Before I can provide a complete answer, it's essential to address the missing information in your question. Specifically, how are numbers like 4000 encoded in hexadecimal format within the result?
Despite the lack of details, I believe I can deduce it from the example you've given.
Deciphering the Unique Numeric Encoding
From what I could gather, it appears that numbers are encoded with two bytes (equivalent to four hex characters) each. Interestingly, the most significant bits of these two bytes (bits 7 and 15) do not affect the value—they are consistently zero.
Moreover, the remaining 14 bits follow an offset binary encoding scheme, where the top bit of those 14 represents the inverted sign bit.
This implies that '4000' translates to zero, '0000' corresponds to -8192 (the minimum value), and '7F7F' signifies 8191 (the maximum value). Notably, the second-to-last character must always be less than or equal to 7 since exceeding this threshold would set a unused bit in this particular custom encoding.
Deriving this information was notably challenging due to the limited data provided in your inquiry.
Analysis of the Provided Example
Your provided input example may be segmented as shown below:
opcode | argument(s)
-------+----------------------------
"F0" |
"A0" | "4000" "417F" "4000" "417F"
"C0" | "4000" "4000"
"80" | "4001"
"C0" | "5F20" "5F20"
"80" | "4000"
Following the numeric conversion explanation above, we can convert this breakdown into:
opcode | argument(s)
-------+------------
240 |
160 | 0 255 0 255
192 | 0 0
128 | 1
192 | 4000 4000
128 | 0
Subsequently, translating these values is simply a matter of executing the necessary instructions.
The underlying algorithm first deciphers the input string into commands. Each command comprises an opcode alongside any numerical arguments present.
These commands then instruct the generation of the required output by monitoring the status of the pen (whether it's up or down) and the current coordinates:
function decode(hex) {
let commands = [];
let command;
for (let i = 0, len; i < hex.length; i+=len) {
// Opcodes occupy 1 byte (two hex characters), whereas
// numbers span 2 bytes (four characters)
len = hex[i] >= "8" ? 2 : 4;
let num = parseInt(hex.slice(i, i+len), 16);
if (len === 2) { // opcode
command = []; // initialize a new command
commands.push(command);
} else { // number
// The prescribed format differs slightly—this seems to clarify it:
num = ((num & 0x7F00) >> 1) + (num & 0x7F) - 0x2000;
}
command.push(num); // append opcode or argument to the current command
}
return commands;
}
function disassemble(hex) {
let isPenDown = false;
let x = 0, y = 0;
let output = "";
let commands = decode(hex);
for (let [opcode, ...args] of commands) {
if (opcode === 0xF0) {
x = y = 0;
isPenDown = false;
output += "CLR;\n";
} else if (opcode === 0x80) {
isPenDown = args[0] > 0;
output += "PEN " + (isPenDown ? "DOWN" : "UP") + ";\n";
} else if (opcode === 0xA0) {
output += "CO " + args.join(" ") + ";\n";
} else if (opcode === 0xC0) {
let allPos = "", lastPos;
for (let i = 0; i < args.length; i+=2) {
x += args[i];
y += args[i+1];
lastPos = ` (${x}, ${y})`;
if (isPenDown) allPos += lastPos;
}
output += "MV" + (allPos || lastPos) + ";\n";
} // else: ignore unknown commands
}
return output;
}
// Sample:
console.log(disassemble("F0A04000417F4000417FC040004000804001C05F205F20804000"));
Further Considerations
In the problem screenshot towards the conclusion, there's a reference to clipping movements within a bounding box. This extends beyond your query relating to decoding the hexadecimal input. For those interested, exploring Q&A focused on computing line segment intersections, such as discussions found here, might prove insightful.