== sigma^2: direct (1.2) / cosh-parametrization / Lemma 2.1(i) ==
d=  4 k=  3  direct=3.141592653590e+00 cosh=3.141592653590e+00 formula=3.141592653590e+00 maxrel=5.2e-14
d=  5 k=  4  direct=2.000000000000e+00 cosh=2.000000000000e+00 formula=2.000000000000e+00 maxrel=5.1e-14
d=  6 k=  4  direct=6.283185307180e+00 cosh=6.283185307180e+00 formula=6.283185307180e+00 maxrel=8.9e-16
d=  7 k=  5  direct=3.141592653590e+00 cosh=3.141592653590e+00 formula=3.141592653590e+00 maxrel=1.1e-16
d=  9 k=  6  direct=4.188790204786e+00 cosh=4.188790204786e+00 formula=4.188790204786e+00 maxrel=2.0e-15
d= 10 k=  7  direct=2.467401100272e+00 cosh=2.467401100272e+00 formula=2.467401100272e+00 maxrel=2.2e-16
d= 12 k=  8  direct=2.631894506957e+00 cosh=2.631894506957e+00 formula=2.631894506957e+00 maxrel=5.6e-16
d= 15 k=  9  direct=5.167712780050e+00 cosh=5.167712780050e+00 formula=5.167712780050e+00 maxrel=8.9e-16
d= 20 k= 12  direct=1.649254451369e+00 cosh=1.649254451369e+00 formula=1.649254451369e+00 maxrel=2.4e-15
d= 25 k= 14  direct=1.884103879390e+00 cosh=1.884103879390e+00 formula=1.884103879390e+00 maxrel=3.1e-15
d= 30 k= 20  direct=2.983874922111e-02 cosh=2.983874922111e-02 formula=2.983874922111e-02 maxrel=7.8e-16
d= 40 k= 22  direct=2.331080051504e-02 cosh=2.331080051504e-02 formula=2.331080051504e-02 maxrel=1.3e-15
d= 50 k= 27  direct=9.647871547020e-04 cosh=9.647871547020e-04 formula=9.647871547020e-04 maxrel=2.4e-15
worst relative deviation sigma^2: 5.2e-14
== cumulants n=3,4,6: direct / Lemma 2.1(iii) / BH25 (6.2) ==
d=  4 k=  3  cum3=2.820948e-01 cum4=1.193662e-01 cum6=2.770501e-02
d=  5 k=  4  cum3=4.165203e-01 cum4=2.285714e-01 cum6=8.524809e-02
d=  6 k=  4  cum3=9.973557e-02 cum4=2.273642e-02 cum6=1.948484e-03
d=  7 k=  5  cum3=1.880632e-01 cum4=6.366198e-02 cum6=1.125791e-02
d=  9 k=  6  cum3=8.994082e-02 cum4=2.035148e-02 cum6=2.009085e-03
d= 10 k=  7  cum3=1.392606e-01 cum4=4.244803e-02 cum6=7.273291e-03
d= 12 k=  8  cum3=7.705056e-02 cum4=1.764494e-02 cum6=2.116794e-03
d= 15 k=  9  cum3=1.256848e-02 cum4=1.172783e-03 cum6=3.864377e-05
d= 20 k= 12  cum3=9.125096e-03 cum4=9.442363e-04 cum6=5.581057e-05
d= 25 k= 14  cum3=8.193376e-04 cum4=3.431895e-05 cum6=7.058760e-07
d= 30 k= 20  cum3=5.774545e-02 cum4=3.212443e-02 cum6=6.617668e-02
d= 40 k= 22  cum3=1.374182e-04 cum4=8.886537e-06 cum6=1.937423e-06
d= 50 k= 27  cum3=2.590587e-05 cum4=2.234243e-06 cum6=2.680860e-06
worst relative deviation cumulants: 6.6e-13
== characteristic function of Z*: direct vs Beta representation ==
worst |cf_direct - cf_beta| over pairs x t in {0.5,1,2,4}: 3.3e-11
