[[TOC]] = Remez tutorial 1/5: exp(x) the quick way = This is a hands-on example of the Lol Remez toolkit. In this section we are going to approximate the ''exp(x)'' function on [-1,1] using a polynomial. == Getting started == If you do not have the full Lol Engine source code, download and unpack the latest [wiki:oss/lolremez LolRemez] tarball. The file you should edit is remez.cpp. == What does Remez do? == Given a function ''f'' and a range ''[a,b]'', the Remez algorithm looks for the polynomial ''P(x)'' that minimises the following error value ''E'': {{{ #!latex $\max_{x \in [a,b]}{\big\vert f(x) - P(x)\big\vert} = E$ }}} Note that ''E'' is not a parameter. It is a value that the algorithm computes together with the polynomial. Though we will see ways to fine-tune the error, a general rule is: if you want a smaller error, ask for a polynomial of higher degree. == 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, 40); return 0; } }}} What does this mean? * We declare function f which returns the exponential of x: this is the function we want to approximate. * We create a RemezSolver object for '''4th-degree polynomials''' and '''real numbers'''. As of now, no other number types are supported. * We run the solver on the '''[-1,1] range''', approximating '''function f''' to ''40 decimals of precision''. The larger the precision, the more iterations are necessary, but the process usually takes only a few seconds for small functions. == 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: {{{ Step 7 error: 5.466676005137979474524666548947155992203e-4 Polynomial estimate: x**0*1.000090000102127639946253082819502265543 +x**1*9.973092516744464320538318907902496576588e-1 +x**2*4.988351170902359155314941477995868737492e-1 +x**3*1.773452743688412268810974931504564418976e-1 +x**4*4.415551762288022300015839013797254330891e-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 = 1.000090000102127639946253082819502265543; const double a1 = 9.973092516744464320538318907902496576588e-1; const double a2 = 4.988351170902359155314941477995868737492e-1; const double a3 = 1.773452743688412268810974931504564418976e-1; const double a4 = 4.415551762288022300015839013797254330891e-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 5.46668e-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].