when we subtract the quantities ac and bdwhich we have calculated in the previous recursive stepwe get the quantity we neednamely (ad bc)this shows that we can indeed compute the sum of ad bc without separately computing each of the individual quantities in summarywe can improve on equation by reducing from four recursive steps to three these three steps are as follows recursively calculate ac recursively calculate bd recursively calculate ( + )( dand subtract ac and bd the following example shows python implementation of the karatsuba algorithmfrom math import log def karatsuba( , )the base case for recursion if or return * #sets nthe number of digits in the highest input number max(int(log ( )+ )int(log ( )+ )rounds up / n_ int(math ceil( )#adds if is uneven if = else #splits the input numbers ab divmod( **n_ cd divmod( **n_ #applies the three recursive steps ac karatsuba( ,cbd karatsuba( ,dad_bc karatsuba(( + ),( + )ac bd #performs the multiplication return ((( ** )*acbd (( **n_ )*(ad_bc))