ExpRoot+Log: A Linear and Universal Basis for Function Approximation

ExpRoot+Log is a fast and interpretable function approximation method based on a hybrid linear basis. It combines exponential square-root, polynomial, and logarithmic terms to efficiently approximate a wide range of functions, including smooth, discontinuous, and decaying ones.

Features

Fast and accurate: Uses a minimal set of basis functions for efficient function approximation.
Interpretable: Each term in the basis has a clear mathematical interpretation.
Flexible: Can handle smooth, discontinuous, and asymptotically decaying functions.
Linear regression: Uses standard least-squares fitting for optimal performance.

Learn more

The full write‑up (motivation, math derivation, and numeric experiments) is available on dev.to:

👉 ExpRoot + Log: A Linear and Universal Basis for Function Approximation
https://dev.to/andysay/exprootlog-a-linear-and-universal-basis-for-function-approximation-2e9d

Usage

Add the dependency to `Cargo.toml`:

[dependencies]
exp_root_log = "0.1.0"

📂 `examples/demo.rs`:

use exp_root_log::approx_exp_root_log;

fn main() {
    // Generate test data
    let x: Vec<f64> = (0..100).map(|i| i as f64 / 100.0).collect();
    let y: Vec<f64> = x.iter().map(|&x| (2.0 * std::f64::consts::PI * x).sin()).collect();

    // Create the approximation function using ExpRoot+Log
    let approx_fn = approx_exp_root_log(
        &x,
        &y,
        &[0.5, 2.0, 5.0, 10.0, 20.0],    //  b_i
        5,                                  //  x^5
        &[1.0, 5.0, 10.0, 20.0],           // log params
    );


    // Evaluate the approximation
    let y_pred: Vec<f64> = x.iter().map(|&xi| approx_fn(xi)).collect();

    // Print the result
    println!("Approximated values: {:?}", y_pred);
}

Benchmark

Function	ExpRoot + Log▪ MSE	Polynomial deg 10▪ MSE	Take‑away
`Sin`	3.67 × 10⁻⁸	1.34 × 10⁻¹¹	Poly‑10 is a hair better on a pure sine; ExpRoot + Log is still < 10⁻⁷.
`ExpDecay`	1.46 × 10⁻¹³	1.14 × 10⁻¹⁵	Both are essentially machine‑precision; ExpRoot + Log keeps up.
`Step`	1.52 × 10⁻²	1.51 × 10⁻²	Equal accuracy on a hard discontinuity, no Gibbs ringing.
`Spike`	4.23 × 10⁻³	2.55 × 10⁻³	Narrow Gaussian spike: poly‑10 wins on raw MSE, but ExpRoot + Log is ~2× better than a 6‑knot cubic spline.

⏱ Average runtime on 2 000 points (Apple M1, cargo run --example benchmark):
ExpRoot + Log ≈ 47 ms | Poly deg 10 ≈ 32 ms
Two SVDs of comparable size; speed improves proportionally if you reduce basis size or enable rayon.

Why choose ExpRoot + Log?

Handles exponential tails without the blow‑up polynomials suffer.
No Gibbs oscillations on steps—log terms give smooth edge control.
Linear least‑squares → works in WASM, embedded, no external BLAS.
Interpretable coefficients: each term is a clear exponential or log “spring” shaping the curve.

git clone https://github.com/andysay1/exp_root_log
cd exp_root_log
cargo run --example benchmark

exp_root_log 0.1.2