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Table 22.3. Error rates for beta
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Beta Function: Small Values |
Max = 0ε (Mean = 0ε) |
Max = 2.86ε (Mean = 1.22ε) |
Max = 2.86ε (Mean = 1.22ε) |
Max = 2.23ε (Mean = 1.14ε) |
Beta Function: Medium Values |
Max = 0.978ε (Mean = 0.0595ε) |
Max = 61.4ε (Mean = 19.4ε) |
Max = 107ε (Mean = 24.5ε) |
Max = 96.5ε (Mean = 22.4ε) |
Beta Function: Divergent Values |
Max = 0ε (Mean = 0ε) |
Max = 8.99ε (Mean = 2.44ε) |
Max = 18.8ε (Mean = 2.71ε) |
Max = 11.4ε (Mean = 2.19ε) |
Table 22.4. Error rates for beta (incomplete)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Incomplete Beta Function: Small Values |
Max = 0ε (Mean = 0ε) |
Max = 11.1ε (Mean = 2.32ε) |
Max = 18.7ε (Mean = 3.19ε) |
Max = 9.94ε (Mean = 2.17ε) |
Incomplete Beta Function: Medium Values |
Max = 0.568ε (Mean = 0.0254ε) |
Max = 69.2ε (Mean = 13.4ε) |
Max = 174ε (Mean = 25ε) |
Max = 90ε (Mean = 12.7ε) |
Incomplete Beta Function: Large and Diverse Values |
Max = 0.999ε (Mean = 0.0325ε) |
Max = 6.84e+04ε (Mean = 2.76e+03ε) |
Max = 6.86e+04ε (Mean = 2.79e+03ε) |
Max = 633ε (Mean = 29.7ε) |
Incomplete Beta Function: Small Integer Values |
Max = 0.786ε (Mean = 0.0323ε) |
Max = 11.6ε (Mean = 3.6ε) |
Max = 51.8ε (Mean = 11ε) |
Max = 26ε (Mean = 6.28ε) |
Table 22.5. Error rates for betac
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Incomplete Beta Function: Small Values |
Max = 0.676ε (Mean = 0.0302ε) |
Max = 9.92ε (Mean = 2.3ε) |
Max = 11.2ε (Mean = 2.94ε) |
Max = 8.94ε (Mean = 2.06ε) |
Incomplete Beta Function: Medium Values |
Max = 0.949ε (Mean = 0.098ε) |
Max = 63.5ε (Mean = 13.5ε) |
Max = 97.6ε (Mean = 24.3ε) |
Max = 90.6ε (Mean = 14.8ε) |
Incomplete Beta Function: Large and Diverse Values |
Max = 1.12ε (Mean = 0.0458ε) |
Max = 1.05e+05ε (Mean = 5.45e+03ε) |
Max = 1.04e+05ε (Mean = 5.46e+03ε) |
Max = 3.72e+03ε (Mean = 113ε) |
Incomplete Beta Function: Small Integer Values |
Max = 0.586ε (Mean = 0.0314ε) |
Max = 11.1ε (Mean = 3.65ε) |
Max = 103ε (Mean = 17.4ε) |
Max = 26.2ε (Mean = 6.36ε) |
Table 22.6. Error rates for binomial_coefficient
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Binomials: small arguments |
Max = 1ε (Mean = 0.369ε) |
Max = 1.5ε (Mean = 0.339ε) |
Max = 1.5ε (Mean = 0.339ε) |
Max = 1ε (Mean = 0.369ε) |
Binomials: large arguments |
Max = 0.939ε (Mean = 0.314ε) |
Max = 26.6ε (Mean = 6.13ε) |
Max = 53.2ε (Mean = 10.8ε) |
Max = 37.2ε (Mean = 7.4ε) |
Table 22.7. Error rates for boost::math::powm1
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
powm1 |
(<math.h>: Max = 2.04ε (Mean = 0.493ε)) |
(Rmath 3.2.3: Max = 1.06ε (Mean = 0.425ε)) |
Max = 1.88ε (Mean = 0.49ε) |
(<math.h>: Max = 1.84ε (Mean = 0.486ε)) |
Table 22.8. Error rates for cbrt
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
cbrt Function |
Max = 0ε (Mean = 0ε) |
Max = 1.34ε (Mean = 0.471ε) |
Max = 1.34ε (Mean = 0.471ε) |
Max = 1.7ε (Mean = 0.565ε) |
Table 22.9. Error rates for cos_pi
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
sin_pi and cos_pi |
Max = 0ε (Mean = 0ε) |
Max = 0.991ε (Mean = 0.302ε) |
Max = 0.991ε (Mean = 0.302ε) |
Max = 0.996ε (Mean = 0.284ε) |
sin_pi and cos_pi near integers and half integers |
Max = 0ε (Mean = 0ε) |
Max = 0.976ε (Mean = 0.28ε) |
Max = 0.976ε (Mean = 0.28ε) |
Max = 0.996ε (Mean = 0.298ε) |
Table 22.10. Error rates for cyl_bessel_i
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel I0: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 1.95ε (Mean = 0.738ε) |
Max = 1.95ε (Mean = 0.661ε) |
Max = 0.762ε (Mean = 0.329ε) |
Bessel I1: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 0.64ε (Mean = 0.202ε) |
Max = 0.64ε (Mean = 0.202ε) |
Max = 0.767ε (Mean = 0.398ε) |
Bessel In: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 1.8ε (Mean = 1.33ε) |
Max = 463ε (Mean = 140ε) |
Max = 3.46ε (Mean = 1.32ε) |
Bessel Iv: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 4.12ε (Mean = 1.85ε) |
Max = 4.12ε (Mean = 1.95ε) |
Max = 2.97ε (Mean = 1.24ε) |
Bessel In: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 4.62ε (Mean = 1.06ε) |
Max = 176ε (Mean = 39.1ε) |
Max = 9.67ε (Mean = 1.88ε) |
Bessel Iv: Random Data |
Max = 0.661ε (Mean = 0.0441ε) |
Max = 8.35ε (Mean = 1.62ε) |
Max = 283ε (Mean = 88.4ε) |
Max = 7.46ε (Mean = 1.71ε) |
Bessel Iv: Mathworld Data (large values) |
Max = 0ε (Mean = 0ε) |
Max = 14.7ε (Mean = 6.66ε) |
Max = 14.7ε (Mean = 6.59ε) |
Max = 3.67ε (Mean = 1.64ε) |
Table 22.11. Error rates for cyl_bessel_i (integer orders)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel I0: Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 1.95ε (Mean = 0.738ε) |
Max = 1.95ε (Mean = 0.661ε) |
Max = 0.762ε (Mean = 0.329ε) |
Bessel I1: Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 0.64ε (Mean = 0.202ε) |
Max = 0.64ε (Mean = 0.202ε) |
Max = 0.767ε (Mean = 0.398ε) |
Bessel In: Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 1.8ε (Mean = 1.33ε) |
Max = 463ε (Mean = 140ε) |
Max = 3.46ε (Mean = 1.32ε) |
Table 22.12. Error rates for cyl_bessel_i_prime
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel I'0: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 0.82ε (Mean = 0.259ε) |
Max = 0.82ε (Mean = 0.259ε) |
Max = 0.82ε (Mean = 0.354ε) |
Bessel I'1: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 1.97ε (Mean = 0.757ε) |
Max = 1.97ε (Mean = 0.757ε) |
Max = 1.36ε (Mean = 0.782ε) |
Bessel I'n: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 2.31ε (Mean = 1.41ε) |
Max = 701ε (Mean = 212ε) |
Max = 3.61ε (Mean = 1.22ε) |
Bessel I'v: Mathworld Data |
Max = 1.62ε (Mean = 0.512ε) |
Max = 2.89e+03ε (Mean = 914ε) |
Max = 2.89e+03ε (Mean = 914ε) |
Max = 3.76e+03ε (Mean = 1.19e+03ε) |
Bessel I'n: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 3.95ε (Mean = 1.06ε) |
Max = 195ε (Mean = 37.1ε) |
Max = 9.85ε (Mean = 1.82ε) |
Bessel I'v: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 14.1ε (Mean = 2.93ε) |
Max = 336ε (Mean = 68.5ε) |
Max = 14ε (Mean = 2.5ε) |
Bessel I'v: Mathworld Data (large values) |
Max = 0ε (Mean = 0ε) |
Max = 42.6ε (Mean = 20.2ε) |
Max = 42.6ε (Mean = 20.2ε) |
Max = 59.5ε (Mean = 26.6ε) |
Table 22.13. Error rates for cyl_bessel_i_prime (integer orders)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel I'0: Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 0.82ε (Mean = 0.259ε) |
Max = 0.82ε (Mean = 0.259ε) |
Max = 0.82ε (Mean = 0.354ε) |
Bessel I'1: Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 1.97ε (Mean = 0.757ε) |
Max = 1.97ε (Mean = 0.757ε) |
Max = 1.36ε (Mean = 0.782ε) |
