Inputs: posex1 is a float that represents the x-axis coordinate of your robot, posey1 is a float that represents your robot's y-axis coordinate. posex2 and posey2 are floats that represent the x and y of your robot's goal. Lastly, theta represents your robot's current pose angle.
Firstly, the angle_goal from your robot's coordinate (not taking its current angle into account) is calculated by finding the arc tangent of the difference between the robot's coordinates and the goal coordinates.
In order to decide whether your robot should go left or right, we must determine where the angle_goal is relative to its current rotational direction. If the angle_goal is on the robot's left rotational hemisphere, the robot should rotate left, otherwise it should rotate right. Since we are working in Radians, π is equivilant to 180 degrees. To check whether the angle_goal is within the left hemisphere of the robot, we must add π to theta (the robot's current direction) to get the upperbound of the range of values we want to check the target may be included in. If the angle_goal is between theta and that upper bound, then the robot must turn in that direction to most efficiently reach its goal.
If your robot is at (0,0), its rotational direction is 0, and it's target is at (2,2), then its angle_goal would equal = 0.785. First we check whether its current angle's deviation from the angle_goal is significant by finding the difference and seeing if its larger than 0.1. If the difference between the angles is insignificant the robot should go straight towards its goal. In this case however, angle_goal - theta (0.785 - 0) is greater than 0.1, so we know that we must turn left or right to near our angle_goal. To find out whether this angle is to the left or the right of the robot's current angle, we must add π to its current angle to discover the point between its left and right hemispheres. In this case, if the angle_goal is between theta and its goal_range, 3.14 (0(theta) + π), then we would know that the robot must turn left to reach its goal.
However, if theta (your robot's current direction) + π is greater than 2π (maximum radians in a circle) then the left hemisphere of your robot is partially across the 0 radian point of the circle. To account for that case, we must calculate how far the goal range wraps around the circle passed the origin. If there is a remainder, we check whether the angle_goal is between theta and 2π or if the angle_goal is present within the remainder of the range that wraps around the origin. If either of these conditions are met then we know that your robot should turn left to most efficiently arrive at its goal, otherwise it should turn right.