Rational approximation abs with MSE = ~1.6e-7

abs-approx.js

const fj = [ 1,        0.01123596,  1,          0.19101124,  0.05617978,  0.39325843, 0.01123596,  0.59550562,  0.21348315]
const wj = [-0.232961,-0.1212703,  -0.29214973,-0.12772172,  0.21085705,  0.6024462,  0.07024019,  0.3625982,  -0.53436423]
const zj = [-1,       -0.01123596,  1,          0.19101124, -0.05617978, -0.39325843, 0.01123596,  0.59550562, -0.21348315]

function approxAbs(x) {
  const B0 = wj[0] / (x + 1 - 1e-15);
  const B1 = wj[1] / (x - zj[1]);
  const B2 = wj[2] / (x - 1 + 1e-15);
  const B3 = wj[3] / (x - zj[3]);
  const B4 = wj[4] / (x - zj[4]);
  const B5 = wj[5] / (x - zj[5]);
  const B6 = wj[6] / (x - zj[6]);
  const B7 = wj[7] / (x - zj[7]);
  const B8 = wj[8] / (x - zj[8]);
  return (
   (B0 + fj[1] * B1 + B2 + fj[3] * B3 + fj[4] * B4 + fj[5] * B5 + fj[6] * B6 + fj[7] * B7 + fj[8] * B8) /
   (B0 + B1 + B2 + B3 + B4 + B5 + B6 + B7 + B8)
  );
}

console.log('~abs(+0.5) =', approxAbs(+0.5));
console.log('~abs(-0.5) =', approxAbs(-0.5));

console.log('~abs(+1.0) =', approxAbs(+1.0));
console.log('~abs(-1.0) =', approxAbs(-1.0));

console.log('~abs(+0.005) =', approxAbs(+0.005));
console.log('~abs(-0.005) =', approxAbs(-0.005));

const length = 1001;
const range = Array.from({ length }, (_,i) => 2 * i / (length - 1) - 1); // range = [-1..1]
console.log('MSE =', range.reduce((acc, v) => acc + (approxAbs(v) - Math.abs(v)) ** 2, 0) / (range.length - 1));

Result:

~abs(+0.5) = 0.5000962924973791
~abs(-0.5) = 0.49988571537395604
~abs(+1.0) = 0.9999999999999993
~abs(-1.0) = 1.000000000000001
~abs(+0.005) = 0.00726190332319611
~abs(-0.005) = 0.007353169539225273

MSE = 1.577348684371353e-7