Egyptian estimates of PI

I saw a neat tweet on how the estimate the Egyptians used for PI.

This is all my speculation, and maybe a math history buff can enlighten me, but the D^2 dependence they should discover pretty naturally.   Including the constant before squaring is, I would argue, just as natural as having it outside the parentheses, so let's go with that for now.   So was there a nearby better fraction?   How well did the Egyptians do?   A brute force program should tell us.

We will use the ancient programming language C:

#include
#include
int main() {
    double min_error = 10;
    for (int denom = 1; denom < 10000; denom++) {
        for (int num = 1; num < denom; num++) {
            double approx = 2*double(num)/double(denom);
            approx = approx*approx;
            double error2 = M_PI-approx;
            error2 = error2*error2;
            if (error2 < min_error) {
                min_error = error2;
                printf("%d/%d %f\n", num, denom, 4*float(num*num)/float(denom*denom));
            }
        }
    }
}

This produces output:

1/2 1.000000
2/3 1.777778
3/4 2.250000
4/5 2.560000
5/6 2.777778
6/7 2.938776
7/8 3.062500
8/9 3.160494
23/26 3.130177
31/35 3.137959
39/44 3.142562
109/123 3.141252
148/167 3.141597
4401/4966 3.141588
4549/5133 3.141589
4697/5300 3.141589
4845/5467 3.141589
4993/5634 3.141589
5141/5801 3.141590
5289/5968 3.141590
5437/6135 3.141590
5585/6302 3.141590
5733/6469 3.141590
5881/6636 3.141591
6029/6803 3.141591
6177/6970 3.141591
6325/7137 3.141591
6473/7304 3.141591
6621/7471 3.141591
6769/7638 3.141591
6917/7805 3.141592
7065/7972 3.141592
7213/8139 3.141592
7361/8306 3.141592
7509/8473 3.141592
7657/8640 3.141592
7805/8807 3.141592
7953/8974 3.141593
8101/9141 3.141593
8249/9308 3.141593
8397/9475 3.141593
8545/9642 3.141593

So 7/8 was already pretty good, and you need to get to 23/26 before you do any better!   I'd say the Egyptians did extremely well.

What if they had put the constants outside the parens?   How well could they have done?   We can change two of the lines above to:

double approx = 4*double(num)/double(denom);//approx = approx*approx;

and the printf to:

printf("%d/%d %f\n", num, denom, 4*float(num)/float(denom));

And we get:

1/2 2.000000
2/3 2.666667
3/4 3.000000
4/5 3.200000
7/9 3.111111
11/14 3.142857
95/121 3.140496
106/135 3.140741
117/149 3.140940
128/163 3.141104
139/177 3.141243
150/191 3.141361
161/205 3.141464
172/219 3.141552
183/233 3.141631
355/452 3.141593

So 7/9 is not bad!  And 11/14 even better.   So no clear winner here on whether the rational constant should be inside the parens or not.