Boost C++ Libraries

...one of the most highly regarded and expertly designed C++ library projects in the world. Herb Sutter and Andrei Alexandrescu, C++ Coding Standards

This is the documentation for a snapshot of the develop branch, built from commit 11d01ebb93.
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Tables of Error Rates for all Functions

Table 23.3. Error rates for beta

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Beta Function: Small Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = +INFε (Mean = +INFε) And other failures.)
(Rmath 3.2.3: Max = 1.14ε (Mean = 0.574ε))

Max = 2.86ε (Mean = 1.22ε)

(<cmath>: Max = 364ε (Mean = 76.2ε))

Max = 2.86ε (Mean = 1.22ε)

Max = 2.23ε (Mean = 1.14ε)

Beta Function: Medium Values

Max = 0.978ε (Mean = 0.0595ε)

(GSL 2.1: Max = 1.18e+03ε (Mean = 238ε))
(Rmath 3.2.3: Max = 1.09e+03ε (Mean = 265ε))

Max = 61.4ε (Mean = 19.4ε)

(<cmath>: Max = 1.07e+03ε (Mean = 264ε))

Max = 107ε (Mean = 24.5ε)

Max = 96.5ε (Mean = 22.4ε)

Beta Function: Divergent Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 12.1ε (Mean = 1.99ε))
(Rmath 3.2.3: Max = 176ε (Mean = 28ε))

Max = 8.99ε (Mean = 2.44ε)

(<cmath>: Max = 128ε (Mean = 23.8ε))

Max = 18.8ε (Mean = 2.71ε)

Max = 11.4ε (Mean = 2.19ε)


Table 23.4. Error rates for beta (incomplete)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Incomplete Beta Function: Small Values

Max = 0ε (Mean = 0ε)

Max = 11.1ε (Mean = 2.32ε)

Max = 18.7ε (Mean = 3.19ε)

Max = 9.94ε (Mean = 2.17ε)

Incomplete Beta Function: Medium Values

Max = 0.568ε (Mean = 0.0254ε)

Max = 69.2ε (Mean = 13.4ε)

Max = 174ε (Mean = 25ε)

Max = 90ε (Mean = 12.7ε)

Incomplete Beta Function: Large and Diverse Values

Max = 0.999ε (Mean = 0.0325ε)

Max = 6.84e+04ε (Mean = 2.76e+03ε)

Max = 6.86e+04ε (Mean = 2.79e+03ε)

Max = 633ε (Mean = 29.7ε)

Incomplete Beta Function: Small Integer Values

Max = 0.786ε (Mean = 0.0323ε)

Max = 11.6ε (Mean = 3.6ε)

Max = 51.8ε (Mean = 11ε)

Max = 26ε (Mean = 6.28ε)


Table 23.5. Error rates for betac

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Incomplete Beta Function: Small Values

Max = 0.676ε (Mean = 0.0302ε)

Max = 9.92ε (Mean = 2.3ε)

Max = 11.2ε (Mean = 2.94ε)

Max = 8.94ε (Mean = 2.06ε)

Incomplete Beta Function: Medium Values

Max = 0.949ε (Mean = 0.098ε)

Max = 63.5ε (Mean = 13.5ε)

Max = 97.6ε (Mean = 24.3ε)

Max = 90.6ε (Mean = 14.8ε)

Incomplete Beta Function: Large and Diverse Values

Max = 1.12ε (Mean = 0.0458ε)

Max = 1.05e+05ε (Mean = 5.45e+03ε)

Max = 1.04e+05ε (Mean = 5.46e+03ε)

Max = 3.72e+03ε (Mean = 113ε)

Incomplete Beta Function: Small Integer Values

Max = 0.586ε (Mean = 0.0314ε)

Max = 11.1ε (Mean = 3.65ε)

Max = 103ε (Mean = 17.4ε)

Max = 26.2ε (Mean = 6.36ε)


Table 23.6. Error rates for binomial_coefficient

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Binomials: small arguments

Max = 1ε (Mean = 0.369ε)

Max = 1.5ε (Mean = 0.339ε)

Max = 1.5ε (Mean = 0.339ε)

Max = 1ε (Mean = 0.369ε)

Binomials: large arguments

Max = 0.939ε (Mean = 0.314ε)

Max = 26.6ε (Mean = 6.13ε)

Max = 53.2ε (Mean = 10.8ε)

Max = 37.2ε (Mean = 7.4ε)


Table 23.7. Error rates for boost::math::powm1

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

powm1

(<math.h>: Max = 2.04ε (Mean = 0.493ε))

Max = 2.04ε (Mean = 0.493ε)

(<cmath>: Max = 2.04ε (Mean = 0.493ε))

(Rmath 3.2.3: Max = 1.06ε (Mean = 0.425ε))

(GSL 2.1: Max = 1.06ε (Mean = 0.425ε))

Max = 1.06ε (Mean = 0.425ε)

Max = 1.88ε (Mean = 0.49ε)

(<math.h>: Max = 1.88ε (Mean = 0.49ε))

(<math.h>: Max = 1.84ε (Mean = 0.486ε))

Max = 1.84ε (Mean = 0.486ε)


Table 23.8. Error rates for cbrt

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

cbrt Function

Max = 0ε (Mean = 0ε)

Max = 1.34ε (Mean = 0.471ε)

(<cmath>: Max = 1.34ε (Mean = 0.471ε))
(<math.h>: Max = 1.34ε (Mean = 0.471ε))

Max = 1.34ε (Mean = 0.471ε)

(<math.h>: Max = 1.34ε (Mean = 0.471ε))

Max = 1.7ε (Mean = 0.565ε)

(<math.h>: Max = 1.7ε (Mean = 0.565ε))


Table 23.9. Error rates for cos_pi

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

sin_pi and cos_pi

Max = 0ε (Mean = 0ε)

Max = 0.991ε (Mean = 0.302ε)

Max = 0.991ε (Mean = 0.302ε)

Max = 0.996ε (Mean = 0.284ε)

sin_pi and cos_pi near integers and half integers

Max = 0ε (Mean = 0ε)

Max = 0.976ε (Mean = 0.28ε)

Max = 0.976ε (Mean = 0.28ε)

Max = 0.996ε (Mean = 0.298ε)


Table 23.10. Error rates for cyl_bessel_i

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel I0: Mathworld Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 270ε (Mean = 91.6ε) And other failures.)
(Rmath 3.2.3: Max = 1.52ε (Mean = 0.622ε) And other failures.)

Max = 1.95ε (Mean = 0.738ε)

(<cmath>: Max = 8.49ε (Mean = 3.46ε) And other failures.)

Max = 1.95ε (Mean = 0.661ε)

Max = 0.762ε (Mean = 0.329ε)

Bessel I1: Mathworld Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 128ε (Mean = 41ε) And other failures.)
(Rmath 3.2.3: Max = 1.53ε (Mean = 0.483ε) And other failures.)

Max = 0.64ε (Mean = 0.202ε)

(<cmath>: Max = 5ε (Mean = 2.15ε) And other failures.)

Max = 0.64ε (Mean = 0.202ε)

Max = 0.767ε (Mean = 0.398ε)

Bessel In: Mathworld Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.31ε (Mean = 0.838ε) And other failures.)
(Rmath 3.2.3: Max = 1.73ε (Mean = 0.601ε) And other failures.)

Max = 1.8ε (Mean = 1.33ε)

(<cmath>: Max = 430ε (Mean = 163ε) And other failures.)

