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Table 20.3. Error rates for beta
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Beta Function: Small Values | Max = 1.75ε (Mean = 0.828ε) | 
                  Max = 2.65ε (Mean = 1.22ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2.86ε (Mean = 1.31ε) | 
| Beta Function: Medium Values | Max = 99.1ε (Mean = 22.4ε) | 
                  Max = 60.4ε (Mean = 19.4ε) | 
                  Max = 0.978ε (Mean = 0.0578ε) | Max = 107ε (Mean = 24.5ε) | 
| Beta Function: Divergent Values | Max = 10.7ε (Mean = 2.22ε) | 
                  Max = 10.3ε (Mean = 2.41ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 19.7ε (Mean = 2.75ε) | 
Table 20.4. Error rates for beta (incomplete)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Incomplete Beta Function: Small Values | Max = 11.1ε (Mean = 2.28ε) | Max = 0ε (Mean = 0ε) | Max = 12.5ε (Mean = 2.01ε) | Max = 18.7ε (Mean = 3.19ε) | 
| Incomplete Beta Function: Medium Values | Max = 91.1ε (Mean = 12.7ε) | Max = 0.568ε (Mean = 0.0342ε) | Max = 67.1ε (Mean = 13.3ε) | Max = 174ε (Mean = 25ε) | 
| Incomplete Beta Function: Large and Diverse Values | Max = 635ε (Mean = 29.5ε) | Max = 0.999ε (Mean = 0.0399ε) | Max = 6.84e+04ε (Mean = 2.76e+03ε) | Max = 6.86e+04ε (Mean = 2.79e+03ε) | 
| Incomplete Beta Function: Small Integer Values | Max = 26.8ε (Mean = 6.61ε) | Max = 0.786ε (Mean = 0.0248ε) | Max = 10.2ε (Mean = 3.49ε) | Max = 51.8ε (Mean = 11ε) | 
Table 20.5. Error rates for betac
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Incomplete Beta Function: Small Values | Max = 12ε (Mean = 2.43ε) | Max = 0.676ε (Mean = 0.0302ε) | Max = 7.75ε (Mean = 1.91ε) | Max = 12.6ε (Mean = 3.02ε) | 
| Incomplete Beta Function: Medium Values | Max = 91.3ε (Mean = 14.8ε) | Max = 0.949ε (Mean = 0.0944ε) | Max = 63.5ε (Mean = 13.4ε) | Max = 97.6ε (Mean = 24.3ε) | 
| Incomplete Beta Function: Large and Diverse Values | Max = 3.72e+003ε (Mean = 113ε) | Max = 1.12ε (Mean = 0.0458ε) | Max = 1.05e+05ε (Mean = 5.45e+03ε) | Max = 1.04e+05ε (Mean = 5.46e+03ε) | 
| Incomplete Beta Function: Small Integer Values | Max = 26.7ε (Mean = 6.67ε) | Max = 0.586ε (Mean = 0.0314ε) | Max = 10.4ε (Mean = 3.54ε) | Max = 103ε (Mean = 17.4ε) | 
Table 20.6. Error rates for binomial_coefficient
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Binomials: small arguments | Max = 1ε (Mean = 0.369ε) | Max = 1ε (Mean = 0.369ε) | Max = 1.5ε (Mean = 0.339ε) | Max = 1.5ε (Mean = 0.339ε) | 
| Binomials: large arguments | Max = 24.3ε (Mean = 6.3ε) | Max = 0.939ε (Mean = 0.314ε) | Max = 26ε (Mean = 6.09ε) | Max = 54.1ε (Mean = 10.9ε) | 
Table 20.7. Error rates for cbrt
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| cbrt Function | 
                  Max = 1.7ε (Mean = 0.565ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 1.34ε (Mean = 0.471ε) | 
                  Max = 1.11ε (Mean = 0.424ε) | 
Table 20.8. Error rates for cos_pi
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| sin_pi and cos_pi | Max = 0.996ε (Mean = 0.281ε) | Max = 0ε (Mean = 0ε) | Max = 0.991ε (Mean = 0.302ε) | Max = 0.991ε (Mean = 0.302ε) | 
| sin_pi and cos_pi near integers and half integers | Max = 0.996ε (Mean = 0.298ε) | Max = 0ε (Mean = 0ε) | Max = 0.976ε (Mean = 0.28ε) | Max = 0.976ε (Mean = 0.28ε) | 
Table 20.9. Error rates for cyl_bessel_i
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel I0: Mathworld Data | Max = 0.877ε (Mean = 0.549ε) | 
                  Max = 4.57ε (Mean = 2.1ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 4.54ε (Mean = 2.11ε) | 
| Bessel I1: Mathworld Data | Max = 0.885ε (Mean = 0.55ε) | 
                  Max = 7.83ε (Mean = 2.79ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 6.52ε (Mean = 2.25ε) | 
| Bessel In: Mathworld Data | Max = 3.46ε (Mean = 1.32ε) | 
                  Max = 1.8ε (Mean = 1.33ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 463ε (Mean = 140ε) | 
| Bessel Iv: Mathworld Data | Max = 2.97ε (Mean = 1.33ε) | 
                  Max = 4.12ε (Mean = 1.85ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 4.12ε (Mean = 1.95ε) | 
| Bessel In: Random Data | Max = 9.67ε (Mean = 1.89ε) | 
                  Max = 6.79ε (Mean = 1.15ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 176ε (Mean = 39.2ε) | 
| Bessel Iv: Random Data | Max = 7.46ε (Mean = 1.54ε) | 
                  Max = 8.35ε (Mean = 1.49ε) | 
                  Max = 0.661ε (Mean = 0.0441ε) | Max = 283ε (Mean = 88.4ε) | 
| Bessel Iv: Mathworld Data (large values) | Max = 3.67ε (Mean = 1.64ε) | 
                  Max = 14.7ε (Mean = 6.57ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 14.7ε (Mean = 6.59ε) | 
Table 20.10. Error rates for cyl_bessel_i (integer orders)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel I0: Mathworld Data (Integer Version) | Max = 0.877ε (Mean = 0.549ε) | 
                  Max = 4.57ε (Mean = 2.1ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 4.54ε (Mean = 2.11ε) | 
| Bessel I1: Mathworld Data (Integer Version) | Max = 0.885ε (Mean = 0.55ε) | 
                  Max = 7.83ε (Mean = 2.79ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 6.52ε (Mean = 2.25ε) | 
| Bessel In: Mathworld Data (Integer Version) | Max = 3.46ε (Mean = 1.32ε) | 
                  Max = 1.8ε (Mean = 1.33ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 463ε (Mean = 140ε) | 
Table 20.11. Error rates for cyl_bessel_i_prime
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel I'0: Mathworld Data | Max = 0.885ε (Mean = 0.567ε) | Max = 0ε (Mean = 0ε) | Max = 7.83ε (Mean = 2.84ε) | Max = 6.52ε (Mean = 2.26ε) | 
| Bessel I'1: Mathworld Data | Max = 1.61ε (Mean = 0.786ε) | Max = 0ε (Mean = 0ε) | Max = 2.3ε (Mean = 1.28ε) | Max = 2.3ε (Mean = 1.29ε) | 
| Bessel I'n: Mathworld Data | Max = 3.61ε (Mean = 1.22ε) | Max = 0ε (Mean = 0ε) | Max = 2.31ε (Mean = 1.41ε) | Max = 701ε (Mean = 212ε) | 
| Bessel I'v: Mathworld Data | Max = 3.76e+003ε (Mean = 1.19e+003ε) | Max = 1.62ε (Mean = 0.512ε) | Max = 2.89e+03ε (Mean = 914ε) | Max = 2.89e+03ε (Mean = 914ε) | 
| Bessel I'n: Random Data | Max = 9.85ε (Mean = 1.83ε) | Max = 0ε (Mean = 0ε) | Max = 6.79ε (Mean = 1.13ε) | Max = 195ε (Mean = 37.1ε) | 
| Bessel I'v: Random Data | Max = 14ε (Mean = 2.46ε) | Max = 0ε (Mean = 0ε) | Max = 14.1ε (Mean = 2.93ε) | Max = 336ε (Mean = 68.5ε) | 
| Bessel I'v: Mathworld Data (large values) | Max = 59.5ε (Mean = 26.8ε) | Max = 0ε (Mean = 0ε) | Max = 42.6ε (Mean = 20.2ε) | Max = 43.9ε (Mean = 20.7ε) | 
Table 20.12. Error rates for cyl_bessel_i_prime (integer orders)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel I'0: Mathworld Data (Integer Version) | Max = 0.885ε (Mean = 0.567ε) | Max = 0ε (Mean = 0ε) | Max = 7.83ε (Mean = 2.84ε) | Max = 6.52ε (Mean = 2.26ε) | 
| Bessel I'1: Mathworld Data (Integer Version) | Max = 1.61ε (Mean = 0.786ε) | Max = 0ε (Mean = 0ε) | Max = 2.3ε (Mean = 1.28ε) | Max = 2.3ε (Mean = 1.29ε) | 
| Bessel I'n: Mathworld Data (Integer Version) | Max = 3.61ε (Mean = 1.22ε) | Max = 0ε (Mean = 0ε) | Max = 2.31ε (Mean = 1.41ε) | Max = 701ε (Mean = 212ε) | 
Table 20.13. Error rates for cyl_bessel_j
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel J0: Mathworld Data | Max = 2.52ε (Mean = 1.2ε) | 
                  Max = 6.55ε (Mean = 2.89ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 6.55ε (Mean = 2.86ε) | 
| Bessel J0: Mathworld Data (Tricky cases) | Max = 1e+007ε (Mean = 4.09e+006ε) | 
