Part 1: exact rational arithmetic
[PASS] e < 2.7183 (partial sum to 1/12! plus the remainder bound 2/13!) (upper bound 2.718281828607)
[PASS] 1.134^2 > 1.2859 (= 1.285956)
[PASS] 1.2859^2 > 1.6535 (= 1.65353881)
[PASS] 1.6535^2 > 2.734 (= 2.73406225)
[PASS] 2.734 > 2.7183 > e, hence e^(1/8) < 1.134 and (1/2) e^(1/8) < 0.567
[PASS] 1.134^8 > e directly (1.134^8 = 2.734667)
[PASS] (1/2)(1+1/(8m))^(m-1) < 0.567 for m = 1..5000 (exact) (max over the range = 0.566559177; limit (1/2)e^(1/8) = 0.566574...)
[PASS] the bound is increasing in m for m = 1..400 (so its supremum is the limit (1/2)e^(1/8))

Part 2: rigorous enclosures (mpmath.iv, monotone Riemann sums)
[PASS] m=  1 (N=20000): a < F < (a+b)/2, h(b) = 1+(b-a)/2 inside its enclosure, sup m f h^(m-1) < 1 
        F in [0.3741759542, 0.3741887368];  sup_[a,b] m f h^(m-1) <= 0.37418874  (analytic bound of the note 0.500000);  1 - sup >= 0.625811
[PASS] m=  2 (N=20000): a < F < (a+b)/2, h(b) = 1+(b-a)/2 inside its enclosure, sup m f h^(m-1) < 1 
        F in [0.1870851493, 0.187091542];  sup_[a,b] m f h^(m-1) <= 0.3975707  (analytic bound of the note 0.531250);  1 - sup >= 0.602429
[PASS] m=  3 (N=4000): a < F < (a+b)/2, h(b) = 1+(b-a)/2 inside its enclosure, sup m f h^(m-1) < 1 
        F in [0.1247142691, 0.1247355798];  sup_[a,b] m f h^(m-1) <= 0.40604842  (analytic bound of the note 0.542535);  1 - sup >= 0.593952
[PASS] m=  4 (N=4000): a < F < (a+b)/2, h(b) = 1+(b-a)/2 inside its enclosure, sup m f h^(m-1) < 1 
        F in [0.09353546017, 0.09355144382];  sup_[a,b] m f h^(m-1) <= 0.41040462  (analytic bound of the note 0.548355);  1 - sup >= 0.589595
[PASS] m=  6 (N=4000): a < F < (a+b)/2, h(b) = 1+(b-a)/2 inside its enclosure, sup m f h^(m-1) < 1 
        F in [0.06235681164, 0.06236746782];  sup_[a,b] m f h^(m-1) <= 0.4148534  (analytic bound of the note 0.554299);  1 - sup >= 0.585147
[PASS] m= 10 (N=4000): a < F < (a+b)/2, h(b) = 1+(b-a)/2 inside its enclosure, sup m f h^(m-1) < 1 
        F in [0.03741400901, 0.03742040291];  sup_[a,b] m f h^(m-1) <= 0.41848106  (analytic bound of the note 0.559146);  1 - sup >= 0.581519
[PASS] m= 25 (N=4000): a < F < (a+b)/2, h(b) = 1+(b-a)/2 inside its enclosure, sup m f h^(m-1) < 1 
        F in [0.01496557544, 0.01496813307];  sup_[a,b] m f h^(m-1) <= 0.42179953  (analytic bound of the note 0.563580);  1 - sup >= 0.5782
[PASS] m=100 (N=4000): a < F < (a+b)/2, h(b) = 1+(b-a)/2 inside its enclosure, sup m f h^(m-1) < 1 
        F in [0.003741390329, 0.003742029747];  sup_[a,b] m f h^(m-1) <= 0.42347818  (analytic bound of the note 0.565823);  1 - sup >= 0.576522

