%%%%%%%%% Cancellation %%%%%%%%%%%%%%%%%%%
% try to evaluate this as x-->0
%
% f(x)=(1-cos(x))/x^2 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% we know that \lim_{x\to 0} (1-cos(x))/x^2 = 1/2 
% So, we take a small value of x: 
x=1.2*10^(-5)
% we check that cos(x) is close to 1:  
cos(x)
% therefore, we expect massive cancellation in (1-cos(x))
(1-cos(x))/x^2
% the result is not bad: 0.49999973297490
% this is correct up to  
abs(1/2-(1-cos(x))/x^2)
% approximately 0.000000267025, so the error is about 10^{-6}
%
% But the thing is that arithmetics is done and numbers 
% are kept in the double precision: ~10^{-18} There is a huge loss of
% accuracy due to cancellation:
%%%%%%%
% To make the problem really obvious, let us imagine that 
% we can only compute cos(x) correct up to 10 significant digits, so 
c=0.9999999999
% with ten correct digits in c, the result is wrong in the first digit:
(1-c)/x^2