Bessel I'n: Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 2.31ε (Mean = 1.41ε) |
Max = 701ε (Mean = 212ε) |
Max = 3.61ε (Mean = 1.22ε) |
Table 22.14. Error rates for cyl_bessel_j
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel J0: Mathworld Data |
Max = 6.55ε (Mean = 2.86ε) |
Max = 0ε (Mean = 0ε) |
Max = 6.55ε (Mean = 2.86ε) |
Max = 2.52ε (Mean = 1.2ε) |
Bessel J0: Mathworld Data (Tricky cases) |
Max = 1.64e+08ε (Mean = 6.69e+07ε) |
Max = 8e+04ε (Mean = 3.27e+04ε) |
Max = 1.64e+08ε (Mean = 6.69e+07ε) |
Max = 1e+07ε (Mean = 4.09e+06ε) |
Bessel J1: Mathworld Data |
Max = 3.59ε (Mean = 1.33ε) |
Max = 0ε (Mean = 0ε) |
Max = 1.44ε (Mean = 0.637ε) |
Max = 1.73ε (Mean = 0.976ε) |
Bessel J1: Mathworld Data (tricky cases) |
Max = 2.18e+05ε (Mean = 9.76e+04ε) |
Max = 106ε (Mean = 47.5ε) |
Max = 2.18e+05ε (Mean = 9.76e+04ε) |
Max = 3.23e+04ε (Mean = 1.45e+04ε) |
Bessel JN: Mathworld Data |
Max = 6.85ε (Mean = 3.35ε) |
Max = 0ε (Mean = 0ε) |
Max = 463ε (Mean = 112ε) |
Max = 14.7ε (Mean = 5.4ε) |
Bessel J: Mathworld Data |
Max = 14.7ε (Mean = 4.11ε) |
Max = 10ε (Mean = 2.24ε) |
Max = 14.7ε (Mean = 4.22ε) |
Max = 14.9ε (Mean = 3.89ε) |
Bessel J: Mathworld Data (large values) |
Max = 607ε (Mean = 305ε) |
Max = 0.536ε (Mean = 0.268ε) |
Max = 607ε (Mean = 305ε) |
Max = 9.31ε (Mean = 5.52ε) |
Bessel JN: Random Data |
Max = 50.8ε (Mean = 3.69ε) |
Max = 0ε (Mean = 0ε) |
Max = 99.6ε (Mean = 22ε) |
Max = 17.5ε (Mean = 1.46ε) |
Bessel J: Random Data |
Max = 11.4ε (Mean = 1.68ε) |
Max = 0ε (Mean = 0ε) |
Max = 260ε (Mean = 34ε) |
Max = 9.24ε (Mean = 1.17ε) |
Bessel J: Random Data (Tricky large values) |
Max = 785ε (Mean = 94.2ε) |
Max = 0ε (Mean = 0ε) |
Max = 785ε (Mean = 97.4ε) |
Max = 59.2ε (Mean = 8.67ε) |
Table 22.15. Error rates for cyl_bessel_j (integer orders)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel J0: Mathworld Data (Integer Version) |
Max = 6.55ε (Mean = 2.86ε) |
Max = 0ε (Mean = 0ε) |
Max = 6.55ε (Mean = 2.86ε) |
Max = 2.52ε (Mean = 1.2ε) |
Bessel J0: Mathworld Data (Tricky cases) (Integer Version) |
Max = 1.64e+08ε (Mean = 6.69e+07ε) |
Max = 8e+04ε (Mean = 3.27e+04ε) |
Max = 1.64e+08ε (Mean = 6.69e+07ε) |
Max = 1e+07ε (Mean = 4.09e+06ε) |
Bessel J1: Mathworld Data (Integer Version) |
Max = 3.59ε (Mean = 1.33ε) |
Max = 0ε (Mean = 0ε) |
Max = 1.44ε (Mean = 0.637ε) |
Max = 1.73ε (Mean = 0.976ε) |
Bessel J1: Mathworld Data (tricky cases) (Integer Version) |
Max = 2.18e+05ε (Mean = 9.76e+04ε) |
Max = 106ε (Mean = 47.5ε) |
Max = 2.18e+05ε (Mean = 9.76e+04ε) |
Max = 3.23e+04ε (Mean = 1.45e+04ε) |
Bessel JN: Mathworld Data (Integer Version) |
Max = 6.85ε (Mean = 3.35ε) |
Max = 0ε (Mean = 0ε) |
Max = 463ε (Mean = 112ε) |
Max = 14.7ε (Mean = 5.4ε) |
Table 22.16. Error rates for cyl_bessel_j_prime
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel J0': Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 18.9ε (Mean = 6.82ε) |
Max = 18.9ε (Mean = 6.72ε) |
Max = 6.62ε (Mean = 2.55ε) |
Bessel J0': Mathworld Data (Tricky cases) |
Max = 0ε (Mean = 0ε) |
Max = 7.44ε (Mean = 3.34ε) |
Max = 7.44ε (Mean = 3.31ε) |
Max = 3.67ε (Mean = 1.74ε) |
Bessel J1': Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 7.9ε (Mean = 3.37ε) |
Max = 7.9ε (Mean = 3.37ε) |
Max = 0.999ε (Mean = 0.627ε) |
Bessel J1': Mathworld Data (tricky cases) |
Max = 287ε (Mean = 129ε) |
Max = 5.88e+05ε (Mean = 2.63e+05ε) |
Max = 5.88e+05ε (Mean = 2.63e+05ε) |
Max = 288ε (Mean = 129ε) |
Bessel JN': Mathworld Data |
Max = 0.527ε (Mean = 0.128ε) |
Max = 1.29e+03ε (Mean = 312ε) |
Max = 1.29e+03ε (Mean = 355ε) |
Max = 14ε (Mean = 6.13ε) |
Bessel J': Mathworld Data |
Max = 21.5ε (Mean = 4.7ε) |
Max = 42.5ε (Mean = 9.31ε) |
Max = 42.5ε (Mean = 9.32ε) |
Max = 23.7ε (Mean = 8ε) |
Bessel J': Mathworld Data (large values) |
Max = 0ε (Mean = 0ε) |
Max = 989ε (Mean = 495ε) |
Max = 989ε (Mean = 495ε) |
Max = 2.9ε (Mean = 1.61ε) |
Bessel JN': Random Data |
Max = 0.593ε (Mean = 0.0396ε) |
Max = 11.3ε (Mean = 1.85ε) |
Max = 79.4ε (Mean = 16.2ε) |
Max = 6.34ε (Mean = 0.999ε) |
Bessel J': Random Data |
Max = 0.885ε (Mean = 0.033ε) |
Max = 139ε (Mean = 6.47ε) |
Max = 279ε (Mean = 27.2ε) |
Max = 176ε (Mean = 9.75ε) |
Bessel J': Random Data (Tricky large values) |
Max = 0ε (Mean = 0ε) |
Max = 474ε (Mean = 62.2ε) |
Max = 474ε (Mean = 64.5ε) |
Max = 379ε (Mean = 45.4ε) |
Table 22.17. Error rates for cyl_bessel_j_prime (integer orders)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel J0': Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 18.9ε (Mean = 6.82ε) |
Max = 18.9ε (Mean = 6.72ε) |
Max = 6.62ε (Mean = 2.55ε) |
Bessel J0': Mathworld Data (Tricky cases) (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 7.44ε (Mean = 3.34ε) |
Max = 7.44ε (Mean = 3.31ε) |
Max = 3.67ε (Mean = 1.74ε) |
Bessel J1': Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 7.9ε (Mean = 3.37ε) |
Max = 7.9ε (Mean = 3.37ε) |
Max = 0.999ε (Mean = 0.627ε) |
Bessel J1': Mathworld Data (tricky cases) (Integer Version) |
Max = 287ε (Mean = 129ε) |
Max = 5.88e+05ε (Mean = 2.63e+05ε) |
Max = 5.88e+05ε (Mean = 2.63e+05ε) |
Max = 288ε (Mean = 129ε) |
Bessel JN': Mathworld Data (Integer Version) |
Max = 0.527ε (Mean = 0.128ε) |
Max = 1.29e+03ε (Mean = 312ε) |
Max = 1.29e+03ε (Mean = 355ε) |
Max = 14ε (Mean = 6.13ε) |
Table 22.18. Error rates for cyl_bessel_k
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel K0: Mathworld Data |
Max = 0.833ε (Mean = 0.436ε) |
Max = 0ε (Mean = 0ε) |
Max = 0.833ε (Mean = 0.436ε) |
Max = 0.833ε (Mean = 0.552ε) |
Bessel K1: Mathworld Data |
Max = 0.786ε (Mean = 0.329ε) |
Max = 0ε (Mean = 0ε) |
Max = 0.786ε (Mean = 0.329ε) |
Max = 0.786ε (Mean = 0.39ε) |
Bessel Kn: Mathworld Data |
Max = 2.6ε (Mean = 1.21ε) |
Max = 0ε (Mean = 0ε) |
Max = 2.6ε (Mean = 1.21ε) |
Max = 3.63ε (Mean = 1.46ε) |
Bessel Kv: Mathworld Data |
Max = 3.58ε (Mean = 2.39ε) |
Max = 0ε (Mean = 0ε) |
Max = 5.21ε (Mean = 2.53ε) |
Max = 4.78ε (Mean = 2.19ε) |
Bessel Kv: Mathworld Data (large values) |
Max = 42.3ε (Mean = 21ε) |
Max = 0ε (Mean = 0ε) |
Max = 42.3ε (Mean = 21ε) |
Max = 59.8ε (Mean = 26.9ε) |
Bessel Kn: Random Data |
Max = 4.55ε (Mean = 1.12ε) |
Max = 0.764ε (Mean = 0.0348ε) |
Max = 4.55ε (Mean = 1.12ε) |
Max = 9.34ε (Mean = 1.7ε) |
Bessel Kv: Random Data |
Max = 7.88ε (Mean = 1.48ε) |
Max = 0.507ε (Mean = 0.0313ε) |
Max = 7.88ε (Mean = 1.47ε) |
Max = 8.33ε (Mean = 1.62ε) |
Table 22.19. Error rates for cyl_bessel_k (integer orders)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel K0: Mathworld Data (Integer Version) |
Max = 0.833ε (Mean = 0.436ε) |
Max = 0ε (Mean = 0ε) |
