Fourier coefficient generator

This repository generates Fourier Extension coefficients for a user supplied function. fourier_extension.py uses SciPy's adaptive quad integrator. --degree fixes the Fourier degree; the script searches for the Hermite order that gives the best precision at that degree.

Install

python -m pip install -r requirements.txt

Usage

python fourier_extension.py --func "FUNCTION_IN_X" \
  --left LOWER_BOUND --right UPPER_BOUND --degree DEGREE

--degree is the highest Fourier index, so degree 32 produces coefficients 0 through 32. Precision is not an input: for the fixed degree, the script searches Hermite orders (ke) and reports the one with the smallest mean Fourier reconstruction error.

Function Examples

Hyperbolic tangent on [-4, 4]:

python fourier_extension.py \
  --func "tanh(x)" --left -4 --right 4 --degree 32

Sigmoid on [-8, 8]:

python fourier_extension.py \
  --func "1/(1+exp(-x))" --left -8 --right 8 --degree 32

GELU using the tanh approximation on [-4, 4]:

python fourier_extension.py \
  --func "0.5*x*(1+tanh(sqrt(2/pi)*(x+0.044715*x**3)))" \
  --left -4 --right 4 --degree 32

Pass the original target function in x; interval mapping is performed internally. Do not manually replace x with a scaled variable such as 64*x.

C++ Output

The command always prints a paste-ready C++ declaration and a comment recording the selected ke and achieved precision:

// ke: 16, precision: 26.6488 bits
static const inline std::vector<std::complex<double>> coeff_exp_2_double_20{
        std::complex<double>(...),
        // ...
};

The variable name follows coeff_<function>_<interval>_double_<degree>, for example coeff_tanh_4_double_32, coeff_sigmoid_8_double_32, and coeff_gelu_4_double_32.

The output coefficients use the a_n - i*b_n convention. The same API is available from Python:

from fourier_extension import calculate_fourier_coefficients
ke, precision, coefficients = calculate_fourier_coefficients("exp(-x**2)", -2, 2, 16)

Use --debug to print per-ke timing and progress to stderr. It does not change the search or pollute the paste-ready C++ declaration on stdout.

Expressions use Python/NumPy syntax and may reference x, for example sin(x), exp(-x**2), and sqrt(abs(x)). The search stops after eight consecutive Hermite orders fail to improve the measured precision.