                  Max = 1.63e+08ε (Mean = 6.67e+07ε) | 
                  Max = 7.98e+04ε (Mean = 3.26e+04ε) | Max = 1.64e+08ε (Mean = 6.69e+07ε) | 
| Bessel J1: Mathworld Data | Max = 1.73ε (Mean = 0.976ε) | 
                  Max = 2.66ε (Mean = 1.38ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.44ε (Mean = 0.637ε) | 
| Bessel J1: Mathworld Data (tricky cases) | Max = 3.23e+004ε (Mean = 1.45e+004ε) | 
                  Max = 2.18e+05ε (Mean = 9.76e+04ε) | 
                  Max = 106ε (Mean = 47.5ε) | Max = 2.18e+05ε (Mean = 9.76e+04ε) | 
| Bessel JN: Mathworld Data | Max = 14.7ε (Mean = 5.4ε) | 
                  Max = 6.85ε (Mean = 3.41ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 463ε (Mean = 112ε) | 
| Bessel J: Mathworld Data | Max = 14.9ε (Mean = 3.82ε) | 
                  Max = 14.7ε (Mean = 4.05ε) | 
                  Max = 10ε (Mean = 2.19ε) | Max = 14.7ε (Mean = 4.12ε) | 
| Bessel J: Mathworld Data (large values) | Max = 9.31ε (Mean = 5.52ε) | 
                  Max = 607ε (Mean = 305ε) | 
                  Max = 0.536ε (Mean = 0.268ε) | Max = 607ε (Mean = 305ε) | 
| Bessel JN: Random Data | Max = 17.5ε (Mean = 1.46ε) | 
                  Max = 50.8ε (Mean = 4.15ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 99.6ε (Mean = 22ε) | 
| Bessel J: Random Data | Max = 9.24ε (Mean = 1.36ε) | 
                  Max = 9.81ε (Mean = 1.59ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 260ε (Mean = 34ε) | 
| Bessel J: Random Data (Tricky large values) | Max = 59.2ε (Mean = 8.67ε) | 
                  Max = 785ε (Mean = 94.2ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 785ε (Mean = 97.4ε) | 
Table 20.14. Error rates for cyl_bessel_j (integer orders)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel J0: Mathworld Data (Integer Version) | 
                  Max = 2.52ε (Mean = 1.2ε) | 
                  Max = 6.55ε (Mean = 2.89ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 6.55ε (Mean = 2.86ε) | 
| Bessel J0: Mathworld Data (Tricky cases) (Integer Version) | 
                  Max = 1e+007ε (Mean = 4.09e+006ε) | 
                  Max = 1.63e+08ε (Mean = 6.67e+07ε) | 
                  Max = 7.98e+04ε (Mean = 3.26e+04ε) | Max = 1.64e+08ε (Mean = 6.69e+07ε) | 
| Bessel J1: Mathworld Data (Integer Version) | 
                  Max = 1.73ε (Mean = 0.976ε) | 
                  Max = 2.66ε (Mean = 1.38ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.44ε (Mean = 0.637ε) | 
| Bessel J1: Mathworld Data (tricky cases) (Integer Version) | 
                  Max = 3.23e+004ε (Mean = 1.45e+004ε) | 
                  Max = 2.18e+05ε (Mean = 9.76e+04ε) | 
                  Max = 106ε (Mean = 47.5ε) | Max = 2.18e+05ε (Mean = 9.76e+04ε) | 
| Bessel JN: Mathworld Data (Integer Version) | 
                  Max = 14.7ε (Mean = 5.4ε) | 
                  Max = 6.85ε (Mean = 3.41ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 463ε (Mean = 112ε) | 
Table 20.15. Error rates for cyl_bessel_j_prime
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel J0': Mathworld Data | Max = 6.62ε (Mean = 2.55ε) | Max = 0ε (Mean = 0ε) | Max = 18.9ε (Mean = 6.81ε) | Max = 18.9ε (Mean = 6.72ε) | 
| Bessel J0': Mathworld Data (Tricky cases) | Max = 3.67ε (Mean = 1.74ε) | Max = 0ε (Mean = 0ε) | Max = 8.08ε (Mean = 4.57ε) | Max = 7.44ε (Mean = 3.31ε) | 
| Bessel J1': Mathworld Data | Max = 0.999ε (Mean = 0.627ε) | Max = 0ε (Mean = 0ε) | Max = 7.9ε (Mean = 3.37ε) | Max = 7.9ε (Mean = 3.37ε) | 
| Bessel J1': Mathworld Data (tricky cases) | Max = 288ε (Mean = 129ε) | Max = 287ε (Mean = 129ε) | Max = 5.88e+05ε (Mean = 2.63e+05ε) | Max = 5.88e+05ε (Mean = 2.63e+05ε) | 
| Bessel JN': Mathworld Data | Max = 14ε (Mean = 6.13ε) | Max = 0.527ε (Mean = 0.128ε) | Max = 1.29e+03ε (Mean = 312ε) | Max = 1.29e+03ε (Mean = 355ε) | 
| Bessel J': Mathworld Data | Max = 23.7ε (Mean = 8.01ε) | Max = 21.5ε (Mean = 4.7ε) | Max = 42.5ε (Mean = 9.31ε) | Max = 42.5ε (Mean = 9.32ε) | 
| Bessel J': Mathworld Data (large values) | Max = 2.9ε (Mean = 1.61ε) | Max = 0ε (Mean = 0ε) | Max = 989ε (Mean = 495ε) | Max = 989ε (Mean = 495ε) | 
| Bessel JN': Random Data | Max = 6.34ε (Mean = 0.997ε) | Max = 0.593ε (Mean = 0.0396ε) | Max = 17.3ε (Mean = 2.47ε) | Max = 79.4ε (Mean = 16.2ε) | 
| Bessel J': Random Data | Max = 176ε (Mean = 9.76ε) | Max = 0.885ε (Mean = 0.0457ε) | Max = 139ε (Mean = 6.47ε) | Max = 279ε (Mean = 27.2ε) | 
| Bessel J': Random Data (Tricky large values) | Max = 379ε (Mean = 45.4ε) | Max = 0ε (Mean = 0ε) | Max = 474ε (Mean = 62.2ε) | Max = 474ε (Mean = 64.5ε) | 
Table 20.16. Error rates for cyl_bessel_j_prime (integer orders)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel J0': Mathworld Data (Integer Version) | Max = 6.62ε (Mean = 2.55ε) | Max = 0ε (Mean = 0ε) | Max = 18.9ε (Mean = 6.81ε) | Max = 18.9ε (Mean = 6.72ε) | 
| Bessel J0': Mathworld Data (Tricky cases) (Integer Version) | Max = 3.67ε (Mean = 1.74ε) | Max = 0ε (Mean = 0ε) | Max = 8.08ε (Mean = 4.57ε) | Max = 7.44ε (Mean = 3.31ε) | 
| Bessel J1': Mathworld Data (Integer Version) | Max = 0.999ε (Mean = 0.627ε) | Max = 0ε (Mean = 0ε) | Max = 7.9ε (Mean = 3.37ε) | Max = 7.9ε (Mean = 3.37ε) | 
| Bessel J1': Mathworld Data (tricky cases) (Integer Version) | Max = 288ε (Mean = 129ε) | Max = 287ε (Mean = 129ε) | Max = 5.88e+05ε (Mean = 2.63e+05ε) | Max = 5.88e+05ε (Mean = 2.63e+05ε) | 
| Bessel JN': Mathworld Data (Integer Version) | Max = 14ε (Mean = 6.13ε) | Max = 0.527ε (Mean = 0.128ε) | Max = 1.29e+03ε (Mean = 312ε) | Max = 1.29e+03ε (Mean = 355ε) | 
Table 20.17. Error rates for cyl_bessel_k
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel K0: Mathworld Data | Max = 1.55ε (Mean = 0.837ε) | 
                  Max = 4.16ε (Mean = 1.46ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 4.16ε (Mean = 1.49ε) | 
| Bessel K1: Mathworld Data | Max = 1ε (Mean = 0.573ε) | 
                  Max = 1.8ε (Mean = 1.02ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.8ε (Mean = 1.02ε) | 
| Bessel Kn: Mathworld Data | Max = 3.63ε (Mean = 1.46ε) | 
                  Max = 4.48ε (Mean = 2.14ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 4.48ε (Mean = 1.98ε) | 
| Bessel Kv: Mathworld Data | Max = 4.78ε (Mean = 2.2ε) | 
                  Max = 3.58ε (Mean = 2.44ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 3.58ε (Mean = 2.29ε) | 
| Bessel Kv: Mathworld Data (large values) | Max = 59.8ε (Mean = 26.9ε) | 
                  Max = 42.3ε (Mean = 21ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 43.1ε (Mean = 21.3ε) | 
| Bessel Kn: Random Data | Max = 7.47ε (Mean = 1.4ε) | 
                  Max = 4.55ε (Mean = 1.09ε) | 
                  Max = 0.764ε (Mean = 0.0348ε) | Max = 4.55ε (Mean = 1.21ε) | 
| Bessel Kv: Random Data | Max = 8.33ε (Mean = 1.62ε) | 
                  Max = 7.88ε (Mean = 1.48ε) | 
                  Max = 0.507ε (Mean = 0.0313ε) | Max = 7.88ε (Mean = 1.49ε) | 
Table 20.18. Error rates for cyl_bessel_k (integer orders)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel K0: Mathworld Data (Integer Version) | Max = 1.55ε (Mean = 0.837ε) | 
                  Max = 4.16ε (Mean = 1.46ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 4.16ε (Mean = 1.49ε) | 
| Bessel K1: Mathworld Data (Integer Version) | Max = 1ε (Mean = 0.573ε) | 
                  Max = 1.8ε (Mean = 1.02ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.8ε (Mean = 1.02ε) | 
