How would you convert circular joystick input to a square overlay?

I'm having an issue converting joystick input to match a square screen. If you directly translate input to the screen, you will get an oval, what I want is a dynamic equation or method for also filling in the corners. I've attempted to go about this several ways and was wondering if anyone has encountered this before. joystick input is debugged, I have the correct values from joystick.value.

 // Perfect Oval matching rectangular screen
 public void DirectTranslation()
 {
     // Input Extremes: (1,0) (0.7, 0.7) (0, 1) (-0.7, 0.7) (-1,0) (-0.7, -0.7) (0,-1) (0.7, -0.7)
     float x = joystick.HorizontalInput;
     float y = joystick.VerticalInput;

     float xPos = x * Screen.width / 2;
     float yPos = y * Screen.height / 2;

     reticle.rectTransform.anchoredPosition = new Vector2(xPos, yPos);
 }

This was my first attempt to make up for the lost space when in a corner, but ended up with a cross cutout. It perfectly reaches the corners, but skips over any input value from -0.3 to 0.3 (I understand why, but could not get it to be a perfect fix):

 // Perfect to the corners, but a cross of null space when x || y are < 0.3
 public void Attempt1Fix()
 {
     float x = joystick.HorizontalInput;
     float y = joystick.VerticalInput;

     float inputMagnitude = Mathf.Abs(x) + Mathf.Abs(y);

     float xRemainder = 0;
     float yRemainder = 0;
     
     xRemainder = 1 - Mathf.Abs(x);
     xRemainder = Mathf.Clamp(xRemainder, 0, 0.3f);

     yRemainder = 1 - Mathf.Abs(y);
     yRemainder = Mathf.Clamp(yRemainder, 0, 0.3f);

     float xPos = x * Screen.width / 2;
     float yPos = y * Screen.height / 2;

     float xExtenstion = xRemainder * (Screen.width / 2);
     float yExtenstion = yRemainder * (Screen.height / 2);

     // Q1: Top Right
     if (x > 0 && y > 0) reticle.rectTransform.anchoredPosition = new Vector2(xPos + xExtenstion, yPos + yExtenstion);

     // Q2: Top Left
     if (x < 0 && y > 0) reticle.rectTransform.anchoredPosition = new Vector2(xPos - xExtenstion, yPos + yExtenstion);

     // Q3: Bottom Left
     if (x < 0 && y < 0) reticle.rectTransform.anchoredPosition = new Vector2(xPos - xExtenstion, yPos - yExtenstion);

     // Q4: Bottom Right
     if (x > 0 && y < 0) reticle.rectTransform.anchoredPosition = new Vector2(xPos + xExtenstion, yPos - yExtenstion);
 }

So now I started searching for a solution, and I came across this https://stackoverflow.com/questions/13211595/how-can-i-convert-coordinates-on-a-circle-to-coordinates-on-a-square which seems to be a perfect solution my issue. I attempted to translate the mathematics equation to C# code, but it returns some NAN values in certain joystick positions:

 // NAN values in some cases
 public void Attemp2Fix()
 {
     float x = joystick.HorizontalInput;
     float y = joystick.VerticalInput;

     // Equation found at URL in question
     // x = ½ √(2 + u² -v² +2u√2 ) - ½ √(2 + u² -v² -2u√2 )
     // y = ½ √(2 - u² +v² +2v√2 ) - ½ √(2 - u² +v² -2v√2 )

     // Attemp to recreate equation
     float xCircle = x * Screen.width / 2;
     float yCircle = y * Screen.height / 2;
     float xSquare = 0.5f * Mathf.Sqrt(2 + (xCircle * xCircle) - (yCircle * yCircle) + 2 * xCircle * Mathf.Sqrt(2)) - 0.5f * Mathf.Sqrt(2 + (xCircle * xCircle) - (yCircle * yCircle) - 2 * xCircle * Mathf.Sqrt(2));
     float ySquare = 0.5f * Mathf.Sqrt(2 - (xCircle * xCircle) + (yCircle * yCircle) + 2 * xCircle * Mathf.Sqrt(2)) - 0.5f * Mathf.Sqrt(2 - (xCircle * xCircle) + (yCircle * yCircle) - 2 * xCircle * Mathf.Sqrt(2));

     // Then apply math to reticle position
     reticle.rectTransform.anchoredPosition = new Vector2(xSquare, ySquare);
 }