Exercise "LaTeX"

• Debian based systems: install Tex Live distribution with the command

  sudo apt-get install texlive texlive-latex-extra texlive-font-utils
  

• MacOS with HomeBrew: install MacTeX distribution. It seems that you first need to install homebrew cask and then

  brew cask install mactex
  

  or

  brew cask install basictex # allegedly a small but functional version of MacTeX
  

1. (Mandatory) Consider the following "quick-and-dirty" implementation of the exponential function,

   double ex(double x){
   if(x<0)return 1/ex(-x);
   if(x>1./8)return pow(ex(x/2),2);
   return 1+x*(1+x/2*(1+x/3*(1+x/4*(1+x/5*(1+x/6*(1+x/7*(1+x/8*(1+x/9*(1+x/10)))))))));
   }
   

   Make a one-/two- page report, in LaTeX, about this implementation along the following lines: i) introduce the exponential function; ii) explain why this implementation might actually work; iii) test whether it works in practice. There should be plots in your report.

2. (Optional) Is there any numerical advantage in using this convoluted expression for the Taylor series,

   1+x*(1+x/2*(1+x/3*(1+x/4*(1+x/5*(1+x/6*(1+x/7*(1+x/8*(1+x/9*(1+x/10)))))))));
   

   instead of the ordinary

   1+x+x2/2!+x3/3!+x4/4!+...
   

   Is there any numerical advantage in inverting the sign of the argument?

3. (Optional) Make a complex version of this implementation,

   complex cex(complex z){
   ...
   }