| Bessel Kn: Mathworld Data (Integer Version) | Max = 3.63ε (Mean = 1.46ε) | 
                  Max = 4.48ε (Mean = 2.14ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 4.48ε (Mean = 1.98ε) | 
Table 20.19. Error rates for cyl_bessel_k_prime
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel K'0: Mathworld Data | Max = 1ε (Mean = 0.573ε) | Max = 0ε (Mean = 0ε) | Max = 1.8ε (Mean = 1.02ε) | Max = 1.8ε (Mean = 1.02ε) | 
| Bessel K'1: Mathworld Data | Max = 1.09ε (Mean = 0.607ε) | Max = 0ε (Mean = 0ε) | Max = 1.96ε (Mean = 1ε) | Max = 1.96ε (Mean = 1.04ε) | 
| Bessel K'n: Mathworld Data | Max = 4.17ε (Mean = 1.74ε) | Max = 0ε (Mean = 0ε) | Max = 4.47ε (Mean = 2.05ε) | Max = 4.47ε (Mean = 1.93ε) | 
| Bessel K'v: Mathworld Data | Max = 3.94ε (Mean = 1.43ε) | Max = 0ε (Mean = 0ε) | Max = 3.94ε (Mean = 2.49ε) | Max = 3.94ε (Mean = 2.35ε) | 
| Bessel K'v: Mathworld Data (large values) | Max = 18.6ε (Mean = 12.1ε) | Max = 0ε (Mean = 0ε) | Max = 59.2ε (Mean = 42.9ε) | Max = 59.8ε (Mean = 43.4ε) | 
| Bessel K'n: Random Data | Max = 8.18ε (Mean = 1.45ε) | Max = 0ε (Mean = 0ε) | Max = 4.53ε (Mean = 1.12ε) | Max = 4.45ε (Mean = 1.23ε) | 
| Bessel K'v: Random Data | Max = 8.32ε (Mean = 1.65ε) | Max = 0ε (Mean = 0ε) | Max = 7.95ε (Mean = 1.53ε) | Max = 7.95ε (Mean = 1.55ε) | 
Table 20.20. Error rates for cyl_bessel_k_prime (integer orders)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel K'0: Mathworld Data (Integer Version) | Max = 1ε (Mean = 0.573ε) | Max = 0ε (Mean = 0ε) | Max = 1.8ε (Mean = 1.02ε) | Max = 1.8ε (Mean = 1.02ε) | 
| Bessel K'1: Mathworld Data (Integer Version) | Max = 1.09ε (Mean = 0.607ε) | Max = 0ε (Mean = 0ε) | Max = 1.96ε (Mean = 1ε) | Max = 1.96ε (Mean = 1.04ε) | 
| Bessel K'n: Mathworld Data (Integer Version) | Max = 4.17ε (Mean = 1.74ε) | Max = 0ε (Mean = 0ε) | Max = 4.47ε (Mean = 2.05ε) | Max = 4.47ε (Mean = 1.93ε) | 
Table 20.21. Error rates for cyl_neumann
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Y0: Mathworld Data | Max = 4.61ε (Mean = 2.29ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 5.59ε (Mean = 2.54ε) | Max = 5.53ε (Mean = 2.4ε) | 
| Y1: Mathworld Data | Max = 4.75ε (Mean = 1.72ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 12.7ε (Mean = 4.34ε) | Max = 6.33ε (Mean = 2.29ε) | 
| Yn: Mathworld Data | Max = 35ε (Mean = 11.8ε) | 
                  Max = 0.993ε (Mean = 0.314ε) | 
                  Max = 55.2ε (Mean = 17.7ε) | Max = 55.2ε (Mean = 17.8ε) | 
| Yv: Mathworld Data | Max = 7.89ε (Mean = 3.27ε) | 
                  Max = 10ε (Mean = 3.02ε) | 
                  Max = 10.7ε (Mean = 4.92ε) | Max = 10.7ε (Mean = 5.1ε) | 
| Yv: Mathworld Data (large values) | Max = 0.682ε (Mean = 0.35ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 1.57ε (Mean = 1.17ε) | Max = 1.57ε (Mean = 1.24ε) | 
| Y0 and Y1: Random Data | Max = 4.17ε (Mean = 1.24ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 11.8ε (Mean = 3.28ε) | Max = 10.8ε (Mean = 3.04ε) | 
| Yn: Random Data | Max = 117ε (Mean = 10.2ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 338ε (Mean = 28.2ε) | Max = 338ε (Mean = 27.5ε) | 
| Yv: Random Data | Max = 1.23e+003ε (Mean = 69.9ε) | 
                  Max = 1.53ε (Mean = 0.102ε) | 
                  Max = 2.08e+03ε (Mean = 149ε) | Max = 2.08e+03ε (Mean = 149ε) | 
Table 20.22. Error rates for cyl_neumann (integer orders)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Y0: Mathworld Data (Integer Version) | 
                  Max = 4.61ε (Mean = 2.29ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 5.59ε (Mean = 2.54ε) | Max = 5.53ε (Mean = 2.4ε) | 
| Y1: Mathworld Data (Integer Version) | 
                  Max = 4.75ε (Mean = 1.72ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 12.7ε (Mean = 4.34ε) | Max = 6.33ε (Mean = 2.29ε) | 
| Yn: Mathworld Data (Integer Version) | 
                  Max = 35ε (Mean = 11.8ε) | 
                  Max = 0.993ε (Mean = 0.314ε) | 
                  Max = 55.2ε (Mean = 17.7ε) | Max = 55.2ε (Mean = 17.8ε) | 
Table 20.23. Error rates for cyl_neumann_prime
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Y'0: Mathworld Data | Max = 4.75ε (Mean = 1.75ε) | Max = 0ε (Mean = 0ε) | Max = 12.7ε (Mean = 4.85ε) | Max = 6.33ε (Mean = 3.14ε) | 
| Y'1: Mathworld Data | Max = 3.08ε (Mean = 1.2ε) | Max = 0.58ε (Mean = 0.193ε) | Max = 36.6ε (Mean = 12.7ε) | Max = 34ε (Mean = 11.8ε) | 
| Y'n: Mathworld Data | Max = 563ε (Mean = 178ε) | Max = 2.05ε (Mean = 0.677ε) | Max = 56ε (Mean = 18.2ε) | Max = 56ε (Mean = 18.2ε) | 
| Y'v: Mathworld Data | Max = 23.7ε (Mean = 10.1ε) | Max = 21.5ε (Mean = 6.49ε) | Max = 42.5ε (Mean = 13.4ε) | Max = 42.5ε (Mean = 13.6ε) | 
| Y'v: Mathworld Data (large values) | Max = 0.627ε (Mean = 0.237ε) | Max = 0ε (Mean = 0ε) | Max = 1.57ε (Mean = 1.24ε) | Max = 1.57ε (Mean = 1.24ε) | 
| Y'0 and Y'1: Random Data | Max = 5.95ε (Mean = 1.36ε) | Max = 0ε (Mean = 0ε) | Max = 23.8ε (Mean = 3.84ε) | Max = 23.8ε (Mean = 3.69ε) | 
| Y'n: Random Data | Max = 621ε (Mean = 36ε) | Max = 1.53ε (Mean = 0.0885ε) | Max = 2.35e+03ε (Mean = 136ε) | Max = 2.35e+03ε (Mean = 136ε) | 
| Y'v: Random Data | Max = 3.23e+004ε (Mean = 1.13e+003ε) | Max = 56.8ε (Mean = 2.59ε) | Max = 1.16e+05ε (Mean = 5.28e+03ε) | Max = 1.16e+05ε (Mean = 5.28e+03ε) | 
Table 20.24. Error rates for cyl_neumann_prime (integer orders)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Y'0: Mathworld Data (Integer Version) | Max = 4.75ε (Mean = 1.75ε) | Max = 0ε (Mean = 0ε) | Max = 12.7ε (Mean = 4.85ε) | Max = 6.33ε (Mean = 3.14ε) | 
| Y'1: Mathworld Data (Integer Version) | Max = 3.08ε (Mean = 1.2ε) | Max = 0.58ε (Mean = 0.193ε) | Max = 36.6ε (Mean = 12.7ε) | Max = 34ε (Mean = 11.8ε) | 
| Y'n: Mathworld Data (Integer Version) | Max = 563ε (Mean = 178ε) | Max = 2.05ε (Mean = 0.677ε) | Max = 56ε (Mean = 18.2ε) | Max = 56ε (Mean = 18.2ε) | 
Table 20.25. Error rates for digamma
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Digamma Function: Large Values | Max = 0.98ε (Mean = 0.369ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.39ε (Mean = 0.413ε) | Max = 1.39ε (Mean = 0.413ε) | 
| Digamma Function: Near the Positive Root | Max = 0.997ε (Mean = 0.527ε) | 
                  Max = 0.891ε (Mean = 0.0995ε) | Max = 1.37ε (Mean = 0.477ε) | Max = 1.31ε (Mean = 0.451ε) | 
| Digamma Function: Near Zero | Max = 0.953ε (Mean = 0.337ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.984ε (Mean = 0.361ε) | Max = 0.984ε (Mean = 0.361ε) | 
| Digamma Function: Negative Values | Max = 214ε (Mean = 16.1ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 180ε (Mean = 13ε) | Max = 180ε (Mean = 13ε) | 
| Digamma Function: Values near 0 | Max = 0ε (Mean = 0ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1ε (Mean = 0.592ε) | Max = 1ε (Mean = 0.592ε) | 
| Digamma Function: Integer arguments | Max = 0.992ε (Mean = 0.452ε) | 
                  Max = 0.992ε (Mean = 0.215ε) | Max = 0.888ε (Mean = 0.403ε) | Max = 0.888ε (Mean = 0.403ε) | 
| Digamma Function: Half integer arguments | Max = 0.78ε (Mean = 0.314ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.906ε (Mean = 0.409ε) | Max = 0.906ε (Mean = 0.409ε) | 
Table 20.26. Error rates for ellint_1
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Elliptic Integral F: Mathworld Data | Max = 0.919ε (Mean = 0.542ε) | 
                  Max = 0.94ε (Mean = 0.509ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.94ε (Mean = 0.509ε) | 
| Elliptic Integral F: Random Data | Max = 2.26ε (Mean = 0.631ε) | 
