Iter[x_]  := 
       {y = x, oldx = 0, 
          While[Abs[oldx - y] > 10^(-5), {oldx = y, 
              y = y/2  +  1/y}], y}[[4]]

newx = 0

newx = 0; x = 1.5

Iter[x]

x = 1.5

oldx = 0

While[Abs[oldx - x] > 10^(-5), {oldx = x, x = x/2  +  1/x}]

x

oldx

Abs[oldx - x]

Iter[1.5]

x = 1.5

Iter[x]

x

y