math java-processing arcball quaternion trig, float rounding error

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I am modifying an arcball class to rotate 1 degree each time rollforward() is called. I am having difficulty reading the code, but I believe I need to write an alternative to XY_to_sphere() where; let point 1 = v1, point 2 = v2, such that

pi/180 = (v1.x * v2.x + v1.y * v2.y + v1.z * v2.z) / (abs(v1.x) * abs(v2.x) + abs(v1.y) * abs(v2.y) + abs(v1.z) * abs(v2.z))

formula for arcball angle

<-update edit->

I tried simply adding one degree((sin(PI/180) * radius)/2 converted to the y axis with; (in rollforward())

v_drag = XY_to_sphere(center_x, center_y - ((sin(PI/180) * radius))/2);

and after 100 rolls it is off by a few degrees(this might be a floating point rounding error ill try and find it but I am going to have to learn some things) here is the difference at 200 full rotations (should be mirror image of top cube)

enter image description here

(new question) It disappears for about 1 degree when it reaches a full rotation and I do not know why. (solved! i set radius to 1 while testing thanks @laancelot)

I am not sure if my original idea, the solution above will work better

now I am going to try and set it up so it resets after it reaches 360 degrees to circumvent the floating point error

</-update edit->

arcball class

// Ariel and V3ga's arcball class with a couple tiny mods by Robert Hodgin and smaller mods by cubes

class Arcball {
  float center_x, center_y, radius;
  Vec3 v_down, v_drag;
  Quat q_now, q_down, q_drag;
  Vec3[] axisSet;
  int axis;
  float mxv, myv;
  float x, y;
  
  Arcball(float center_x, float center_y, float radius){
    this.center_x = center_x;
    this.center_y = center_y;
    this.radius = radius;

    v_down = new Vec3();
    v_drag = new Vec3();

    q_now = new Quat();
    q_down = new Quat();
    q_drag = new Quat();

    axisSet = new Vec3[] {new Vec3(1.0f, 0.0f, 0.0f), new Vec3(0.0f, 1.0f, 0.0f), new Vec3(0.0f, 0.0f, 1.0f)};
    axis = -1;  // no constraints...    
  }

  void rollforward(){
    q_down.set(q_now);
    v_down = XY_to_sphere(center_x, center_y);
    q_down.set(q_now);
    q_drag.reset();
    
    v_drag = XY_to_sphere(center_x, center_y - ((sin(PI/180) * radius))/2);
    q_drag.set(Vec3.dot(v_down, v_drag), Vec3.cross(v_down, v_drag)); 
  }

/*
  void mousePressed(){
    v_down = XY_to_sphere(mouseX, mouseY);  
    q_down.set(q_now);
    q_drag.reset();
  }

  void mouseDragged(){
    v_drag = XY_to_sphere(mouseX, mouseY);
    q_drag.set(Vec3.dot(v_down, v_drag), Vec3.cross(v_down, v_drag));
  }
*/
  void run(){
    q_now = Quat.mul(q_drag, q_down);
    applyQuat2Matrix(q_now);
    
    x += mxv;
    y += myv;
    mxv -= mxv * .01;
    myv -= myv * .01;
  }
  
  Vec3 XY_to_sphere(float x, float y){
    Vec3 v = new Vec3();
    v.x = (x - center_x) / radius;
    v.y = (y - center_y) / radius;

    float mag = v.x * v.x + v.y * v.y;
    if (mag > 1.0f){
      v.normalize();
    } else {
      v.z = sqrt(1.0f - mag);
    }

    return (axis == -1) ? v : constrain_vector(v, axisSet[axis]);
  }

  Vec3 constrain_vector(Vec3 vector, Vec3 axis){
    Vec3 res = new Vec3();
    res.sub(vector, Vec3.mul(axis, Vec3.dot(axis, vector)));
    res.normalize();
    return res;
  }

  void applyQuat2Matrix(Quat q){
    // instead of transforming q into a matrix and applying it...

    float[] aa = q.getValue();
    rotate(aa[0], aa[1], aa[2], aa[3]);
  }
}

static class Vec3{
  float x, y, z;

