Example

**Input**: 420

**Output**: 20.493901531919196

Welcome to Project Lovelace! We're still super new so there are still tons of bugs to find and improvements to make. If you have any suggestions, complaints, or any comments at all please let us know on Discourse!

Babylonian square roots

**You will learn about**:
calculating square roots, iterative algorithms, and Babylonian math.

The square root of a number S is denoted $\sqrt{S}$ with the property that $\sqrt{S} \times \sqrt{S} = S$. It's easy to calculate for square numbers, like $3 \times 3 = 9$ so $\sqrt{9} = 3$. For other numbers it's not as easy but the ancient Babylonians seemed to know how to calculate them pretty accurately (see the figure and caption).

We have no idea how they calculated square roots so accurately, but a method they might have used has been called the Babylonian method. The basic idea is to start with an initial guess $x_0$ for $\sqrt{S}$. Then we can keep improve on this guess: if our current guess, $x_n$, is an overestimate to $\sqrt{S}$ then $S/x_n$ will be an underestimate and vice versa, so averaging the two should produce a better guess: $$ x_{n+1} = \frac{1}{2} \left( x_n + \frac{S}{x_n} \right) $$ We can use this formula to calculate better and better guesses. Given a number $S$, calculate and return its square root using the Babylonian method.

**Input**:
A number $S \ge 0$.

**Output**:
The square root of $S$ approximated using the babylonian method. Your answer should be accurate to 10 significant
figures. That is, it should be correct to within 1 part in 10 billion, or the relative error should be lower than
$10^{-10}$). You can calculate the relative error of your guess $x_n$ using $|x_n^2 - S| / S$ where $|\cdotp|$ is
the absolute value (or `abs`) function.

Difficulty | Timesink | ||
---|---|---|---|

Maximum runtime | 60 s | Max. memory usage | 250 MiB |

Function signature | babylonian_sqrt(n) |

Write a function that accepts the input as function parameters and returns the correct output. Make sure to read the description above to produce the correct output in the correct format and use the correct function signature so we can run your code. A good first step is to try reproducing the example(s). Your code must not take longer than the maximum runtime to run and must not use more memory than the allowed limit.

You must be logged in to view your submissions.

- The method we used here is also called Heron's method after the Hero of Alexandria who explicitly described it in 60 AD.
- The Babylonians did lots of other cool stuff that was way ahead of their time like calculate Jupiter’s position from the area under a time-velocity graph .

- Babylon and the square root of 2, The Azimuth Project
- Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context , David Fowler & Eleanor Robson, Historia Mathematica, Volume 25, Issue 4, pp. 366-378 (1998).

Let us know what you think about this problem! Was it too hard? Difficult to understand? Also feel free to
discuss the problem, ask questions, and post cool stuff on Discourse. You should be able see a discussion
thread below. Feel free to post your solutions but if you do please **organize and document your code
well** so others can learn from it.