                  Max = 1.57ε (Mean = 0.56ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.57ε (Mean = 0.561ε) | 
Table 20.27. Error rates for ellint_1 (complete)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Elliptic Integral K: Mathworld Data | Max = 0.915ε (Mean = 0.547ε) | 
                  Max = 0.887ε (Mean = 0.296ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.887ε (Mean = 0.296ε) | 
| Elliptic Integral K: Random Data | Max = 0.958ε (Mean = 0.408ε) | 
                  Max = 1.27ε (Mean = 0.473ε) | 
                  Max = 0.851ε (Mean = 0.0851ε) | Max = 1.27ε (Mean = 0.473ε) | 
Table 20.28. Error rates for ellint_2
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Elliptic Integral E: Mathworld Data | Max = 1.31ε (Mean = 0.727ε) | 
                  Max = 0.656ε (Mean = 0.317ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.656ε (Mean = 0.317ε) | 
| Elliptic Integral E: Random Data | Max = 2.23ε (Mean = 0.639ε) | 
                  Max = 2.05ε (Mean = 0.632ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2.05ε (Mean = 0.632ε) | 
Table 20.29. Error rates for ellint_2 (complete)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Elliptic Integral E: Mathworld Data | Max = 1.3ε (Mean = 0.615ε) | 
                  Max = 0.836ε (Mean = 0.469ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.836ε (Mean = 0.469ε) | 
| Elliptic Integral E: Random Data | Max = 1.71ε (Mean = 0.553ε) | 
                  Max = 1.97ε (Mean = 0.629ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.97ε (Mean = 0.629ε) | 
Table 20.30. Error rates for ellint_3
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Elliptic Integral PI: Mathworld Data | Max = 565ε (Mean = 102ε) | 
                  Max = 475ε (Mean = 86.3ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 475ε (Mean = 86.3ε) | 
| Elliptic Integral PI: Random Data | Max = 9.08ε (Mean = 0.99ε) | 
                  Max = 4.54ε (Mean = 0.895ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 4.49ε (Mean = 0.891ε) | 
| Elliptic Integral PI: Large Random Data | Max = 2.86ε (Mean = 0.944ε) | 
                  Max = 3.7ε (Mean = 0.893ε) | 
                  Max = 0.557ε (Mean = 0.0389ε) | Max = 3.7ε (Mean = 0.892ε) | 
Table 20.31. Error rates for ellint_3 (complete)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Complete Elliptic Integral PI: Mathworld Data | Max = 0.971ε (Mean = 0.464ε) | 
                  Max = 1.4ε (Mean = 0.575ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.4ε (Mean = 0.575ε) | 
| Complete Elliptic Integral PI: Random Data | Max = 2.46ε (Mean = 0.654ε) | 
                  Max = 2.45ε (Mean = 0.696ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2.4ε (Mean = 0.681ε) | 
Table 20.32. Error rates for ellint_d
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Elliptic Integral E: Mathworld Data | Max = 0.862ε (Mean = 0.457ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.3ε (Mean = 0.813ε) | Max = 1.3ε (Mean = 0.813ε) | 
| Elliptic Integral D: Random Data | Max = 2.87ε (Mean = 0.805ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2.51ε (Mean = 0.883ε) | Max = 2.51ε (Mean = 0.883ε) | 
Table 20.33. Error rates for ellint_d (complete)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Elliptic Integral E: Mathworld Data | Max = 0.637ε (Mean = 0.368ε) | Max = 0.637ε (Mean = 0.368ε) | Max = 1.27ε (Mean = 0.735ε) | Max = 1.27ε (Mean = 0.735ε) | 
| Elliptic Integral D: Random Data | Max = 1.27ε (Mean = 0.355ε) | Max = 0ε (Mean = 0ε) | Max = 1.27ε (Mean = 0.334ε) | Max = 1.27ε (Mean = 0.334ε) | 
Table 20.34. Error rates for ellint_rc
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| RC: Random data | Max = 0.962ε (Mean = 0.407ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.995ε (Mean = 0.433ε) | Max = 0.995ε (Mean = 0.438ε) | 
Table 20.35. Error rates for ellint_rd
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| RD: Random data | Max = 2.16ε (Mean = 0.803ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2.73ε (Mean = 0.831ε) | Max = 2.73ε (Mean = 0.829ε) | 
| RD: y = z | Max = 16.5ε (Mean = 0.843ε) | 
                  Max = 0.896ε (Mean = 0.022ε) | Max = 2.65ε (Mean = 0.82ε) | Max = 2.65ε (Mean = 0.819ε) | 
| RD: x = y | Max = 3.51ε (Mean = 0.816ε) | 
                  Max = 0.824ε (Mean = 0.0272ε) | Max = 2.85ε (Mean = 0.865ε) | Max = 2.85ε (Mean = 0.865ε) | 
| RD: x = 0, y = z | Max = 1.16ε (Mean = 0.493ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.19ε (Mean = 0.522ε) | Max = 1.19ε (Mean = 0.522ε) | 
| RD: x = y = z | Max = 1.03ε (Mean = 0.418ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.998ε (Mean = 0.387ε) | Max = 0.998ε (Mean = 0.387ε) | 
| RD: x = 0 | Max = 2.64ε (Mean = 0.894ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2.79ε (Mean = 0.883ε) | Max = 2.79ε (Mean = 0.883ε) | 
Table 20.36. Error rates for ellint_rf
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| RF: Random data | Max = 2.02ε (Mean = 0.677ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2.54ε (Mean = 0.674ε) | Max = 2.54ε (Mean = 0.674ε) | 
| RF: x = y = z | Max = 0.999ε (Mean = 0.335ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.991ε (Mean = 0.345ε) | Max = 0.991ε (Mean = 0.345ε) | 
| RF: x = y or y = z or x = z | Max = 1.21ε (Mean = 0.394ε) | 
                  Max = 0.536ε (Mean = 0.00658ε) | Max = 1.95ε (Mean = 0.418ε) | Max = 1.57ε (Mean = 0.418ε) | 
| RF: x = 0, y = z | Max = 0.999ε (Mean = 0.407ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.894ε (Mean = 0.338ε) | Max = 0.894ε (Mean = 0.338ε) | 
| RF: z = 0 | Max = 1.89ε (Mean = 0.587ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.7ε (Mean = 0.539ε) | Max = 1.7ε (Mean = 0.539ε) | 
Table 20.37. Error rates for ellint_rg
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| RG: Random Data | Max = 3.65ε (Mean = 0.929ε) | 
                  Max = 0.983ε (Mean = 0.0172ε) | Max = 3.95ε (Mean = 0.951ε) | Max = 3.95ε (Mean = 0.951ε) | 
| RG: two values 0 | Max = 0ε (Mean = 0ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0ε (Mean = 0ε) | Max = 0ε (Mean = 0ε) | 
| RG: All values the same or zero | Max = 1.06ε (Mean = 0.348ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.992ε (Mean = 0.288ε) | Max = 0.992ε (Mean = 0.288ε) | 
| RG: two values the same | Max = 1.96ε (Mean = 0.374ε) | 
                  Max = 0.594ε (Mean = 0.0103ε) | Max = 1.51ε (Mean = 0.404ε) | Max = 1.51ε (Mean = 0.404ε) | 
| RG: one value zero | Max = 1.96ε (Mean = 0.674ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2.14ε (Mean = 0.722ε) | Max = 2.14ε (Mean = 0.722ε) | 
Table 20.38. Error rates for ellint_rj
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| RJ: Random data | Max = 119ε (Mean = 4.32ε) | 
                  Max = 0.52ε (Mean = 0.0184ε) | Max = 186ε (Mean = 6.67ε) | Max = 186ε (Mean = 6.7ε) | 
| RJ: 4 Equal Values | Max = 1.03ε (Mean = 0.418ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.998ε (Mean = 0.387ε) | Max = 0.998ε (Mean = 0.387ε) | 
| RJ: 3 Equal Values | Max = 39.9ε (Mean = 1.12ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 20.8ε (Mean = 0.986ε) | Max = 18.2ε (Mean = 0.917ε) | 
| RJ: 2 Equal Values | Max = 214ε (Mean = 5.05ε) | 
                  Max = 0.6ε (Mean = 0.0228ε) | Max = 220ε (Mean = 6.64ε) | Max = 135ε (Mean = 5.3ε) | 
| RJ: Equal z and p | Max = 15.4ε (Mean = 1.05ε) | 