Max = 0.833ε (Mean = 0.436ε) |
Max = 0.833ε (Mean = 0.552ε) |
Bessel K1: Mathworld Data (Integer Version) |
Max = 0.786ε (Mean = 0.329ε) |
Max = 0ε (Mean = 0ε) |
Max = 0.786ε (Mean = 0.329ε) |
Max = 0.786ε (Mean = 0.39ε) |
Bessel Kn: Mathworld Data (Integer Version) |
Max = 2.6ε (Mean = 1.21ε) |
Max = 0ε (Mean = 0ε) |
Max = 2.6ε (Mean = 1.21ε) |
Max = 3.63ε (Mean = 1.46ε) |
Table 22.20. Error rates for cyl_bessel_k_prime
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel K'0: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 0.786ε (Mean = 0.329ε) |
Max = 0.786ε (Mean = 0.329ε) |
Max = 0.786ε (Mean = 0.39ε) |
Bessel K'1: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 0.736ε (Mean = 0.389ε) |
Max = 0.736ε (Mean = 0.389ε) |
Max = 0.761ε (Mean = 0.444ε) |
Bessel K'n: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 2.16ε (Mean = 1.08ε) |
Max = 2.16ε (Mean = 1.08ε) |
Max = 4.17ε (Mean = 1.75ε) |
Bessel K'v: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 3.94ε (Mean = 2.44ε) |
Max = 3.94ε (Mean = 2.34ε) |
Max = 3.94ε (Mean = 1.47ε) |
Bessel K'v: Mathworld Data (large values) |
Max = 0ε (Mean = 0ε) |
Max = 59.2ε (Mean = 42.9ε) |
Max = 58.7ε (Mean = 42.6ε) |
Max = 18.6ε (Mean = 11.8ε) |
Bessel K'n: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 4.45ε (Mean = 1.19ε) |
Max = 4.45ε (Mean = 1.19ε) |
Max = 9.67ε (Mean = 1.73ε) |
Bessel K'v: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 7.95ε (Mean = 1.53ε) |
Max = 7.95ε (Mean = 1.52ε) |
Max = 8.32ε (Mean = 1.65ε) |
Table 22.21. Error rates for cyl_bessel_k_prime (integer orders)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel K'0: Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 0.786ε (Mean = 0.329ε) |
Max = 0.786ε (Mean = 0.329ε) |
Max = 0.786ε (Mean = 0.39ε) |
Bessel K'1: Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 0.736ε (Mean = 0.389ε) |
Max = 0.736ε (Mean = 0.389ε) |
Max = 0.761ε (Mean = 0.444ε) |
Bessel K'n: Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 2.16ε (Mean = 1.08ε) |
Max = 2.16ε (Mean = 1.08ε) |
Max = 4.17ε (Mean = 1.75ε) |
Table 22.22. Error rates for cyl_neumann
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Y0: Mathworld Data |
Max = 5.53ε (Mean = 2.4ε) |
Max = 0ε (Mean = 0ε) |
Max = 5.53ε (Mean = 2.4ε) |
Max = 4.61ε (Mean = 2.29ε) |
Y1: Mathworld Data |
Max = 6.33ε (Mean = 2.25ε) |
Max = 0ε (Mean = 0ε) |
Max = 6.33ε (Mean = 2.29ε) |
Max = 4.75ε (Mean = 1.72ε) |
Yn: Mathworld Data |
Max = 55.2ε (Mean = 17.8ε) |
Max = 0.993ε (Mean = 0.314ε) |
Max = 55.2ε (Mean = 17.8ε) |
Max = 35ε (Mean = 11.9ε) |
Yv: Mathworld Data |
Max = 10.7ε (Mean = 4.93ε) |
Max = 10ε (Mean = 3.02ε) |
Max = 10.7ε (Mean = 5.1ε) |
Max = 7.89ε (Mean = 3.27ε) |
Yv: Mathworld Data (large values) |
Max = 1.7ε (Mean = 1.33ε) |
Max = 0ε (Mean = 0ε) |
Max = 1.7ε (Mean = 1.33ε) |
Max = 0.682ε (Mean = 0.423ε) |
Y0 and Y1: Random Data |
Max = 10.8ε (Mean = 3.04ε) |
Max = 0ε (Mean = 0ε) |
Max = 10.8ε (Mean = 3.04ε) |
Max = 4.17ε (Mean = 1.24ε) |
Yn: Random Data |
Max = 338ε (Mean = 27.5ε) |
Max = 0ε (Mean = 0ε) |
Max = 338ε (Mean = 27.5ε) |
Max = 117ε (Mean = 10.2ε) |
Yv: Random Data |
Max = 2.08e+03ε (Mean = 149ε) |
Max = 1.53ε (Mean = 0.102ε) |
Max = 2.08e+03ε (Mean = 149ε) |
Max = 1.23e+03ε (Mean = 69.9ε) |
Table 22.23. Error rates for cyl_neumann (integer orders)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Y0: Mathworld Data (Integer Version) |
Max = 5.53ε (Mean = 2.4ε) |
Max = 0ε (Mean = 0ε) |
Max = 5.53ε (Mean = 2.4ε) |
Max = 4.61ε (Mean = 2.29ε) |
Y1: Mathworld Data (Integer Version) |
Max = 6.33ε (Mean = 2.25ε) |
Max = 0ε (Mean = 0ε) |
Max = 6.33ε (Mean = 2.29ε) |
Max = 4.75ε (Mean = 1.72ε) |
Yn: Mathworld Data (Integer Version) |
Max = 55.2ε (Mean = 17.8ε) |
Max = 0.993ε (Mean = 0.314ε) |
Max = 55.2ε (Mean = 17.8ε) |
Max = 35ε (Mean = 11.9ε) |
Table 22.24. Error rates for cyl_neumann_prime
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Y'0: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 6.33ε (Mean = 3.12ε) |
Max = 6.33ε (Mean = 3.14ε) |
Max = 4.75ε (Mean = 1.75ε) |
Y'1: Mathworld Data |
Max = 0.58ε (Mean = 0.193ε) |
Max = 37.1ε (Mean = 12.8ε) |
Max = 34ε (Mean = 11.8ε) |
Max = 3.08ε (Mean = 1.2ε) |
Y'n: Mathworld Data |
Max = 2.05ε (Mean = 0.677ε) |
Max = 56ε (Mean = 18.2ε) |
Max = 56ε (Mean = 21.3ε) |
Max = 563ε (Mean = 178ε) |
Y'v: Mathworld Data |
Max = 21.5ε (Mean = 6.49ε) |
Max = 42.5ε (Mean = 13.4ε) |
Max = 42.5ε (Mean = 13.6ε) |
Max = 23.7ε (Mean = 10.1ε) |
Y'v: Mathworld Data (large values) |
Max = 0ε (Mean = 0ε) |
Max = 1.57ε (Mean = 1.24ε) |
Max = 1.57ε (Mean = 1.24ε) |
Max = 0.627ε (Mean = 0.237ε) |
Y'0 and Y'1: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 23.8ε (Mean = 3.69ε) |
Max = 23.8ε (Mean = 3.69ε) |
Max = 5.95ε (Mean = 1.36ε) |
Y'n: Random Data |
Max = 1.53ε (Mean = 0.0885ε) |
Max = 2.35e+03ε (Mean = 136ε) |
Max = 2.35e+03ε (Mean = 136ε) |
Max = 621ε (Mean = 36ε) |
Y'v: Random Data |
Max = 56.8ε (Mean = 2.59ε) |
Max = 1.16e+05ε (Mean = 5.28e+03ε) |
Max = 1.16e+05ε (Mean = 5.28e+03ε) |
Max = 3.23e+04ε (Mean = 1.13e+03ε) |
Table 22.25. Error rates for cyl_neumann_prime (integer orders)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Y'0: Mathworld Data (Integer Version) |
Max = 0ε (Mean = 0ε) |
Max = 6.33ε (Mean = 3.12ε) |
Max = 6.33ε (Mean = 3.14ε) |
Max = 4.75ε (Mean = 1.75ε) |
Y'1: Mathworld Data (Integer Version) |
Max = 0.58ε (Mean = 0.193ε) |
Max = 37.1ε (Mean = 12.8ε) |
Max = 34ε (Mean = 11.8ε) |
Max = 3.08ε (Mean = 1.2ε) |
Y'n: Mathworld Data (Integer Version) |
Max = 2.05ε (Mean = 0.677ε) |
Max = 56ε (Mean = 18.2ε) |
Max = 56ε (Mean = 21.3ε) |
Max = 563ε (Mean = 178ε) |
Table 22.26. Error rates for digamma
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Digamma Function: Large Values |
Max = 0ε (Mean = 0ε) |
Max = 1.39ε (Mean = 0.413ε) |
Max = 1.39ε (Mean = 0.413ε) |
Max = 0.98ε (Mean = 0.369ε) |
Digamma Function: Near the Positive Root |
Max = 0.891ε (Mean = 0.0995ε) |
Max = 1.37ε (Mean = 0.477ε) |
Max = 1.31ε (Mean = 0.471ε) |
Max = 0.997ε (Mean = 0.527ε) |
Digamma Function: Near Zero |
Max = 0ε (Mean = 0ε) |
Max = 0.984ε (Mean = 0.361ε) |
Max = 0.984ε (Mean = 0.361ε) |
Max = 0.953ε (Mean = 0.337ε) |
Digamma Function: Negative Values |
Max = 0ε (Mean = 0ε) |
Max = 180ε (Mean = 13ε) |
Max = 180ε (Mean = 13ε) |
Max = 214ε (Mean = 16.1ε) |
Digamma Function: Values near 0 |
Max = 0ε (Mean = 0ε) |
Max = 1ε (Mean = 0.592ε) |
Max = 1ε (Mean = 0.592ε) |
Max = 0ε (Mean = 0ε) |
Digamma Function: Integer arguments |
Max = 0.992ε (Mean = 0.215ε) |
Max = 0.888ε (Mean = 0.403ε) |
