arctan

Show how to culculate arctan

1. By Taylor Series

Accurate but a bit slow to use in embedded system

arctan.js

let arctan = (x, taylorSeriesSize = 10) => {
  const pow = (x, n) => {
    if (n === 0) return 1;
    if (n === 1) return x;
    if (n % 2 === 0) {
      return pow(x * x, n / 2);
    } else {
      return x * pow(x * x, (n - 1) / 2);
    }
  };
  const n2n1 = (n) => 2 * n + 1;
  const x2n1 = (n) => pow(x, n2n1(n));
  const minus1n = (n) => (n % 2 === 0 ? 1 : -1);
  const sigma = (n, xk) => [...Array(taylorSeriesSize).keys()].map(xk).reduce((xk0, xk1) => xk0 + xk1);
  const pi = Math.PI;

  let tan_rad;

  if (x >= 1) {
    tan_rad = pi / 2 - sigma(taylorSeriesSize, (n) => minus1n(n) / (n2n1(n) * x2n1(n)));
  } else if (x <= -1) {
    tan_rad = -pi / 2 - sigma(taylorSeriesSize, (n) => minus1n(n) / (n2n1(n) * x2n1(n)));
  } else {
    tan_rad = sigma(taylorSeriesSize, (n) => (minus1n(n) * x2n1(n)) / n2n1(n));
  }

  return (tan_rad / pi) * 180;
};

2. Suggested in Stack Overflow

Fast but inaccurate (e.g. arctan2(1.732) outputs 51°)

.js

let arctan2 = (x) => {
  const pi = Math.PI;
  const abs = Math.abs;
  const rad = (pi / 4) * x - x * (abs(x) - 1) * (0.2447 + 0.0663 * abs(x));
  return (rad / pi) * 180;
};

Comparision

Compare required time in milliseconds

.js

let time = (fn) => {
  const t0 = performance.now();
  fn();
  const t1 = performance.now();
  console.log(t1 - t0);
};
time(() => arctan(1.7320508))
time(() => arctan2(1.7320508))