                  Max = 0.742ε (Mean = 0.0166ε) | Max = 17.2ε (Mean = 1.16ε) | Max = 16.6ε (Mean = 1.15ε) | 
Table 20.39. Error rates for erf
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Erf Function: Small Values | 
                  Max = 0.996ε (Mean = 0.182ε) | 
                  Max = 0.925ε (Mean = 0.193ε) | 
                  Max = 0.841ε (Mean = 0.0687ε) | 
                  Max = 0.925ε (Mean = 0.193ε) | 
| Erf Function: Medium Values | 
                  Max = 1ε (Mean = 0.169ε) | 
                  Max = 1.5ε (Mean = 0.193ε) | 
                  Max = 1ε (Mean = 0.119ε) | 
                  Max = 1.5ε (Mean = 0.202ε) | 
| Erf Function: Large Values | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0ε (Mean = 0ε) | 
Table 20.40. Error rates for erf_inv
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Inverse Erf Function | Max = 1.09ε (Mean = 0.502ε) | Max = 0ε (Mean = 0ε) | Max = 0.996ε (Mean = 0.389ε) | Max = 0.996ε (Mean = 0.385ε) | 
Table 20.41. Error rates for erfc
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Erf Function: Small Values | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0.658ε (Mean = 0.0537ε) | 
                  Max = 0ε (Mean = 0ε) | 
| Erf Function: Medium Values | 
                  Max = 1.65ε (Mean = 0.373ε) | 
                  Max = 1.76ε (Mean = 0.365ε) | 
                  Max = 0.983ε (Mean = 0.213ε) | 
                  Max = 1.76ε (Mean = 0.383ε) | 
| Erf Function: Large Values | 
                  Max = 1.14ε (Mean = 0.248ε) | 
                  Max = 1.57ε (Mean = 0.542ε) | 
                  Max = 0.868ε (Mean = 0.147ε) | 
                  Max = 1.57ε (Mean = 0.564ε) | 
Table 20.42. Error rates for erfc_inv
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Inverse Erfc Function | Max = 1ε (Mean = 0.491ε) | Max = 0ε (Mean = 0ε) | Max = 0.996ε (Mean = 0.397ε) | Max = 0.996ε (Mean = 0.397ε) | 
| Inverse Erfc Function: extreme values | Max = 1.62ε (Mean = 0.383ε) | Max = 1.62ε (Mean = 0.385ε) | 
Table 20.43. Error rates for expint (Ei)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Exponential Integral Ei | Max = 1.43ε (Mean = 0.541ε) | 
                  Max = 5.05ε (Mean = 0.821ε) | 
                  Max = 0.994ε (Mean = 0.142ε) | Max = 3.34ε (Mean = 0.631ε) | 
| Exponential Integral Ei: double exponent range | Max = 1.7ε (Mean = 0.66ε) | 
                  Max = 1.72ε (Mean = 0.593ε) | 
                  Max = 0.998ε (Mean = 0.156ε) | Max = 1.72ε (Mean = 0.618ε) | 
| Exponential Integral Ei: long exponent range | 
                  Max = 1.98ε (Mean = 0.595ε) | Max = 1.98ε (Mean = 0.575ε) | 
Table 20.44. Error rates for expint (En)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Exponential Integral En | Max = 7.16ε (Mean = 1.85ε) | 
                  Max = 0.589ε (Mean = 0.0331ε) | Max = 9.97ε (Mean = 2.13ε) | Max = 9.97ε (Mean = 2.13ε) | 
| Exponential Integral En: small z values | Max = 2.62ε (Mean = 0.531ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.99ε (Mean = 0.559ε) | Max = 1.99ε (Mean = 0.559ε) | 
| Exponential Integral E1 | Max = 0.988ε (Mean = 0.486ε) | 
                  Max = 0.556ε (Mean = 0.0625ε) | Max = 0.965ε (Mean = 0.414ε) | Max = 0.965ε (Mean = 0.409ε) | 
Table 20.45. Error rates for expm1
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Random test data | 
                  Max = 0.996ε (Mean = 0.283ε) | 
                  Max = 0.793ε (Mean = 0.126ε) | 
                  Max = 0.992ε (Mean = 0.402ε) | 
                  Max = 1.31ε (Mean = 0.406ε) | 
Table 20.46. Error rates for gamma_p
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| tgamma(a, z) medium values | Max = 35.1ε (Mean = 6.97ε) | 
                  Max = 0.955ε (Mean = 0.05ε) | Max = 41ε (Mean = 8.09ε) | Max = 239ε (Mean = 30.2ε) | 
| tgamma(a, z) small values | Max = 1.54ε (Mean = 0.439ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2ε (Mean = 0.461ε) | Max = 2ε (Mean = 0.472ε) | 
| tgamma(a, z) large values | Max = 244ε (Mean = 20.2ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 3.08e+04ε (Mean = 1.86e+03ε) | Max = 3.02e+04ε (Mean = 1.91e+03ε) | 
| tgamma(a, z) integer and half integer values | Max = 13ε (Mean = 2.93ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 11.8ε (Mean = 2.65ε) | Max = 71.6ε (Mean = 9.47ε) | 
Table 20.47. Error rates for gamma_p_inv
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| incomplete gamma inverse(a, z) medium values | Max = 1.01ε (Mean = 0.307ε) | 
                  Max = 0.993ε (Mean = 0.15ε) | Max = 1.62ε (Mean = 0.365ε) | Max = 1.86ε (Mean = 0.405ε) | 
| incomplete gamma inverse(a, z) large values | Max = 0.924ε (Mean = 0.118ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 0.509ε (Mean = 0.0447ε) | Max = 0.509ε (Mean = 0.0447ε) | 
| incomplete gamma inverse(a, z) small values | Max = 1.1e+003ε (Mean = 108ε) | 
                  Max = 441ε (Mean = 53.9ε) | Max = 9.17e+03ε (Mean = 1.32e+03ε) | Max = 1.09e+04ε (Mean = 1.46e+03ε) | 
Table 20.48. Error rates for gamma_p_inva
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Incomplete gamma inverses. | Max = 3.52ε (Mean = 0.997ε) | Max = 0ε (Mean = 0ε) | Max = 6.44ε (Mean = 1.1ε) | Max = 4.08ε (Mean = 1.12ε) | 
Table 20.49. Error rates for gamma_q
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| tgamma(a, z) medium values | Max = 23.7ε (Mean = 4.03ε) | 
                  Max = 0.927ε (Mean = 0.035ε) | Max = 31.3ε (Mean = 6.56ε) | Max = 199ε (Mean = 26.6ε) | 
| tgamma(a, z) small values | Max = 2.26ε (Mean = 0.732ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2.45ε (Mean = 0.832ε) | Max = 2.25ε (Mean = 0.81ε) | 
| tgamma(a, z) large values | Max = 470ε (Mean = 31.5ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 6.82e+03ε (Mean = 414ε) | Max = 1.15e+04ε (Mean = 733ε) | 
| tgamma(a, z) integer and half integer values | Max = 8.48ε (Mean = 1.42ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 11.1ε (Mean = 2.09ε) | Max = 54.7ε (Mean = 6.16ε) | 
Table 20.50. Error rates for gamma_q_inv
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| incomplete gamma inverse(a, z) medium values | Max = 3.46ε (Mean = 0.475ε) | 
                  Max = 0.912ε (Mean = 0.154ε) | Max = 6.2ε (Mean = 0.659ε) | Max = 6.2ε (Mean = 0.661ε) | 
| incomplete gamma inverse(a, z) large values | Max = 0.814ε (Mean = 0.0856ε) | 
                  Max = 0.894ε (Mean = 0.0915ε) | Max = 0ε (Mean = 0ε) | Max = 0ε (Mean = 0ε) | 
| incomplete gamma inverse(a, z) small values | Max = 451ε (Mean = 65ε) | 
                  Max = 292ε (Mean = 36.4ε) | Max = 8.28e+03ε (Mean = 963ε) | Max = 8.98e+03ε (Mean = 877ε) | 
Table 20.51. Error rates for gamma_q_inva
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Incomplete gamma inverses. | Max = 5.64ε (Mean = 1.09ε) | Max = 0ε (Mean = 0ε) | Max = 6.91ε (Mean = 1.17ε) | Max = 7.86ε (Mean = 1.25ε) | 
Table 20.52. Error rates for hermite
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Hermite Polynomials | Max = 4.46ε (Mean = 1.41ε) | Max = 0ε (Mean = 0ε) | Max = 6.24ε (Mean = 2.07ε) | Max = 6.24ε (Mean = 2.07ε) | 
Table 20.53. Error rates for heuman_lambda
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Elliptic Integral Jacobi Zeta: Mathworld Data | Max = 1.08ε (Mean = 0.734ε) | Max = 0ε (Mean = 0ε) | Max = 1.89ε (Mean = 0.887ε) | Max = 1.89ε (Mean = 0.887ε) | 
| Elliptic Integral Heuman Lambda: Random Data | Max = 2.12ε (Mean = 0.59ε) | Max = 0ε (Mean = 0ε) | Max = 3.82ε (Mean = 0.609ε) | Max = 3.82ε (Mean = 0.608ε) | 
