Polynomial Approximations using the Remez Algorithm
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README.md

LolRemez

A Remez algorithm implementation to approximate functions using polynomials.

A tutorial is available in the wiki section.

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)

Exponent shortcuts:

  • ,

Constants:

Math functions:

  • abs() (absolute value)
  • sqrt() (square root), cbrt() (cubic root)
  • exp(), exp2(), erf(), log(), log2(), log10()
  • sin(), cos(), tan()
  • asin(), acos(), atan()
  • sinh(), cosh(), tanh()

Parsing rules:

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

Changes

In Git trunk:

  • Implement erf().
  • 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