Max = 0.888ε (Mean = 0.403ε) |
Max = 0.992ε (Mean = 0.452ε) |
Digamma Function: Half integer arguments |
Max = 0ε (Mean = 0ε) |
Max = 0.906ε (Mean = 0.409ε) |
Max = 0.906ε (Mean = 0.409ε) |
Max = 0.78ε (Mean = 0.314ε) |
Table 22.27. Error rates for ellint_1
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Elliptic Integral F: Mathworld Data |
Max = 0.94ε (Mean = 0.509ε) |
Max = 0ε (Mean = 0ε) |
Max = 0.94ε (Mean = 0.509ε) |
Max = 0.919ε (Mean = 0.542ε) |
Elliptic Integral F: Random Data |
Max = 1.57ε (Mean = 0.56ε) |
Max = 0ε (Mean = 0ε) |
Max = 1.57ε (Mean = 0.561ε) |
Max = 2.26ε (Mean = 0.631ε) |
Table 22.28. Error rates for ellint_1 (complete)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Elliptic Integral K: Mathworld Data |
Max = 0.887ε (Mean = 0.296ε) |
Max = 0ε (Mean = 0ε) |
Max = 0.887ε (Mean = 0.296ε) |
Max = 0.915ε (Mean = 0.547ε) |
Elliptic Integral K: Random Data |
Max = 1.27ε (Mean = 0.473ε) |
Max = 0.851ε (Mean = 0.0851ε) |
Max = 1.27ε (Mean = 0.473ε) |
Max = 0.958ε (Mean = 0.408ε) |
Table 22.29. Error rates for ellint_2
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Elliptic Integral E: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 0.656ε (Mean = 0.317ε) |
Max = 0.656ε (Mean = 0.317ε) |
Max = 1.31ε (Mean = 0.727ε) |
Elliptic Integral E: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 2.05ε (Mean = 0.632ε) |
Max = 2.05ε (Mean = 0.632ε) |
Max = 2.23ε (Mean = 0.639ε) |
Elliptic Integral E: Small Angles |
Max = 0ε (Mean = 0ε) |
Max = 1ε (Mean = 0.283ε) |
Max = 1ε (Mean = 0.283ε) |
Max = 1ε (Mean = 0.421ε) |
Table 22.30. Error rates for ellint_2 (complete)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Elliptic Integral E: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 0.836ε (Mean = 0.469ε) |
Max = 0.836ε (Mean = 0.469ε) |
Max = 1.3ε (Mean = 0.615ε) |
Elliptic Integral E: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 1.97ε (Mean = 0.629ε) |
Max = 1.97ε (Mean = 0.629ε) |
Max = 1.71ε (Mean = 0.553ε) |
Table 22.31. Error rates for ellint_3
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Elliptic Integral PI: Mathworld Data |
Max = 475ε (Mean = 86.3ε) |
Max = 0ε (Mean = 0ε) |
Max = 475ε (Mean = 86.3ε) |
Max = 565ε (Mean = 102ε) |
Elliptic Integral PI: Random Data |
Max = 4.54ε (Mean = 0.895ε) |
Max = 0ε (Mean = 0ε) |
Max = 4.49ε (Mean = 0.885ε) |
Max = 8.33ε (Mean = 0.971ε) |
Elliptic Integral PI: Large Random Data |
Max = 3.7ε (Mean = 0.893ε) |
Max = 0.557ε (Mean = 0.0389ε) |
Max = 3.7ε (Mean = 0.892ε) |
Max = 2.86ε (Mean = 0.944ε) |
Table 22.32. Error rates for ellint_3 (complete)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Complete Elliptic Integral PI: Mathworld Data |
Max = 1.4ε (Mean = 0.575ε) |
Max = 0ε (Mean = 0ε) |
Max = 1.4ε (Mean = 0.575ε) |
Max = 0.971ε (Mean = 0.464ε) |
Complete Elliptic Integral PI: Random Data |
Max = 2.45ε (Mean = 0.696ε) |
Max = 0ε (Mean = 0ε) |
Max = 2.4ε (Mean = 0.677ε) |
Max = 2.46ε (Mean = 0.657ε) |
Table 22.33. Error rates for ellint_d
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Elliptic Integral E: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 1.3ε (Mean = 0.813ε) |
Max = 1.3ε (Mean = 0.813ε) |
Max = 0.862ε (Mean = 0.457ε) |
Elliptic Integral D: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 2.51ε (Mean = 0.883ε) |
Max = 2.51ε (Mean = 0.883ε) |
Max = 2.87ε (Mean = 0.805ε) |
Table 22.34. Error rates for ellint_d (complete)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Elliptic Integral E: Mathworld Data |
Max = 0.637ε (Mean = 0.368ε) |
Max = 1.27ε (Mean = 0.735ε) |
Max = 1.27ε (Mean = 0.735ε) |
Max = 0.637ε (Mean = 0.368ε) |
Elliptic Integral D: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 1.27ε (Mean = 0.334ε) |
Max = 1.27ε (Mean = 0.334ε) |
Max = 1.27ε (Mean = 0.355ε) |
Table 22.35. Error rates for ellint_rc
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|
RC: Random data |
Max = 0ε (Mean = 0ε) |
Max = 0.995ε (Mean = 0.433ε) |
Max = 0.962ε (Mean = 0.407ε) |
Table 22.36. Error rates for ellint_rd
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|
RD: Random data |
Max = 0ε (Mean = 0ε) |
Max = 2.73ε (Mean = 0.831ε) |
Max = 2.16ε (Mean = 0.803ε) |
RD: y = z |
Max = 0.896ε (Mean = 0.022ε) |
Max = 2.65ε (Mean = 0.82ε) |
Max = 16.5ε (Mean = 0.843ε) |
RD: x = y |
Max = 0.824ε (Mean = 0.0272ε) |
Max = 2.85ε (Mean = 0.865ε) |
Max = 3.51ε (Mean = 0.816ε) |
RD: x = 0, y = z |
Max = 0ε (Mean = 0ε) |
Max = 1.19ε (Mean = 0.522ε) |
Max = 1.16ε (Mean = 0.497ε) |
RD: x = y = z |
Max = 0ε (Mean = 0ε) |
Max = 0.998ε (Mean = 0.387ε) |
Max = 1.03ε (Mean = 0.418ε) |
RD: x = 0 |
Max = 0ε (Mean = 0ε) |
Max = 2.79ε (Mean = 0.883ε) |
Max = 2.64ε (Mean = 0.894ε) |
Table 22.37. Error rates for ellint_rf
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|
RF: Random data |
Max = 0ε (Mean = 0ε) |
Max = 2.54ε (Mean = 0.674ε) |
Max = 2.02ε (Mean = 0.677ε) |
RF: x = y = z |
Max = 0ε (Mean = 0ε) |
Max = 0.991ε (Mean = 0.345ε) |
Max = 0.999ε (Mean = 0.34ε) |
RF: x = y or y = z or x = z |
Max = 0.536ε (Mean = 0.00658ε) |
Max = 1.95ε (Mean = 0.418ε) |
Max = 1.21ε (Mean = 0.394ε) |
RF: x = 0, y = z |
Max = 0ε (Mean = 0ε) |
Max = 0.894ε (Mean = 0.338ε) |
Max = 0.999ε (Mean = 0.407ε) |
RF: z = 0 |
Max = 0ε (Mean = 0ε) |
Max = 1.7ε (Mean = 0.539ε) |
Max = 1.89ε (Mean = 0.587ε) |
Table 22.38. Error rates for ellint_rg
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|
RG: Random Data |
Max = 0.983ε (Mean = 0.0172ε) |
Max = 3.95ε (Mean = 0.951ε) |
Max = 3.65ε (Mean = 0.929ε) |
RG: two values 0 |
Max = 0ε (Mean = 0ε) |
Max = 0ε (Mean = 0ε) |
Max = 0ε (Mean = 0ε) |
RG: All values the same or zero |
Max = 0ε (Mean = 0ε) |
Max = 0.992ε (Mean = 0.288ε) |
Max = 1.06ε (Mean = 0.348ε) |
RG: two values the same |
Max = 0.594ε (Mean = 0.0103ε) |
Max = 1.51ε (Mean = 0.404ε) |
Max = 1.96ε (Mean = 0.374ε) |
RG: one value zero |
Max = 0ε (Mean = 0ε) |
Max = 2.14ε (Mean = 0.722ε) |
Max = 1.96ε (Mean = 0.674ε) |
Table 22.39. Error rates for ellint_rj
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|
RJ: Random data |
Max = 0.52ε (Mean = 0.0184ε) |
Max = 186ε (Mean = 6.67ε) |
Max = 215ε (Mean = 7.66ε) |
RJ: 4 Equal Values |
Max = 0ε (Mean = 0ε) |
Max = 0.998ε (Mean = 0.387ε) |
Max = 1.03ε (Mean = 0.418ε) |
RJ: 3 Equal Values |
Max = 0ε (Mean = 0ε) |
Max = 20.8ε (Mean = 0.986ε) |
Max = 39.9ε (Mean = 1.17ε) |
RJ: 2 Equal Values |
Max = 0.6ε (Mean = 0.0228ε) |
Max = 220ε (Mean = 6.64ε) |
Max = 214ε (Mean = 5.28ε) |
RJ: Equal z and p |
Max = 0.742ε (Mean = 0.0166ε) |
Max = 17.2ε (Mean = 1.16ε) |
Max = 16.1ε (Mean = 1.14ε) |
