Is faster better?: Consider the map

x$1->0\mathrm{.}375*$x$1+1\mathrm{.}5/$x$1-0\mathrm{.}5/($x$1*$x$1*$x$1).$

Iterate this map $($derived by Alkalsadi around $1450$AD$)$ with the starting value $2\mathrm{.}0$ until $14$ digits settle down. How many iterations are required? Now, count the number of all the additions, subtractions, multiplications and divisions you have made to obtain your approximation to sqrt$(2).$

Next do the same with Newton's method using x$1->0\mathrm{.}5*($x$1+2\mathrm{.}0/$x$1).$

Which method requires fewer number of iterations?

Which method requires fewer number of arithmetical operations to compute sqrt$(2)$ to the same precision?