Max = 463ε (Mean = 140ε)

Max = 3.46ε (Mean = 1.32ε)

Bessel Iv: Mathworld Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 5.95ε (Mean = 2.08ε) And other failures.)
(Rmath 3.2.3: Max = 3.53ε (Mean = 1.39ε))

Max = 4.12ε (Mean = 1.85ε)

(<cmath>: Max = 616ε (Mean = 221ε) And other failures.)

Max = 4.12ε (Mean = 1.95ε)

Max = 2.97ε (Mean = 1.24ε)

Bessel In: Random Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 261ε (Mean = 53.2ε) And other failures.)
(Rmath 3.2.3: Max = 7.37ε (Mean = 2.4ε))

Max = 4.62ε (Mean = 1.06ε)

(<cmath>: Max = 645ε (Mean = 132ε))

Max = 176ε (Mean = 39.1ε)

Max = 9.67ε (Mean = 1.88ε)

Bessel Iv: Random Data

Max = 0.661ε (Mean = 0.0441ε)

(GSL 2.1: Max = 6.18e+03ε (Mean = 1.55e+03ε) And other failures.)
(Rmath 3.2.3: Max = 4.28e+08ε (Mean = 2.85e+07ε))

Max = 8.35ε (Mean = 1.62ε)

(<cmath>: Max = 1.05e+03ε (Mean = 224ε) And other failures.)

Max = 283ε (Mean = 88.4ε)

Max = 7.46ε (Mean = 1.71ε)

Bessel Iv: Mathworld Data (large values)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 37ε (Mean = 18ε) And other failures.)
(Rmath 3.2.3: Max = 3.77e+168ε (Mean = 2.39e+168ε) And other failures.)

Max = 14.7ε (Mean = 6.66ε)

(<cmath>: Max = 118ε (Mean = 57.2ε) And other failures.)

Max = 14.7ε (Mean = 6.59ε)

Max = 3.67ε (Mean = 1.64ε)


Table 23.11. Error rates for cyl_bessel_i (integer orders)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel I0: Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.79ε (Mean = 0.482ε))
(Rmath 3.2.3: Max = 1.52ε (Mean = 0.622ε) And other failures.)

Max = 1.95ε (Mean = 0.738ε)

(<cmath>: Max = 8.49ε (Mean = 3.46ε) And other failures.)

Max = 1.95ε (Mean = 0.661ε)

Max = 0.762ε (Mean = 0.329ε)

Bessel I1: Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.82ε (Mean = 0.456ε))
(Rmath 3.2.3: Max = 1.53ε (Mean = 0.483ε) And other failures.)

Max = 0.64ε (Mean = 0.202ε)

(<cmath>: Max = 5ε (Mean = 2.15ε) And other failures.)

Max = 0.64ε (Mean = 0.202ε)

Max = 0.767ε (Mean = 0.398ε)

Bessel In: Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 5.15ε (Mean = 2.13ε) And other failures.)
(Rmath 3.2.3: Max = 1.73ε (Mean = 0.601ε) And other failures.)

Max = 1.8ε (Mean = 1.33ε)

(<cmath>: Max = 430ε (Mean = 163ε) And other failures.)

Max = 463ε (Mean = 140ε)

Max = 3.46ε (Mean = 1.32ε)


Table 23.12. Error rates for cyl_bessel_i_prime

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel I'0: Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 0.82ε (Mean = 0.259ε)

Max = 0.82ε (Mean = 0.259ε)

Max = 0.82ε (Mean = 0.354ε)

Bessel I'1: Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 1.97ε (Mean = 0.757ε)

Max = 1.97ε (Mean = 0.757ε)

Max = 1.36ε (Mean = 0.782ε)

Bessel I'n: Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 2.31ε (Mean = 1.41ε)

Max = 701ε (Mean = 212ε)

Max = 3.61ε (Mean = 1.22ε)

Bessel I'v: Mathworld Data

Max = 1.62ε (Mean = 0.512ε)

Max = 2.89e+03ε (Mean = 914ε)

Max = 2.89e+03ε (Mean = 914ε)

Max = 3.76e+03ε (Mean = 1.19e+03ε)

Bessel I'n: Random Data

Max = 0ε (Mean = 0ε)

Max = 3.95ε (Mean = 1.06ε)

Max = 195ε (Mean = 37.1ε)

Max = 9.85ε (Mean = 1.82ε)

Bessel I'v: Random Data

Max = 0ε (Mean = 0ε)

Max = 14.1ε (Mean = 2.93ε)

Max = 336ε (Mean = 68.5ε)

Max = 14ε (Mean = 2.5ε)

Bessel I'v: Mathworld Data (large values)

Max = 0ε (Mean = 0ε)

Max = 42.6ε (Mean = 20.2ε)

Max = 42.6ε (Mean = 20.2ε)

Max = 59.5ε (Mean = 26.6ε)


Table 23.13. Error rates for cyl_bessel_i_prime (integer orders)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel I'0: Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

Max = 0.82ε (Mean = 0.259ε)

Max = 0.82ε (Mean = 0.259ε)

Max = 0.82ε (Mean = 0.354ε)

Bessel I'1: Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

Max = 1.97ε (Mean = 0.757ε)

Max = 1.97ε (Mean = 0.757ε)

Max = 1.36ε (Mean = 0.782ε)

Bessel I'n: Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

Max = 2.31ε (Mean = 1.41ε)

Max = 701ε (Mean = 212ε)

Max = 3.61ε (Mean = 1.22ε)


Table 23.14. Error rates for cyl_bessel_j

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel J0: Mathworld Data

Max = 6.55ε (Mean = 2.86ε)

(<cmath>: Max = 5.04ε (Mean = 1.78ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.629ε (Mean = 0.223ε) And other failures.)
(Rmath 3.2.3: Max = 0.629ε (Mean = 0.223ε) And other failures.)

Max = 6.55ε (Mean = 2.86ε)

Max = 2.52ε (Mean = 1.2ε)

Bessel J0: Mathworld Data (Tricky cases)

Max = 1.64e+08ε (Mean = 6.69e+07ε)

(<cmath>: Max = 4.79e+08ε (Mean = 1.96e+08ε))

Max = 8e+04ε (Mean = 3.27e+04ε)

(GSL 2.1: Max = 6.5e+07ε (Mean = 2.66e+07ε))
(Rmath 3.2.3: Max = 1.04e+07ε (Mean = 4.29e+06ε))

Max = 1.64e+08ε (Mean = 6.69e+07ε)

Max = 1e+07ε (Mean = 4.09e+06ε)

Bessel J1: Mathworld Data

Max = 3.59ε (Mean = 1.33ε)

(<cmath>: Max = 6.1ε (Mean = 2.95ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 6.62ε (Mean = 2.35ε) And other failures.)
(Rmath 3.2.3: Max = 0.946ε (Mean = 0.39ε) And other failures.)

Max = 1.44ε (Mean = 0.637ε)

Max = 1.73ε (Mean = 0.976ε)

Bessel J1: Mathworld Data (tricky cases)

Max = 2.18e+05ε (Mean = 9.76e+04ε)

(<cmath>: Max = 2.15e+06ε (Mean = 1.58e+06ε))

Max = 106ε (Mean = 47.5ε)

(GSL 2.1: Max = 8.75e+05ε (Mean = 5.32e+05ε))
(Rmath 3.2.3: Max = 2.93e+06ε (Mean = 1.7e+06ε))

Max = 2.18e+05ε (Mean = 9.76e+04ε)

Max = 3.23e+04ε (Mean = 1.45e+04ε)

Bessel JN: Mathworld Data

Max = 6.85ε (Mean = 3.35ε)

(<cmath>: Max = 2.13e+19ε (Mean = 5.16e+18ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 6.9e+05ε (Mean = 2.15e+05ε) And other failures.)
(Rmath 3.2.3: Max = +INFε (Mean = +INFε) And other failures.)