Table 20.54. Error rates for ibeta
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Incomplete Beta Function: Small Values | Max = 9.42ε (Mean = 2.24ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 6.27ε (Mean = 1.7ε) | Max = 21.3ε (Mean = 2.74ε) | 
| Incomplete Beta Function: Medium Values | Max = 108ε (Mean = 16.3ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 51.4ε (Mean = 12ε) | Max = 124ε (Mean = 18.4ε) | 
| Incomplete Beta Function: Large and Diverse Values | Max = 1.19e+003ε (Mean = 59.5ε) | 
                  Max = 1.26ε (Mean = 0.063ε) | Max = 1.96e+04ε (Mean = 997ε) | Max = 4.98e+04ε (Mean = 2.07e+03ε) | 
| Incomplete Beta Function: Small Integer Values | Max = 3.69ε (Mean = 0.765ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 4.45ε (Mean = 0.814ε) | Max = 44.5ε (Mean = 10.1ε) | 
Table 20.55. Error rates for ibeta_inv
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Inverse incomplete beta | Max = 7.08e+003ε (Mean = 244ε) | 
                  Max = 3.21ε (Mean = 0.158ε) | Max = 4.53e+04ε (Mean = 2.93e+03ε) | Max = 4.22e+04ε (Mean = 2.8e+03ε) | 
Table 20.56. Error rates for ibeta_inva
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Inverse incomplete beta | Max = 395ε (Mean = 24.7ε) | Max = 0.602ε (Mean = 0.0239ε) | Max = 377ε (Mean = 25.1ε) | Max = 438ε (Mean = 31.6ε) | 
Table 20.57. Error rates for ibeta_invb
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Inverse incomplete beta | Max = 409ε (Mean = 17.9ε) | Max = 0.836ε (Mean = 0.0491ε) | Max = 407ε (Mean = 27.2ε) | Max = 407ε (Mean = 24.4ε) | 
Table 20.58. Error rates for ibetac
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Incomplete Beta Function: Small Values | Max = 6.56ε (Mean = 1.88ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 8.47ε (Mean = 1.68ε) | Max = 13.8ε (Mean = 2.69ε) | 
| Incomplete Beta Function: Medium Values | Max = 56.2ε (Mean = 14.1ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 75.3ε (Mean = 11.3ε) | Max = 132ε (Mean = 19.8ε) | 
| Incomplete Beta Function: Large and Diverse Values | Max = 1.87e+003ε (Mean = 93.8ε) | 
                  Max = 0.981ε (Mean = 0.0638ε) | Max = 3.45e+04ε (Mean = 1.32e+03ε) | Max = 4.26e+04ε (Mean = 1.54e+03ε) | 
| Incomplete Beta Function: Small Integer Values | Max = 6.37ε (Mean = 1.03ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 5.34ε (Mean = 1.11ε) | Max = 107ε (Mean = 17.1ε) | 
Table 20.59. Error rates for ibetac_inv
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Inverse incomplete beta | Max = 5.53e+003ε (Mean = 220ε) | 
                  Max = 1.71ε (Mean = 0.108ε) | Max = 6.17e+04ε (Mean = 3.77e+03ε) | Max = 5.15e+04ε (Mean = 3.51e+03ε) | 
Table 20.60. Error rates for ibetac_inva
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Inverse incomplete beta | Max = 408ε (Mean = 27.8ε) | Max = 0.683ε (Mean = 0.0271ε) | Max = 382ε (Mean = 22.2ε) | Max = 315ε (Mean = 23.7ε) | 
Table 20.61. Error rates for ibetac_invb
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Inverse incomplete beta | Max = 329ε (Mean = 18.2ε) | Max = 0.724ε (Mean = 0.0303ε) | Max = 317ε (Mean = 19.7ε) | Max = 369ε (Mean = 21.7ε) | 
Table 20.62. Error rates for jacobi_cn
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Jacobi Elliptic: Mathworld Data | Max = 45.8ε (Mean = 11.4ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 71.6ε (Mean = 19.3ε) | Max = 71.6ε (Mean = 19.4ε) | 
| Jacobi Elliptic: Random Data | Max = 1.83ε (Mean = 0.455ε) | 
                  Max = 0.816ε (Mean = 0.0563ε) | Max = 1.68ε (Mean = 0.443ε) | Max = 1.68ε (Mean = 0.454ε) | 
| Jacobi Elliptic: Random Small Values | Max = 26.2ε (Mean = 1.17ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 10.4ε (Mean = 0.594ε) | Max = 10.4ε (Mean = 0.602ε) | 
| Jacobi Elliptic: Modulus near 1 | Max = 513ε (Mean = 126ε) | 
                  Max = 0.919ε (Mean = 0.127ε) | Max = 675ε (Mean = 87.1ε) | Max = 675ε (Mean = 86.8ε) | 
| Jacobi Elliptic: Large Phi | Max = 3.27e+004ε (Mean = 1.93e+003ε) | 
                  Max = 14.2ε (Mean = 0.927ε) | Max = 2.97e+04ε (Mean = 1.9e+03ε) | Max = 2.97e+04ε (Mean = 1.9e+03ε) | 
Table 20.63. Error rates for jacobi_dn
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Jacobi Elliptic: Mathworld Data | Max = 34.3ε (Mean = 8.71ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 49ε (Mean = 14ε) | Max = 49ε (Mean = 14ε) | 
| Jacobi Elliptic: Random Data | Max = 1.52ε (Mean = 0.466ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.53ε (Mean = 0.473ε) | Max = 1.53ε (Mean = 0.481ε) | 
| Jacobi Elliptic: Random Small Values | Max = 16.1ε (Mean = 0.685ε) | 
                  Max = 0.5ε (Mean = 0.0122ε) | Max = 22.4ε (Mean = 0.777ε) | Max = 22.4ε (Mean = 0.763ε) | 
| Jacobi Elliptic: Modulus near 1 | Max = 6.24e+003ε (Mean = 482ε) | 
                  Max = 2.28ε (Mean = 0.194ε) | Max = 3.75e+03ε (Mean = 293ε) | Max = 3.75e+03ε (Mean = 293ε) | 
| Jacobi Elliptic: Large Phi | Max = 1.67e+004ε (Mean = 1e+003ε) | 
                  Max = 14.1ε (Mean = 0.897ε) | Max = 2.82e+04ε (Mean = 1.79e+03ε) | Max = 2.82e+04ε (Mean = 1.79e+03ε) | 
Table 20.64. Error rates for jacobi_sn
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Jacobi Elliptic: Mathworld Data | Max = 481ε (Mean = 113ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 341ε (Mean = 80.7ε) | Max = 341ε (Mean = 80.7ε) | 
| Jacobi Elliptic: Random Data | Max = 1.92ε (Mean = 0.567ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 2.01ε (Mean = 0.584ε) | Max = 2.01ε (Mean = 0.593ε) | 
| Jacobi Elliptic: Random Small Values | Max = 2.11ε (Mean = 0.385ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.99ε (Mean = 0.347ε) | Max = 1.99ε (Mean = 0.347ε) | 
| Jacobi Elliptic: Modulus near 1 | Max = 23.2ε (Mean = 1.85ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 109ε (Mean = 7.35ε) | Max = 109ε (Mean = 7.38ε) | 
| Jacobi Elliptic: Large Phi | Max = 4.36e+004ε (Mean = 2.54e+003ε) | 
                  Max = 12ε (Mean = 0.771ε) | Max = 2.45e+04ε (Mean = 1.51e+03ε) | Max = 2.45e+04ε (Mean = 1.51e+03ε) | 
Table 20.65. Error rates for jacobi_zeta
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Elliptic Integral Jacobi Zeta: Mathworld Data | Max = 0.505ε (Mean = 0.119ε) | Max = 0ε (Mean = 0ε) | Max = 1.66ε (Mean = 0.48ε) | Max = 1.66ε (Mean = 0.48ε) | 
| Elliptic Integral Jacobi Zeta: Random Data | Max = 3.89ε (Mean = 0.818ε) | Max = 0ε (Mean = 0ε) | Max = 2.99ε (Mean = 0.824ε) | Max = 3.96ε (Mean = 1.06ε) | 
| Elliptic Integral Jacobi Zeta: Large Phi Values | Max = 2.92ε (Mean = 0.964ε) | Max = 0ε (Mean = 0ε) | Max = 2.92ε (Mean = 0.951ε) | Max = 3.05ε (Mean = 1.13ε) | 
Table 20.66. Error rates for laguerre(n, m, x)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Associated Laguerre Polynomials | Max = 434ε (Mean = 11.1ε) | 
                  Max = 0.84ε (Mean = 0.0358ε) | 
                  Max = 167ε (Mean = 6.38ε) | Max = 167ε (Mean = 6.38ε) | 
Table 20.67. Error rates for laguerre(n, x)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Laguerre Polynomials | Max = 3.1e+003ε (Mean = 185ε) | 
                  Max = 6.82ε (Mean = 0.408ε) | 
                  Max = 1.39e+04ε (Mean = 828ε) | Max = 1.39e+04ε (Mean = 828ε) | 
Table 20.68. Error rates for legendre_p
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Legendre Polynomials: Small Values | Max = 211ε (Mean = 20.4ε) | 
