Computation Contest #7 Results and Solution

Solution

The problem of this contest was to test two methods to approximate pi and measure how accurate they get compared to how many digits it gets right.
I wrote a simple code which uses the following two methods:

1. Count how many grid dots are in a circle(can be done in linear time when using the circle function).
2. Use the following sequence to apprixmate it:

I decided to go with a maximum accuracy of twenty binary digits. Here is my code

import javax.swing.JFrame;
import java.awt.Graphics;
import java.awt.Color;

public class abcd extends JFrame {
    int[] firstMethod;
    int[] secondMethod;
    public abcd() {
        setSize(800, 400);
        setVisible(true);
        
        doFirstMethod();
        doSecondMethod();
        repaint();
    }
    // return accuracy in bits:
    public int getAccuracy(double approx) {
        if((int)approx != 3)
            return 0;
        long approxL = Double.doubleToLongBits(approx);
        long piL = Double.doubleToLongBits(Math.PI);
        long mantissaStart = 0x0008000000000000L;
        int prec = 0;   
        for(long i = mantissaStart; i >= 0; i >>= 1) {
            if((approxL & i) != (piL & i))
                return prec;
            prec++;
        }
        return prec;
    }
    public void doFirstMethod() {
        firstMethod = new int[20];
        int index = 0;
        for(int i = 0; index < 20; i += 1+(i>>3)) {
            double π = countDots(i);
            int numDigits = getAccuracy(π);
            if(numDigits >= index+1) {
                // Repeat the calculation a couple of times to increase time accuracy
                long t1 = System.nanoTime();
                for(int j = 0; j < 1000000; j++)
                    getAccuracy(π);
                long t2 = System.nanoTime();
                firstMethod[index] = (int)((t2-t1)/1000000.0);
                index++;
            }
        }
    }
    public void doSecondMethod() {
        secondMethod = new int[20];
        int index = 0;
        for(int i = 0; index < 20; i += 1+(i>>3)) {
            double π = sumUp(i);
            int numDigits = getAccuracy(π);
            if(numDigits >= index+1) {
                // Repeat the calculation a couple of times to increase time accuracy
                long t1 = System.nanoTime();
                for(int j = 0; j < 1000000; j++)
                    getAccuracy(π);
                long t2 = System.nanoTime();
                secondMethod[index] = (int)((t2-t1)/1000000.0);
                index++;
            }
        }
    }
    
    // Use the partial sums of the infinite sequence sum 1/n² = π²/6:
    public double sumUp(int n) {
        double sum = 0;
        for(int i = 1; i <= n; i++) {
            sum += 1.0/i/i;
        }
        return Math.sqrt(sum*6);
    }
    
    // Counts how many grid points lie in-/outside a quarter circle. Kind of similar to integrating the circle function.
    public double countDots(int r) {
        long num = 0;
        long rSqr = r*r;
        for(int i = 0; i < r; i++) {
            num += (long)(Math.sqrt(rSqr-i*i)); // Use the floor function to find the number of grid points inside the circle at this position.
        }
        return 4*num/(double)rSqr;
    }
    
    public void paint(Graphics g) {
        g.setColor(Color.BLACK);
        for(int i = 1; i < 20; i++) {
            g.drawLine(i*10, 200-firstMethod[i-1], i*10+10, 200-firstMethod[i]);
            g.drawLine(200+i*10, 200-secondMethod[i-1], 200+i*10+10, 200-secondMethod[i]);
        }
    }
    
    public static void main(String[] args) {
        new abcd();
    }
}

This code outputs the following two graphs:

Not much to see except from the fact that both graphs are growing kind of linear.
↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓

List of participants with their entries:

Name	Solution found	Comment
@lallo	Integrating the circle function	Strange that your function seems to grow exponentially, while mine are at least on the scale I observed mostly linear. Probably python is doing something strange?

↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓

Winners:

Congratulations @lallo, you won 2 SBI!

↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓

The next contest starts tomorrow. Don't miss it!