Tables of Error Rates for all Functions

Table 17.3. Error rates for beta

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Beta Function: Small Values	Max = 1.75ε (Mean = 0.828ε)	Max = 2.65ε (Mean = 1.22ε) (<tr1/cmath>: Max = 11.9ε (Mean = 7.03ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 3.94ε (Mean = 1.44ε)) (Rmath 3.0.2: Max = 1.14ε (Mean = 0.601ε)) (Cephes: Max = 1.97ε (Mean = 0.934ε))	Max = 2.86ε (Mean = 1.31ε)
Beta Function: Medium Values	Max = 99.1ε (Mean = 22.4ε)	Max = 60.4ε (Mean = 19.4ε) (<tr1/cmath>: Max = 1.07e+03ε (Mean = 264ε))	Max = 0.978ε (Mean = 0.0578ε) (GSL 1.16: Max = 1.18e+03ε (Mean = 238ε)) (Rmath 3.0.2: Max = 1.09e+03ε (Mean = 265ε)) (Cephes: Max = 1.5e+03ε (Mean = 329ε))	Max = 107ε (Mean = 24.5ε)
Beta Function: Divergent Values	Max = 10.7ε (Mean = 2.22ε)	Max = 10.3ε (Mean = 2.41ε) (<tr1/cmath>: Max = 128ε (Mean = 23.8ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 12.1ε (Mean = 1.99ε)) (Rmath 3.0.2: Max = 176ε (Mean = 28ε)) (Cephes: Max = 175ε (Mean = 22ε))	Max = 19.7ε (Mean = 2.75ε)

Table 17.4. Error rates for beta (incomplete)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.5. Error rates for betac

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.6. Error rates for binomial_coefficient

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.7. Error rates for cbrt

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
cbrt Function	Max = 1.7ε (Mean = 0.565ε) (<math.h>: Max = 1.7ε (Mean = 0.565ε))	Max = 0ε (Mean = 0ε) (Cephes: Max = 0ε (Mean = 0ε))	Max = 1.34ε (Mean = 0.471ε) (<tr1/cmath>: Max = 1.34ε (Mean = 0.471ε)) (<math.h>: Max = 1.34ε (Mean = 0.471ε))	Max = 1.11ε (Mean = 0.424ε) (<math.h>: Max = 1.11ε (Mean = 0.424ε))

Table 17.8. Error rates for cos_pi

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.9. Error rates for cyl_bessel_i

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Bessel I0: Mathworld Data	Max = 0.877ε (Mean = 0.549ε)	Max = 4.57ε (Mean = 2.1ε) (<tr1/cmath>: Max = 8.49ε (Mean = 3.46ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 270ε (Mean = 91.6ε) And other failures.) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 2.55e+43ε (Mean = 8.06e+42ε) And other failures.)	Max = 4.54ε (Mean = 2.11ε)
Bessel I1: Mathworld Data	Max = 0.885ε (Mean = 0.55ε)	Max = 7.83ε (Mean = 2.79ε) (<tr1/cmath>: Max = 5ε (Mean = 2.15ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 128ε (Mean = 41ε) And other failures.) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 1.28e+43ε (Mean = 4.05e+42ε) And other failures.)	Max = 6.52ε (Mean = 2.25ε)
Bessel In: Mathworld Data	Max = 3.46ε (Mean = 1.32ε)	Max = 1.8ε (Mean = 1.33ε) (<tr1/cmath>: Max = 430ε (Mean = 163ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 2.31ε (Mean = 0.838ε) And other failures.) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 3.67e+177ε (Mean = +INFε) And other failures.)	Max = 463ε (Mean = 140ε)
Bessel Iv: Mathworld Data	Max = 2.97ε (Mean = 1.33ε)	Max = 4.12ε (Mean = 1.85ε) (<tr1/cmath>: Max = 616ε (Mean = 221ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 5.95ε (Mean = 2.08ε) And other failures.) (Rmath 3.0.2: Max = 1e+04ε (Mean = 3.18e+03ε)) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 4.12ε (Mean = 1.95ε)
Bessel In: Random Data	Max = 9.67ε (Mean = 1.89ε)	Max = 6.79ε (Mean = 1.15ε) (<tr1/cmath>: Max = 645ε (Mean = 132ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 261ε (Mean = 53.2ε) And other failures.) (Rmath 3.0.2: Max = 7.37ε (Mean = 2.4ε)) (Cephes: Max = 4.22e+06ε (Mean = 2.26e+05ε))	Max = 176ε (Mean = 39.2ε)
Bessel Iv: Random Data	Max = 7.46ε (Mean = 1.54ε)	Max = 8.35ε (Mean = 1.49ε) (<tr1/cmath>: Max = 1.05e+03ε (Mean = 224ε) And other failures.)	Max = 0.661ε (Mean = 0.0441ε) (GSL 1.16: Max = 6.18e+03ε (Mean = 1.55e+03ε) And other failures.) (Rmath 3.0.2: Max = 4.28e+08ε (Mean = 2.85e+07ε)) (Cephes: Max = 6e+30ε (Mean = 4e+29ε) And other failures.)	Max = 283ε (Mean = 88.4ε)
Bessel Iv: Mathworld Data (large values)	Max = 3.67ε (Mean = 1.64ε)	Max = 14.7ε (Mean = 6.57ε) (<tr1/cmath>: Max = 118ε (Mean = 57.2ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 37ε (Mean = 18ε) And other failures.) (Rmath 3.0.2: Max = 3.77e+168ε (Mean = 2.39e+168ε) And other failures.) (Cephes: Max = 73.7ε (Mean = 58.5ε))	Max = 14.7ε (Mean = 6.59ε)

Table 17.10. Error rates for cyl_bessel_i (integer orders)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Bessel I0: Mathworld Data (Integer Version)	Max = 0.877ε (Mean = 0.549ε)	Max = 4.57ε (Mean = 2.1ε) (<tr1/cmath>: Max = 8.49ε (Mean = 3.46ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0.79ε (Mean = 0.482ε)) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 2.55e+43ε (Mean = 8.06e+42ε) And other failures.)	Max = 4.54ε (Mean = 2.11ε)
Bessel I1: Mathworld Data (Integer Version)	Max = 0.885ε (Mean = 0.55ε)	Max = 7.83ε (Mean = 2.79ε) (<tr1/cmath>: Max = 5ε (Mean = 2.15ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0.82ε (Mean = 0.456ε)) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 1.28e+43ε (Mean = 4.05e+42ε) And other failures.)	Max = 6.52ε (Mean = 2.25ε)
Bessel In: Mathworld Data (Integer Version)	Max = 3.46ε (Mean = 1.32ε)	Max = 1.8ε (Mean = 1.33ε) (<tr1/cmath>: Max = 430ε (Mean = 163ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 5.15ε (Mean = 2.13ε) And other failures.) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 3.67e+177ε (Mean = +INFε) And other failures.)	Max = 463ε (Mean = 140ε)

Table 17.11. Error rates for cyl_bessel_i_prime

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.12. Error rates for cyl_bessel_i_prime (integer orders)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.13. Error rates for cyl_bessel_j