Max = 463ε (Mean = 112ε)

Max = 14.7ε (Mean = 5.4ε)

Bessel J: Mathworld Data

Max = 14.7ε (Mean = 4.11ε)

(<cmath>: Max = 3.49e+05ε (Mean = 8.09e+04ε) And other failures.)

Max = 10ε (Mean = 2.24ε)

(GSL 2.1: Max = 2.39e+05ε (Mean = 5.37e+04ε) And other failures.)
(Rmath 3.2.3: Max = +INFε (Mean = +INFε) And other failures.)

Max = 14.7ε (Mean = 4.22ε)

Max = 14.9ε (Mean = 3.89ε)

Bessel J: Mathworld Data (large values)

Max = 607ε (Mean = 305ε)

(<cmath>: Max = 34.9ε (Mean = 17.4ε) And other failures.)

Max = 0.536ε (Mean = 0.268ε)

(GSL 2.1: Max = 4.91e+03ε (Mean = 2.46e+03ε) And other failures.)
(Rmath 3.2.3: Max = 5.9ε (Mean = 3.76ε))

Max = 607ε (Mean = 305ε)

Max = 9.31ε (Mean = 5.52ε)

Bessel JN: Random Data

Max = 50.8ε (Mean = 3.69ε)

(<cmath>: Max = 1.12e+03ε (Mean = 88.7ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 75.7ε (Mean = 5.36ε))
(Rmath 3.2.3: Max = 3.93ε (Mean = 1.22ε))

Max = 99.6ε (Mean = 22ε)

Max = 17.5ε (Mean = 1.46ε)

Bessel J: Random Data

Max = 11.4ε (Mean = 1.68ε)

(<cmath>: Max = 501ε (Mean = 52.3ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 15.5ε (Mean = 3.33ε) And other failures.)
(Rmath 3.2.3: Max = 6.74ε (Mean = 1.3ε))

Max = 260ε (Mean = 34ε)

Max = 9.24ε (Mean = 1.17ε)

Bessel J: Random Data (Tricky large values)

Max = 785ε (Mean = 94.2ε)

(<cmath>: Max = 5.01e+17ε (Mean = 6.23e+16ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.48e+05ε (Mean = 5.11e+04ε))
(Rmath 3.2.3: Max = 71.6ε (Mean = 11.7ε))

Max = 785ε (Mean = 97.4ε)

Max = 59.2ε (Mean = 8.67ε)


Table 23.15. Error rates for cyl_bessel_j (integer orders)

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel J0: Mathworld Data (Integer Version)

Max = 6.55ε (Mean = 2.86ε)

(<cmath>: Max = 5.04ε (Mean = 1.78ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.12ε (Mean = 0.488ε))
(Rmath 3.2.3: Max = 0.629ε (Mean = 0.223ε) And other failures.)

Max = 6.55ε (Mean = 2.86ε)

Max = 2.52ε (Mean = 1.2ε)

(<math.h>: Max = 1.89ε (Mean = 0.988ε))

Bessel J0: Mathworld Data (Tricky cases) (Integer Version)

Max = 1.64e+08ε (Mean = 6.69e+07ε)

(<cmath>: Max = 4.79e+08ε (Mean = 1.96e+08ε))

Max = 8e+04ε (Mean = 3.27e+04ε)

(GSL 2.1: Max = 1e+07ε (Mean = 4.11e+06ε))
(Rmath 3.2.3: Max = 1.04e+07ε (Mean = 4.29e+06ε))

Max = 1.64e+08ε (Mean = 6.69e+07ε)

Max = 1e+07ε (Mean = 4.09e+06ε)

(<math.h>: Max = 2.54e+08ε (Mean = 1.04e+08ε))

Bessel J1: Mathworld Data (Integer Version)

Max = 3.59ε (Mean = 1.33ε)

(<cmath>: Max = 6.1ε (Mean = 2.95ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.89ε (Mean = 0.721ε))
(Rmath 3.2.3: Max = 0.946ε (Mean = 0.39ε) And other failures.)

Max = 1.44ε (Mean = 0.637ε)

Max = 1.73ε (Mean = 0.976ε)

(<math.h>: Max = 11.4ε (Mean = 4.15ε))

Bessel J1: Mathworld Data (tricky cases) (Integer Version)

Max = 2.18e+05ε (Mean = 9.76e+04ε)

(<cmath>: Max = 2.15e+06ε (Mean = 1.58e+06ε))

Max = 106ε (Mean = 47.5ε)

(GSL 2.1: Max = 1.26e+06ε (Mean = 6.28e+05ε))
(Rmath 3.2.3: Max = 2.93e+06ε (Mean = 1.7e+06ε))

Max = 2.18e+05ε (Mean = 9.76e+04ε)

Max = 3.23e+04ε (Mean = 1.45e+04ε)

(<math.h>: Max = 1.44e+07ε (Mean = 6.5e+06ε))

Bessel JN: Mathworld Data (Integer Version)

Max = 6.85ε (Mean = 3.35ε)

(<cmath>: Max = 2.13e+19ε (Mean = 5.16e+18ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 6.9e+05ε (Mean = 2.53e+05ε) And other failures.)
(Rmath 3.2.3: Max = +INFε (Mean = +INFε) And other failures.)

Max = 463ε (Mean = 112ε)

Max = 14.7ε (Mean = 5.4ε)

(<math.h>: Max = +INFε (Mean = +INFε) And other failures.)


Table 23.16. Error rates for cyl_bessel_j_prime

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel J0': Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 18.9ε (Mean = 6.82ε)

Max = 18.9ε (Mean = 6.72ε)

Max = 6.62ε (Mean = 2.55ε)

Bessel J0': Mathworld Data (Tricky cases)

Max = 0ε (Mean = 0ε)

Max = 7.44ε (Mean = 3.34ε)

Max = 7.44ε (Mean = 3.31ε)

Max = 3.67ε (Mean = 1.74ε)

Bessel J1': Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 7.9ε (Mean = 3.37ε)

Max = 7.9ε (Mean = 3.37ε)

Max = 0.999ε (Mean = 0.627ε)

Bessel J1': Mathworld Data (tricky cases)

Max = 287ε (Mean = 129ε)

Max = 5.88e+05ε (Mean = 2.63e+05ε)

Max = 5.88e+05ε (Mean = 2.63e+05ε)

Max = 288ε (Mean = 129ε)

Bessel JN': Mathworld Data

Max = 0.527ε (Mean = 0.128ε)

Max = 1.29e+03ε (Mean = 312ε)

Max = 1.29e+03ε (Mean = 355ε)

Max = 14ε (Mean = 6.13ε)

Bessel J': Mathworld Data

Max = 21.5ε (Mean = 4.7ε)

Max = 42.5ε (Mean = 9.31ε)

Max = 42.5ε (Mean = 9.32ε)

Max = 23.7ε (Mean = 8ε)

Bessel J': Mathworld Data (large values)

Max = 0ε (Mean = 0ε)

Max = 989ε (Mean = 495ε)

Max = 989ε (Mean = 495ε)

Max = 2.9ε (Mean = 1.61ε)

Bessel JN': Random Data

Max = 0.593ε (Mean = 0.0396ε)

Max = 11.3ε (Mean = 1.85ε)

Max = 79.4ε (Mean = 16.2ε)

Max = 6.34ε (Mean = 0.999ε)

Bessel J': Random Data

Max = 0.885ε (Mean = 0.033ε)

