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848d4b7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | 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())) |