% Updated 200918 by Wei-Cheng Wang
% What is the purpose of this program?
% Which one of f1, f2 is a more accurate approximation of
% exp(x) - cos(x) - x near x=0? why?
% floating point algorithm in matlab is double precision by default.
% Nevertheless, newer versions of matlab contains vpa, 
% Vaiable Precision Arithmetic,
% which is based on symbolic calculation so you can specify almost
% any degree of precision upon numerical evaluation.
% It is probably built in some newer versions of octave as well
clf
a = 3.e-8; % 3.0*10^(-8)
h = a/16;
x = -a:h:a;
f1 = exp(x) - cos(x) - x;
f2 = x.^2 + x.^3/6;

x_more_accurate = vpa(x,24);
f_more_accurate = exp(x_more_accurate)-cos(x_more_accurate) - x_more_accurate;
f_double_precision = double(f_more_accurate);

figure(1), plot(x,f_double_precision,x,f1,'x',x,f2,'o')