Table 22.40. Error rates for erf
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Erf Function: Small Values |
Max = 0.925ε (Mean = 0.193ε) |
Max = 0.841ε (Mean = 0.0687ε) |
Max = 0.925ε (Mean = 0.193ε) |
Max = 0.996ε (Mean = 0.182ε) |
Erf Function: Medium Values |
Max = 1.5ε (Mean = 0.193ε) |
Max = 1ε (Mean = 0.119ε) |
Max = 1.5ε (Mean = 0.197ε) |
Max = 1ε (Mean = 0.171ε) |
Erf Function: Large Values |
Max = 0ε (Mean = 0ε) |
Max = 0ε (Mean = 0ε) |
Max = 0ε (Mean = 0ε) |
Max = 0ε (Mean = 0ε) |
Table 22.41. Error rates for erf_inv
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Inverse Erf Function |
Max = 0ε (Mean = 0ε) |
Max = 0.996ε (Mean = 0.389ε) |
Max = 1.08ε (Mean = 0.395ε) |
Max = 1.09ε (Mean = 0.502ε) |
Table 22.42. Error rates for erfc
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Erf Function: Small Values |
Max = 0ε (Mean = 0ε) |
Max = 0.658ε (Mean = 0.0537ε) |
Max = 0ε (Mean = 0ε) |
Max = 0ε (Mean = 0ε) |
Erf Function: Medium Values |
Max = 1.76ε (Mean = 0.365ε) |
Max = 0.983ε (Mean = 0.213ε) |
Max = 1.76ε (Mean = 0.38ε) |
Max = 1.65ε (Mean = 0.373ε) |
Erf Function: Large Values |
Max = 1.57ε (Mean = 0.542ε) |
Max = 0.868ε (Mean = 0.147ε) |
Max = 1.57ε (Mean = 0.564ε) |
Max = 1.14ε (Mean = 0.248ε) |
Table 22.43. Error rates for erfc_inv
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Inverse Erfc Function |
Max = 0ε (Mean = 0ε) |
Max = 0.996ε (Mean = 0.397ε) |
Max = 1.08ε (Mean = 0.403ε) |
Max = 1ε (Mean = 0.491ε) |
Inverse Erfc Function: extreme values |
Max = 1.62ε (Mean = 0.383ε) |
Max = 1.62ε (Mean = 0.383ε) |
Table 22.44. Error rates for expint (Ei)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Exponential Integral Ei |
Max = 5.05ε (Mean = 0.821ε) |
Max = 0.994ε (Mean = 0.142ε) |
Max = 5.05ε (Mean = 0.835ε) |
Max = 1.43ε (Mean = 0.54ε) |
Exponential Integral Ei: double exponent range |
Max = 1.72ε (Mean = 0.593ε) |
Max = 0.998ε (Mean = 0.156ε) |
Max = 1.72ε (Mean = 0.607ε) |
Max = 1.7ε (Mean = 0.66ε) |
Exponential Integral Ei: long exponent range |
Max = 1.98ε (Mean = 0.595ε) |
Max = 1.98ε (Mean = 0.575ε) |
Table 22.45. Error rates for expint (En)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Exponential Integral En |
Max = 0.589ε (Mean = 0.0331ε) |
Max = 9.97ε (Mean = 2.13ε) |
Max = 9.97ε (Mean = 2.13ε) |
Max = 7.16ε (Mean = 1.85ε) |
Exponential Integral En: small z values |
Max = 0ε (Mean = 0ε) |
Max = 1.99ε (Mean = 0.559ε) |
Max = 1.99ε (Mean = 0.559ε) |
Max = 2.62ε (Mean = 0.531ε) |
Exponential Integral E1 |
Max = 0.556ε (Mean = 0.0625ε) |
Max = 0.965ε (Mean = 0.414ε) |
Max = 0.965ε (Mean = 0.408ε) |
Max = 0.988ε (Mean = 0.486ε) |
Table 22.46. Error rates for expm1
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Random test data |
Max = 0.992ε (Mean = 0.402ε) |
Max = 0.793ε (Mean = 0.126ε) |
Max = 1.31ε (Mean = 0.428ε) |
Max = 1.31ε (Mean = 0.496ε) |
Table 22.47. Error rates for gamma_p
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
tgamma(a, z) medium values |
Max = 0.955ε (Mean = 0.05ε) |
Max = 41.6ε (Mean = 8.09ε) |
Max = 239ε (Mean = 30.2ε) |
Max = 35.1ε (Mean = 6.98ε) |
tgamma(a, z) small values |
Max = 0ε (Mean = 0ε) |
Max = 2ε (Mean = 0.464ε) |
Max = 2ε (Mean = 0.461ε) |
Max = 1.54ε (Mean = 0.439ε) |
tgamma(a, z) large values |
Max = 0ε (Mean = 0ε) |
Max = 3.08e+04ε (Mean = 1.86e+03ε) |
Max = 3.02e+04ε (Mean = 1.91e+03ε) |
Max = 243ε (Mean = 20.2ε) |
tgamma(a, z) integer and half integer values |
Max = 0ε (Mean = 0ε) |
Max = 11.8ε (Mean = 2.66ε) |
Max = 71.6ε (Mean = 9.47ε) |
Max = 13ε (Mean = 2.97ε) |
Table 22.48. Error rates for gamma_p_inv
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
incomplete gamma inverse(a, z) medium values |
Max = 0.993ε (Mean = 0.15ε) |
Max = 1.8ε (Mean = 0.406ε) |
Max = 1.89ε (Mean = 0.466ε) |
Max = 1.71ε (Mean = 0.34ε) |
incomplete gamma inverse(a, z) large values |
Max = 0ε (Mean = 0ε) |
Max = 0.509ε (Mean = 0.0447ε) |
Max = 0.509ε (Mean = 0.0447ε) |
Max = 0.924ε (Mean = 0.108ε) |
incomplete gamma inverse(a, z) small values |
Max = 441ε (Mean = 53.9ε) |
Max = 9.17e+03ε (Mean = 1.45e+03ε) |
Max = 1.09e+04ε (Mean = 1.3e+03ε) |
Max = 1.1e+03ε (Mean = 131ε) |
Table 22.49. Error rates for gamma_p_inva
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Incomplete gamma inverses. |
Max = 0ε (Mean = 0ε) |
Max = 7.87ε (Mean = 1.15ε) |
Max = 4.08ε (Mean = 1.12ε) |
Max = 4.92ε (Mean = 1.03ε) |
Table 22.50. Error rates for gamma_q
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
tgamma(a, z) medium values |
Max = 0.927ε (Mean = 0.035ε) |
Max = 32.3ε (Mean = 6.61ε) |
Max = 199ε (Mean = 26.6ε) |
Max = 23.7ε (Mean = 4ε) |
tgamma(a, z) small values |
Max = 0ε (Mean = 0ε) |
Max = 2.45ε (Mean = 0.885ε) |
Max = 2.45ε (Mean = 0.819ε) |
Max = 2.26ε (Mean = 0.74ε) |
tgamma(a, z) large values |
Max = 0ε (Mean = 0ε) |
Max = 6.82e+03ε (Mean = 414ε) |
Max = 1.15e+04ε (Mean = 733ε) |
Max = 469ε (Mean = 31.5ε) |
tgamma(a, z) integer and half integer values |
Max = 0ε (Mean = 0ε) |
Max = 11.1ε (Mean = 2.07ε) |
Max = 54.7ε (Mean = 6.16ε) |
Max = 8.72ε (Mean = 1.48ε) |
Table 22.51. Error rates for gamma_q_inv
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
incomplete gamma inverse(a, z) medium values |
Max = 0.912ε (Mean = 0.154ε) |
Max = 6.2ε (Mean = 0.627ε) |
Max = 6.2ε (Mean = 0.683ε) |
Max = 2.88ε (Mean = 0.469ε) |
incomplete gamma inverse(a, z) large values |
Max = 0.894ε (Mean = 0.0915ε) |
Max = 0ε (Mean = 0ε) |
Max = 0ε (Mean = 0ε) |
Max = 0.814ε (Mean = 0.0856ε) |
incomplete gamma inverse(a, z) small values |
Max = 292ε (Mean = 36.4ε) |
Max = 8.28e+03ε (Mean = 1.09e+03ε) |
Max = 8.98e+03ε (Mean = 877ε) |
Max = 451ε (Mean = 64.7ε) |
Table 22.52. Error rates for gamma_q_inva
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Incomplete gamma inverses. |
Max = 0ε (Mean = 0ε) |
Max = 8.42ε (Mean = 1.3ε) |
Max = 7.86ε (Mean = 1.24ε) |
Max = 5.05ε (Mean = 1.08ε) |
Table 22.53. Error rates for hermite
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Hermite Polynomials |
Max = 0ε (Mean = 0ε) |
Max = 6.24ε (Mean = 2.07ε) |
Max = 6.24ε (Mean = 2.07ε) |
Max = 4.46ε (Mean = 1.41ε) |
Table 22.54. Error rates for heuman_lambda
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Elliptic Integral Jacobi Zeta: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 1.89ε (Mean = 0.887ε) |
Max = 1.89ε (Mean = 0.887ε) |
Max = 1.08ε (Mean = 0.734ε) |
Elliptic Integral Heuman Lambda: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 3.82ε (Mean = 0.609ε) |
Max = 3.82ε (Mean = 0.608ε) |
Max = 2.12ε (Mean = 0.588ε) |
Table 22.55. Error rates for ibeta
