| from mpmath import mp | |
| mp.dps = 110 | |
| def hyper3f2_half_series(z, tol=None, max_terms=200000): | |
| if tol is None: | |
| tol = mp.eps | |
| s = mp.mpf(1) | |
| term = mp.mpf(1) | |
| for n in range(1, max_terms + 1): | |
| term *= ((n - mp.mpf('0.5'))**3) * z / (n**3) | |
| s_new = s + term | |
| if abs(term) <= tol * abs(s_new): | |
| return s_new | |
| s = s_new | |
| raise RuntimeError("Series did not converge within max_terms") | |
| def compute(): | |
| # Non-trivial algebraic argument | |
| z = mp.sqrt(2) - 1 | |
| with mp.workdps(140): | |
| # Clausen identity: 3F2(1/2,1/2,1/2;1,1;z) = [2F1(1/4,1/4;1;z)]^2 | |
| f2 = mp.hyper([mp.mpf(1)/4, mp.mpf(1)/4], [mp.mpf(1)], z) | |
| val_clausen = f2 * f2 | |
| # Independent computation by direct series for 3F2 | |
| val_series = hyper3f2_half_series(z, tol=mp.mpf('1e-130')) | |
| # Return the more stable average if they agree closely | |
| if abs(val_clausen - val_series) <= mp.mpf('1e-120') * max(1, abs(val_clausen), abs(val_series)): | |
| return mp.mpf((val_clausen + val_series) / 2) | |
| return mp.mpf(val_clausen) | |
| if __name__ == "__main__": | |
| print(str(compute())) |