Max = 139ε (Mean = 6.47ε)

Max = 279ε (Mean = 27.2ε)

Max = 176ε (Mean = 9.75ε)

Bessel J': Random Data (Tricky large values)

Max = 0ε (Mean = 0ε)

Max = 474ε (Mean = 62.2ε)

Max = 474ε (Mean = 64.5ε)

Max = 379ε (Mean = 45.4ε)


Table 23.17. Error rates for cyl_bessel_j_prime (integer orders)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel J0': Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

Max = 18.9ε (Mean = 6.82ε)

Max = 18.9ε (Mean = 6.72ε)

Max = 6.62ε (Mean = 2.55ε)

Bessel J0': Mathworld Data (Tricky cases) (Integer Version)

Max = 0ε (Mean = 0ε)

Max = 7.44ε (Mean = 3.34ε)

Max = 7.44ε (Mean = 3.31ε)

Max = 3.67ε (Mean = 1.74ε)

Bessel J1': Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

Max = 7.9ε (Mean = 3.37ε)

Max = 7.9ε (Mean = 3.37ε)

Max = 0.999ε (Mean = 0.627ε)

Bessel J1': Mathworld Data (tricky cases) (Integer Version)

Max = 287ε (Mean = 129ε)

Max = 5.88e+05ε (Mean = 2.63e+05ε)

Max = 5.88e+05ε (Mean = 2.63e+05ε)

Max = 288ε (Mean = 129ε)

Bessel JN': Mathworld Data (Integer Version)

Max = 0.527ε (Mean = 0.128ε)

Max = 1.29e+03ε (Mean = 312ε)

Max = 1.29e+03ε (Mean = 355ε)

Max = 14ε (Mean = 6.13ε)


Table 23.18. Error rates for cyl_bessel_k

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel K0: Mathworld Data

Max = 0.833ε (Mean = 0.436ε)

(<cmath>: Max = 9.33ε (Mean = 3.25ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 6.04ε (Mean = 2.16ε))
(Rmath 3.2.3: Max = 0.833ε (Mean = 0.601ε))

Max = 0.833ε (Mean = 0.436ε)

Max = 0.833ε (Mean = 0.552ε)

Bessel K1: Mathworld Data

Max = 0.786ε (Mean = 0.329ε)

(<cmath>: Max = 8.94ε (Mean = 3.19ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 6.26ε (Mean = 2.21ε))
(Rmath 3.2.3: Max = 0.894ε (Mean = 0.516ε))

Max = 0.786ε (Mean = 0.329ε)

Max = 0.786ε (Mean = 0.39ε)

Bessel Kn: Mathworld Data

Max = 2.6ε (Mean = 1.21ε)

(<cmath>: Max = 12.9ε (Mean = 4.91ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 3.36ε (Mean = 1.43ε) And other failures.)
(Rmath 3.2.3: Max = 8.48ε (Mean = 2.98ε))

Max = 2.6ε (Mean = 1.21ε)

Max = 3.63ε (Mean = 1.46ε)

Bessel Kv: Mathworld Data

Max = 3.58ε (Mean = 2.39ε)

(<cmath>: Max = 13ε (Mean = 4.81ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 5.47ε (Mean = 2.04ε) And other failures.)
(Rmath 3.2.3: Max = 3.15ε (Mean = 1.35ε))

Max = 5.21ε (Mean = 2.53ε)

Max = 4.78ε (Mean = 2.19ε)

Bessel Kv: Mathworld Data (large values)

Max = 42.3ε (Mean = 21ε)

(<cmath>: Max = 42.3ε (Mean = 19.8ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 308ε (Mean = 142ε) And other failures.)
(Rmath 3.2.3: Max = 84.6ε (Mean = 37.8ε))

Max = 42.3ε (Mean = 21ε)

Max = 59.8ε (Mean = 26.9ε)

Bessel Kn: Random Data

Max = 4.55ε (Mean = 1.12ε)

(<cmath>: Max = 13.9ε (Mean = 2.91ε))

Max = 0.764ε (Mean = 0.0348ε)

(GSL 2.1: Max = 8.71ε (Mean = 1.76ε) And other failures.)
(Rmath 3.2.3: Max = 7.47ε (Mean = 1.34ε))

Max = 4.55ε (Mean = 1.12ε)

Max = 9.34ε (Mean = 1.7ε)

Bessel Kv: Random Data

Max = 7.88ε (Mean = 1.48ε)

(<cmath>: Max = 13.6ε (Mean = 2.68ε) And other failures.)

Max = 0.507ε (Mean = 0.0313ε)

(GSL 2.1: Max = 9.71ε (Mean = 1.47ε) And other failures.)
(Rmath 3.2.3: Max = 7.37ε (Mean = 1.49ε))

Max = 7.88ε (Mean = 1.47ε)

Max = 8.33ε (Mean = 1.62ε)


Table 23.19. Error rates for cyl_bessel_k (integer orders)

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel K0: Mathworld Data (Integer Version)

Max = 0.833ε (Mean = 0.436ε)

(<cmath>: Max = 9.33ε (Mean = 3.25ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.2ε (Mean = 0.733ε))
(Rmath 3.2.3: Max = 0.833ε (Mean = 0.601ε))

Max = 0.833ε (Mean = 0.436ε)

Max = 0.833ε (Mean = 0.552ε)

Bessel K1: Mathworld Data (Integer Version)

Max = 0.786ε (Mean = 0.329ε)

(<cmath>: Max = 8.94ε (Mean = 3.19ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.626ε (Mean = 0.333ε))
(Rmath 3.2.3: Max = 0.894ε (Mean = 0.516ε))

Max = 0.786ε (Mean = 0.329ε)

Max = 0.786ε (Mean = 0.39ε)

Bessel Kn: Mathworld Data (Integer Version)

Max = 2.6ε (Mean = 1.21ε)

(<cmath>: Max = 12.9ε (Mean = 4.91ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 168ε (Mean = 59.5ε))
(Rmath 3.2.3: Max = 8.48ε (Mean = 2.98ε))

Max = 2.6ε (Mean = 1.21ε)

Max = 3.63ε (Mean = 1.46ε)


Table 23.20. Error rates for cyl_bessel_k_prime

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel K'0: Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 0.786ε (Mean = 0.329ε)

Max = 0.786ε (Mean = 0.329ε)

Max = 0.786ε (Mean = 0.39ε)

Bessel K'1: Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 0.736ε (Mean = 0.389ε)

Max = 0.736ε (Mean = 0.389ε)

Max = 0.761ε (Mean = 0.444ε)

Bessel K'n: Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 2.16ε (Mean = 1.08ε)

Max = 2.16ε (Mean = 1.08ε)

Max = 4.17ε (Mean = 1.75ε)

Bessel K'v: Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 3.94ε (Mean = 2.44ε)

Max = 3.94ε (Mean = 2.34ε)

Max = 3.94ε (Mean = 1.47ε)

Bessel K'v: Mathworld Data (large values)

Max = 0ε (Mean = 0ε)

Max = 59.2ε (Mean = 42.9ε)

Max = 58.7ε (Mean = 42.6ε)

Max = 18.6ε (Mean = 11.8ε)

Bessel K'n: Random Data

Max = 0ε (Mean = 0ε)

Max = 4.45ε (Mean = 1.19ε)

Max = 4.45ε (Mean = 1.19ε)

Max = 9.67ε (Mean = 1.73ε)

Bessel K'v: Random Data

Max = 0ε (Mean = 0ε)

Max = 7.95ε (Mean = 1.53ε)

Max = 7.95ε (Mean = 1.52ε)