                  Max = 0.732ε (Mean = 0.0619ε) | 
                  Max = 69.2ε (Mean = 9.58ε) | Max = 69.2ε (Mean = 9.58ε) | 
| Legendre Polynomials: Large Values | Max = 300ε (Mean = 33.2ε) | 
                  Max = 0.632ε (Mean = 0.0693ε) | 
                  Max = 699ε (Mean = 59.6ε) | Max = 699ε (Mean = 59.6ε) | 
Table 20.69. Error rates for legendre_p (associated)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Associated Legendre Polynomials: Small Values | Max = 121ε (Mean = 7.14ε) | 
                  Max = 0.999ε (Mean = 0.05ε) | 
                  Max = 175ε (Mean = 9.88ε) | Max = 77.7ε (Mean = 5.59ε) | 
Table 20.70. Error rates for legendre_q
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Legendre Polynomials: Small Values | Max = 46.4ε (Mean = 7.32ε) | 
                  Max = 0.612ε (Mean = 0.0517ε) | Max = 50.9ε (Mean = 9ε) | Max = 50.9ε (Mean = 8.98ε) | 
| Legendre Polynomials: Large Values | Max = 4.6e+003ε (Mean = 366ε) | 
                  Max = 2.49ε (Mean = 0.202ε) | Max = 5.98e+03ε (Mean = 478ε) | Max = 5.98e+03ε (Mean = 478ε) | 
Table 20.71. Error rates for lgamma
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| factorials | 
                  Max = 0.914ε (Mean = 0.167ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0.991ε (Mean = 0.311ε) | 
                  Max = 0.991ε (Mean = 0.383ε) | 
| near 0 | 
                  Max = 0.964ε (Mean = 0.462ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 1.42ε (Mean = 0.566ε) | 
                  Max = 1.42ε (Mean = 0.566ε) | 
| near 1 | 
                  Max = 0.867ε (Mean = 0.468ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0.948ε (Mean = 0.36ε) | 
                  Max = 0.866ε (Mean = 0.355ε) | 
| near 2 | 
                  Max = 0.591ε (Mean = 0.159ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0.878ε (Mean = 0.242ε) | 
                  Max = 0.878ε (Mean = 0.241ε) | 
| near -10 | 
                  Max = 4.22ε (Mean = 1.33ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 3.81ε (Mean = 1.01ε) | 
                  Max = 3.81ε (Mean = 1.01ε) | 
| near -55 | 
                  Max = 0.821ε (Mean = 0.419ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0.821ε (Mean = 0.513ε) | 
                  Max = 1.59ε (Mean = 0.587ε) | 
Table 20.72. Error rates for log1p
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Random test data | 
                  Max = 0.509ε (Mean = 0.057ε) | 
                  Max = 0.846ε (Mean = 0.153ε) | 
                  Max = 0.818ε (Mean = 0.227ε) | 
                  Max = 1.53ε (Mean = 0.627ε) | 
Table 20.73. Error rates for non central beta CDF
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Non Central Beta, medium parameters | Max = 240ε (Mean = 31ε) | 
                  Max = 0.998ε (Mean = 0.0659ε) | Max = 825ε (Mean = 27.4ε) | Max = 832ε (Mean = 38.1ε) | 
| Non Central Beta, large parameters | Max = 3.41e+003ε (Mean = 475ε) | 
                  Max = 1.18ε (Mean = 0.175ε) | Max = 2.5e+04ε (Mean = 3.78e+03ε) | Max = 2.57e+04ε (Mean = 4.43e+03ε) | 
Table 20.74. Error rates for non central beta CDF complement
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Non Central Beta, medium parameters | Max = 619ε (Mean = 62.7ε) | 
                  Max = 0.998ε (Mean = 0.0957ε) | Max = 396ε (Mean = 50.7ε) | Max = 554ε (Mean = 57.3ε) | 
| Non Central Beta, large parameters | Max = 8.67e+003ε (Mean = 1.04e+003ε) | 
                  Max = 0.986ε (Mean = 0.188ε) | Max = 6.83e+03ε (Mean = 993ε) | Max = 3.56e+03ε (Mean = 704ε) | 
Table 20.75. Error rates for non central chi squared CDF
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Non Central Chi Squared, medium parameters | Max = 48.9ε (Mean = 10ε) | 
                  Max = 0.99ε (Mean = 0.0529ε) | Max = 46.5ε (Mean = 10.3ε) | Max = 115ε (Mean = 13.9ε) | 
| Non Central Chi Squared, large parameters | Max = 9.79e+003ε (Mean = 723ε) | 
                  Max = 1.07ε (Mean = 0.102ε) | Max = 3.07e+03ε (Mean = 336ε) | Max = 6.17e+03ε (Mean = 677ε) | 
Table 20.76. Error rates for non central chi squared CDF complement
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Non Central Chi Squared, medium parameters | Max = 98.6ε (Mean = 15.8ε) | 
                  Max = 0.96ε (Mean = 0.0635ε) | Max = 107ε (Mean = 17.1ε) | Max = 171ε (Mean = 22.8ε) | 
| Non Central Chi Squared, large parameters | Max = 5.43e+003ε (Mean = 705ε) | 
                  Max = 2.11ε (Mean = 0.278ε) | Max = 5.02e+03ε (Mean = 630ε) | Max = 5.1e+03ε (Mean = 577ε) | 
Table 20.77. Error rates for non central t CDF
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Non Central T | Max = 138ε (Mean = 31.5ε) | 
                  Max = 0.796ε (Mean = 0.0691ε) | Max = 141ε (Mean = 31.1ε) | Max = 145ε (Mean = 30.2ε) | 
| Non Central T (small non-centrality) | Max = 3.61ε (Mean = 1.03ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 7.86ε (Mean = 1.69ε) | Max = 9.15ε (Mean = 2.25ε) | 
| Non Central T (large parameters) | Max = 286ε (Mean = 62.8ε) | 
                  Max = 257ε (Mean = 72.1ε) | Max = 5.26e+05ε (Mean = 1.48e+05ε) | Max = 5.24e+05ε (Mean = 1.47e+05ε) | 
Table 20.78. Error rates for non central t CDF complement
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Non Central T | Max = 150ε (Mean = 32.3ε) | 
                  Max = 0.707ε (Mean = 0.0497ε) | Max = 203ε (Mean = 31.8ε) | Max = 340ε (Mean = 43.6ε) | 
| Non Central T (small non-centrality) | Max = 5.21ε (Mean = 1.43ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 7.48ε (Mean = 1.86ε) | Max = 10.9ε (Mean = 2.43ε) | 
| Non Central T (large parameters) | Max = 227ε (Mean = 50.4ε) | 
                  Max = 478ε (Mean = 96.3ε) | Max = 9.79e+05ε (Mean = 1.97e+05ε) | Max = 9.79e+05ε (Mean = 1.97e+05ε) | 
Table 20.79. Error rates for owens_t
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Owens T (medium small values) | Max = 4.37ε (Mean = 0.973ε) | Max = 0ε (Mean = 0ε) | Max = 3.34ε (Mean = 0.942ε) | Max = 3.34ε (Mean = 0.91ε) | 
| Owens T (large and diverse values) | Max = 3.78ε (Mean = 0.621ε) | Max = 0ε (Mean = 0ε) | Max = 49ε (Mean = 2.16ε) | Max = 24.5ε (Mean = 1.38ε) | 
Table 20.80. Error rates for polygamma
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Mathematica Data | Max = 6.34ε (Mean = 1.53ε) | 
                  Max = 0.824ε (Mean = 0.0574ε) | Max = 7.38ε (Mean = 1.84ε) | Max = 18.3ε (Mean = 4.16ε) | 
| Mathematica Data - large arguments | Max = 150ε (Mean = 15.1ε) | 
                  Max = 0.998ε (Mean = 0.0592ε) | Max = 2.23ε (Mean = 0.323ε) | Max = 2.35ε (Mean = 0.34ε) | 
| Mathematica Data - negative arguments | Max = 497ε (Mean = 129ε) | 
                  Max = 0.516ε (Mean = 0.022ε) | Max = 269ε (Mean = 87.7ε) | Max = 269ε (Mean = 87.9ε) | 
| Mathematica Data - large negative arguments | Max = 162ε (Mean = 101ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 155ε (Mean = 96.4ε) | Max = 155ε (Mean = 96.4ε) | 
| Mathematica Data - small arguments | Max = 3ε (Mean = 0.496ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 3.33ε (Mean = 0.75ε) | Max = 3.33ε (Mean = 0.75ε) | 
| Mathematica Data - Large orders and other bug cases | Max = 200ε (Mean = 57.2ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 54.5ε (Mean = 13.3ε) | Max = 90.1ε (Mean = 30.6ε) | 
Table 20.81. Error rates for powm1
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| powm1 | Max = 1.99ε (Mean = 0.461ε) | Max = 1.26ε (Mean = 0.428ε) | Max = 2.04ε (Mean = 0.521ε) | Max = 2.14ε (Mean = 0.498ε) | 
