p = N[3, 3000];   f[x_]  :=  Cos[x] - x;
For[i = 1,  
      i <= 25,  i++,  { p = 
          p - f[p]/f'[p], }]
exactvalue = p;

p = N[3, 3000];   f[x_]  :=  Cos[x] - x;
For[i = 1,  
      i <= 13,  i++,  
{ 
        p = p - f[p]/f'[p],  
error  =  
          Abs[p  -  exactvalue],  
Print[i],  
        Print[Log[
            1.0*error]]
}
]

p0 = N[0, 3000];   p1  =  N[2, 3000];   f[x_]  :=  
      Cos[x] - x;
For[i = 1,  
      i <= 16,  i++,  
{p = 
          p1 - f[p1]*(p1 - p0)/(f[p1] - 
                    f[p0]),  
error  =  
          Abs[p  -  exactvalue],  
Print[i],  
        Print[Log[1.0*error]],  
p0  =  p1,  
        p1  =  p
}
]

p = N[1, 3000];   g[x_]  :=  Cos[x];
For[i = 1,  
      i <= 100,  i++,  
{p = 
          g[p],  
error  =  
          Abs[p  -  exactvalue],   
Print[i],  
        Print[Log[
            1.0*error]]
}
]

p = N[1, 3000];   g[x_]  :=  Cos[x];
p1  =  g[p];   p2 = g[p1];
For[i = 1,  
      i ≤ 
        100,  i++,  
{pa = 
          p - (p1 - p)^2/(p2 - 2*p1 + 
                  p),  
error = 
          Abs[pa - exactvalue], 
Print[i],   Print[
          Log[1.0*error]], 
p = p1,  p1 = p2,  
        p2 = g[p1],  
}
]

p = N[1, 3000];   g[x_]  :=  Cos[x];
For[i = 1,  
      i ≤ 
        11,  i++,  
{p1  =  g[p];   p2 = 
          g[p1]; 
pa = 
          p - (p1 - p)^2/(p2 - 2*p1 + 
                  p),  
error = 
          Abs[pa - exactvalue], 
Print[i],   Print[
          Log[1.0*error]], 
p = 
            pa; 
}
]