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Bessel J0: Mathworld Data	Max = 2.52ε (Mean = 1.2ε)	Max = 6.55ε (Mean = 2.89ε) (<tr1/cmath>: Max = 5.04ε (Mean = 1.78ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0.629ε (Mean = 0.223ε) And other failures.) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 1.12ε (Mean = 0.568ε))	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) (<tr1/cmath>: Max = 4.79e+08ε (Mean = 1.96e+08ε))	Max = 7.98e+04ε (Mean = 3.26e+04ε) (GSL 1.16: Max = 6.5e+07ε (Mean = 2.66e+07ε)) (Rmath 3.0.2: Max = 1.04e+07ε (Mean = 4.29e+06ε)) (Cephes: Max = 2.54e+08ε (Mean = 1.04e+08ε))	Max = 1.64e+08ε (Mean = 6.69e+07ε)
Bessel J1: Mathworld Data	Max = 1.73ε (Mean = 0.976ε)	Max = 2.66ε (Mean = 1.38ε) (<tr1/cmath>: Max = 6.1ε (Mean = 2.95ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 6.62ε (Mean = 2.35ε) And other failures.) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 2.88ε (Mean = 1.12ε))	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) (<tr1/cmath>: Max = 2.15e+06ε (Mean = 1.58e+06ε))	Max = 106ε (Mean = 47.5ε) (GSL 1.16: Max = 8.75e+05ε (Mean = 5.32e+05ε)) (Rmath 3.0.2: Max = 2.93e+06ε (Mean = 1.7e+06ε)) (Cephes: Max = 9.56e+05ε (Mean = 4.99e+05ε))	Max = 2.18e+05ε (Mean = 9.76e+04ε)
Bessel JN: Mathworld Data	Max = 14.7ε (Mean = 5.4ε)	Max = 6.85ε (Mean = 3.41ε) (<tr1/cmath>: Max = 2.13e+19ε (Mean = 5.16e+18ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 6.9e+05ε (Mean = 2.15e+05ε) And other failures.) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 5.53e+05ε (Mean = 1.9e+05ε))	Max = 463ε (Mean = 112ε)
Bessel J: Mathworld Data	Max = 14.9ε (Mean = 3.82ε)	Max = 14.7ε (Mean = 4.05ε) (<tr1/cmath>: Max = 3.49e+05ε (Mean = 7.89e+04ε) And other failures.)	Max = 10ε (Mean = 2.19ε) (GSL 1.16: Max = 2.39e+05ε (Mean = 5.24e+04ε) And other failures.) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 5.47e+05ε (Mean = 1.3e+05ε))	Max = 14.7ε (Mean = 4.12ε)
Bessel J: Mathworld Data (large values)	Max = 9.31ε (Mean = 5.52ε)	Max = 607ε (Mean = 305ε) (<tr1/cmath>: Max = 34.9ε (Mean = 17.4ε) And other failures.)	Max = 0.536ε (Mean = 0.268ε) (GSL 1.16: Max = 4.91e+03ε (Mean = 2.46e+03ε) And other failures.) (Rmath 3.0.2: Max = 35.9ε (Mean = 18.1ε)) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 607ε (Mean = 305ε)
Bessel JN: Random Data	Max = 17.5ε (Mean = 1.46ε)	Max = 50.8ε (Mean = 4.15ε) (<tr1/cmath>: Max = 1.12e+03ε (Mean = 88.7ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 75.7ε (Mean = 5.36ε)) (Rmath 3.0.2: Max = 3.93ε (Mean = 1.22ε)) (Cephes: Max = 91.4ε (Mean = 6.47ε))	Max = 99.6ε (Mean = 22ε)
Bessel J: Random Data	Max = 9.24ε (Mean = 1.36ε)	Max = 9.81ε (Mean = 1.59ε) (<tr1/cmath>: Max = 501ε (Mean = 52.3ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 15.5ε (Mean = 3.33ε) And other failures.) (Rmath 3.0.2: Max = 6.74ε (Mean = 1.3ε)) (Cephes: Max = 16.7ε (Mean = 2.5ε))	Max = 260ε (Mean = 34ε)
Bessel J: Random Data (Tricky large values)	Max = 59.2ε (Mean = 8.67ε)	Max = 785ε (Mean = 94.2ε) (<tr1/cmath>: Max = 5.01e+17ε (Mean = 6.23e+16ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 2.48e+05ε (Mean = 5.11e+04ε)) (Rmath 3.0.2: Max = 71.6ε (Mean = 11.7ε)) (Cephes: Max = 2.48e+05ε (Mean = 3.02e+04ε))	Max = 785ε (Mean = 97.4ε)

Table 17.14. Error rates for cyl_bessel_j (integer orders)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Bessel J0: Mathworld Data (Integer Version)	Max = 2.52ε (Mean = 1.2ε) (<math.h>: Max = 1.89ε (Mean = 0.988ε))	Max = 6.55ε (Mean = 2.89ε) (<tr1/cmath>: Max = 5.04ε (Mean = 1.78ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 1.12ε (Mean = 0.488ε)) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 1.12ε (Mean = 0.568ε))	Max = 6.55ε (Mean = 2.86ε)
Bessel J0: Mathworld Data (Tricky cases) (Integer Version)	Max = 1e+007ε (Mean = 4.09e+006ε) (<math.h>: Max = 2.54e+008ε (Mean = 1.04e+008ε))	Max = 1.63e+08ε (Mean = 6.67e+07ε) (<tr1/cmath>: Max = 4.79e+08ε (Mean = 1.96e+08ε))	Max = 7.98e+04ε (Mean = 3.26e+04ε) (GSL 1.16: Max = 1e+07ε (Mean = 4.11e+06ε)) (Rmath 3.0.2: Max = 1.04e+07ε (Mean = 4.29e+06ε)) (Cephes: Max = 2.54e+08ε (Mean = 1.04e+08ε))	Max = 1.64e+08ε (Mean = 6.69e+07ε)
Bessel J1: Mathworld Data (Integer Version)	Max = 1.73ε (Mean = 0.976ε) (<math.h>: Max = 11.4ε (Mean = 4.15ε))	Max = 2.66ε (Mean = 1.38ε) (<tr1/cmath>: Max = 6.1ε (Mean = 2.95ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 1.89ε (Mean = 0.721ε)) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 2.88ε (Mean = 1.12ε))	Max = 1.44ε (Mean = 0.637ε)
Bessel J1: Mathworld Data (tricky cases) (Integer Version)	Max = 3.23e+004ε (Mean = 1.45e+004ε) (<math.h>: Max = 1.44e+007ε (Mean = 6.5e+006ε))	Max = 2.18e+05ε (Mean = 9.76e+04ε) (<tr1/cmath>: Max = 2.15e+06ε (Mean = 1.58e+06ε))	Max = 106ε (Mean = 47.5ε) (GSL 1.16: Max = 1.26e+06ε (Mean = 6.28e+05ε)) (Rmath 3.0.2: Max = 2.93e+06ε (Mean = 1.7e+06ε)) (Cephes: Max = 9.56e+05ε (Mean = 4.99e+05ε))	Max = 2.18e+05ε (Mean = 9.76e+04ε)
Bessel JN: Mathworld Data (Integer Version)	Max = 14.7ε (Mean = 5.4ε) (<math.h>: Max = +INFε (Mean = +INFε) And other failures.)	Max = 6.85ε (Mean = 3.41ε) (<tr1/cmath>: Max = 2.13e+19ε (Mean = 5.16e+18ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 6.9e+05ε (Mean = 2.53e+05ε) And other failures.) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 463ε (Mean = 112ε)

Table 17.15. Error rates for cyl_bessel_j_prime

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.16. Error rates for cyl_bessel_j_prime (integer orders)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.17. Error rates for cyl_bessel_k

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Bessel K0: Mathworld Data	Max = 1.55ε (Mean = 0.837ε)	Max = 4.16ε (Mean = 1.46ε) (<tr1/cmath>: Max = 9.33ε (Mean = 3.25ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 6.04ε (Mean = 2.16ε)) (Rmath 3.0.2: Max = 0.833ε (Mean = 0.601ε))	Max = 4.16ε (Mean = 1.49ε)
Bessel K1: Mathworld Data	Max = 1ε (Mean = 0.573ε)	Max = 1.8ε (Mean = 1.02ε) (<tr1/cmath>: Max = 8.94ε (Mean = 3.19ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 6.26ε (Mean = 2.21ε)) (Rmath 3.0.2: Max = 0.894ε (Mean = 0.516ε))	Max = 1.8ε (Mean = 1.02ε)
Bessel Kn: Mathworld Data	Max = 3.63ε (Mean = 1.46ε)	Max = 4.48ε (Mean = 2.14ε) (<tr1/cmath>: Max = 12.9ε (Mean = 4.91ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 3.36ε (Mean = 1.43ε) And other failures.) (Rmath 3.0.2: Max = 8.48ε (Mean = 2.98ε))	Max = 4.48ε (Mean = 1.98ε)
Bessel Kv: Mathworld Data	Max = 4.78ε (Mean = 2.2ε)	Max = 3.58ε (Mean = 2.44ε) (<tr1/cmath>: Max = 13ε (Mean = 4.81ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 5.47ε (Mean = 2.04ε) And other failures.) (Rmath 3.0.2: Max = 3.15ε (Mean = 1.35ε))	Max = 3.58ε (Mean = 2.29ε)
Bessel Kv: Mathworld Data (large values)	Max = 59.8ε (Mean = 26.9ε)	Max = 42.3ε (Mean = 21ε) (<tr1/cmath>: Max = 42.3ε (Mean = 19.8ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 308ε (Mean = 142ε) And other failures.) (Rmath 3.0.2: Max = 84.6ε (Mean = 37.8ε))	Max = 43.1ε (Mean = 21.3ε)
Bessel Kn: Random Data	Max = 7.47ε (Mean = 1.4ε)	Max = 4.55ε (Mean = 1.09ε) (<tr1/cmath>: Max = 13.9ε (Mean = 2.91ε))	Max = 0.764ε (Mean = 0.0348ε) (GSL 1.16: Max = 8.71ε (Mean = 1.76ε) And other failures.) (Rmath 3.0.2: Max = 7.47ε (Mean = 1.34ε))	Max = 4.55ε (Mean = 1.21ε)
Bessel Kv: Random Data	Max = 8.33ε (Mean = 1.62ε)	Max = 7.88ε (Mean = 1.48ε) (<tr1/cmath>: Max = 13.6ε (Mean = 2.68ε) And other failures.)	Max = 0.507ε (Mean = 0.0313ε) (GSL 1.16: Max = 9.71ε (Mean = 1.47ε) And other failures.) (Rmath 3.0.2: Max = 7.37ε (Mean = 1.49ε))	Max = 7.88ε (Mean = 1.49ε)

