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

The culprit is loss of significance (wikipedia). It occurs when you take difference (y - x) of two close numbers x ≈ y. For example,

The difference has only one significant digit although you started with numbers of five significant digits!

This type of numerical error can happen in two parts. One in inside of , the other in the function (θ - sinθ cos θ ) when θ is small .

Inside of the is just arithmetic operations so we can represent numbers as fractions to avoid the error. (d is a floating point number but I still convert it into a fraction as well.)

(θ - sinθ cos θ ) involves non-arithmetic operation, so I use Taylor expansion. Wolfram|Alpha tells me like this.

The first three terms is good enough and I replaced the function when .