭠|0x2b60| X decreases by 1.|(-1,0)
|⭡|0x2b61| Y increases by 1.|(0,+1)|
|⭢|0x2b62| X increases by 1.|(+1,0)|
|⭣|0x2b63| Y decreases by 1.|(0,-1)|
|⭦|0x2b66| Perform ⭡ and ⭠.|(-1,+1)|
|⭧|0x2b67| Perform ⭡ and ⭢.|(+1,+1)|
|⭨|0x2b68| Perform ⭣ and ⭢.|(+1,-1)|
|⭩|0x2b69| Perform ⭣ and ⭠.|(-1,-1)|

The output should print where this pointer ends up.

## Sample Test Cases

| Input | Output|
| :-: | :-: |
|⭠⭠⭣⭧⭨⭢⭢⭩⭡ | (1,-1)|
|⭡⭣⭨⭨⭦⭩⭦⭩⭩⭠⭡⭡ | (-4,-1)|
|⭡⭡⭡ | (0,3)|
| [blank string] | (0,0) |
Top Answer
# [APL (Dyalog Unicode)], 28 bytes ([SBCS])


    +/∘×9 11∘.○0j¯1*.5⊥7 10⊤⎕UCS

[Try it online!][TIO-k69z6lrw]

[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-k69z6lrw]: https://tio.run/##SyzI0U2pTMzJT///X1v/UceMw9MtFQwNgQy9R9O7DbIOrTfU0jN91LXUXMHQ4FHXkkd9U0Odg/@nPWqb8Ki3D8jz9H/U1XxovfGjtolAXnCQM5AM8fAEKlFQf7R2waO1Cx@tXfRo7eJHa5c9Wrv80doVj9auVOeCSS4Ay0CEF4HRSqAOdQWFR71zFQwVgLZDlC4Eq1sBRsvAiiDkSpgVMD2H1pug6EKSMlAwBgnDeQYA "APL (Dyalog Unicode) – Try It Online"

### How it works

First, observe that the decimal values have a pattern:

| Input | Hex value | Decimal value | `a=(n/10)%7` | `b=n%10` | `c=a/2+b` | `(-i)**c` |
| :-: | :-: | :-: | :-: | :-: | :-: | :-: |
| ⭠ | 0x2b60 | 11104 | 4 | 4 | 6 | $-1$ |
| ⭡ | 0x2b61 | 11105 | 4 | 5 | 7 | $i$ |
| ⭢ | 0x2b62 | 11106 | 4 | 6 | 8 | $1$ |
| ⭣ | 0x2b63 | 11107 | 4 | 7 | 9 | $-i$ |
| ⭦ | 0x2b66 | 11110 | 5 | 0 | 2.5 | $(-1+i)/\sqrt2$ |
| ⭧ | 0x2b67 | 11111 | 5 | 1 | 3.5 | $(1+i)/\sqrt2$ |
| ⭨ | 0x2b68 | 11112 | 5 | 2 | 4.5 | $(1-i)/\sqrt2$ |
| ⭩ | 0x2b69 | 11113 | 5 | 3 | 5.5 | $(-1-i)/\sqrt2$ |

If we represent the movements with a complex number, we can see that clockwise rotation (power of `-i`) on ⭠ and ⭦ gives the other arrows in the respective group. And ⭦ can also be derived from ⭠ by 45° rotation followed by scaling.

    +/∘×9 11∘.○0j¯1*.5⊥7 10⊤⎕UCS
                            ⎕UCS  ⍝ Convert to Unicode codepoints
                       7 10⊤      ⍝ a=(n/10)%7 and b=n%10 for each n
                    .5⊥           ⍝ .5×a+b for each n
               0j¯1*              ⍝ Power of -i
        9 11∘.○  ⍝ Extract real and imaginary parts
       ×         ⍝ Signum of each number; effectively scale 45deg-rotations
    +/∘          ⍝ Sum of real and imaginary part respectively
Answer #2
# [Golfscript](https://http://www.golfscript.com/), ~~71 bytes~~ 58 bytes

Line breaks are stylistic and for readability, they do not contribute to character count.

~~This is not a particularly elegant solution, but it works and is likely very shrinkable.~~

Shrunk by 13 characters using a rotating array.

My guess is a nested block loop that can somehow generate these particular arrays without needing to spell them out.

~~In a perfect world, you wouldn't need to duplicate the original array, meaning we could cut out the whole third line and another few characters.~~

Kinda half-done. Still wanna get rid of the loop, I think this can be brought down to 40 characters.

Will continue optimizing - just wanted to get a concrete solution out.

~~One character can be shaved by swapping the order of the arrays, but it's less clear, and I'd like it to be clear until I've golfed it much lower.~~

This is no longer true.

