EXERCISES concerning computer numbers
ITU KCS Introductory Programming 2019-09-23 * sestoft@itu.dk


1. Find Java types for the variables x, y, z, v, w, s and t, and values
   for them, such that:
   It holds that x+1 < x
   It holds that y != y
   It holds that z+1 == z
   It holds that v+w-w != v
   It holds that s==t && 1.0/s != 1.0/t
   
2. Write a Java program to check that your solutions are correct.

3. Find the binary representation

   of +3 and -3 as 8-bit byte, 32-bit int and 64-bit long
   of +128 and -127 as byte, int and long
   of 0.0625 and 0.625 and 62.5 and 625 as float
   of 0.01 as a 64-bit double

4. Check the results using displayInt etc from file Numbers.java.
   This file can be found at the address
   http://www.itu.dk/people/sestoft/bachelor/Numbers.java

5. Compute the following in a Java program:
   1.0/Double.POSITIVE_INFINITY and
   Math.exp(Double.NEGATIVE_INFINITY) and 
   1.0/(1.0+1.0/0.0) and
   Math.pow(0.0, -1.0)

6. Write a Java program to compute the sum of the first N reciprocals 
   both forwards, as in 
   1.0/1.0 + 1.0/2.0 + 1.0/3.0 + 1.0/4.0 + ... + 1.0/N
   and backwards, as in 
   1.0/N + 1.0/(N-1) + ... + 1.0/3.0 + 1.0/2.0 + 1.0/1.0 

   Use float (32-bit floating-point numbers) to represent the sum, and
   try with N=1000000, N=10000000 and N=100000000, that is, 1 million,
   10 million and 100 million.  

   In mathematics, the results of the two orders of summation would be
   the same, of course.  What do you observe?  Explain the results.
   Which result is more likely to be correct?

7. A naive programmer tries to compute the sum of the first N=count
   reciprocals of squares 
     1.0/(N*N) + 1.0/((N-1)*(N-1)) + ... + 1.0/9.0 + 1.0/4.0 + 1.0/1.0 
   using this Java program:

      int count = 100000;
      double sum = 0.0;
      for (int i=count; i>=1; i--)
	sum += 1.0/(i*i);
      System.out.println(sum);

   Disappointingly, this Java program prints Infinity rather than the
   expected result, which is roughly (pi^2)/6 = 1.6449.  Explain why
   the result is so wrong.

   Hint: Integer computations are to blame, not floating-point. 

8. Change the above Java program so that it computes the correct result.