You aren't updating the value itself, so the calculation is the exact same on every iteration.
function out = babyloniansqrt(S,threshold)
%Write lead line here
%
% write usage instructions and explanation here
x = 1;
if nargin<2
%set default threshold if not provided
threshold = 10^-5;
end
E = inf; %set initial value to enter the loop
while E>threshold
x = (x+S./x)/2;
E = abs(S-x.^2);
end
out=x;
end
