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https://github.com/samhocevar/lolremez

📈 Polynomial Approximations using the Remez Algorithm
https://github.com/samhocevar/lolremez

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📈 Polynomial Approximations using the Remez Algorithm

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README 

        
# LolRemez



A Remez algorithm implementation to approximate functions using polynomials.



A tutorial is available [in the wiki section](https://github.com/samhocevar/lolremez/wiki).



Build instructions are available below.



## Example



Approximate `atan(sqrt(3+x³)-exp(1+x))` over the range `[sqrt(2),pi²]` with a 5th degree polynomial for `double` floats:



```sh

lolremez --double -d 5 -r "sqrt(2):pi²" "atan(sqrt(3+x³)-exp(1+x))"

```



Result:



```c++

/* Approximation of f(x) = atan(sqrt(3+x³)-exp(1+x))

 * on interval [ sqrt(2), pi² ]

 * with a polynomial of degree 5. */

double f(double x)

{

    double u = -3.9557569330471555e-5;

    u = u * x + 1.2947712130833294e-3;

    u = u * x + -1.6541397035559147e-2;

    u = u * x + 1.0351664953941214e-1;

    u = u * x + -3.2051562487328135e-1;

    return u * x + -1.1703528319321961;

}

```



## Available functions



Binary functions/operators:



 - \+ \- \* / %

 - *atan2(y, x)*, *pow(x, y)*

 - *min(x, y)*, *max(x, y)*

 - *fmod(x, y)*



Exponent shortcuts:



 - *x²*, *x³*…



Constants:



 - *e*,

 - *pi*, *π*

 - *tau*, *τ* (Tau is [great for trolling](https://tauday.com/tau-manifesto))



Math functions:



 - *abs()* (absolute value)

 - *sqrt()* (square root), *cbrt()* (cubic root)

 - *exp()*, *exp2()*, *expm1()*, *erf()*, *erfc()*, *erfcx()*,

 - *log()*, *log2()*, *log10()*, *log1p()*

 - *sin()*, *cos()*, *tan()*

 - *asin()*, *acos()*, *atan()*

 - *sinh()*, *cosh()*, *tanh()*



Parsing rules:



 - *-a^b* is *-(a^b)*

 - *a^b^c* is *(a^b)^c*



## Limitations



As of now, the `erf()` family of function is inaccurate in the [7,19] range.

See [this issue](https://github.com/lolengine/lol/issues/2)



## Build LolRemez



### Setup



If you got the source code from Git, make sure the submodules are properly initialised:



    git submodule update --init --recursive



On Windows, just open `lolremez.sln` in Visual Studio.



On Linux, make sure the following packages are installed:



    automake autoconf libtool pkg-config



### Compile



On Windows, just build the solution in Visual Studio.



On Linux, bootstrap the project and configure it:



    ./bootstrap

    ./configure



Finally, build the project:



    make



The resulting executable is `lolremez`. You can manually copy it to any

installation location, or run the following:



    sudo make install



## Docker



A docker image can easily be built using the provided [Dockerfile](./Dockerfile)



```bash

docker build -t lolremez .

```

This command will create a local Docker image "lolremez", you can the invoke `lolremez` as follows:



```

docker run --rm lolremez --double -d 5 -r "sqrt(2):pi²" "atan(sqrt(3+x³)-exp(1+x))"

// Approximation of f(x) = atan(sqrt(3+x³)-exp(1+x))

// on interval [ sqrt(2), pi² ]

// with a polynomial of degree 5.

// p(x)=((((-3.9557569330471555e-5*x+1.2947712130833294e-3)*x-1.6541397035559147e-2)*x+1.0351664953941214e-1)*x-3.2051562487328135e-1)*x-1.1703528319321961

double f(double x)

{

    double u = -3.9557569330471555e-5;

    u = u * x + 1.2947712130833294e-3;

    u = u * x + -1.6541397035559147e-2;

    u = u * x + 1.0351664953941214e-1;

    u = u * x + -3.2051562487328135e-1;

    return u * x + -1.1703528319321961;

}

```



## Changes



### News for LolRemez 0.7:



 - Fix a problem making hyperbolic functions unavailable.

 - A Windows build is provided with the release.



### News for LolRemez 0.6:



 - Fix a grave problem with extrema finding when using a weight function;

   results were suboptimal.

 - Switch to a header-only big float implementation that tremendously

   improves build times.

 - Print gnuplot-friendly formulas.



### News for LolRemez 0.5:



 - Fix a severe bug in *cbrt()*.

 - Implement *erf()*.

 - Implement *%* and *fmod()*.

 - Make the executable size even smaller.

 - lolremez can now also act as a simple command line calculator.



### News for LolRemez 0.4:



 - Allow expressions in the range specification: `-r -pi/4:pi/4` is now valid.

 - Allow using “pi” and “e” in expressions, as well as “π”.

 - Bugfixes in the expression parser.

 - Support for hexadecimal floats, *e.g.* `0x1.999999999999999ap-4`.

 - New `--float`, `--double` and `--long-double` options to choose precision.

 - Trim down DLL dependencies to allow for lighter binaries.



### News for LolRemez 0.3:



 - implemented an expression parser so that the user does not have to

   recompile the software before each use.

 - C/C++ function output.

 - use threading to find zeros and extrema.

 - use successive parabolic interpolation to find extrema.



### News for LolRemez 0.2:



 - significant performance and accuracy improvements thanks to various

   bugfixes and a better extrema finder for the error function.

 - user can now define accuracy of the final result.

 - exp(), sin(), cos() and tan() are now about 20% faster.

 - multiplying a real number by an integer power of two is now a virtually

   free operation.

 - fixed a rounding bug in the real number printing routine.



### Initial release: LolRemez 0.1