Max = 8.32ε (Mean = 1.65ε)


Table 23.21. Error rates for cyl_bessel_k_prime (integer orders)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel K'0: Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

Max = 0.786ε (Mean = 0.329ε)

Max = 0.786ε (Mean = 0.329ε)

Max = 0.786ε (Mean = 0.39ε)

Bessel K'1: Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

Max = 0.736ε (Mean = 0.389ε)

Max = 0.736ε (Mean = 0.389ε)

Max = 0.761ε (Mean = 0.444ε)

Bessel K'n: Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

Max = 2.16ε (Mean = 1.08ε)

Max = 2.16ε (Mean = 1.08ε)

Max = 4.17ε (Mean = 1.75ε)


Table 23.22. Error rates for cyl_neumann

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Y0: Mathworld Data

Max = 5.53ε (Mean = 2.4ε)

(<cmath>: Max = 2.05e+05ε (Mean = 6.87e+04ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 60.9ε (Mean = 20.4ε))
(Rmath 3.2.3: Max = 167ε (Mean = 56.5ε))

Max = 5.53ε (Mean = 2.4ε)

Max = 4.61ε (Mean = 2.29ε)

Y1: Mathworld Data

Max = 6.33ε (Mean = 2.25ε)

(<cmath>: Max = 9.71e+03ε (Mean = 4.08e+03ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 23.4ε (Mean = 8.1ε))
(Rmath 3.2.3: Max = 193ε (Mean = 64.4ε))

Max = 6.33ε (Mean = 2.29ε)

Max = 4.75ε (Mean = 1.72ε)

Yn: Mathworld Data

Max = 55.2ε (Mean = 17.8ε)

(<cmath>: Max = 2.2e+20ε (Mean = 6.97e+19ε) And other failures.)

Max = 0.993ε (Mean = 0.314ε)

(GSL 2.1: Max = 2.41e+05ε (Mean = 7.62e+04ε) And other failures.)
(Rmath 3.2.3: Max = 1.24e+04ε (Mean = 4e+03ε))

Max = 55.2ε (Mean = 17.8ε)

Max = 35ε (Mean = 11.9ε)

Yv: Mathworld Data

Max = 10.7ε (Mean = 4.93ε)

(<cmath>: Max = 3.49e+15ε (Mean = 1.05e+15ε) And other failures.)

Max = 10ε (Mean = 3.02ε)

(GSL 2.1: Max = 1.07e+05ε (Mean = 3.22e+04ε) And other failures.)
(Rmath 3.2.3: Max = 243ε (Mean = 73.9ε))

Max = 10.7ε (Mean = 5.1ε)

Max = 7.89ε (Mean = 3.27ε)

Yv: Mathworld Data (large values)

Max = 1.7ε (Mean = 1.33ε)

(<cmath>: Max = 43.2ε (Mean = 16.3ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 60.8ε (Mean = 23ε) And other failures.)
(Rmath 3.2.3: Max = 0.682ε (Mean = 0.335ε))

Max = 1.7ε (Mean = 1.33ε)

Max = 0.682ε (Mean = 0.423ε)

Y0 and Y1: Random Data

Max = 10.8ε (Mean = 3.04ε)

(<cmath>: Max = 2.59e+03ε (Mean = 500ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 34.4ε (Mean = 8.9ε))
(Rmath 3.2.3: Max = 83ε (Mean = 14.2ε))

Max = 10.8ε (Mean = 3.04ε)

Max = 4.17ε (Mean = 1.24ε)

Yn: Random Data

Max = 338ε (Mean = 27.5ε)

(<cmath>: Max = 4.01e+03ε (Mean = 348ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 500ε (Mean = 47.8ε))
(Rmath 3.2.3: Max = 691ε (Mean = 67.9ε))

Max = 338ε (Mean = 27.5ε)

Max = 117ε (Mean = 10.2ε)

Yv: Random Data

Max = 2.08e+03ε (Mean = 149ε)

(<cmath>: Max = +INFε (Mean = +INFε) And other failures.)

Max = 1.53ε (Mean = 0.102ε)

(GSL 2.1: Max = 1.41e+06ε (Mean = 7.67e+04ε))
(Rmath 3.2.3: Max = 1.79e+05ε (Mean = 9.64e+03ε))

Max = 2.08e+03ε (Mean = 149ε)

Max = 1.23e+03ε (Mean = 69.9ε)


Table 23.23. Error rates for cyl_neumann (integer orders)

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Y0: Mathworld Data (Integer Version)

Max = 5.53ε (Mean = 2.4ε)

(<cmath>: Max = 2.05e+05ε (Mean = 6.87e+04ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 6.46ε (Mean = 2.38ε))
(Rmath 3.2.3: Max = 167ε (Mean = 56.5ε))

Max = 5.53ε (Mean = 2.4ε)

Max = 4.61ε (Mean = 2.29ε)

(<math.h>: Max = 5.37e+03ε (Mean = 1.81e+03ε))

Y1: Mathworld Data (Integer Version)

Max = 6.33ε (Mean = 2.25ε)

(<cmath>: Max = 9.71e+03ε (Mean = 4.08e+03ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.51ε (Mean = 0.839ε))
(Rmath 3.2.3: Max = 193ε (Mean = 64.4ε))

Max = 6.33ε (Mean = 2.29ε)

Max = 4.75ε (Mean = 1.72ε)

(<math.h>: Max = 1.86e+04ε (Mean = 6.2e+03ε))

Yn: Mathworld Data (Integer Version)

Max = 55.2ε (Mean = 17.8ε)

(<cmath>: Max = 2.2e+20ε (Mean = 6.97e+19ε) And other failures.)

Max = 0.993ε (Mean = 0.314ε)

(GSL 2.1: Max = 2.41e+05ε (Mean = 7.62e+04ε))
(Rmath 3.2.3: Max = 1.24e+04ε (Mean = 4e+03ε))

Max = 55.2ε (Mean = 17.8ε)

Max = 35ε (Mean = 11.9ε)

(<math.h>: Max = 2.49e+05ε (Mean = 8.14e+04ε))


Table 23.24. Error rates for cyl_neumann_prime

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Y'0: Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 6.33ε (Mean = 3.12ε)

Max = 6.33ε (Mean = 3.14ε)

Max = 4.75ε (Mean = 1.75ε)

Y'1: Mathworld Data

Max = 0.58ε (Mean = 0.193ε)

Max = 37.1ε (Mean = 12.8ε)

Max = 34ε (Mean = 11.8ε)

Max = 3.08ε (Mean = 1.2ε)

Y'n: Mathworld Data

Max = 2.05ε (Mean = 0.677ε)

Max = 56ε (Mean = 18.2ε)

Max = 56ε (Mean = 21.3ε)

Max = 563ε (Mean = 178ε)

Y'v: Mathworld Data

Max = 21.5ε (Mean = 6.49ε)

Max = 42.5ε (Mean = 13.4ε)

Max = 42.5ε (Mean = 13.6ε)

Max = 23.7ε (Mean = 10.1ε)

Y'v: Mathworld Data (large values)

Max = 0ε (Mean = 0ε)

Max = 1.57ε (Mean = 1.24ε)

Max = 1.57ε (Mean = 1.24ε)

Max = 0.627ε (Mean = 0.237ε)

Y'0 and Y'1: Random Data

Max = 0ε (Mean = 0ε)

Max = 23.8ε (Mean = 3.69ε)

Max = 23.8ε (Mean = 3.69ε)

Max = 5.95ε (Mean = 1.36ε)

Y'n: Random Data

Max = 1.53ε (Mean = 0.0885ε)

