<div align="center"> ### Babylonian Square Root <img src="https://latex.codecogs.com/svg.image?\dpi{100}%20\%20x_0%20\approx%20\sqrt{S}%20\\x_{n%20+%201}%20=%20\frac{x_n%20+%20\frac{S}{x_n}}{2}%20\\\sqrt{S}%20=%20\displaystyle%20\lim_{n%20\to%20\infty}x_n" height="250" /> #### LaTeX ``` x_0 \approx \sqrt{S} \\ x_{n + 1} = \frac{x_n + \frac{S}{x_n}}{2} \\ \sqrt{S} = \displaystyle \lim_{n \to \infty}x_n ``` ### Karatsuba Algorithm [Wikipedia](https://en.wikipedia.org/wiki/Karatsuba_algorithm) [An amazing video on the topic](https://youtu.be/cCKOl5li6YM) ``` function karatsuba (num1, num2) if (num1 < 10) or (num2 < 10) return num1 × num2 /* fall back to traditional multiplication */ /* Calculates the size of the numbers. */ m = min (size_base10(num1), size_base10(num2)) m2 = floor (m / 2) /* m2 = ceil (m / 2) will also work */ /* Split the digit sequences in the middle. */ high1, low1 = split_at (num1, m2) high2, low2 = split_at (num2, m2) /* 3 recursive calls made to numbers approximately half the size. */ z0 = karatsuba (low1, low2) z1 = karatsuba (low1 + high1, low2 + high2) z2 = karatsuba (high1, high2) return (z2 × 10 ^ (m2 × 2)) + ((z1 - z2 - z0) × 10 ^ m2) + z0 ``` </div>