public class SqrtDemo {
	public static void main(String[] args){
		System.out.println("Sqrt via bisection");
		System.out.println(ourSqrt1(75,8,9));
		System.out.println("Sqrt via Newton's method");
		System.out.println(ourSqrt2(75,8));
		System.out.println("Derivative of sin via differences");
		System.out.println(getSinDeriv(1));
		System.out.println("12345678901234567890123456789012");
		System.out.println(Integer.toBinaryString(Float.floatToIntBits(2.0F)));
		
	}
	// use a bisection method to find as accurately possible sqrt(x) 
	// given initial lower and upper bounds
	public static float ourSqrt1(float x, float lower, float upper){
		if(lower*lower > x){
			throw new IllegalArgumentException("lower is too large");
		}
		else if (upper*upper < x){
			throw new IllegalArgumentException("upper is too small");
		}
		float mid = 0;
		while(true){
			mid = (lower + upper)/2;
			System.out.println("lower: " + lower);
			System.out.println("mid " + mid);
			System.out.println("upper: " + upper + "\n");
			// quit if cannot make any further improvement
			// because lower and upper are consecutive floats
			if(mid == lower|mid == upper){
				break;
			}
			float midsq = mid*mid;
			if(midsq < x){ //search to right
				lower = mid;
			}
			else if(midsq > x){//search to left
				upper = mid;
			}
		} 
		return mid;
	}
	// this method converges much faster
	// since we can very accurate answer very fast,
	// we use double instead of float
	// init is the initial estimate
	public static double ourSqrt2(double x, double init){
		double s = init;
		double snew = 0;
		while(true){
			// think of s and xos as being two sides of a 
			// rectangle whose area is s*xos = x
			// we want the rectangle to be "more square"
			// so we set the new x to their average
			double xos = x/s;
			System.out.println(s);
			snew = (s + xos)/2;
			if(snew == s){ 
				break;
			}
			s = snew;
		}
		return s;
	}
	// this illustrates cancellation
	public static double getSinDeriv(double x){
		// suppose we have Math.sin but not cos
		double h = 1;
		double divDif = 0;
		for(int j=0;j<20;j++){
			h = h/10;

			double d1 = Math.sin(x + h);
			double d2 = Math.sin(x);
			divDif = (d1 - d2)/h;
			System.out.println("h = " + h + "\n" + d1 + "\n"+ d2 + "\n" + divDif);
		}
		System.out.println("right answer is\n"+ Math.cos(x));
		return divDif;
	}

}