Table 17.18. Error rates for cyl_bessel_k (integer orders)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Bessel K0: Mathworld Data (Integer Version)	Max = 1.55ε (Mean = 0.837ε)	Max = 4.16ε (Mean = 1.46ε) (<tr1/cmath>: Max = 9.33ε (Mean = 3.25ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 1.2ε (Mean = 0.733ε)) (Rmath 3.0.2: Max = 0.833ε (Mean = 0.601ε)) (Cephes: Max = 1.1e+06ε (Mean = 3.68e+05ε))	Max = 4.16ε (Mean = 1.49ε)
Bessel K1: Mathworld Data (Integer Version)	Max = 1ε (Mean = 0.573ε)	Max = 1.8ε (Mean = 1.02ε) (<tr1/cmath>: Max = 8.94ε (Mean = 3.19ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0.626ε (Mean = 0.333ε)) (Rmath 3.0.2: Max = 0.894ε (Mean = 0.516ε)) (Cephes: Max = 5.38e+05ε (Mean = 1.79e+05ε))	Max = 1.8ε (Mean = 1.02ε)
Bessel Kn: Mathworld Data (Integer Version)	Max = 3.63ε (Mean = 1.46ε)	Max = 4.48ε (Mean = 2.14ε) (<tr1/cmath>: Max = 12.9ε (Mean = 4.91ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 168ε (Mean = 59.5ε)) (Rmath 3.0.2: Max = 8.48ε (Mean = 2.98ε)) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 4.48ε (Mean = 1.98ε)

Table 17.19. Error rates for cyl_bessel_k_prime

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.20. Error rates for cyl_bessel_k_prime (integer orders)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.21. Error rates for cyl_neumann

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Y0: Mathworld Data	Max = 4.61ε (Mean = 2.29ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 60.9ε (Mean = 20.4ε)) (Rmath 3.0.2: Max = 167ε (Mean = 56.5ε))	Max = 5.59ε (Mean = 2.54ε) (<tr1/cmath>: Max = 2.05e+05ε (Mean = 6.87e+04ε))	Max = 5.53ε (Mean = 2.4ε)
Y1: Mathworld Data	Max = 4.75ε (Mean = 1.72ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 23.4ε (Mean = 8.1ε)) (Rmath 3.0.2: Max = 193ε (Mean = 64.4ε))	Max = 12.7ε (Mean = 4.34ε) (<tr1/cmath>: Max = 9.71e+03ε (Mean = 4.08e+03ε))	Max = 6.33ε (Mean = 2.29ε)
Yn: Mathworld Data	Max = 35ε (Mean = 11.8ε)	Max = 0.993ε (Mean = 0.314ε) (GSL 1.16: Max = 2.41e+05ε (Mean = 7.62e+04ε) And other failures.) (Rmath 3.0.2: Max = 1.24e+04ε (Mean = 4e+03ε))	Max = 55.2ε (Mean = 17.7ε) (<tr1/cmath>: Max = 2.2e+20ε (Mean = 6.97e+19ε) And other failures.)	Max = 55.2ε (Mean = 17.8ε)
Yv: Mathworld Data	Max = 7.89ε (Mean = 3.27ε)	Max = 10ε (Mean = 3.02ε) (GSL 1.16: Max = 1.07e+05ε (Mean = 3.22e+04ε) And other failures.) (Rmath 3.0.2: Max = 1.05e+03ε (Mean = 326ε))	Max = 10.7ε (Mean = 4.92ε) (<tr1/cmath>: Max = 3.49e+15ε (Mean = 1.05e+15ε) And other failures.)	Max = 10.7ε (Mean = 5.1ε)
Yv: Mathworld Data (large values)	Max = 0.682ε (Mean = 0.35ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 60.8ε (Mean = 23ε) And other failures.) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.)	Max = 1.57ε (Mean = 1.17ε) (<tr1/cmath>: Max = 43.2ε (Mean = 16.3ε) And other failures.)	Max = 1.57ε (Mean = 1.24ε)
Y0 and Y1: Random Data	Max = 4.17ε (Mean = 1.24ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 34.4ε (Mean = 8.9ε)) (Rmath 3.0.2: Max = 83ε (Mean = 14.2ε))	Max = 11.8ε (Mean = 3.28ε) (<tr1/cmath>: Max = 2.59e+03ε (Mean = 500ε))	Max = 10.8ε (Mean = 3.04ε)
Yn: Random Data	Max = 117ε (Mean = 10.2ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 500ε (Mean = 47.8ε)) (Rmath 3.0.2: Max = 691ε (Mean = 67.9ε))	Max = 338ε (Mean = 28.2ε) (<tr1/cmath>: Max = 4.01e+03ε (Mean = 348ε))	Max = 338ε (Mean = 27.5ε)
Yv: Random Data	Max = 1.23e+003ε (Mean = 69.9ε)	Max = 1.53ε (Mean = 0.102ε) (GSL 1.16: Max = 1.41e+06ε (Mean = 7.67e+04ε)) (Rmath 3.0.2: Max = 1.79e+05ε (Mean = 9.64e+03ε))	Max = 2.08e+03ε (Mean = 149ε) (<tr1/cmath>: Max = +INFε (Mean = +INFε) And other failures.)	Max = 2.08e+03ε (Mean = 149ε)

Table 17.22. Error rates for cyl_neumann (integer orders)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Y0: Mathworld Data (Integer Version)	Max = 4.61ε (Mean = 2.29ε) (<math.h>: Max = 5.37e+003ε (Mean = 1.81e+003ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 6.46ε (Mean = 2.38ε)) (Rmath 3.0.2: Max = 167ε (Mean = 56.5ε)) (Cephes: Max = 5.37e+03ε (Mean = 1.81e+03ε))	Max = 5.59ε (Mean = 2.54ε) (<tr1/cmath>: Max = 2.05e+05ε (Mean = 6.87e+04ε))	Max = 5.53ε (Mean = 2.4ε)
Y1: Mathworld Data (Integer Version)	Max = 4.75ε (Mean = 1.72ε) (<math.h>: Max = 1.86e+004ε (Mean = 6.2e+003ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 1.51ε (Mean = 0.839ε)) (Rmath 3.0.2: Max = 193ε (Mean = 64.4ε)) (Cephes: Max = 1.86e+04ε (Mean = 6.2e+03ε))	Max = 12.7ε (Mean = 4.34ε) (<tr1/cmath>: Max = 9.71e+03ε (Mean = 4.08e+03ε))	Max = 6.33ε (Mean = 2.29ε)
Yn: Mathworld Data (Integer Version)	Max = 35ε (Mean = 11.8ε) (<math.h>: Max = 2.49e+005ε (Mean = 8.14e+004ε))	Max = 0.993ε (Mean = 0.314ε) (GSL 1.16: Max = 2.41e+05ε (Mean = 7.62e+04ε)) (Rmath 3.0.2: Max = 1.24e+04ε (Mean = 4e+03ε)) (Cephes: Max = 2.49e+05ε (Mean = 8.14e+04ε))	Max = 55.2ε (Mean = 17.7ε) (<tr1/cmath>: Max = 2.2e+20ε (Mean = 6.97e+19ε) And other failures.)	Max = 55.2ε (Mean = 17.8ε)

Table 17.23. Error rates for cyl_neumann_prime

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.24. Error rates for cyl_neumann_prime (integer orders)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.25. Error rates for digamma

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Digamma Function: Large Values	Max = 0.98ε (Mean = 0.369ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 1.84ε (Mean = 0.71ε)) (Rmath 3.0.2: Max = 1.18ε (Mean = 0.331ε)) (Cephes: Max = 0.919ε (Mean = 0.394ε))	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) (GSL 1.16: Max = 135ε (Mean = 11.9ε)) (Rmath 3.0.2: Max = 2.02e+03ε (Mean = 256ε)) (Cephes: Max = 1.42e+04ε (Mean = 1.14e+03ε))	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) (GSL 1.16: Max = 0.953ε (Mean = 0.348ε)) (Rmath 3.0.2: Max = 1.17ε (Mean = 0.564ε)) (Cephes: Max = 3.5ε (Mean = 1.04ε))	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) (GSL 1.16: Max = 4.56e+04ε (Mean = 3.91e+03ε)) (Rmath 3.0.2: Max = 4.6e+04ε (Mean = 3.94e+03ε)) (Cephes: Max = 214ε (Mean = 16.4ε))	Max = 180ε (Mean = 13ε)	Max = 180ε (Mean = 13ε)
Digamma Function: Values near 0	Max = 0ε (Mean = 0ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0.866ε (Mean = 0.387ε)) (Rmath 3.0.2: Max = 3.58e+05ε (Mean = 1.6e+05ε)) (Cephes: Max = 0.5ε (Mean = 0.224ε))	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) (GSL 1.16: Max = 1.18ε (Mean = 0.607ε)) (Rmath 3.0.2: Max = 4.33ε (Mean = 0.982ε)) (Cephes: Max = 0.992ε (Mean = 0.383ε))	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) (GSL 1.16: Max = 1.09ε (Mean = 0.531ε)) (Rmath 3.0.2: Max = 46.2ε (Mean = 7.24ε)) (Cephes: Max = 8.56ε (Mean = 1.44ε))	Max = 0.906ε (Mean = 0.409ε)	Max = 0.906ε (Mean = 0.409ε)

