= Remez tutorial first example: exp(x) = This is a hands-on example of the Lol Remez toolkit. In this section we are going to approximate the ''exp(x)'' function using a polynomial. == Getting started == If you do not have the full Lol Engine source code, download and unpack the latest [/raw-attachment/wiki/oss/lolremez/ LolRemez] tarball. The file you should edit is `remez.cpp`. == Source code == {{{ #!cpp #include "lol/math/real.h" #include "lol/math/remez.h" using lol::real; using lol::RemezSolver; real f(real const &x) { return exp(x); } int main(int argc, char **argv) { RemezSolver<4, real> solver; solver.Run(-1, 1, f, 30); return 0; } }}} What does this mean? * we declare function `f` which returns the exponential of 'x'. * we create a `RemezSolver` object for '''4th-degree polynomials''' and '''real numbers'''. * we run the solver on the '''[-1,1] range''', approximating '''function `f`''' for '''30 iterations'''. == Compilation == If you are using LolRemez, just put the above source code in `remez.cpp` and type: {{{ make }}} == Execution == To launch the test, type: {{{ ./remez }}} After all the iterations the output should be as follows: {{{ Final error: 5.462771976237482581009771665937582411463e-4 Polynomial estimate: x**0*1.000090756764725753887362987792025308996 +x**1*9.973086551667860566788019540269306006270e-1 +x**2*4.988332174505582284710918757571761729419e-1 +x**3*1.773462612793916519454714108029230813767e-1 +x**4*4.415666059995979611944324860870682575219e-2 }}} == Using the results == The above results can be used in a more CPU-friendly implementation such as the following one: {{{ #!cpp double fastexp(double x) { const double a0 = 1.000090756764725753887362987792025308996; const double a1 = 9.973086551667860566788019540269306006270e-1; const double a2 = 4.988332174505582284710918757571761729419e-1; const double a3 = 1.773462612793916519454714108029230813767e-1; const double a4 = 4.415666059995979611944324860870682575219e-2; return a0 + x * (a1 + x * (a2 + x * (a3 + x * a4))); } }}} == Analysing the results == Plotting the real exponential function and our `fastexp` function gives the following curves: [[Image(fastexp.png, nolink)]] The curves are undistinguishable. Actually they differ by no more than 5.462772e-4, which is the value the `./remez` output gave. It can be verified on the following error curve: [[Image(fastexp-error.png, nolink)]] == Conclusion == You should now be all set up for your own minimax polynomial computation! Please report any trouble you may have had with this document to sam@hocevar.net. You may then carry on to the next section: [wiki:doc/maths/remez/tutorial-relative-error switching to relative error].