Max = 2.35e+03ε (Mean = 136ε)

Max = 2.35e+03ε (Mean = 136ε)

Max = 621ε (Mean = 36ε)

Y'v: Random Data

Max = 56.8ε (Mean = 2.59ε)

Max = 1.16e+05ε (Mean = 5.28e+03ε)

Max = 1.16e+05ε (Mean = 5.28e+03ε)

Max = 3.23e+04ε (Mean = 1.13e+03ε)


Table 23.25. Error rates for cyl_neumann_prime (integer orders)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Y'0: Mathworld Data (Integer Version)

Max = 0ε (Mean = 0ε)

Max = 6.33ε (Mean = 3.12ε)

Max = 6.33ε (Mean = 3.14ε)

Max = 4.75ε (Mean = 1.75ε)

Y'1: Mathworld Data (Integer Version)

Max = 0.58ε (Mean = 0.193ε)

Max = 37.1ε (Mean = 12.8ε)

Max = 34ε (Mean = 11.8ε)

Max = 3.08ε (Mean = 1.2ε)

Y'n: Mathworld Data (Integer Version)

Max = 2.05ε (Mean = 0.677ε)

Max = 56ε (Mean = 18.2ε)

Max = 56ε (Mean = 21.3ε)

Max = 563ε (Mean = 178ε)


Table 23.26. Error rates for digamma

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Digamma Function: Large Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.84ε (Mean = 0.71ε))
(Rmath 3.2.3: Max = 1.18ε (Mean = 0.331ε))

Max = 1.39ε (Mean = 0.413ε)

Max = 1.39ε (Mean = 0.413ε)

Max = 0.98ε (Mean = 0.369ε)

Digamma Function: Near the Positive Root

Max = 0.891ε (Mean = 0.0995ε)

(GSL 2.1: Max = 135ε (Mean = 11.9ε))
(Rmath 3.2.3: Max = 2.02e+03ε (Mean = 256ε))

Max = 1.37ε (Mean = 0.477ε)

Max = 1.31ε (Mean = 0.471ε)

Max = 0.997ε (Mean = 0.527ε)

Digamma Function: Near Zero

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.953ε (Mean = 0.348ε))
(Rmath 3.2.3: Max = 1.17ε (Mean = 0.564ε))

Max = 0.984ε (Mean = 0.361ε)

Max = 0.984ε (Mean = 0.361ε)

Max = 0.953ε (Mean = 0.337ε)

Digamma Function: Negative Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 4.56e+04ε (Mean = 3.91e+03ε))
(Rmath 3.2.3: Max = 4.6e+04ε (Mean = 3.94e+03ε))

Max = 180ε (Mean = 13ε)

Max = 180ε (Mean = 13ε)

Max = 214ε (Mean = 16.1ε)

Digamma Function: Values near 0

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.866ε (Mean = 0.387ε))
(Rmath 3.2.3: Max = 3.58e+05ε (Mean = 1.6e+05ε))

Max = 1ε (Mean = 0.592ε)

Max = 1ε (Mean = 0.592ε)

Max = 0ε (Mean = 0ε)

Digamma Function: Integer arguments

Max = 0.992ε (Mean = 0.215ε)

(GSL 2.1: Max = 1.18ε (Mean = 0.607ε))
(Rmath 3.2.3: Max = 4.33ε (Mean = 0.982ε))

Max = 0.888ε (Mean = 0.403ε)

Max = 0.888ε (Mean = 0.403ε)

Max = 0.992ε (Mean = 0.452ε)

Digamma Function: Half integer arguments

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.09ε (Mean = 0.531ε))
(Rmath 3.2.3: Max = 46.2ε (Mean = 7.24ε))

Max = 0.906ε (Mean = 0.409ε)

Max = 0.906ε (Mean = 0.409ε)

Max = 0.78ε (Mean = 0.314ε)


Table 23.27. Error rates for ellint_1

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Elliptic Integral F: Mathworld Data

Max = 0.94ε (Mean = 0.509ε)

(<cmath>: Max = +INFε (Mean = +INFε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.919ε (Mean = 0.544ε))

Max = 0.94ε (Mean = 0.509ε)

Max = 0.919ε (Mean = 0.542ε)

Elliptic Integral F: Random Data

Max = 1.57ε (Mean = 0.56ε)

(<cmath>: Max = 2.56ε (Mean = 0.816ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.99ε (Mean = 0.797ε))

Max = 1.57ε (Mean = 0.561ε)

Max = 2.26ε (Mean = 0.631ε)


Table 23.28. Error rates for ellint_1 (complete)

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Elliptic Integral K: Mathworld Data

Max = 0.887ε (Mean = 0.296ε)

(<cmath>: Max = 1.19ε (Mean = 0.765ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.623ε (Mean = 0.393ε))

Max = 0.887ε (Mean = 0.296ε)

Max = 0.915ε (Mean = 0.547ε)

Elliptic Integral K: Random Data

Max = 1.27ε (Mean = 0.473ε)

(<cmath>: Max = 2.19ε (Mean = 0.694ε))

Max = 0.851ε (Mean = 0.0851ε)

(GSL 2.1: Max = 2.32ε (Mean = 0.688ε))

Max = 1.27ε (Mean = 0.473ε)

Max = 0.958ε (Mean = 0.408ε)


Table 23.29. Error rates for ellint_2

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Elliptic Integral E: Mathworld Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.63ε (Mean = 0.325ε))

Max = 0.656ε (Mean = 0.317ε)

(<cmath>: Max = +INFε (Mean = +INFε) And other failures.)

Max = 0.656ε (Mean = 0.317ε)

Max = 1.31ε (Mean = 0.727ε)

Elliptic Integral E: Random Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 4.4ε (Mean = 1.16ε))

Max = 2.05ε (Mean = 0.632ε)

(<cmath>: Max = 3.08e+04ε (Mean = 3.84e+03ε))

Max = 2.05ε (Mean = 0.632ε)

Max = 2.23ε (Mean = 0.639ε)

Elliptic Integral E: Small Angles

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.5ε (Mean = 0.118ε))

Max = 1ε (Mean = 0.283ε)

(<cmath>: Max = 2ε (Mean = 0.333ε))

Max = 1ε (Mean = 0.283ε)

Max = 1ε (Mean = 0.421ε)


Table 23.30. Error rates for ellint_2 (complete)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Elliptic Integral E: Mathworld Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 3.09ε (Mean = 1.04ε) And other failures.)

Max = 0.836ε (Mean = 0.469ε)

(<cmath>: Max = 170ε (Mean = 55.1ε))

Max = 0.836ε (Mean = 0.469ε)

Max = 1.3ε (Mean = 0.615ε)

Elliptic Integral E: Random Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 4.34ε (Mean = 1.18ε))

Max = 1.97ε (Mean = 0.629ε)

(<cmath>: Max = 2.49e+04ε (Mean = 3.39e+03ε))

Max = 1.97ε (Mean = 0.629ε)

Max = 1.71ε (Mean = 0.553ε)


Table 23.31. Error rates for ellint_3

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Elliptic Integral PI: Mathworld Data

Max = 475ε (Mean = 86.3ε)

(<cmath>: Max = +INFε (Mean = +INFε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.48e+05ε (Mean = 2.54e+04ε) And other failures.)

