// For: https://forum.arduino.cc/t/basic-mathematical-equationcoding/1073852
// "Find" x by searching for it.

float bottom_x;
float bottom_y;

void setup() 
{
  Serial.begin(115200);
  Serial.setTimeout(100);  // 100 ms timeout for serial input

  // Find the bottom between -1000 and +1000 with step 1.0
  bottom_x = -1000.0;
  bottom_y = Calc(bottom_x);
  for( float x=-1000.0; x<1000.0; x+=1.0)
  {
    float y = Calc(x);
    if( y < bottom_y)
    {
      bottom_y = y;
      bottom_x = x;
    }
  }

  // Find a more accurate bottom with step 0.001
  for( float x=bottom_x-1.0; x<bottom_x+1.0; x+=0.0001)
  {
    float y = Calc(x);
    if( y < bottom_y)     // new lower bottom ?
    {
      bottom_y = y;
      bottom_x = x;
    }
  }
  
  Serial.print( "The bottom of the curve is at x=");
  Serial.print(bottom_x);
  Serial.print(" (y=");
  Serial.print(bottom_y);
  Serial.println(")");

  Serial.println("Enter a value for y (integer value)");
}

void loop() 
{
  if( Serial.available() > 0)
  {
    // using parseInt, even though I don't like that function
    long y_int = Serial.parseInt();
    
    while( Serial.available() > 0)
      Serial.read();        // remove trailing CR or LF

    float y_target = float( y_int);

    if( y_target < bottom_y)
    {
      Serial.println("That value is not in the curve");
      Serial.println("Enter a good value for y (integer value)");
    }
    else
    {
      // search upward
      Serial.print("Searching upward ...");
      delay(1000);
      Serial.println("... not implemented yet");

      // search downward
      Serial.print("Searching downward ...");
      delay(1000);
      Serial.println("... not implemented yet");

      Serial.println("Enter a new value for y (integer value)");
    }
  }
}

float Calc( float x)
{
  float y = (0.0227*x*x) - (3.1401 * x) + 177.35;
  return( y);
}