Exercise "epsilon"

1. Maximum/minimum representable integers.
   1. The maximum representable integer is the largest integer i for which i+1>i holds true.

      Using the while loop determine your maximum integer and compare it with int.MaxValue.

      Hint: something like

      int i=1; while(i+1>i) {i++;}
      Write("my max int = {0}\n",i);
      

      It can take some seconds to calculate.
   2. The minimum representable integer is the most negative integer i for which i-1<i holds true.

      Using the while loop determine your minimum integer and compare with int.MinValue.

2. The machine epsilon is the difference between 1.0 and the next representable floating point number. Using the while loop calculate the machine epsilon for the types float and double. Something like

   double x=1; while(1+x!=1){x/=2;} x*=2;
   float y=1F; while((float)(1F+y) != 1F){y/=2F;} y*=2F;
   

   There seem to be no predefined values for this numbers in csharp (I couldn't find it in any case). However, in a IEEE 64-bit floating-point number (double), where 1bit is reserved for the sign and 11bits for exponent, there are 52bits remaining for the fraction, therefore the double machine epsilon must be about System.Math.Pow(2,-52). For single precision (float) the machine epsilon should be about System.Math.Pow(2,-23). Check this.

3. Suppose tiny=epsilon/2. Calculate the two sums,

   sumA=1+tiny+tiny+...+tiny;
   sumB=tiny+tiny+...+tiny+1;
   

   which should seemingly be the same and print out the values sumA-1 and sumB-1. Explain the difference. Someting like

   int n=(int)1e6;
   double epsilon=Pow(2,-52);
   double tiny=epsilon/2;
   double sumA=0,sumB=0;
   
   sumA+=1; for(int i=0;i<n;i++){sumA+=tiny;}
   WriteLine($"sumA-1 = {sumA-1:e} should be {n*tiny:e}");
   
   for(int i=0;i<n;i++){sumB+=tiny;} sumB+=1;
   WriteLine($"sumB-1 = {sumB-1:e} should be {n*tiny:e}");
   

4. Implement a function with the signature

   bool approx(double a, double b, double tau=1e-9, double epsilon=1e-9)
   

   that returns true if the numbers 'a' and 'b' are equal with absolute precision τ,

   |a-b|<τ
   

   or are equal with relative precision 'epsilon',

   |a-b|/(|a|+|b|)<ε
   

   and returns false otherwise. Test it.