Table 17.26. Error rates for ellint_1

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Elliptic Integral F: Mathworld Data	Max = 0.919ε (Mean = 0.542ε)	Max = 0.94ε (Mean = 0.509ε) (<tr1/cmath>: Max = +INFε (Mean = +INFε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0.919ε (Mean = 0.544ε)) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 0.94ε (Mean = 0.509ε)
Elliptic Integral F: Random Data	Max = 2.26ε (Mean = 0.631ε)	Max = 1.57ε (Mean = 0.56ε) (<tr1/cmath>: Max = 2.56ε (Mean = 0.816ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 2.99ε (Mean = 0.797ε)) (Cephes: Max = 2.18ε (Mean = 0.535ε))	Max = 1.57ε (Mean = 0.561ε)

Table 17.27. Error rates for ellint_1 (complete)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Elliptic Integral K: Mathworld Data	Max = 0.915ε (Mean = 0.547ε)	Max = 0.887ε (Mean = 0.296ε) (<tr1/cmath>: Max = 1.19ε (Mean = 0.765ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0.623ε (Mean = 0.393ε))	Max = 0.887ε (Mean = 0.296ε)
Elliptic Integral K: Random Data	Max = 0.958ε (Mean = 0.408ε)	Max = 1.27ε (Mean = 0.473ε) (<tr1/cmath>: Max = 2.19ε (Mean = 0.694ε))	Max = 0.851ε (Mean = 0.0851ε) (GSL 1.16: Max = 2.32ε (Mean = 0.688ε))	Max = 1.27ε (Mean = 0.473ε)

Table 17.28. Error rates for ellint_2

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Elliptic Integral E: Mathworld Data	Max = 1.31ε (Mean = 0.727ε)	Max = 0.656ε (Mean = 0.317ε) (<tr1/cmath>: Max = +INFε (Mean = +INFε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0.63ε (Mean = 0.325ε)) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 0.656ε (Mean = 0.317ε)
Elliptic Integral E: Random Data	Max = 2.23ε (Mean = 0.639ε)	Max = 2.05ε (Mean = 0.632ε) (<tr1/cmath>: Max = 3.08e+04ε (Mean = 3.84e+03ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 4.4ε (Mean = 1.16ε)) (Cephes: Max = 8.92ε (Mean = 1.48ε))	Max = 2.05ε (Mean = 0.632ε)

Table 17.29. Error rates for ellint_2 (complete)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Elliptic Integral E: Mathworld Data	Max = 1.3ε (Mean = 0.615ε)	Max = 0.836ε (Mean = 0.469ε) (<tr1/cmath>: Max = 170ε (Mean = 55.1ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 3.09ε (Mean = 1.04ε) And other failures.)	Max = 0.836ε (Mean = 0.469ε)
Elliptic Integral E: Random Data	Max = 1.71ε (Mean = 0.553ε)	Max = 1.97ε (Mean = 0.629ε) (<tr1/cmath>: Max = 2.49e+04ε (Mean = 3.39e+03ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 4.34ε (Mean = 1.18ε))	Max = 1.97ε (Mean = 0.629ε)

Table 17.30. Error rates for ellint_3

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Elliptic Integral PI: Mathworld Data	Max = 565ε (Mean = 102ε)	Max = 475ε (Mean = 86.3ε) (<tr1/cmath>: Max = +INFε (Mean = +INFε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 1.48e+05ε (Mean = 2.54e+04ε) And other failures.)	Max = 475ε (Mean = 86.3ε)
Elliptic Integral PI: Random Data	Max = 9.08ε (Mean = 0.99ε)	Max = 4.54ε (Mean = 0.895ε) (<tr1/cmath>: Max = 3.37e+20ε (Mean = 3.47e+19ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 633ε (Mean = 50.1ε))	Max = 4.49ε (Mean = 0.891ε)
Elliptic Integral PI: Large Random Data	Max = 2.86ε (Mean = 0.944ε)	Max = 3.7ε (Mean = 0.893ε) (<tr1/cmath>: Max = 2.52e+18ε (Mean = 4.83e+17ε) And other failures.)	Max = 0.557ε (Mean = 0.0389ε) (GSL 1.16: Max = 40.1ε (Mean = 7.77ε))	Max = 3.7ε (Mean = 0.892ε)

Table 17.31. Error rates for ellint_3 (complete)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Complete Elliptic Integral PI: Mathworld Data	Max = 0.971ε (Mean = 0.464ε)	Max = 1.4ε (Mean = 0.575ε) (<tr1/cmath>: Max = 6.31e+20ε (Mean = 1.53e+20ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 6.33e+04ε (Mean = 1.54e+04ε) And other failures.)	Max = 1.4ε (Mean = 0.575ε)
Complete Elliptic Integral PI: Random Data	Max = 2.46ε (Mean = 0.654ε)	Max = 2.45ε (Mean = 0.696ε) (<tr1/cmath>: Max = 8.78e+20ε (Mean = 1.02e+20ε) And other failures.)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 24ε (Mean = 2.99ε))	Max = 2.4ε (Mean = 0.681ε)

Table 17.32. Error rates for ellint_d

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Elliptic Integral E: Mathworld Data	Max = 0.862ε (Mean = 0.457ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0.862ε (Mean = 0.568ε))	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) (GSL 1.16: Max = 3.01ε (Mean = 0.928ε))	Max = 2.51ε (Mean = 0.883ε)	Max = 2.51ε (Mean = 0.883ε)

Table 17.33. Error rates for ellint_d (complete)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.34. Error rates for ellint_rc

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
RC: Random data	Max = 0.962ε (Mean = 0.407ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 2.4ε (Mean = 0.624ε))	Max = 0.995ε (Mean = 0.433ε)	Max = 0.995ε (Mean = 0.438ε)

Table 17.35. Error rates for ellint_rd

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
RD: Random data	Max = 2.16ε (Mean = 0.803ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 2.59ε (Mean = 0.878ε))	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) (GSL 1.16: Max = 2.88ε (Mean = 0.839ε))	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) (GSL 1.16: Max = 3.74ε (Mean = 0.84ε))	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) (GSL 1.16: Max = 2ε (Mean = 0.656ε))	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) (GSL 1.16: Max = 1.03ε (Mean = 0.418ε))	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) (GSL 1.16: Max = 2.85ε (Mean = 0.781ε))	Max = 2.79ε (Mean = 0.883ε)	Max = 2.79ε (Mean = 0.883ε)

Table 17.36. Error rates for ellint_rf

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
RF: Random data	Max = 2.02ε (Mean = 0.677ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 2.73ε (Mean = 0.804ε))	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) (GSL 1.16: Max = 0.999ε (Mean = 0.34ε))	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) (GSL 1.16: Max = 2.89ε (Mean = 0.749ε))	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) (GSL 1.16: Max = 1.29ε (Mean = 0.527ε))	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) (GSL 1.16: Max = 2.54ε (Mean = 0.781ε))	Max = 1.7ε (Mean = 0.539ε)	Max = 1.7ε (Mean = 0.539ε)

Table 17.37. Error rates for ellint_rg

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
RG: Random Data	Max = 3.65ε (Mean = 0.929ε)	Max = 0.983ε (Mean = 0.0172ε) (GSL 1.16: 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) (GSL 1.16: 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) (GSL 1.16: 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) (GSL 1.16: 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) (GSL 1.16: Max = 0ε (Mean = 0ε))	Max = 2.14ε (Mean = 0.722ε)	Max = 2.14ε (Mean = 0.722ε)

Table 17.38. Error rates for ellint_rj

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
RJ: Random data	Max = 119ε (Mean = 4.32ε)	Max = 0.52ε (Mean = 0.0184ε) (GSL 1.16: Max = 3.57ε (Mean = 0.704ε) And other failures.)	Max = 186ε (Mean = 6.67ε)	Max = 186ε (Mean = 6.7ε)
RJ: 4 Equal Values	Max = 1.03ε (Mean = 0.418ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 1.03ε (Mean = 0.418ε))	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) (GSL 1.16: Max = 3.96ε (Mean = 1.06ε))	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) (GSL 1.16: Max = 2.57ε (Mean = 0.754ε))	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) (GSL 1.16: Max = 2.62ε (Mean = 0.699ε))	Max = 17.2ε (Mean = 1.16ε)	Max = 16.6ε (Mean = 1.15ε)

Table 17.39. Error rates for erf

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Erf Function: Small Values	Max = 0.996ε (Mean = 0.182ε) (<math.h>: Max = 1.57ε (Mean = 0.317ε))	Max = 0.925ε (Mean = 0.193ε) (<tr1/cmath>: Max = 0.944ε (Mean = 0.191ε)) (<math.h>: Max = 0.944ε (Mean = 0.191ε))	Max = 0.841ε (Mean = 0.0687ε) (GSL 1.16: Max = 2.06ε (Mean = 0.319ε)) (Cephes: Max = 1.13ε (Mean = 0.442ε))	Max = 0.925ε (Mean = 0.193ε) (<math.h>: Max = 0.944ε (Mean = 0.194ε))
Erf Function: Medium Values	Max = 1ε (Mean = 0.169ε) (<math.h>: Max = 1.19ε (Mean = 0.244ε))	Max = 1.5ε (Mean = 0.193ε) (<tr1/cmath>: Max = 0.921ε (Mean = 0.0723ε)) (<math.h>: Max = 0.921ε (Mean = 0.0723ε))	Max = 1ε (Mean = 0.119ε) (GSL 1.16: Max = 2.31ε (Mean = 0.368ε)) (Cephes: Max = 1.34ε (Mean = 0.279ε))	Max = 1.5ε (Mean = 0.202ε) (<math.h>: Max = 0.921ε (Mean = 0.071ε))
Erf Function: Large Values	Max = 0ε (Mean = 0ε) (<math.h>: Max = 0ε (Mean = 0ε))	Max = 0ε (Mean = 0ε) (<tr1/cmath>: Max = 0ε (Mean = 0ε)) (<math.h>: Max = 0ε (Mean = 0ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0ε (Mean = 0ε)) (Cephes: Max = 0ε (Mean = 0ε))	Max = 0ε (Mean = 0ε) (<math.h>: Max = 0ε (Mean = 0ε))

