TL;DR

600D - Area of Two Circles' Intersection requires careful treatment of loss of significance, more than the geometry itself. The test is plagued by the numerical errors, and rejects "more accurate" solution.

Geometry -------- Let's review the geometry, first. We need three numbers (r₁, r₂, d); where d is the distance between two centers,

Intersection of two circles is determined from two circular sectors subtracted by two triangles, yielding

where angles θ₁ and θ₂ are determined from the cosine formula

So far, so good. I wrote Python code (with double representation), and failed. Rewrote the code in C++ using long double, and succeeded. But..., what was wrong?

Loss of significance

Loss of significance (wikipedia) occurs when you take difference (y - x) of two close numbers x ≈ y. This can happen in two parts in our calculation. One in inside of cos^- 1(), the other in the function (θ - sin{θ}cos{θ}) when x is small .