newtons method applied to sqrt.

sqrt.js

/**
 * @param {number} k
 * @returns {number}
 */
export function sqrt(k) {
  if (k < 0) {
    throw new Error("Cannot compute square root of a negative number");
  }
  if (k === 0 || k === 1) {
    return k;
  }
  // initial guess
  let x = k > 1 ? k / 2 : 2 * k;
  let i = 0;
  while (true) {
    console.log(`Iteration ${i++}: x = ${x}`);
    let y = x * x - k;
    if (Math.abs(y) < 1e-14) {
      return x;
    }
    // 2x is the derivative of x^2
    // this is Newton's method for finding roots
    // x - f(x)/f'(x) = x - (x^2 - k) / (2x)
    const newX = (k / x + x) / 2;
    const delta = Math.abs(newX - x);
    x = newX;
    // check if we made significant progress
    if (Math.abs(delta) < 1e-14) {
      break;
    }
  }
  console.log(`Iteration ${i++}: x = ${x}`);
  return x;
}