#### Table of Contents

# 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 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*:

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

#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**to`f`

*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:

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:

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:

## 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.

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