Almost Pi

Let $f_{n} (k) = e^{k / n} - 1$ , for all non-negative integers $k$ .

Remarkably, $f_{200} (6) + f_{200} (75) + f_{200} (89) + f_{200} (226) = \underset{―}{3.1415926} 44529 \dots \approx π$ .

In fact, it is the best approximation of $π$ of the form $f_{n} (a) + f_{n} (b) + f_{n} (c) + f_{n} (d)$ for $n = 200$ .

Let $g (n) = a^{2} + b^{2} + c^{2} + d^{2}$ for $a, b, c, d$ that minimize the error: $| f_{n} (a) + f_{n} (b) + f_{n} (c) + f_{n} (d) - π |$
(where $| x |$ denotes the absolute value of $x$ ).

You are given $g (200) = 6^{2} + 75^{2} + 89^{2} + 226^{2} = 64658$ .

Find $g (10000)$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.