Table 17.40. Error rates for erf_inv

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.41. Error rates for erfc

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Erf Function: Small Values	Max = 0ε (Mean = 0ε) (<math.h>: Max = 0ε (Mean = 0ε))	Max = 0ε (Mean = 0ε) (<tr1/cmath>: Max = 0ε (Mean = 0ε)) (<math.h>: Max = 0ε (Mean = 0ε))	Max = 0.658ε (Mean = 0.0537ε) (GSL 1.16: Max = 1.01ε (Mean = 0.485ε)) (Cephes: Max = 0.786ε (Mean = 0.0642ε))	Max = 0ε (Mean = 0ε) (<math.h>: Max = 0ε (Mean = 0ε))
Erf Function: Medium Values	Max = 1.65ε (Mean = 0.373ε) (<math.h>: Max = 2.36ε (Mean = 0.539ε))	Max = 1.76ε (Mean = 0.365ε) (<tr1/cmath>: Max = 1.35ε (Mean = 0.307ε)) (<math.h>: Max = 1.35ε (Mean = 0.307ε))	Max = 0.983ε (Mean = 0.213ε) (GSL 1.16: Max = 2.64ε (Mean = 0.662ε)) (Cephes: Max = 3.59ε (Mean = 0.779ε))	Max = 1.76ε (Mean = 0.383ε) (<math.h>: Max = 2.81ε (Mean = 0.739ε))
Erf Function: Large Values	Max = 1.14ε (Mean = 0.248ε) (<math.h>: Max = 1.84ε (Mean = 0.331ε))	Max = 1.57ε (Mean = 0.542ε) (<tr1/cmath>: Max = 1.26ε (Mean = 0.441ε)) (<math.h>: Max = 1.26ε (Mean = 0.441ε))	Max = 0.868ε (Mean = 0.147ε) (GSL 1.16: Max = 3.9ε (Mean = 0.472ε)) (Cephes: Max = 2.74ε (Mean = 0.413ε))	Max = 1.57ε (Mean = 0.564ε) (<math.h>: Max = 4.91ε (Mean = 1.54ε))

Table 17.42. Error rates for erfc_inv

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.43. Error rates for expint (Ei)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Exponential Integral Ei	Max = 1.43ε (Mean = 0.541ε)	Max = 5.05ε (Mean = 0.821ε) (<tr1/cmath>: Max = 14.1ε (Mean = 2.43ε) And other failures.)	Max = 0.994ε (Mean = 0.142ε) (GSL 1.16: Max = 8.96ε (Mean = 0.703ε)) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 3.34ε (Mean = 0.631ε)
Exponential Integral Ei: double exponent range	Max = 1.7ε (Mean = 0.66ε)	Max = 1.72ε (Mean = 0.593ε) (<tr1/cmath>: Max = 3.11ε (Mean = 1.13ε))	Max = 0.998ε (Mean = 0.156ε) (GSL 1.16: Max = 1.5ε (Mean = 0.612ε)) (Cephes: Max = 1.77ε (Mean = 0.644ε))	Max = 1.72ε (Mean = 0.618ε)
Exponential Integral Ei: long exponent range		Max = 1.98ε (Mean = 0.595ε) (<tr1/cmath>: Max = 1.93ε (Mean = 0.855ε))		Max = 1.98ε (Mean = 0.575ε)

Table 17.44. Error rates for expint (En)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Exponential Integral En	Max = 7.16ε (Mean = 1.85ε)	Max = 0.589ε (Mean = 0.0331ε) (GSL 1.16: Max = 58.5ε (Mean = 17.1ε))	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) (GSL 1.16: Max = 115ε (Mean = 23.6ε))	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) (GSL 1.16: Max = 0.988ε (Mean = 0.469ε))	Max = 0.965ε (Mean = 0.414ε)	Max = 0.965ε (Mean = 0.409ε)

Table 17.45. Error rates for expm1

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Random test data	Max = 0.996ε (Mean = 0.283ε) (<math.h>: Max = 1.31ε (Mean = 0.496ε))	Max = 0.793ε (Mean = 0.126ε) (Rmath 3.0.2: Max = 0.793ε (Mean = 0.126ε)) (Cephes: Max = 1.53ε (Mean = 0.535ε))	Max = 0.992ε (Mean = 0.402ε) (<tr1/cmath>: Max = 1.26e+19ε (Mean = 4.89e+18ε) And other failures.) (<math.h>: Max = 0.992ε (Mean = 0.402ε))	Max = 1.31ε (Mean = 0.406ε) (<math.h>: Max = 0.996ε (Mean = 0.426ε))

Table 17.46. Error rates for gamma_p

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
tgamma(a, z) medium values	Max = 35.1ε (Mean = 6.97ε)	Max = 0.955ε (Mean = 0.05ε) (GSL 1.16: Max = 342ε (Mean = 45.8ε)) (Rmath 3.0.2: Max = 389ε (Mean = 44ε)) (Cephes: Max = 492ε (Mean = 101ε))	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) (GSL 1.16: Max = 4.82ε (Mean = 0.758ε)) (Rmath 3.0.2: Max = 1.01ε (Mean = 0.306ε)) (Cephes: Max = 21ε (Mean = 5.65ε))	Max = 2ε (Mean = 0.461ε)	Max = 2ε (Mean = 0.472ε)
tgamma(a, z) large values	Max = 244ε (Mean = 20.2ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 1.02e+03ε (Mean = 105ε)) (Rmath 3.0.2: Max = 1.11e+03ε (Mean = 67.5ε)) (Cephes: Max = 8.18e+06ε (Mean = 7.69e+05ε))	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) (GSL 1.16: Max = 128ε (Mean = 22.6ε)) (Rmath 3.0.2: Max = 66.2ε (Mean = 12.2ε)) (Cephes: Max = 83.6ε (Mean = 22.2ε))	Max = 11.8ε (Mean = 2.65ε)	Max = 71.6ε (Mean = 9.47ε)

Table 17.47. Error rates for gamma_p_inv

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
incomplete gamma inverse(a, z) medium values	Max = 1.01ε (Mean = 0.307ε)	Max = 0.993ε (Mean = 0.15ε) (Rmath 3.0.2: Max = 4.88ε (Mean = 0.868ε))	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) (Rmath 3.0.2: Max = 0.816ε (Mean = 0.0874ε))	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) (Rmath 3.0.2: Max = 547ε (Mean = 61.6ε))	Max = 9.17e+03ε (Mean = 1.32e+03ε)	Max = 1.09e+04ε (Mean = 1.46e+03ε)

Table 17.48. Error rates for gamma_p_inva

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.49. Error rates for gamma_q

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
tgamma(a, z) medium values	Max = 23.7ε (Mean = 4.03ε)	Max = 0.927ε (Mean = 0.035ε) (GSL 1.16: Max = 201ε (Mean = 13.5ε)) (Rmath 3.0.2: Max = 131ε (Mean = 12.7ε)) (Cephes: Max = 388ε (Mean = 93.8ε))	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) (GSL 1.16: Max = 1.38e+10ε (Mean = 1.05e+09ε)) (Rmath 3.0.2: Max = 65.6ε (Mean = 11ε)) (Cephes: Max = 3.42e+11ε (Mean = 4.1e+10ε))	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) (GSL 1.16: Max = 2.71e+04ε (Mean = 2.16e+03ε)) (Rmath 3.0.2: Max = 1.02e+03ε (Mean = 62.7ε)) (Cephes: Max = 8.17e+06ε (Mean = 7.7e+05ε))	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) (GSL 1.16: Max = 118ε (Mean = 12.5ε)) (Rmath 3.0.2: Max = 138ε (Mean = 16.9ε)) (Cephes: Max = 129ε (Mean = 26.5ε))	Max = 11.1ε (Mean = 2.09ε)	Max = 54.7ε (Mean = 6.16ε)

Table 17.50. Error rates for gamma_q_inv

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
incomplete gamma inverse(a, z) medium values	Max = 3.46ε (Mean = 0.475ε)	Max = 0.912ε (Mean = 0.154ε) (Rmath 3.0.2: Max = 4.66ε (Mean = 0.792ε))	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) (Rmath 3.0.2: Max = 0.894ε (Mean = 0.106ε))	Max = 0ε (Mean = 0ε)	Max = 0ε (Mean = 0ε)
incomplete gamma inverse(a, z) small values	Max = 451ε (Mean = 65ε)	Max = 292ε (Mean = 36.4ε) (Rmath 3.0.2: Max = 415ε (Mean = 48.7ε))	Max = 8.28e+03ε (Mean = 963ε)	Max = 8.98e+03ε (Mean = 877ε)