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Incomplete Beta Function: Small Values |
Max = 0ε (Mean = 0ε) |
Max = 8.97ε (Mean = 2.09ε) |
Max = 21.3ε (Mean = 2.75ε) |
Max = 8.4ε (Mean = 1.93ε) |
Incomplete Beta Function: Medium Values |
Max = 0ε (Mean = 0ε) |
Max = 50ε (Mean = 12.1ε) |
Max = 124ε (Mean = 18.4ε) |
Max = 106ε (Mean = 16.3ε) |
Incomplete Beta Function: Large and Diverse Values |
Max = 1.26ε (Mean = 0.063ε) |
Max = 1.96e+04ε (Mean = 997ε) |
Max = 4.98e+04ε (Mean = 2.07e+03ε) |
Max = 1.32e+03ε (Mean = 68.5ε) |
Incomplete Beta Function: Small Integer Values |
Max = 0ε (Mean = 0ε) |
Max = 4.45ε (Mean = 0.814ε) |
Max = 44.5ε (Mean = 10.1ε) |
Max = 3.85ε (Mean = 0.791ε) |
Table 22.56. Error rates for ibeta_inv
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Inverse incomplete beta |
Max = 11ε (Mean = 0.345ε) |
Max = 3.8e+04ε (Mean = 2.66e+03ε) |
Max = 4.07e+04ε (Mean = 2.86e+03ε) |
Max = 8.59e+03ε (Mean = 277ε) |
Table 22.57. Error rates for ibeta_inva
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Inverse incomplete beta |
Max = 0.602ε (Mean = 0.0239ε) |
Max = 377ε (Mean = 24.4ε) |
Max = 438ε (Mean = 31.3ε) |
Max = 242ε (Mean = 22.9ε) |
Table 22.58. Error rates for ibeta_invb
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Inverse incomplete beta |
Max = 0.765ε (Mean = 0.0422ε) |
Max = 407ε (Mean = 27.2ε) |
Max = 407ε (Mean = 24.4ε) |
Max = 409ε (Mean = 19.3ε) |
Table 22.59. Error rates for ibetac
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Incomplete Beta Function: Small Values |
Max = 0ε (Mean = 0ε) |
Max = 10.6ε (Mean = 2.22ε) |
Max = 13.8ε (Mean = 2.68ε) |
Max = 6.94ε (Mean = 1.71ε) |
Incomplete Beta Function: Medium Values |
Max = 0ε (Mean = 0ε) |
Max = 73.9ε (Mean = 11.2ε) |
Max = 132ε (Mean = 19.8ε) |
Max = 56.7ε (Mean = 14.3ε) |
Incomplete Beta Function: Large and Diverse Values |
Max = 0.981ε (Mean = 0.0573ε) |
Max = 3.45e+04ε (Mean = 1.32e+03ε) |
Max = 6.31e+04ε (Mean = 2.04e+03ε) |
Max = 1.88e+03ε (Mean = 82.7ε) |
Incomplete Beta Function: Small Integer Values |
Max = 0ε (Mean = 0ε) |
Max = 5.34ε (Mean = 1.11ε) |
Max = 107ε (Mean = 17.1ε) |
Max = 6.37ε (Mean = 1.03ε) |
Table 22.60. Error rates for ibetac_inv
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Inverse incomplete beta |
Max = 0.977ε (Mean = 0.0976ε) |
Max = 4.88e+04ε (Mean = 3.16e+03ε) |
Max = 5.05e+04ε (Mean = 3.33e+03ε) |
Max = 2.93e+03ε (Mean = 198ε) |
Table 22.61. Error rates for ibetac_inva
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Inverse incomplete beta |
Max = 0.683ε (Mean = 0.0314ε) |
Max = 382ε (Mean = 22.2ε) |
Max = 315ε (Mean = 23.7ε) |
Max = 408ε (Mean = 26.7ε) |
Table 22.62. Error rates for ibetac_invb
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Inverse incomplete beta |
Max = 0.724ε (Mean = 0.0303ε) |
Max = 317ε (Mean = 19.8ε) |
Max = 369ε (Mean = 22.6ε) |
Max = 271ε (Mean = 16.4ε) |
Table 22.63. Error rates for jacobi_cn
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Jacobi Elliptic: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 71.6ε (Mean = 19.3ε) |
Max = 71.6ε (Mean = 19.4ε) |
Max = 45.8ε (Mean = 11.4ε) |
Jacobi Elliptic: Random Data |
Max = 0.816ε (Mean = 0.0563ε) |
Max = 1.68ε (Mean = 0.443ε) |
Max = 1.68ε (Mean = 0.454ε) |
Max = 1.83ε (Mean = 0.455ε) |
Jacobi Elliptic: Random Small Values |
Max = 0ε (Mean = 0ε) |
Max = 10.4ε (Mean = 0.594ε) |
Max = 10.4ε (Mean = 0.602ε) |
Max = 26.2ε (Mean = 1.17ε) |
Jacobi Elliptic: Modulus near 1 |
Max = 0.919ε (Mean = 0.127ε) |
Max = 675ε (Mean = 87.1ε) |
Max = 675ε (Mean = 86.8ε) |
Max = 513ε (Mean = 126ε) |
Jacobi Elliptic: Large Phi |
Max = 14.2ε (Mean = 0.927ε) |
Max = 2.97e+04ε (Mean = 1.9e+03ε) |
Max = 2.97e+04ε (Mean = 1.9e+03ε) |
Max = 3.27e+04ε (Mean = 1.93e+03ε) |
Table 22.64. Error rates for jacobi_dn
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Jacobi Elliptic: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 49ε (Mean = 14ε) |
Max = 49ε (Mean = 14ε) |
Max = 34.3ε (Mean = 8.71ε) |
Jacobi Elliptic: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 1.53ε (Mean = 0.473ε) |
Max = 1.53ε (Mean = 0.481ε) |
Max = 1.52ε (Mean = 0.466ε) |
Jacobi Elliptic: Random Small Values |
Max = 0.5ε (Mean = 0.0122ε) |
Max = 22.4ε (Mean = 0.777ε) |
Max = 22.4ε (Mean = 0.763ε) |
Max = 16.1ε (Mean = 0.685ε) |
Jacobi Elliptic: Modulus near 1 |
Max = 2.28ε (Mean = 0.194ε) |
Max = 3.75e+03ε (Mean = 293ε) |
Max = 3.75e+03ε (Mean = 293ε) |
Max = 6.24e+03ε (Mean = 482ε) |
Jacobi Elliptic: Large Phi |
Max = 14.1ε (Mean = 0.897ε) |
Max = 2.82e+04ε (Mean = 1.79e+03ε) |
Max = 2.82e+04ε (Mean = 1.79e+03ε) |
Max = 1.67e+04ε (Mean = 1e+03ε) |
Table 22.65. Error rates for jacobi_sn
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Jacobi Elliptic: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 341ε (Mean = 80.7ε) |
Max = 341ε (Mean = 80.7ε) |
Max = 481ε (Mean = 113ε) |
Jacobi Elliptic: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 2.01ε (Mean = 0.584ε) |
Max = 2.01ε (Mean = 0.593ε) |
Max = 1.92ε (Mean = 0.567ε) |
Jacobi Elliptic: Random Small Values |
Max = 0ε (Mean = 0ε) |
Max = 1.99ε (Mean = 0.347ε) |
Max = 1.99ε (Mean = 0.347ε) |
Max = 2.11ε (Mean = 0.385ε) |
Jacobi Elliptic: Modulus near 1 |
Max = 0ε (Mean = 0ε) |
Max = 109ε (Mean = 7.35ε) |
Max = 109ε (Mean = 7.38ε) |
Max = 23.2ε (Mean = 1.85ε) |
Jacobi Elliptic: Large Phi |
Max = 12ε (Mean = 0.771ε) |
Max = 2.45e+04ε (Mean = 1.51e+03ε) |
Max = 2.45e+04ε (Mean = 1.51e+03ε) |
Max = 4.36e+04ε (Mean = 2.54e+03ε) |
Table 22.66. Error rates for jacobi_zeta
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Elliptic Integral Jacobi Zeta: Mathworld Data |
Max = 0ε (Mean = 0ε) |
Max = 1.66ε (Mean = 0.48ε) |
Max = 1.66ε (Mean = 0.48ε) |
Max = 1.52ε (Mean = 0.357ε) |
Elliptic Integral Jacobi Zeta: Random Data |
Max = 0ε (Mean = 0ε) |
Max = 2.99ε (Mean = 0.824ε) |
Max = 3.96ε (Mean = 1.06ε) |
Max = 3.89ε (Mean = 0.824ε) |
Elliptic Integral Jacobi Zeta: Large Phi Values |
Max = 0ε (Mean = 0ε) |
Max = 2.92ε (Mean = 0.951ε) |
Max = 3.05ε (Mean = 1.13ε) |
Max = 2.52ε (Mean = 0.977ε) |
Table 22.67. Error rates for laguerre(n, m, x)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Associated Laguerre Polynomials |
Max = 0.84ε (Mean = 0.0358ε) |
Max = 167ε (Mean = 6.38ε) |
Max = 167ε (Mean = 6.38ε) |
Max = 434ε (Mean = 11.1ε) |
Table 22.68. Error rates for laguerre(n, x)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Laguerre Polynomials |
Max = 6.82ε (Mean = 0.408ε) |
Max = 1.39e+04ε (Mean = 828ε) |
Max = 1.39e+04ε (Mean = 828ε) |
Max = 3.1e+03ε (Mean = 185ε) |