Table 20.82. Error rates for sin_pi
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| sin_pi and cos_pi | Max = 0.99ε (Mean = 0.328ε) | Max = 0ε (Mean = 0ε) | Max = 0.996ε (Mean = 0.335ε) | Max = 0.996ε (Mean = 0.336ε) | 
| sin_pi and cos_pi near integers and half integers | Max = 0.996ε (Mean = 0.343ε) | Max = 0ε (Mean = 0ε) | Max = 0.976ε (Mean = 0.293ε) | Max = 0.976ε (Mean = 0.293ε) | 
Table 20.83. Error rates for sph_bessel
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel j: Random Data | Max = 245ε (Mean = 16.3ε) | 
                  Max = 243ε (Mean = 13.3ε) | 
                  Max = 0.978ε (Mean = 0.0539ε) | Max = 243ε (Mean = 33.7ε) | 
Table 20.84. Error rates for sph_bessel_prime
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Bessel j': Random Data | Max = 307ε (Mean = 25.2ε) | Max = 0.753ε (Mean = 0.0343ε) | Max = 167ε (Mean = 12ε) | Max = 167ε (Mean = 33.2ε) | 
Table 20.85. Error rates for sph_neumann
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| y: Random Data | Max = 281ε (Mean = 31.1ε) | 
                  Max = 0.995ε (Mean = 0.0665ε) | 
                  Max = 234ε (Mean = 19.5ε) | Max = 234ε (Mean = 19.8ε) | 
Table 20.86. Error rates for sph_neumann_prime
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| y': Random Data | Max = 296ε (Mean = 25.6ε) | Max = 0.752ε (Mean = 0.062ε) | Max = 158ε (Mean = 18.8ε) | Max = 158ε (Mean = 20.2ε) | 
Table 20.87. Error rates for spherical_harmonic_i
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Spherical Harmonics | Max = 2.27e+004ε (Mean = 725ε) | Max = 1.36ε (Mean = 0.0765ε) | Max = 2.89e+03ε (Mean = 108ε) | Max = 1.03e+04ε (Mean = 327ε) | 
Table 20.88. Error rates for spherical_harmonic_r
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Spherical Harmonics | Max = 2.27e+004ε (Mean = 725ε) | Max = 1.58ε (Mean = 0.0707ε) | Max = 2.89e+03ε (Mean = 108ε) | Max = 1.03e+04ε (Mean = 327ε) | 
Table 20.89. Error rates for sqrt1pm1
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| sqrt1pm1 | Max = 1.36ε (Mean = 0.44ε) | Max = 1.3ε (Mean = 0.404ε) | Max = 1.33ε (Mean = 0.409ε) | Max = 1.54ε (Mean = 0.568ε) | 
Table 20.90. Error rates for tgamma
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| factorials | 
                  Max = 1.85ε (Mean = 0.491ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 1.96ε (Mean = 0.483ε) | 
                  Max = 172ε (Mean = 41ε) | 
| near 0 | 
                  Max = 1.96ε (Mean = 0.684ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 2ε (Mean = 0.73ε) | 
                  Max = 2ε (Mean = 0.647ε) | 
| near 1 | 
                  Max = 2ε (Mean = 0.865ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 2ε (Mean = 0.85ε) | 
                  Max = 3.01ε (Mean = 1.06ε) | 
| near 2 | 
                  Max = 2ε (Mean = 0.995ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 2ε (Mean = 0.913ε) | 
                  Max = 5.01ε (Mean = 1.89ε) | 
| near -10 | 
                  Max = 1.73ε (Mean = 0.729ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 2.6ε (Mean = 0.985ε) | 
                  Max = 1.75ε (Mean = 0.819ε) | 
| near -55 | 
                  Max = 1.8ε (Mean = 0.817ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 1.8ε (Mean = 0.847ε) | 
                  Max = 98.5ε (Mean = 53.4ε) | 
Table 20.91. Error rates for tgamma1pm1
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| tgamma1pm1(dz) | Max = 0.982ε (Mean = 0.399ε) | Max = 0ε (Mean = 0ε) | Max = 1.12ε (Mean = 0.49ε) | Max = 3.97ε (Mean = 0.713ε) | 
Table 20.92. Error rates for tgamma_delta_ratio
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| tgamma + small delta ratios | Max = 10.1ε (Mean = 1.25ε) | Max = 0ε (Mean = 0ε) | Max = 5.56ε (Mean = 0.969ε) | Max = 15.4ε (Mean = 2.09ε) | 
| tgamma + small delta ratios (negative delta) | Max = 8.04ε (Mean = 1.31ε) | Max = 0ε (Mean = 0ε) | Max = 8.67ε (Mean = 1.29ε) | Max = 18.3ε (Mean = 2.03ε) | 
| tgamma + small integer ratios | Max = 2.74ε (Mean = 0.736ε) | Max = 0ε (Mean = 0ε) | Max = 1.96ε (Mean = 0.677ε) | Max = 1.96ε (Mean = 0.677ε) | 
| tgamma + small integer ratios (negative delta) | Max = 2.15ε (Mean = 0.685ε) | Max = 0ε (Mean = 0ε) | Max = 1.62ε (Mean = 0.451ε) | Max = 1.62ε (Mean = 0.451ε) | 
| integer tgamma ratios | Max = 0.968ε (Mean = 0.386ε) | Max = 0ε (Mean = 0ε) | Max = 0.997ε (Mean = 0.4ε) | Max = 0.997ε (Mean = 0.4ε) | 
| integer tgamma ratios (negative delta) | Max = 0.974ε (Mean = 0.184ε) | Max = 0ε (Mean = 0ε) | Max = 0.853ε (Mean = 0.176ε) | Max = 0.853ε (Mean = 0.176ε) | 
Table 20.93. Error rates for tgamma (incomplete)
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| tgamma(a, z) medium values | Max = 8.14ε (Mean = 1.71ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 7.35ε (Mean = 1.69ε) | Max = 412ε (Mean = 95.5ε) | 
| tgamma(a, z) small values | Max = 2.53ε (Mean = 0.66ε) | 
                  Max = 0.753ε (Mean = 0.0474ε) | Max = 2.13ε (Mean = 0.717ε) | Max = 2.13ε (Mean = 0.712ε) | 
| tgamma(a, z) integer and half integer values | Max = 5.16ε (Mean = 1.44ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 5.52ε (Mean = 1.52ε) | Max = 79.6ε (Mean = 20.9ε) | 
Table 20.94. Error rates for tgamma_lower
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| tgamma(a, z) medium values | Max = 5.62ε (Mean = 1.43ε) | 
                  Max = 0.833ε (Mean = 0.0315ε) | Max = 6.79ε (Mean = 1.38ε) | Max = 363ε (Mean = 63.8ε) | 
| tgamma(a, z) small values | Max = 1.57ε (Mean = 0.527ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 1.97ε (Mean = 0.552ε) | Max = 1.97ε (Mean = 0.567ε) | 
| tgamma(a, z) integer and half integer values | Max = 2.69ε (Mean = 0.866ε) | 
                  Max = 0ε (Mean = 0ε) | Max = 4.83ε (Mean = 1.12ε) | Max = 84.7ε (Mean = 17.5ε) | 
Table 20.95. Error rates for tgamma_ratio
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| tgamma ratios | Max = 3.66ε (Mean = 1.27ε) | Max = 0ε (Mean = 0ε) | Max = 3.09ε (Mean = 1.15ε) | Max = 174ε (Mean = 61.2ε) | 
Table 20.96. Error rates for trigamma
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Mathematica Data | Max = 1ε (Mean = 0.382ε) | 
                  Max = 0.998ε (Mean = 0.105ε) | Max = 1.28ε (Mean = 0.449ε) | Max = 1.28ε (Mean = 0.447ε) | 
Table 20.97. Error rates for zeta
| 
                  Microsoft Visual C++ version 12.0 | 
                  GNU C++ version 5.1.0 | 
                  GNU C++ version 5.1.0 | 
                  Sun compiler version 0x5130 | |
|---|---|---|---|---|
| Zeta: Random values greater than 1 | Max = 0.836ε (Mean = 0.093ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0.846ε (Mean = 0.0833ε) | Max = 0.846ε (Mean = 0.0743ε) | 
| Zeta: Random values less than 1 | Max = 7.03ε (Mean = 2.98ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 7.03ε (Mean = 2.71ε) | Max = 70.1ε (Mean = 17.1ε) | 
| Zeta: Values close to and greater than 1 | Max = 0.994ε (Mean = 0.421ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0.995ε (Mean = 0.5ε) | Max = 0.995ε (Mean = 0.5ε) | 
| Zeta: Values close to and less than 1 | Max = 0.991ε (Mean = 0.375ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 0.998ε (Mean = 0.508ε) | Max = 0.998ε (Mean = 0.568ε) | 
| Zeta: Integer arguments | Max = 6.5ε (Mean = 2.17ε) | 
                  Max = 0ε (Mean = 0ε) | 
                  Max = 9ε (Mean = 3.06ε) | Max = 21ε (Mean = 7.13ε) |