Max = 475ε (Mean = 86.3ε)

Max = 565ε (Mean = 102ε)

Elliptic Integral PI: Random Data

Max = 4.54ε (Mean = 0.895ε)

(<cmath>: Max = 3.37e+20ε (Mean = 3.47e+19ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 633ε (Mean = 50.1ε))

Max = 4.49ε (Mean = 0.885ε)

Max = 8.33ε (Mean = 0.971ε)

Elliptic Integral PI: Large Random Data

Max = 3.7ε (Mean = 0.893ε)

(<cmath>: Max = 2.52e+18ε (Mean = 4.83e+17ε) And other failures.)

Max = 0.557ε (Mean = 0.0389ε)

(GSL 2.1: Max = 40.1ε (Mean = 7.77ε))

Max = 3.7ε (Mean = 0.892ε)

Max = 2.86ε (Mean = 0.944ε)


Table 23.32. Error rates for ellint_3 (complete)

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Complete Elliptic Integral PI: Mathworld Data

Max = 1.4ε (Mean = 0.575ε)

(<cmath>: Max = 6.31e+20ε (Mean = 1.53e+20ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 6.33e+04ε (Mean = 1.54e+04ε) And other failures.)

Max = 1.4ε (Mean = 0.575ε)

Max = 0.971ε (Mean = 0.464ε)

Complete Elliptic Integral PI: Random Data

Max = 2.45ε (Mean = 0.696ε)

(<cmath>: Max = 8.78e+20ε (Mean = 1.02e+20ε) And other failures.)

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 24ε (Mean = 2.99ε))

Max = 2.4ε (Mean = 0.677ε)

Max = 2.46ε (Mean = 0.657ε)


Table 23.33. Error rates for ellint_d

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Elliptic Integral E: Mathworld Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.862ε (Mean = 0.568ε))

Max = 1.3ε (Mean = 0.813ε)

Max = 1.3ε (Mean = 0.813ε)

Max = 0.862ε (Mean = 0.457ε)

Elliptic Integral D: Random Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 3.01ε (Mean = 0.928ε))

Max = 2.51ε (Mean = 0.883ε)

Max = 2.51ε (Mean = 0.883ε)

Max = 2.87ε (Mean = 0.805ε)


Table 23.34. Error rates for ellint_d (complete)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Elliptic Integral E: Mathworld Data

Max = 0.637ε (Mean = 0.368ε)

Max = 1.27ε (Mean = 0.735ε)

Max = 1.27ε (Mean = 0.735ε)

Max = 0.637ε (Mean = 0.368ε)

Elliptic Integral D: Random Data

Max = 0ε (Mean = 0ε)

Max = 1.27ε (Mean = 0.334ε)

Max = 1.27ε (Mean = 0.334ε)

Max = 1.27ε (Mean = 0.355ε)


Table 23.35. Error rates for ellint_rc

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Microsoft Visual C++ version 14.1
Win32
double

RC: Random data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.4ε (Mean = 0.624ε))

Max = 0.995ε (Mean = 0.433ε)

Max = 0.962ε (Mean = 0.407ε)


Table 23.36. Error rates for ellint_rd

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Microsoft Visual C++ version 14.1
Win32
double

RD: Random data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.59ε (Mean = 0.878ε))

Max = 2.73ε (Mean = 0.831ε)

Max = 2.16ε (Mean = 0.803ε)

RD: y = z

Max = 0.896ε (Mean = 0.022ε)

(GSL 2.1: Max = 2.88ε (Mean = 0.839ε))

Max = 2.65ε (Mean = 0.82ε)

Max = 16.5ε (Mean = 0.843ε)

RD: x = y

Max = 0.824ε (Mean = 0.0272ε)

(GSL 2.1: Max = 3.74ε (Mean = 0.84ε))

Max = 2.85ε (Mean = 0.865ε)

Max = 3.51ε (Mean = 0.816ε)

RD: x = 0, y = z

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2ε (Mean = 0.656ε))

Max = 1.19ε (Mean = 0.522ε)

Max = 1.16ε (Mean = 0.497ε)

RD: x = y = z

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.03ε (Mean = 0.418ε))

Max = 0.998ε (Mean = 0.387ε)

Max = 1.03ε (Mean = 0.418ε)

RD: x = 0

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.85ε (Mean = 0.781ε))

Max = 2.79ε (Mean = 0.883ε)

Max = 2.64ε (Mean = 0.894ε)


Table 23.37. Error rates for ellint_rf

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Microsoft Visual C++ version 14.1
Win32
double

RF: Random data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.73ε (Mean = 0.804ε))

Max = 2.54ε (Mean = 0.674ε)

Max = 2.02ε (Mean = 0.677ε)

RF: x = y = z

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.999ε (Mean = 0.34ε))

Max = 0.991ε (Mean = 0.345ε)

Max = 0.999ε (Mean = 0.34ε)

RF: x = y or y = z or x = z

Max = 0.536ε (Mean = 0.00658ε)

(GSL 2.1: Max = 2.89ε (Mean = 0.749ε))

Max = 1.95ε (Mean = 0.418ε)

Max = 1.21ε (Mean = 0.394ε)

RF: x = 0, y = z

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.29ε (Mean = 0.527ε))

Max = 0.894ε (Mean = 0.338ε)

Max = 0.999ε (Mean = 0.407ε)

RF: z = 0

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.54ε (Mean = 0.781ε))

Max = 1.7ε (Mean = 0.539ε)

Max = 1.89ε (Mean = 0.587ε)


Table 23.38. Error rates for ellint_rg

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Microsoft Visual C++ version 14.1
Win32
double

RG: Random Data

Max = 0.983ε (Mean = 0.0172ε)

(GSL 2.1: Max = 0.983ε (Mean = 0.0172ε))

Max = 3.95ε (Mean = 0.951ε)

Max = 3.65ε (Mean = 0.929ε)

RG: two values 0

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0ε (Mean = 0ε))

Max = 0ε (Mean = 0ε)

Max = 0ε (Mean = 0ε)

RG: All values the same or zero

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0ε (Mean = 0ε))

Max = 0.992ε (Mean = 0.288ε)

Max = 1.06ε (Mean = 0.348ε)

RG: two values the same

Max = 0.594ε (Mean = 0.0103ε)

(GSL 2.1: Max = 0.594ε (Mean = 0.0103ε))

Max = 1.51ε (Mean = 0.404ε)

Max = 1.96ε (Mean = 0.374ε)

RG: one value zero

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0ε (Mean = 0ε))

Max = 2.14ε (Mean = 0.722ε)

Max = 1.96ε (Mean = 0.674ε)


Table 23.39. Error rates for ellint_rj

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Microsoft Visual C++ version 14.1
Win32
double

RJ: Random data

Max = 0.52ε (Mean = 0.0184ε)

(GSL 2.1: Max = 3.57ε (Mean = 0.704ε) And other failures.)

