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 using a polynomial.

Getting started

If you do not have the full Lol Engine source code, download and unpack the latest LolRemez tarball.

The file you should edit is remez.cpp.

Source code

#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:

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:

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:

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: switching to relative error.