LolRemez

A Remez algorithm implementation to approximate functions using polynomials.

A tutorial is available in the wiki section.

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:

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

Result:

/* 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)

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

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

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.

samhocevar / lolremez