Babylonian Square Root
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
An amazing video on the topic
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