Table 22.69. Error rates for legendre_p
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Legendre Polynomials: Small Values |
Max = 0.732ε (Mean = 0.0619ε) |
Max = 69.2ε (Mean = 9.58ε) |
Max = 69.2ε (Mean = 9.58ε) |
Max = 211ε (Mean = 20.4ε) |
Legendre Polynomials: Large Values |
Max = 0.632ε (Mean = 0.0693ε) |
Max = 699ε (Mean = 59.6ε) |
Max = 699ε (Mean = 59.6ε) |
Max = 300ε (Mean = 33.2ε) |
Table 22.70. Error rates for legendre_p (associated)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Associated Legendre Polynomials: Small Values |
Max = 0.999ε (Mean = 0.05ε) |
Max = 175ε (Mean = 9.88ε) |
Max = 77.7ε (Mean = 5.59ε) |
Max = 121ε (Mean = 7.14ε) |
Table 22.71. Error rates for legendre_q
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Legendre Polynomials: Small Values |
Max = 0.612ε (Mean = 0.0517ε) |
Max = 50.9ε (Mean = 9ε) |
Max = 50.9ε (Mean = 8.98ε) |
Max = 46.4ε (Mean = 7.32ε) |
Legendre Polynomials: Large Values |
Max = 2.49ε (Mean = 0.202ε) |
Max = 5.98e+03ε (Mean = 478ε) |
Max = 5.98e+03ε (Mean = 478ε) |
Max = 4.6e+03ε (Mean = 366ε) |
Table 22.72. Error rates for lgamma
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
factorials |
Max = 0ε (Mean = 0ε) |
Max = 0.991ε (Mean = 0.308ε) |
Max = 0.991ε (Mean = 0.383ε) |
Max = 0.914ε (Mean = 0.175ε) |
near 0 |
Max = 0ε (Mean = 0ε) |
Max = 1.42ε (Mean = 0.566ε) |
Max = 1.42ε (Mean = 0.566ε) |
Max = 0.964ε (Mean = 0.462ε) |
near 1 |
Max = 0ε (Mean = 0ε) |
Max = 0.948ε (Mean = 0.36ε) |
Max = 0.948ε (Mean = 0.36ε) |
Max = 0.867ε (Mean = 0.468ε) |
near 2 |
Max = 0ε (Mean = 0ε) |
Max = 0.878ε (Mean = 0.242ε) |
Max = 0.878ε (Mean = 0.242ε) |
Max = 0.591ε (Mean = 0.159ε) |
near -10 |
Max = 0ε (Mean = 0ε) |
Max = 3.81ε (Mean = 1.01ε) |
Max = 3.81ε (Mean = 1.01ε) |
Max = 4.22ε (Mean = 1.33ε) |
near -55 |
Max = 0ε (Mean = 0ε) |
Max = 0.821ε (Mean = 0.513ε) |
Max = 1.59ε (Mean = 0.587ε) |
Max = 0.821ε (Mean = 0.419ε) |
Table 22.73. Error rates for log1p
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Random test data |
Max = 0.818ε (Mean = 0.227ε) |
Max = 0.846ε (Mean = 0.153ε) |
Max = 2.3ε (Mean = 0.66ε) |
Max = 0.509ε (Mean = 0.057ε) |
Table 22.74. Error rates for non central beta CDF
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Non Central Beta, medium parameters |
Max = 0.998ε (Mean = 0.0649ε) |
Max = 824ε (Mean = 27.4ε) |
Max = 832ε (Mean = 38.1ε) |
Max = 242ε (Mean = 31ε) |
Non Central Beta, large parameters |
Max = 1.18ε (Mean = 0.175ε) |
Max = 2.5e+04ε (Mean = 3.78e+03ε) |
Max = 2.57e+04ε (Mean = 4.45e+03ε) |
Max = 3.66e+03ε (Mean = 500ε) |
Table 22.75. Error rates for non central beta CDF complement
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Non Central Beta, medium parameters |
Max = 0.998ε (Mean = 0.0936ε) |
Max = 396ε (Mean = 50.7ε) |
Max = 554ε (Mean = 57.2ε) |
Max = 624ε (Mean = 62.7ε) |
Non Central Beta, large parameters |
Max = 0.986ε (Mean = 0.188ε) |
Max = 6.83e+03ε (Mean = 993ε) |
Max = 3.56e+03ε (Mean = 707ε) |
Max = 1.25e+04ε (Mean = 1.49e+03ε) |
Table 22.76. Error rates for non central chi squared CDF
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Non Central Chi Squared, medium parameters |
Max = 0.99ε (Mean = 0.0544ε) |
Max = 46.5ε (Mean = 10.3ε) |
Max = 115ε (Mean = 13.9ε) |
Max = 48.9ε (Mean = 10ε) |
Non Central Chi Squared, large parameters |
Max = 1.07ε (Mean = 0.102ε) |
Max = 3.07e+03ε (Mean = 336ε) |
Max = 6.17e+03ε (Mean = 677ε) |
Max = 9.79e+03ε (Mean = 723ε) |
Table 22.77. Error rates for non central chi squared CDF complement
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Non Central Chi Squared, medium parameters |
Max = 0.96ε (Mean = 0.0635ε) |
Max = 107ε (Mean = 17.2ε) |
Max = 171ε (Mean = 22.8ε) |
Max = 98.6ε (Mean = 15.8ε) |
Non Central Chi Squared, large parameters |
Max = 2.11ε (Mean = 0.278ε) |
Max = 5.02e+03ε (Mean = 630ε) |
Max = 5.1e+03ε (Mean = 577ε) |
Max = 5.43e+03ε (Mean = 705ε) |
Table 22.78. Error rates for non central t CDF
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Non Central T |
Max = 0.796ε (Mean = 0.0691ε) |
Max = 139ε (Mean = 31ε) |
Max = 145ε (Mean = 30.9ε) |
Max = 135ε (Mean = 32.1ε) |
Non Central T (small non-centrality) |
Max = 0ε (Mean = 0ε) |
Max = 3.86ε (Mean = 1.4ε) |
Max = 9.15ε (Mean = 2.13ε) |
Max = 6.17ε (Mean = 1.45ε) |
Non Central T (large parameters) |
Max = 257ε (Mean = 72.1ε) |
Max = 5.26e+05ε (Mean = 1.48e+05ε) |
Max = 5.24e+05ε (Mean = 1.47e+05ε) |
Max = 286ε (Mean = 62.8ε) |
Table 22.79. Error rates for non central t CDF complement
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Non Central T |
Max = 0.707ε (Mean = 0.0497ε) |
Max = 201ε (Mean = 31.7ε) |
Max = 340ε (Mean = 43.2ε) |
Max = 154ε (Mean = 32.1ε) |
Non Central T (small non-centrality) |
Max = 0ε (Mean = 0ε) |
Max = 10.5ε (Mean = 2.13ε) |
Max = 10.5ε (Mean = 2.39ε) |
Max = 4.6ε (Mean = 1.63ε) |
Non Central T (large parameters) |
Max = 478ε (Mean = 96.3ε) |
Max = 9.79e+05ε (Mean = 1.97e+05ε) |
Max = 9.79e+05ε (Mean = 1.97e+05ε) |
Max = 227ε (Mean = 50.4ε) |
Table 22.80. Error rates for owens_t
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Owens T (medium small values) |
Max = 0ε (Mean = 0ε) |
Max = 3.34ε (Mean = 0.944ε) |
Max = 3.34ε (Mean = 0.911ε) |
Max = 4.37ε (Mean = 0.98ε) |
Owens T (large and diverse values) |
Max = 0ε (Mean = 0ε) |
Max = 49ε (Mean = 2.16ε) |
Max = 24.5ε (Mean = 1.39ε) |
Max = 3.78ε (Mean = 0.621ε) |
Table 22.81. Error rates for polygamma
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Mathematica Data |
Max = 0.824ε (Mean = 0.0574ε) |
Max = 7.38ε (Mean = 1.84ε) |
Max = 34.3ε (Mean = 7.65ε) |
Max = 9.32ε (Mean = 1.95ε) |
Mathematica Data - large arguments |
Max = 0.998ε (Mean = 0.0592ε) |
Max = 2.23ε (Mean = 0.323ε) |
Max = 11.1ε (Mean = 0.848ε) |
Max = 150ε (Mean = 13.9ε) |
Mathematica Data - negative arguments |
Max = 0.516ε (Mean = 0.022ε) |
Max = 269ε (Mean = 87.7ε) |
Max = 269ε (Mean = 88.4ε) |
Max = 497ε (Mean = 129ε) |
Mathematica Data - large negative arguments |
Max = 0ε (Mean = 0ε) |
Max = 155ε (Mean = 96.4ε) |
Max = 155ε (Mean = 96.4ε) |
Max = 162ε (Mean = 101ε) |
Mathematica Data - small arguments |
Max = 0ε (Mean = 0ε) |
Max = 3.33ε (Mean = 0.75ε) |
Max = 3.33ε (Mean = 0.75ε) |
Max = 3ε (Mean = 0.496ε) |
Mathematica Data - Large orders and other bug cases |
Max = 0ε (Mean = 0ε) |
Max = 54.5ε (Mean = 13.3ε) |
Max = 145ε (Mean = 55.9ε) |
Max = 200ε (Mean = 57.2ε) |
Table 22.82. Error rates for powm1
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
powm1 |
Max = 1.06ε (Mean = 0.425ε) |
Max = 2.04ε (Mean = 0.493ε) |
Max = 1.88ε (Mean = 0.49ε) |