  Vec3(){
  }

  Vec3(float x, float y, float z){
    this.x = x;
    this.y = y;
    this.z = z;
  }

  void normalize(){
    float length = length();
    x /= length;
    y /= length;
    z /= length;
  }

  float length(){
    return (float) Math.sqrt(x * x + y * y + z * z);
  }

  static Vec3 cross(Vec3 v1, Vec3 v2){
    Vec3 res = new Vec3();
    res.x = v1.y * v2.z - v1.z * v2.y;
    res.y = v1.z * v2.x - v1.x * v2.z;
    res.z = v1.x * v2.y - v1.y * v2.x;
    return res;
  }

  static float dot(Vec3 v1, Vec3 v2){
    return v1.x * v2.x + v1.y * v2.y + v1.z * v2.z;
  }
  
  static Vec3 mul(Vec3 v, float d){
    Vec3 res = new Vec3();
    res.x = v.x * d;
    res.y = v.y * d;
    res.z = v.z * d;
    return res;
  }

  void sub(Vec3 v1, Vec3 v2){
    x = v1.x - v2.x;
    y = v1.y - v2.y;
    z = v1.z - v2.z;
  }
}

static class Quat{
  float w, x, y, z;

  Quat(){
    reset();
  }

  Quat(float w, float x, float y, float z){
    this.w = w;
    this.x = x;
    this.y = y;
    this.z = z;
  }

  void reset(){
    w = 1.0f;
    x = 0.0f;
    y = 0.0f;
    z = 0.0f;
  }

  void set(float w, Vec3 v){
    this.w = w;
    x = v.x;
    y = v.y;
    z = v.z;
  }

  void set(Quat q){
    w = q.w;
    x = q.x;
    y = q.y;
    z = q.z;
  }

  static Quat mul(Quat q1, Quat q2){
    Quat res = new Quat();
    res.w = q1.w * q2.w - q1.x * q2.x - q1.y * q2.y - q1.z * q2.z;
    res.x = q1.w * q2.x + q1.x * q2.w + q1.y * q2.z - q1.z * q2.y;
    res.y = q1.w * q2.y + q1.y * q2.w + q1.z * q2.x - q1.x * q2.z;
    res.z = q1.w * q2.z + q1.z * q2.w + q1.x * q2.y - q1.y * q2.x;
    return res;
  }
  
  float[] getValue(){
    // transforming this quat into an angle and an axis vector...

    float[] res = new float[4];

    float sa = (float) Math.sqrt(1.0f - w * w);
    if (sa < EPSILON){
      sa = 1.0f;
    }

    res[0] = (float) Math.acos(w) * 2.0f;
    res[1] = x / sa;
    res[2] = y / sa;
    res[3] = z / sa;
    return res;
  }
}

my code, use w key to roll the cube (loop automates one rotation)

Arcball arcball;

int i;
int test_count = 0;

boolean[] keys = new boolean[13];
    final int w = 0;


void setup() {
  size(900, 700, P3D); 
  frameRate(90);
  noStroke();
  arcball = new Arcball(width/2, height/2, 100);   //100 is radius
}

void draw() {
  lights();
  background(255,160,122);
  
 print(" \n degree = " + i );
  i++;
  if(i <= (360 * 1)) { arcball.rollforward(); }
  else { print(" break"); }
  
  if(keys[w]) { 
    arcball.rollforward(); 
    test_count = test_count + 1;
    print(" " + test_count);
  }

  translate(width/2, height/2-100, 0);
  box(50);
   
  translate(0, 200, 0);
  arcball.run();
  box(50);  
                           
}

void keyPressed() {
  switch(key) {
    case 119: 
        keys[w] = true;
        break;
  }
}
void keyReleased() {
  switch(key) {
    case 119: 
        keys[w] = false;
        break;
    } 
}
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using my idea in the question to reset every 2*PI

  if(keys[w]) { 
    arcball.rollforward(PI/180);
    degreeW_count = degreeW_count + 1;
  }

  if(degreeW_count == 360) {
    arcball = new Arcball(width/2, height/2, 100);
    degreeW_count = 0;
  }

this totally circumvents any rounding error that would accumulate with any data type using irrational numbers and periodic functions!