Clear[“Global  *
*Assuming  e  =  x  -  α  and     =  (f(j)  (α))/j!,  for  j=1,  2,  ,  8*
f[e_ *e  +   *e2  +   *e3  +   *e4  +   *e5  +   *e6  +   *e7  +   *e8;
*Assuming  b  =  w  -  α*
fe  =  f[e];  b  =  e  +  β  fe;  fw  =  f b ;  d  =   fw-fe β  fe ;
*Assuming  u = y-α*
u = e-Series fe/d,  {e,  0,  8 //Simplify;
(*Assuming  v = z-α*)
fu f u ;  v e- fe2 d fe-fu- fu2 fw //Simplify;
fv f v ;  fyz fu-fv / u-v ;
(*Assuming  e   Subscript x,  new -α*)
e v- fv fv v-e fyz β  d fu fe 2
   + β  d β  d (fu/fw) fv fw fv/fu) //FullSimplify
Algorithm 1: The Mathematica code for finding the asymptotic error constant in Theorem 4.