Max = 1.84ε (Mean = 0.486ε) |
Table 22.83. Error rates for sin_pi
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
sin_pi and cos_pi |
Max = 0ε (Mean = 0ε) |
Max = 0.996ε (Mean = 0.335ε) |
Max = 0.996ε (Mean = 0.336ε) |
Max = 0.99ε (Mean = 0.328ε) |
sin_pi and cos_pi near integers and half integers |
Max = 0ε (Mean = 0ε) |
Max = 0.976ε (Mean = 0.293ε) |
Max = 0.976ε (Mean = 0.293ε) |
Max = 0.996ε (Mean = 0.343ε) |
Table 22.84. Error rates for sph_bessel
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel j: Random Data |
Max = 243ε (Mean = 13.3ε) |
Max = 0.978ε (Mean = 0.0445ε) |
Max = 243ε (Mean = 33.7ε) |
Max = 245ε (Mean = 16.3ε) |
Table 22.85. Error rates for sph_bessel_prime
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Bessel j': Random Data |
Max = 0.753ε (Mean = 0.0343ε) |
Max = 167ε (Mean = 12ε) |
Max = 167ε (Mean = 33.2ε) |
Max = 307ε (Mean = 25.2ε) |
Table 22.86. Error rates for sph_neumann
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
y: Random Data |
Max = 234ε (Mean = 19.5ε) |
Max = 0.995ε (Mean = 0.0665ε) |
Max = 234ε (Mean = 19.8ε) |
Max = 281ε (Mean = 31.1ε) |
Table 22.87. Error rates for sph_neumann_prime
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
y': Random Data |
Max = 0.988ε (Mean = 0.0869ε) |
Max = 158ε (Mean = 18.8ε) |
Max = 158ε (Mean = 20.2ε) |
Max = 296ε (Mean = 25.6ε) |
Table 22.88. Error rates for spherical_harmonic_i
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Spherical Harmonics |
Max = 1.36ε (Mean = 0.0765ε) |
Max = 2.89e+03ε (Mean = 108ε) |
Max = 1.03e+04ε (Mean = 327ε) |
Max = 2.27e+04ε (Mean = 725ε) |
Table 22.89. Error rates for spherical_harmonic_r
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Spherical Harmonics |
Max = 1.58ε (Mean = 0.0707ε) |
Max = 2.89e+03ε (Mean = 108ε) |
Max = 1.03e+04ε (Mean = 327ε) |
Max = 2.27e+04ε (Mean = 725ε) |
Table 22.90. Error rates for sqrt1pm1
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
sqrt1pm1 |
Max = 1.3ε (Mean = 0.404ε) |
Max = 1.33ε (Mean = 0.404ε) |
Max = 1.54ε (Mean = 0.563ε) |
Max = 1.35ε (Mean = 0.497ε) |
Table 22.91. Error rates for tgamma
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
factorials |
Max = 0ε (Mean = 0ε) |
Max = 2.67ε (Mean = 0.617ε) |
Max = 172ε (Mean = 41ε) |
Max = 1.85ε (Mean = 0.566ε) |
near 0 |
Max = 0ε (Mean = 0ε) |
Max = 2ε (Mean = 0.608ε) |
Max = 2ε (Mean = 0.647ε) |
Max = 1.5ε (Mean = 0.635ε) |
near 1 |
Max = 0ε (Mean = 0ε) |
Max = 2.51ε (Mean = 1.02ε) |
Max = 3.01ε (Mean = 1.06ε) |
Max = 1.1ε (Mean = 0.59ε) |
near 2 |
Max = 0ε (Mean = 0ε) |
Max = 4.1ε (Mean = 1.55ε) |
Max = 5.01ε (Mean = 1.89ε) |
Max = 2ε (Mean = 0.733ε) |
near -10 |
Max = 0ε (Mean = 0ε) |
Max = 1.75ε (Mean = 0.895ε) |
Max = 1.75ε (Mean = 0.819ε) |
Max = 1.86ε (Mean = 0.881ε) |
near -55 |
Max = 0ε (Mean = 0ε) |
Max = 2.69ε (Mean = 1.09ε) |
Max = 98.5ε (Mean = 53.4ε) |
Max = 2.7ε (Mean = 1.35ε) |
Table 22.92. Error rates for tgamma1pm1
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
tgamma1pm1(dz) |
Max = 0ε (Mean = 0ε) |
Max = 1.12ε (Mean = 0.49ε) |
Max = 6.61ε (Mean = 0.84ε) |
Max = 3.31ε (Mean = 0.517ε) |
Table 22.93. Error rates for tgamma_delta_ratio
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
tgamma + small delta ratios |
Max = 0ε (Mean = 0ε) |
Max = 5.83ε (Mean = 1.3ε) |
Max = 15.4ε (Mean = 2.09ε) |
Max = 7.56ε (Mean = 1.31ε) |
tgamma + small delta ratios (negative delta) |
Max = 0ε (Mean = 0ε) |
Max = 7.94ε (Mean = 1.4ε) |
Max = 18.3ε (Mean = 2.03ε) |
Max = 7.43ε (Mean = 1.42ε) |
tgamma + small integer ratios |
Max = 0ε (Mean = 0ε) |
Max = 1.96ε (Mean = 0.677ε) |
Max = 1.96ε (Mean = 0.677ε) |
Max = 2.74ε (Mean = 0.736ε) |
tgamma + small integer ratios (negative delta) |
Max = 0ε (Mean = 0ε) |
Max = 1.62ε (Mean = 0.451ε) |
Max = 1.62ε (Mean = 0.451ε) |
Max = 2.15ε (Mean = 0.685ε) |
integer tgamma ratios |
Max = 0ε (Mean = 0ε) |
Max = 0.997ε (Mean = 0.4ε) |
Max = 0.997ε (Mean = 0.4ε) |
Max = 0.968ε (Mean = 0.386ε) |
integer tgamma ratios (negative delta) |
Max = 0ε (Mean = 0ε) |
Max = 0.853ε (Mean = 0.176ε) |
Max = 0.853ε (Mean = 0.176ε) |
Max = 0.974ε (Mean = 0.175ε) |
Table 22.94. Error rates for tgamma (incomplete)
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
tgamma(a, z) medium values |
Max = 0ε (Mean = 0ε) |
Max = 8.47ε (Mean = 1.9ε) |
Max = 412ε (Mean = 95.5ε) |
Max = 8.14ε (Mean = 1.76ε) |
tgamma(a, z) small values |
Max = 0.753ε (Mean = 0.0474ε) |
Max = 2.31ε (Mean = 0.775ε) |
Max = 2.13ε (Mean = 0.717ε) |
Max = 2.53ε (Mean = 0.66ε) |
tgamma(a, z) integer and half integer values |
Max = 0ε (Mean = 0ε) |
Max = 5.52ε (Mean = 1.48ε) |
Max = 79.6ε (Mean = 20.9ε) |
Max = 5.16ε (Mean = 1.33ε) |
Table 22.95. Error rates for tgamma_lower
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
tgamma(a, z) medium values |
Max = 0.833ε (Mean = 0.0315ε) |
Max = 6.79ε (Mean = 1.46ε) |
Max = 363ε (Mean = 63.8ε) |
Max = 5.62ε (Mean = 1.49ε) |
tgamma(a, z) small values |
Max = 0ε (Mean = 0ε) |
Max = 1.97ε (Mean = 0.555ε) |
Max = 1.97ε (Mean = 0.558ε) |
Max = 1.57ε (Mean = 0.525ε) |
tgamma(a, z) integer and half integer values |
Max = 0ε (Mean = 0ε) |
Max = 4.83ε (Mean = 1.15ε) |
Max = 84.7ε (Mean = 17.5ε) |
Max = 2.69ε (Mean = 0.849ε) |
Table 22.96. Error rates for tgamma_ratio
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
tgamma ratios |
Max = 0.694ε (Mean = 0.0347ε) |
Max = 2.99ε (Mean = 1.15ε) |
Max = 174ε (Mean = 61.2ε) |
Max = 3.28ε (Mean = 1.12ε) |
Table 22.97. Error rates for trigamma
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Mathematica Data |
Max = 0.998ε (Mean = 0.105ε) |
Max = 1.28ε (Mean = 0.449ε) |
Max = 1.28ε (Mean = 0.449ε) |
Max = 1ε (Mean = 0.382ε) |
Table 22.98. Error rates for zeta
GNU C++ version 7.1.0 |
GNU C++ version 7.1.0 |
Sun compiler version 0x5150 |
Microsoft Visual C++ version 14.1 |
|
---|---|---|---|---|
Zeta: Random values greater than 1 |
Max = 0.846ε (Mean = 0.0833ε) |
Max = 0ε (Mean = 0ε) |
Max = 0.846ε (Mean = 0.0833ε) |
Max = 0.836ε (Mean = 0.093ε) |
Zeta: Random values less than 1 |
Max = 7.03ε (Mean = 2.93ε) |
Max = 0ε (Mean = 0ε) |
Max = 70.1ε (Mean = 17.1ε) |
Max = 6.84ε (Mean = 3.12ε) |
Zeta: Values close to and greater than 1 |
Max = 0.995ε (Mean = 0.5ε) |
Max = 0ε (Mean = 0ε) |
Max = 0.995ε (Mean = 0.5ε) |
Max = 0.994ε (Mean = 0.421ε) |
Zeta: Values close to and less than 1 |
Max = 0.998ε (Mean = 0.508ε) |
Max = 0ε (Mean = 0ε) |
Max = 0.998ε (Mean = 0.508ε) |
Max = 0.991ε (Mean = 0.375ε) |
Zeta: Integer arguments |
Max = 9ε (Mean = 3.06ε) |
Max = 0ε (Mean = 0ε) |
Max = 28ε (Mean = 5.62ε) |
Max = 9ε (Mean = 3ε) |