(a)
M(a,b)>=a,

(b)
M(a,b)<a+0.5b

(c)
r^2=(M-a)^2=M^2+a^2-2*M*a=2*a^2+b^2-2*M*a;

(d)
b^2=r^2+2*a*r, (0<=r<0.5b)

(e)
(r+d)^2+2*a*(r+d)=(r^2+2*a*r)+(d^2+2*r*d+2*a*d)

Figure 1: Algorithm description.