Part 3: 50-digit composite Gauss-Legendre quadrature (not rigorous)
[PASS] m=  1: Sc from (3) > 0 at all 249 interior points, (3) = (5), f <= r, 1 <= h <= h(b), h(b) exact, F inside the rigorous enclosure 
        F = 0.374182345504147;  min Sc = 5.41993e-103, max Sc = 29.1192;  max m f h^(m-1) = 0.37418235;  max rel. diff (3) vs (5) = 6.28e-51
[PASS] m=  2: Sc from (3) > 0 at all 249 interior points, (3) = (5), f <= r, 1 <= h <= h(b), h(b) exact, F inside the rigorous enclosure 
        F = 0.18708834562096;  min Sc = 2.08054e-102, max Sc = 117.966;  max m f h^(m-1) = 0.39737565;  max rel. diff (3) vs (5) = 6.68e-51
[PASS] m=  3: Sc from (3) > 0 at all 249 interior points, (3) = (5), f <= r, 1 <= h <= h(b), h(b) exact, F inside the rigorous enclosure 
        F = 0.124724924471187;  min Sc = 4.61069e-102, max Sc = 266.563;  max m f h^(m-1) = 0.40574581;  max rel. diff (3) vs (5) = 7.85e-51
[PASS] m=  4: Sc from (3) > 0 at all 249 interior points, (3) = (5), f <= r, 1 <= h <= h(b), h(b) exact, F inside the rigorous enclosure 
        F = 0.0935434520328014;  min Sc = 8.1323e-102, max Sc = 474.907;  max m f h^(m-1) = 0.41006186;  max rel. diff (3) vs (5) = 5.92e-51
[PASS] m=  5: Sc from (3) > 0 at all 249 interior points, (3) = (5), f <= r, 1 <= h <= h(b), h(b) exact 
        F = 0.0748346453803787;  min Sc = 1.26453e-101, max Sc = 742.998;  max m f h^(m-1) = 0.41269497;  max rel. diff (3) vs (5) = 6.41e-51
[PASS] m=  7: Sc from (3) > 0 at all 249 interior points, (3) = (5), f <= r, 1 <= h <= h(b), h(b) exact 
        F = 0.0534532229621431;  min Sc = 2.46457e-101, max Sc = 1458.45;  max m f h^(m-1) = 0.41574512;  max rel. diff (3) vs (5) = 1.09e-50
[PASS] m= 12: Sc from (3) > 0 at all 249 interior points, (3) = (5), f <= r, 1 <= h <= h(b), h(b) exact 
        F = 0.0311809887197896;  min Sc = 7.19961e-101, max Sc = 4292.73;  max m f h^(m-1) = 0.41896971;  max rel. diff (3) vs (5) = 7.67e-51
[PASS] m= 40: Sc from (3) > 0 at all 249 interior points, (3) = (5), f <= r, 1 <= h <= h(b), h(b) exact 
        F = 0.00935427950740452;  min Sc = 7.95188e-100, max Sc = 47769.7;  max m f h^(m-1) = 0.42217773;  max rel. diff (3) vs (5) = 8.51e-51
[PASS] F = 0.374182... for m = 1 and F = 0.187088... for m = 2 (Remark 3.3) (F_1 = 0.374182345504, F_2 = 0.187088345621)

Part 4: other admissible parameters (Remark 3.3) and negative controls
[PASS] admissible (m,a,b) = (1,1/10,9/10): m b (1+(b-a)/2)^(m-1) = 0.9000 < 1 and Sc > 0 on the grid (min Sc = 1.8937e-39)
[PASS] admissible (m,a,b) = (1,9/10,99/100): m b (1+(b-a)/2)^(m-1) = 0.9900 < 1 and Sc > 0 on the grid (min Sc = 1.2681e-39)
[PASS] admissible (m,a,b) = (2,1/5,2/5): m b (1+(b-a)/2)^(m-1) = 0.8800 < 1 and Sc > 0 on the grid (min Sc = 9.7361e-39)
[PASS] admissible (m,a,b) = (3,1/10,1/4): m b (1+(b-a)/2)^(m-1) = 0.8667 < 1 and Sc > 0 on the grid (min Sc = 2.517e-38)
[PASS] admissible (m,a,b) = (5,1/100,3/20): m b (1+(b-a)/2)^(m-1) = 0.9831 < 1 and Sc > 0 on the grid (min Sc = 6.4731e-38)
[PASS] negative control (m,a,b) = (1,2,3): condition violated (3.000 >= 1) and Sc < 0 somewhere (min Sc = -2.2118)
[PASS] negative control (m,a,b) = (2,1,2): condition violated (6.000 >= 1) and Sc < 0 somewhere (min Sc = -3.9057)
[PASS] negative control (m,a,b) = (3,1/2,3/2): condition violated (10.125 >= 1) and Sc < 0 somewhere (min Sc = -5.06)

mpmath 1.3.0
RUN2 PROFILE: ALL CHECKS PASSED
