Published unlisted
Edited
Jul 30, 2022
function sine_arclength(x) {
let i = Math.round(x * (1/Math.PI)); x -= i*Math.PI; // reduce to [-π/2, π/2]
let sx = (x > 0) - (x < 0); x *= sx; // reduce to [0, π/2]
let y = cheb_eval(sine_arclength_coeffs, x * (4/Math.PI) - 1);
const K = 3.820197789027712;
return i * K + sx * y;
}
f = (x) =>
sine_arclength(x) - x*(3.820197789027712/Math.PI);
// Coeffs generated using Chebfun (http://www.chebfun.org) using the Matlab code:
// f = cumsum(chebfun(@(x) sqrt(1 + cos(x)*cos(x)), [0, pi/2])); format longe; f.coeffs
sine_arclength_coeffs = [
1.003779909381942e+00, 9.563086185904272e-01, -5.148678032547906e-02, -1.469577505554187e-03,
2.794604256600180e-03, 2.223165035001421e-04, -3.636212528869967e-05, -1.195702400746026e-05,
-2.042301587639539e-06, 1.421301353688803e-08, 1.159950864889720e-07, 3.388424989626866e-08,
2.975891659682502e-09, -1.314232608846369e-09, -6.149778187104962e-10, -1.019788962754806e-10,
1.210888156113980e-11, 1.153819027032262e-11, 2.849307759807778e-12, 4.221699706939989e-14,
-2.112279791468886e-13, -7.398458675657313e-14, -7.696536772197088e-15, 3.487971687135836e-15,
1.827243706664117e-15, 3.310356733875579e-16,
]
// Clenshaw's method.
cheb_eval = function cheb_eval(coeffs, x) {
const x2 = 2 * x;
let d = coeffs.length - 1, b2 = 0, b1 = (d % 2) ? coeffs[d--] : 0;
for (let i = d; i >= 2; i -= 2) {
b2 = coeffs[i] + x2 * b1 - b2;
b1 = coeffs[i-1] + x2 * b2 - b1;
}
return coeffs[0] + x * b1 - b2;
}
{
const n = 1e6;
const inputs = [...Array(n)].map(_ => Math.random() * 20);
const outputs = Array(n);
const t0 = +new Date;
for (let i = 0; i < n; ++i) {
outputs[i] = sine_arclength(inputs[i]);
}
return (+new Date) - t0;
}
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