Table 17.51. Error rates for gamma_q_inva

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.52. Error rates for hermite

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.53. Error rates for heuman_lambda

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.54. Error rates for ibeta

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Incomplete Beta Function: Small Values	Max = 9.42ε (Mean = 2.24ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 682ε (Mean = 32.6ε)) (Rmath 3.0.2: Max = 22.9ε (Mean = 3.35ε)) (Cephes: Max = 22.7ε (Mean = 1.69ε))	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) (GSL 1.16: Max = 690ε (Mean = 151ε)) (Rmath 3.0.2: Max = 232ε (Mean = 27.9ε)) (Cephes: Max = 1.55e+03ε (Mean = 121ε))	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) (GSL 1.16: Max = 3.9e+05ε (Mean = 1.82e+04ε) And other failures.) (Rmath 3.0.2: Max = 574ε (Mean = 49.4ε)) (Cephes: Max = 5.92e+05ε (Mean = 2.17e+04ε))	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) (GSL 1.16: Max = 254ε (Mean = 50.9ε)) (Rmath 3.0.2: Max = 62.2ε (Mean = 8.95ε)) (Cephes: Max = 7.96ε (Mean = 1.3ε))	Max = 4.45ε (Mean = 0.814ε)	Max = 44.5ε (Mean = 10.1ε)

Table 17.55. Error rates for ibeta_inv

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Inverse incomplete beta	Max = 7.08e+003ε (Mean = 244ε)	Max = 3.21ε (Mean = 0.158ε) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 1.28e+10ε (Mean = 5.17e+08ε))	Max = 4.53e+04ε (Mean = 2.93e+03ε)	Max = 4.22e+04ε (Mean = 2.8e+03ε)

Table 17.56. Error rates for ibeta_inva

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.57. Error rates for ibeta_invb

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.58. Error rates for ibetac

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Incomplete Beta Function: Small Values	Max = 6.56ε (Mean = 1.88ε)	Max = 0ε (Mean = 0ε) (Rmath 3.0.2: Max = 22.4ε (Mean = 3.67ε))	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) (Rmath 3.0.2: Max = 204ε (Mean = 25.8ε))	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) (Rmath 3.0.2: Max = 889ε (Mean = 68.4ε))	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) (Rmath 3.0.2: Max = 84.6ε (Mean = 18ε))	Max = 5.34ε (Mean = 1.11ε)	Max = 107ε (Mean = 17.1ε)

Table 17.59. Error rates for ibetac_inv

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Inverse incomplete beta	Max = 5.53e+003ε (Mean = 220ε)	Max = 1.71ε (Mean = 0.108ε) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (Cephes: Max = 1.71ε (Mean = 0.108ε))	Max = 6.17e+04ε (Mean = 3.77e+03ε)	Max = 5.15e+04ε (Mean = 3.51e+03ε)

Table 17.60. Error rates for ibetac_inva

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.61. Error rates for ibetac_invb

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.62. Error rates for jacobi_cn

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Jacobi Elliptic: Mathworld Data	Max = 45.8ε (Mean = 11.4ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 17.3ε (Mean = 4.29ε) And other failures.) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	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) (GSL 1.16: Max = 2.43ε (Mean = 0.803ε)) (Cephes: Max = 1.83ε (Mean = 0.437ε))	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) (GSL 1.16: Max = 55.2ε (Mean = 1.64ε) And other failures.) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	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) (GSL 1.16: Max = 0ε (Mean = 0ε) And other failures.) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	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) (GSL 1.16: Max = 5.92e+03ε (Mean = 477ε)) (Cephes: Max = 3.27e+04ε (Mean = 1.93e+03ε))	Max = 2.97e+04ε (Mean = 1.9e+03ε)	Max = 2.97e+04ε (Mean = 1.9e+03ε)

Table 17.63. Error rates for jacobi_dn

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Jacobi Elliptic: Mathworld Data	Max = 34.3ε (Mean = 8.71ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 2.82ε (Mean = 1.18ε) And other failures.) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 49ε (Mean = 14ε)	Max = 49ε (Mean = 14ε)
Jacobi Elliptic: Random Data	Max = 1.52ε (Mean = 0.466ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 3ε (Mean = 0.61ε)) (Cephes: Max = 1.49ε (Mean = 0.321ε))	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) (GSL 1.16: Max = 1.5ε (Mean = 0.391ε) And other failures.) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	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) (GSL 1.16: Max = 0ε (Mean = 0ε) And other failures.) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	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) (GSL 1.16: Max = 121ε (Mean = 22ε)) (Cephes: Max = 196ε (Mean = 30.2ε))	Max = 2.82e+04ε (Mean = 1.79e+03ε)	Max = 2.82e+04ε (Mean = 1.79e+03ε)

Table 17.64. Error rates for jacobi_sn

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Jacobi Elliptic: Mathworld Data	Max = 481ε (Mean = 113ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 588ε (Mean = 146ε) And other failures.) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 341ε (Mean = 80.7ε)	Max = 341ε (Mean = 80.7ε)
Jacobi Elliptic: Random Data	Max = 1.92ε (Mean = 0.567ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 4.02ε (Mean = 1.07ε)) (Cephes: Max = 1.92ε (Mean = 0.567ε))	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) (GSL 1.16: Max = 11.7ε (Mean = 1.65ε) And other failures.) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	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) (GSL 1.16: Max = 0ε (Mean = 0ε) And other failures.) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	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) (GSL 1.16: Max = 4.54e+04ε (Mean = 2.63e+03ε)) (Cephes: Max = 4.36e+04ε (Mean = 2.54e+03ε))	Max = 2.45e+04ε (Mean = 1.51e+03ε)	Max = 2.45e+04ε (Mean = 1.51e+03ε)

Table 17.65. Error rates for jacobi_zeta

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.66. Error rates for laguerre(n, m, x)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Associated Laguerre Polynomials	Max = 434ε (Mean = 11.1ε)	Max = 0.84ε (Mean = 0.0358ε) (GSL 1.16: Max = 434ε (Mean = 10.7ε))	Max = 167ε (Mean = 6.38ε) (<tr1/cmath>: Max = 206ε (Mean = 6.86ε))	Max = 167ε (Mean = 6.38ε)

Table 17.67. Error rates for laguerre(n, x)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Laguerre Polynomials	Max = 3.1e+003ε (Mean = 185ε)	Max = 6.82ε (Mean = 0.408ε) (GSL 1.16: Max = 3.1e+03ε (Mean = 185ε))	Max = 1.39e+04ε (Mean = 828ε) (<tr1/cmath>: Max = 4.2e+03ε (Mean = 251ε))	Max = 1.39e+04ε (Mean = 828ε)

Table 17.68. Error rates for legendre_p

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Legendre Polynomials: Small Values	Max = 211ε (Mean = 20.4ε)	Max = 0.732ε (Mean = 0.0619ε) (GSL 1.16: Max = 211ε (Mean = 20.4ε))	Max = 69.2ε (Mean = 9.58ε) (<tr1/cmath>: Max = 124ε (Mean = 13.2ε))	Max = 69.2ε (Mean = 9.58ε)
Legendre Polynomials: Large Values	Max = 300ε (Mean = 33.2ε)	Max = 0.632ε (Mean = 0.0693ε) (GSL 1.16: Max = 300ε (Mean = 33.2ε))	Max = 699ε (Mean = 59.6ε) (<tr1/cmath>: Max = 343ε (Mean = 32.1ε))	Max = 699ε (Mean = 59.6ε)

Table 17.69. Error rates for legendre_p (associated)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Associated Legendre Polynomials: Small Values	Max = 121ε (Mean = 7.14ε)	Max = 0.999ε (Mean = 0.05ε) (GSL 1.16: Max = 121ε (Mean = 6.75ε) And other failures.)	Max = 175ε (Mean = 9.88ε) (<tr1/cmath>: Max = 175ε (Mean = 9.36ε) And other failures.)	Max = 77.7ε (Mean = 5.59ε)

Table 17.70. Error rates for legendre_q

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Legendre Polynomials: Small Values	Max = 46.4ε (Mean = 7.32ε)	Max = 0.612ε (Mean = 0.0517ε) (GSL 1.16: Max = 46.4ε (Mean = 7.46ε))	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) (GSL 1.16: Max = 4.6e+03ε (Mean = 366ε))	Max = 5.98e+03ε (Mean = 478ε)	Max = 5.98e+03ε (Mean = 478ε)

