PASS bands k = 1..199: at the midpoint of [q_{k+1}, q_k] the maximum of 2 sqrt(j)(1-w)^(j-1), j < 400, is at j = k
PASS band 1: the closed form equals the integral 2/sqrt(k) [(1-w)^k] between the band ends
PASS band 2: the closed form equals the integral 2/sqrt(k) [(1-w)^k] between the band ends
PASS band 5: the closed form equals the integral 2/sqrt(k) [(1-w)^k] between the band ends
PASS band 40: the closed form equals the integral 2/sqrt(k) [(1-w)^k] between the band ends
     N = 100000: sum over bands 1..100000 = 1.4179457804199175980, q_(N+1) = 4.9999625e-6
     tail in [0.003836012736, 0.003836041506]
     I in [1.42178179316, 1.42178182193], width 2.88e-8
     pi/2 - I in [0.149014504869, 0.149014533639]
PASS certified width below 1e-7
PASS I = 1.42178 to 5 places (Tomar's midpoint value; his certified lower bound for int R is 1.42177)
PASS pi/2 - I = 0.149... (Ivanisvili-Stone, Remark 1.3)
     float midpoint rule in theta, 400000 points: 1.42178181
PASS the float midpoint rule agrees with the certified value to 2e-6
     time 0.8 s
A40G_PASS
