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Does any fast modern algorithm for approximating $\pi$ have a geometric interpretation as $\int \sqrt{1 - x^2}$ does?

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The most common modern method for approximating $\pi$ are Machin-like formulas. They work by writing $\pi$ as an integer combination of arctangents of small rational numbers, and then computing each arctangent by its power series.

The initial decomposition has a clear geometric flavor -- it amounts to finding a way to fill out a straight angle exactly by the corners of slim right triangles with integer ratios between the cathetes.

But there's nothing clearly geometric about the power series computation in the second phase.

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