Table 17.71. Error rates for lgamma

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
factorials	Max = 0.914ε (Mean = 0.167ε) (<math.h>: Max = 0.958ε (Mean = 0.38ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 33.6ε (Mean = 2.78ε)) (Rmath 3.0.2: Max = 1.55ε (Mean = 0.592ε)) (Cephes: Max = 1.55ε (Mean = 0.512ε))	Max = 0.991ε (Mean = 0.311ε) (<tr1/cmath>: Max = 1.67ε (Mean = 0.487ε)) (<math.h>: Max = 1.67ε (Mean = 0.487ε))	Max = 0.991ε (Mean = 0.383ε) (<math.h>: Max = 1.36ε (Mean = 0.476ε))
near 0	Max = 0.964ε (Mean = 0.462ε) (<math.h>: Max = 0.962ε (Mean = 0.372ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 5.21ε (Mean = 1.57ε)) (Rmath 3.0.2: Max = 0ε (Mean = 0ε)) (Cephes: Max = 1.16ε (Mean = 0.341ε))	Max = 1.42ε (Mean = 0.566ε) (<tr1/cmath>: Max = 0.964ε (Mean = 0.543ε)) (<math.h>: Max = 0.964ε (Mean = 0.543ε))	Max = 1.42ε (Mean = 0.566ε) (<math.h>: Max = 0.964ε (Mean = 0.543ε))
near 1	Max = 0.867ε (Mean = 0.468ε) (<math.h>: Max = 0.906ε (Mean = 0.565ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 442ε (Mean = 88.8ε)) (Rmath 3.0.2: Max = 7.99e+04ε (Mean = 1.68e+04ε)) (Cephes: Max = 1.14e+05ε (Mean = 2.64e+04ε))	Max = 0.948ε (Mean = 0.36ε) (<tr1/cmath>: Max = 0.615ε (Mean = 0.096ε)) (<math.h>: Max = 0.615ε (Mean = 0.096ε))	Max = 0.866ε (Mean = 0.355ε) (<math.h>: Max = 1.71ε (Mean = 0.581ε))
near 2	Max = 0.591ε (Mean = 0.159ε) (<math.h>: Max = 0.741ε (Mean = 0.473ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 1.17e+03ε (Mean = 274ε)) (Rmath 3.0.2: Max = 2.63e+05ε (Mean = 5.84e+04ε)) (Cephes: Max = 5.08e+05ε (Mean = 9.04e+04ε))	Max = 0.878ε (Mean = 0.242ε) (<tr1/cmath>: Max = 0.741ε (Mean = 0.263ε)) (<math.h>: Max = 0.741ε (Mean = 0.263ε))	Max = 0.878ε (Mean = 0.241ε) (<math.h>: Max = 0.598ε (Mean = 0.235ε))
near -10	Max = 4.22ε (Mean = 1.33ε) (<math.h>: Max = 0.997ε (Mean = 0.444ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 24.9ε (Mean = 4.6ε)) (Rmath 3.0.2: Max = 2.41e+05ε (Mean = 4.29e+04ε)) (Cephes: Max = 0.997ε (Mean = 0.429ε))	Max = 3.81ε (Mean = 1.01ε) (<tr1/cmath>: Max = 3.01ε (Mean = 0.86ε)) (<math.h>: Max = 3.01ε (Mean = 0.86ε))	Max = 3.81ε (Mean = 1.01ε) (<math.h>: Max = 3.04ε (Mean = 1.01ε))
near -55	Max = 0.821ε (Mean = 0.419ε) (<math.h>: Max = 249ε (Mean = 43.1ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 7.02ε (Mean = 1.47ε)) (Rmath 3.0.2: Max = 4.08e+04ε (Mean = 7.26e+03ε)) (Cephes: Max = 1.64ε (Mean = 0.693ε))	Max = 0.821ε (Mean = 0.513ε) (<tr1/cmath>: Max = 1.58ε (Mean = 0.672ε)) (<math.h>: Max = 1.58ε (Mean = 0.672ε))	Max = 1.59ε (Mean = 0.587ε) (<math.h>: Max = 0.821ε (Mean = 0.674ε))

Table 17.72. Error rates for log1p

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Random test data	Max = 0.509ε (Mean = 0.057ε) (<math.h>: Max = 0.509ε (Mean = 0.057ε))	Max = 0.846ε (Mean = 0.153ε) (Rmath 3.0.2: Max = 0.846ε (Mean = 0.153ε)) (Cephes: Max = 0.799ε (Mean = 0.122ε))	Max = 0.818ε (Mean = 0.227ε) (<tr1/cmath>: Max = 0.818ε (Mean = 0.227ε)) (<math.h>: Max = 0.818ε (Mean = 0.227ε))	Max = 1.53ε (Mean = 0.627ε) (<math.h>: Max = 0.818ε (Mean = 0.249ε))

Table 17.73. Error rates for non central beta CDF

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Non Central Beta, medium parameters	Max = 240ε (Mean = 31ε)	Max = 0.998ε (Mean = 0.0659ε) (Rmath 3.0.2: Max = 1.46e+26ε (Mean = 3.5e+24ε) And other failures.)	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) (Rmath 3.0.2: Max = 1.01e+36ε (Mean = 1.19e+35ε) And other failures.)	Max = 2.5e+04ε (Mean = 3.78e+03ε)	Max = 2.57e+04ε (Mean = 4.43e+03ε)

Table 17.74. Error rates for non central beta CDF complement

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Non Central Beta, medium parameters	Max = 619ε (Mean = 62.7ε)	Max = 0.998ε (Mean = 0.0957ε) (Rmath 3.0.2: Max = 7.5e+97ε (Mean = 1.37e+96ε) And other failures.)	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) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.)	Max = 6.83e+03ε (Mean = 993ε)	Max = 3.56e+03ε (Mean = 704ε)

Table 17.75. Error rates for non central chi squared CDF

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Non Central Chi Squared, medium parameters	Max = 48.9ε (Mean = 10ε)	Max = 0.99ε (Mean = 0.0529ε) (Rmath 3.0.2: Max = 727ε (Mean = 121ε))	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) (Rmath 3.0.2: Max = 3.27e+08ε (Mean = 2.23e+07ε))	Max = 3.07e+03ε (Mean = 336ε)	Max = 6.17e+03ε (Mean = 677ε)

Table 17.76. Error rates for non central chi squared CDF complement

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Non Central Chi Squared, medium parameters	Max = 98.6ε (Mean = 15.8ε)	Max = 0.96ε (Mean = 0.0635ε) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.)	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) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.)	Max = 5.02e+03ε (Mean = 630ε)	Max = 5.1e+03ε (Mean = 577ε)

Table 17.77. Error rates for non central t CDF

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Non Central T	Max = 138ε (Mean = 31.5ε)	Max = 0.796ε (Mean = 0.0691ε) (Rmath 3.0.2: Max = 5.28e+15ε (Mean = 8.49e+14ε) And other failures.)	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) (Rmath 3.0.2: Max = 2.09e+03ε (Mean = 244ε))	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) (Rmath 3.0.2: Max = 2.46ε (Mean = 0.657ε))	Max = 5.26e+05ε (Mean = 1.48e+05ε)	Max = 5.24e+05ε (Mean = 1.47e+05ε)

Table 17.78. Error rates for non central t CDF complement

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Non Central T	Max = 150ε (Mean = 32.3ε)	Max = 0.707ε (Mean = 0.0497ε) (Rmath 3.0.2: Max = 6.19e+15ε (Mean = 6.72e+14ε) And other failures.)	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) (Rmath 3.0.2: Max = 1.87e+03ε (Mean = 263ε))	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) (Rmath 3.0.2: Max = 2.24ε (Mean = 0.945ε))	Max = 9.79e+05ε (Mean = 1.97e+05ε)	Max = 9.79e+05ε (Mean = 1.97e+05ε)

Table 17.79. Error rates for owens_t

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.80. Error rates for polygamma

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Mathematica Data	Max = 6.34ε (Mean = 1.53ε)	Max = 0.824ε (Mean = 0.0574ε) (Rmath 3.0.2: Max = 108ε (Mean = 15.2ε)) (GSL 1.16: Max = 62.9ε (Mean = 12.8ε))	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) (Rmath 3.0.2: Max = 1.71e+56ε (Mean = 1.01e+55ε) And other failures.) (GSL 1.16: Max = 244ε (Mean = 32.8ε) And other failures.)	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) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (GSL 1.16: Max = 36.6ε (Mean = 3.04ε) And other failures.)	Max = 269ε (Mean = 87.7ε)	Max = 269ε (Mean = 87.9ε)
Mathematica Data - large negative arguments	Max = 162ε (Mean = 101ε)	Max = 0ε (Mean = 0ε) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (GSL 1.16: Max = 1.79ε (Mean = 0.197ε) And other failures.)	Max = 155ε (Mean = 96.4ε)	Max = 155ε (Mean = 96.4ε)
Mathematica Data - small arguments	Max = 3ε (Mean = 0.496ε)	Max = 0ε (Mean = 0ε) (Rmath 3.0.2: Max = 106ε (Mean = 20ε)) (GSL 1.16: Max = 15.2ε (Mean = 5.03ε))	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) (Rmath 3.0.2: Max = +INFε (Mean = +INFε) And other failures.) (GSL 1.16: Max = 151ε (Mean = 39.3ε) And other failures.)	Max = 54.5ε (Mean = 13.3ε)	Max = 90.1ε (Mean = 30.6ε)

Table 17.81. Error rates for powm1

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.82. Error rates for sin_pi

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.83. Error rates for sph_bessel