2\+.[-1 1 1 -1 -1 0 1 0]:a;


2\+. # Put an ASCII 50 at the start of the given array.
[-1 1 1 -1 -1 0 1 0]:a;
     #  Remember this encoding of directionality for later.
{    #  Open a block (X encoding)
10%  #  Mod the arrows by 10
a\=  #  Check that number against a, as the index
+    #  Add it to our running tally at the front (to the 50)
}*   #  Do this to every element in the array, iterating,
     #  and pumping out a value. This is a bit tricky to
     #  explain, but in essence, this is a weird workaround
     #  the usual loop command, since that loop outputs
     #  a bunch of non-arrayed elements, which we can't
     #  handle effectively. This lets us output one number
     #  (The sum of them all), but skips the first element.
     #  This is why we added 2 at the beginning, since it's
     #  read as ASCII 50. We subtract that later.
\    #  Take our duplicated string
     #  Repeat the previous process with the Y array.
     #  Take away that pesky 50 from the sum of both indices

Then our stack is X, Y, which gets automatically outputted.

This is more elegant that my previous solution, but the array itself takes a lot of space, and I'd like to shorten it.
Answer #3
# LuaTeX, ~~191~~ 190 bytes

This is just a simple switch cases answer. The string should be delimited by `~` on the left and `⭤` (U+2b64) on the right end.

Function definition:

\def~#1{\catcode`#1"D}~:~;~.~,~|\let|\def\let:\newcount:.:,|:{\advance.}|;{\advance,}|~#1{\ifcase\numexpr`#1-`⭠:-1|;1|:1|;-1|\2||:-1;1|:1;1|:1;-1|:-1;-1\fi~}|\2#1~{ \fi\the. \the,}\let|\or

Complete script:

\def~#1{\catcode`#1"D}~:~;~.~,~|\let|\def\let:\newcount:.:,|:{\advance.}|;{\advance,}|~#1{\ifcase\numexpr`#1-`⭠:-1|;1|:1|;-1|\2||:-1;1|:1;1|:1;-1|:-1;-1\fi~}|\2#1~{ \fi\the. \the,}\let|\or


.0,0%reset counters

.0,0%reset counters

.0,0%reset counters

Output (as PDF):

Directions on a Cartesian Plane
First question - of course, that's how function input works and is nearly universally allowed.  
Second question - you may freely assume your program may be closed between each call.
@Mathgeek must a function work multiple times, or can it leave some variables unequal to zero, requiring them to be reset prior to the next call?
Skillmon replying to Mathgeek
is a function call like `⭠⭠⭣⭧⭨⭢⭢⭩⭡~` legal (`~` being the function)?
This set is fixed.
@Mathgeek can we use a different set of characters as input, or is this set fixed?
Bubbler replying to Adám
I don't think there's a short and foolproof way to get `0J0` (before text replace) for the empty input.
@Bubbler You might be able to save bytes by generating complex numbers and then `'J'⎕R','⍕`
I don't think so. Punctuation would entail a period, a comma, a slash, asterisk, octothorpe, bang, etc. An alphanumberic character is not punctuation.
Adám replying to Mathgeek
>or any variation thereof with different punctuation, etc.   

Would `X,Y` but with a letter instead of the `,` be acceptable?
On that note, if you somehow made a program using Cantor pairing to be lower than evaluating via Cartesian means, I'd be wholly impressed.
That's a simplification of the actual thing here - a complex plane is a Cartesian plane, but a Cartesian plane isn't necessarily a complex plane. I could use Polar coordinates too, but that isn't representative of "all" planes. The main reason for this is; "Would the output be reasonable if the question was asked in more dimensions"? Polar coordinates might be, but complex numbers are a modified interpretation of the cartesian plane, and act as a vector system, not as a coordinate system (even though in practice they tend to be used similarly). I also wouldn't allow [Cantor Pairing](https://en.wikipedia.org/wiki/Pairing_function#Cantor_pairing_function) on a strictly-positive-plane question either, even though it's "a valid representation of natural coordinates on a Cartesian Plane".
Bubbler replying to Mathgeek
Just a note that [a complex number is a mathematically valid representation of coordinates on the Cartesian plane](https://en.wikipedia.org/wiki/Complex_plane), and, on CGCC.SE,  complex numbers are used so often on 2D-themed challenges that we (coming from SE) usually allow them as one of valid I/O formats. (CGCC.SE has a culture of "flexible I/O as much as possible" so that we can minimize the hassle of parsing/formatting and focus on golfing the task at hand.)
The output must be an ordered pair of integers. You can format the however you like, but complex numbers are not a proper output. Accepted outputs could be  
(X, Y) or (X,Y) or (X Y) or X, Y or X,Y or X.Y or X. Y or X Y or any variation thereof with different punctuation, etc. non-cartesian ordered pairs are not valid outputs.
@Mathgeek May I output a complex number instead of the Cartesian coordinates?