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux long double	GNU C++ version 5.1.0 linux double	Sun compiler version 0x5130 Sun Solaris long double
Bessel j: Random Data	Max = 245ε (Mean = 16.3ε)	Max = 243ε (Mean = 13.3ε) (<tr1/cmath>: Max = 1.91e+06ε (Mean = 1.09e+05ε))	Max = 0.978ε (Mean = 0.0539ε) (GSL 1.16: Max = 1.79e+03ε (Mean = 107ε))	Max = 243ε (Mean = 33.7ε)

Table 17.84. Error rates for sph_bessel_prime

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.85. Error rates for sph_neumann

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
y: Random Data	Max = 281ε (Mean = 31.1ε)	Max = 0.995ε (Mean = 0.0665ε) (GSL 1.16: Max = 8.5e+04ε (Mean = 5.33e+03ε))	Max = 234ε (Mean = 19.5ε) (<tr1/cmath>: Max = 1.6e+06ε (Mean = 1.4e+05ε))	Max = 234ε (Mean = 19.8ε)

Table 17.86. Error rates for sph_neumann_prime

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.87. Error rates for spherical_harmonic_i

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.88. Error rates for spherical_harmonic_r

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.89. Error rates for sqrt1pm1

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.90. Error rates for tgamma

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
factorials	Max = 1.85ε (Mean = 0.491ε) (<math.h>: Max = 3.17ε (Mean = 0.928ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 3.95ε (Mean = 0.783ε)) (Rmath 3.0.2: Max = 314ε (Mean = 93.4ε)) (Cephes: Max = 3.19ε (Mean = 0.884ε))	Max = 1.96ε (Mean = 0.483ε) (<tr1/cmath>: Max = 1.66ε (Mean = 0.584ε)) (<math.h>: Max = 1.66ε (Mean = 0.584ε))	Max = 172ε (Mean = 41ε) (<math.h>: Max = 0ε (Mean = 0ε))
near 0	Max = 1.96ε (Mean = 0.684ε) (<math.h>: Max = 1ε (Mean = 0.405ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 4.51ε (Mean = 1.92ε)) (Rmath 3.0.2: Max = 1ε (Mean = 0.335ε)) (Cephes: Max = 1ε (Mean = 0.548ε))	Max = 2ε (Mean = 0.73ε) (<tr1/cmath>: Max = 1ε (Mean = 0.376ε)) (<math.h>: Max = 1ε (Mean = 0.376ε))	Max = 2ε (Mean = 0.647ε) (<math.h>: Max = 0.5ε (Mean = 0.0791ε))
near 1	Max = 2ε (Mean = 0.865ε) (<math.h>: Max = 1ε (Mean = 0.4ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 4.41ε (Mean = 1.81ε)) (Rmath 3.0.2: Max = 1ε (Mean = 0.32ε)) (Cephes: Max = 1ε (Mean = 0.518ε))	Max = 2ε (Mean = 0.85ε) (<tr1/cmath>: Max = 0.918ε (Mean = 0.203ε)) (<math.h>: Max = 0.918ε (Mean = 0.203ε))	Max = 3.01ε (Mean = 1.06ε) (<math.h>: Max = 1ε (Mean = 0.175ε))
near 2	Max = 2ε (Mean = 0.995ε) (<math.h>: Max = 0ε (Mean = 0ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 7.95ε (Mean = 3.12ε)) (Rmath 3.0.2: Max = 1ε (Mean = 0.191ε)) (Cephes: Max = 1.09ε (Mean = 0.502ε))	Max = 2ε (Mean = 0.913ε) (<tr1/cmath>: Max = 0.558ε (Mean = 0.298ε)) (<math.h>: Max = 0.558ε (Mean = 0.298ε))	Max = 5.01ε (Mean = 1.89ε) (<math.h>: Max = 0ε (Mean = 0ε))
near -10	Max = 1.73ε (Mean = 0.729ε) (<math.h>: Max = 0.866ε (Mean = 0.445ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 2.6ε (Mean = 1.05ε)) (Rmath 3.0.2: Max = 6.34e+05ε (Mean = 1.2e+05ε)) (Cephes: Max = 2.6ε (Mean = 0.956ε))	Max = 2.6ε (Mean = 0.985ε) (<tr1/cmath>: Max = 2.26ε (Mean = 1.08ε)) (<math.h>: Max = 2.26ε (Mean = 1.08ε))	Max = 1.75ε (Mean = 0.819ε) (<math.h>: Max = 0ε (Mean = 0ε))
near -55	Max = 1.8ε (Mean = 0.817ε) (<math.h>: Max = 3.87e+004ε (Mean = 6.71e+003ε))	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 1.8ε (Mean = 0.782ε)) (Rmath 3.0.2: Max = 6.36e+06ε (Mean = 1.13e+06ε)) (Cephes: Max = 2.7ε (Mean = 0.988ε))	Max = 1.8ε (Mean = 0.847ε) (<tr1/cmath>: Max = 1.79ε (Mean = 0.75ε)) (<math.h>: Max = 1.79ε (Mean = 0.75ε))	Max = 98.5ε (Mean = 53.4ε) (<math.h>: Max = 0ε (Mean = 0ε))

Table 17.91. Error rates for tgamma1pm1

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.92. Error rates for tgamma_delta_ratio

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
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 17.93. Error rates for tgamma (incomplete)

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
tgamma(a, z) medium values	Max = 8.14ε (Mean = 1.71ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 200ε (Mean = 13.3ε))	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) (GSL 1.16: Max = 1.38e+10ε (Mean = 1.05e+09ε))	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) (GSL 1.16: Max = 117ε (Mean = 12.5ε))	Max = 5.52ε (Mean = 1.52ε)	Max = 79.6ε (Mean = 20.9ε)

Table 17.94. Error rates for tgamma_lower

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
tgamma(a, z) medium values	Max = 5.62ε (Mean = 1.43ε)	Max = 0.833ε (Mean = 0.0315ε) (GSL 1.16: 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) (GSL 1.16: 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) (GSL 1.16: Max = 0ε (Mean = 0ε))	Max = 4.83ε (Mean = 1.12ε)	Max = 84.7ε (Mean = 17.5ε)

Table 17.95. Error rates for tgamma_ratio

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
tgamma ratios	Max = 3.66ε (Mean = 1.27ε)	Max = 0ε (Mean = 0ε)	Max = 3.09ε (Mean = 1.15ε)	Max = 174ε (Mean = 61.2ε)

Table 17.96. Error rates for trigamma

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Mathematica Data	Max = 1ε (Mean = 0.382ε)	Max = 0.998ε (Mean = 0.105ε) (Rmath 3.0.2: Max = 1.34e+04ε (Mean = 1.51e+03ε)) (GSL 1.16: Max = 1.34e+04ε (Mean = 1.49e+03ε))	Max = 1.28ε (Mean = 0.449ε)	Max = 1.28ε (Mean = 0.447ε)

Table 17.97. Error rates for zeta

	Microsoft Visual C++ version 12.0 Win32 double	GNU C++ version 5.1.0 linux double	GNU C++ version 5.1.0 linux long double	Sun compiler version 0x5130 Sun Solaris long double
Zeta: Random values greater than 1	Max = 0.836ε (Mean = 0.093ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 8.69ε (Mean = 1.03ε)) (Cephes: Max = 4.49e+33ε (Mean = 6.85e+32ε) And other failures.)	Max = 0.846ε (Mean = 0.0833ε) (<tr1/cmath>: Max = 5.45ε (Mean = 1ε))	Max = 0.846ε (Mean = 0.0743ε)
Zeta: Random values less than 1	Max = 7.03ε (Mean = 2.98ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 137ε (Mean = 13.8ε)) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 7.03ε (Mean = 2.71ε) (<tr1/cmath>: Max = 538ε (Mean = 59.3ε))	Max = 70.1ε (Mean = 17.1ε)
Zeta: Values close to and greater than 1	Max = 0.994ε (Mean = 0.421ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 7.73ε (Mean = 4.07ε)) (Cephes: Max = 6.77e+15ε (Mean = 1.52e+15ε) And other failures.)	Max = 0.995ε (Mean = 0.5ε) (<tr1/cmath>: Max = 1.9e+06ε (Mean = 5.11e+05ε))	Max = 0.995ε (Mean = 0.5ε)
Zeta: Values close to and less than 1	Max = 0.991ε (Mean = 0.375ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 0.991ε (Mean = 0.28ε)) (Cephes: Max = 8.66e+15ε (Mean = 1.9e+15ε) And other failures.)	Max = 0.998ε (Mean = 0.508ε) (<tr1/cmath>: Max = 8.53e+06ε (Mean = 1.87e+06ε))	Max = 0.998ε (Mean = 0.568ε)
Zeta: Integer arguments	Max = 6.5ε (Mean = 2.17ε)	Max = 0ε (Mean = 0ε) (GSL 1.16: Max = 3.75ε (Mean = 1.1ε)) (Cephes: Max = +INFε (Mean = +INFε) And other failures.)	Max = 9ε (Mean = 3.06ε) (<tr1/cmath>: Max = 70.3ε (Mean = 17.4ε))	Max = 21ε (Mean = 7.13ε)