Max = 186ε (Mean = 6.67ε)

Max = 215ε (Mean = 7.66ε)

RJ: 4 Equal Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.03ε (Mean = 0.418ε))

Max = 0.998ε (Mean = 0.387ε)

Max = 1.03ε (Mean = 0.418ε)

RJ: 3 Equal Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 3.96ε (Mean = 1.06ε))

Max = 20.8ε (Mean = 0.986ε)

Max = 39.9ε (Mean = 1.17ε)

RJ: 2 Equal Values

Max = 0.6ε (Mean = 0.0228ε)

(GSL 2.1: Max = 2.57ε (Mean = 0.754ε))

Max = 220ε (Mean = 6.64ε)

Max = 214ε (Mean = 5.28ε)

RJ: Equal z and p

Max = 0.742ε (Mean = 0.0166ε)

(GSL 2.1: Max = 2.62ε (Mean = 0.699ε))

Max = 17.2ε (Mean = 1.16ε)

Max = 16.1ε (Mean = 1.14ε)


Table 23.40. Error rates for erf

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Erf Function: Small Values

Max = 0.925ε (Mean = 0.193ε)

(<cmath>: Max = 0.944ε (Mean = 0.191ε))
(<math.h>: Max = 0.944ε (Mean = 0.191ε))

Max = 0.841ε (Mean = 0.0687ε)

(GSL 2.1: Max = 2.06ε (Mean = 0.319ε))

Max = 0.925ε (Mean = 0.193ε)

(<math.h>: Max = 0.944ε (Mean = 0.194ε))

Max = 0.996ε (Mean = 0.182ε)

(<math.h>: Max = 1.57ε (Mean = 0.317ε))

Erf Function: Medium Values

Max = 1.5ε (Mean = 0.193ε)

(<cmath>: Max = 0.921ε (Mean = 0.0723ε))
(<math.h>: Max = 0.921ε (Mean = 0.0723ε))

Max = 1ε (Mean = 0.119ε)

(GSL 2.1: Max = 2.31ε (Mean = 0.368ε))

Max = 1.5ε (Mean = 0.197ε)

(<math.h>: Max = 0.921ε (Mean = 0.071ε))

Max = 1ε (Mean = 0.171ε)

(<math.h>: Max = 1.19ε (Mean = 0.244ε))

Erf Function: Large Values

Max = 0ε (Mean = 0ε)

(<cmath>: Max = 0ε (Mean = 0ε))
(<math.h>: Max = 0ε (Mean = 0ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0ε (Mean = 0ε))

Max = 0ε (Mean = 0ε)

(<math.h>: Max = 0ε (Mean = 0ε))

Max = 0ε (Mean = 0ε)

(<math.h>: Max = 0ε (Mean = 0ε))


Table 23.41. Error rates for erf_inv

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Inverse Erf Function

Max = 0ε (Mean = 0ε)

Max = 0.996ε (Mean = 0.389ε)

Max = 1.08ε (Mean = 0.395ε)

Max = 1.09ε (Mean = 0.502ε)


Table 23.42. Error rates for erfc

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Erf Function: Small Values

Max = 0ε (Mean = 0ε)

(<cmath>: Max = 0ε (Mean = 0ε))
(<math.h>: Max = 0ε (Mean = 0ε))

Max = 0.658ε (Mean = 0.0537ε)

(GSL 2.1: Max = 1.01ε (Mean = 0.485ε))

Max = 0ε (Mean = 0ε)

(<math.h>: Max = 0ε (Mean = 0ε))

Max = 0ε (Mean = 0ε)

(<math.h>: Max = 0ε (Mean = 0ε))

Erf Function: Medium Values

Max = 1.76ε (Mean = 0.365ε)

(<cmath>: Max = 1.35ε (Mean = 0.307ε))
(<math.h>: Max = 1.35ε (Mean = 0.307ε))

Max = 0.983ε (Mean = 0.213ε)

(GSL 2.1: Max = 2.64ε (Mean = 0.662ε))

Max = 1.76ε (Mean = 0.38ε)

(<math.h>: Max = 2.81ε (Mean = 0.739ε))

Max = 1.65ε (Mean = 0.373ε)

(<math.h>: Max = 2.36ε (Mean = 0.539ε))

Erf Function: Large Values

Max = 1.57ε (Mean = 0.542ε)

(<cmath>: Max = 1.26ε (Mean = 0.441ε))
(<math.h>: Max = 1.26ε (Mean = 0.441ε))

Max = 0.868ε (Mean = 0.147ε)

(GSL 2.1: Max = 3.9ε (Mean = 0.472ε))

Max = 1.57ε (Mean = 0.564ε)

(<math.h>: Max = 4.91ε (Mean = 1.54ε))

Max = 1.14ε (Mean = 0.248ε)

(<math.h>: Max = 1.84ε (Mean = 0.331ε))


Table 23.43. Error rates for erfc_inv

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Inverse Erfc Function

Max = 0ε (Mean = 0ε)

Max = 0.996ε (Mean = 0.397ε)

Max = 1.08ε (Mean = 0.403ε)

Max = 1ε (Mean = 0.491ε)

Inverse Erfc Function: extreme values

Max = 1.62ε (Mean = 0.383ε)

Max = 1.62ε (Mean = 0.383ε)


Table 23.44. Error rates for expint (Ei)

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Exponential Integral Ei

Max = 5.05ε (Mean = 0.821ε)

(<cmath>: Max = 14.1ε (Mean = 2.43ε) And other failures.)

Max = 0.994ε (Mean = 0.142ε)

(GSL 2.1: Max = 8.96ε (Mean = 0.703ε))

Max = 5.05ε (Mean = 0.835ε)

Max = 1.43ε (Mean = 0.54ε)

Exponential Integral Ei: double exponent range

Max = 1.72ε (Mean = 0.593ε)

(<cmath>: Max = 3.11ε (Mean = 1.13ε))

Max = 0.998ε (Mean = 0.156ε)

(GSL 2.1: Max = 1.5ε (Mean = 0.612ε))

Max = 1.72ε (Mean = 0.607ε)

Max = 1.7ε (Mean = 0.66ε)

Exponential Integral Ei: long exponent range

Max = 1.98ε (Mean = 0.595ε)

(<cmath>: Max = 1.93ε (Mean = 0.855ε))

Max = 1.98ε (Mean = 0.575ε)


Table 23.45. Error rates for expint (En)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Exponential Integral En

Max = 0.589ε (Mean = 0.0331ε)

(GSL 2.1: Max = 58.5ε (Mean = 17.1ε))

Max = 9.97ε (Mean = 2.13ε)

Max = 9.97ε (Mean = 2.13ε)

Max = 7.16ε (Mean = 1.85ε)

Exponential Integral En: small z values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 115ε (Mean = 23.6ε))

Max = 1.99ε (Mean = 0.559ε)

Max = 1.99ε (Mean = 0.559ε)

Max = 2.62ε (Mean = 0.531ε)

Exponential Integral E1

Max = 0.556ε (Mean = 0.0625ε)

(GSL 2.1: Max = 0.988ε (Mean = 0.469ε))

Max = 0.965ε (Mean = 0.414ε)

Max = 0.965ε (Mean = 0.408ε)

Max = 0.988ε (Mean = 0.486ε)


Table 23.46. Error rates for expm1

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Random test data

Max = 0.992ε (Mean = 0.402ε)

(<cmath>: Max = 0.992ε (Mean = 0.402ε))
(<math.h>: Max = 0.992ε (Mean = 0.402ε))

Max = 0.793ε (Mean = 0.126ε)

(Rmath 3.2.3: Max = 0.793ε (Mean = 0.126ε))

Max = 1.31ε (Mean = 0.428ε)

(<math.h>: Max = 0.996ε (Mean = 0.426ε))

Max = 1.31ε (Mean = 0.496ε)

(<math.h>: Max = 1.31ε (Mean = 0.496ε))


Table 23.47. Error rates for gamma_p

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

tgamma(a, z) medium values

Max = 0.955ε (Mean = 0.05ε)

(GSL 2.1: Max = 342ε (Mean = 45.8ε))
(Rmath 3.2.3: Max = 389ε (Mean = 44ε))

Max = 41.6ε (Mean = 8.09ε)

Max = 239ε (Mean = 30.2ε)

Max = 35.1ε (Mean = 6.98ε)

tgamma(a, z) small values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 4.82ε (Mean = 0.758ε))
(Rmath 3.2.3: Max = 1.01ε (Mean = 0.306ε))

Max = 2ε (Mean = 0.464ε)

Max = 2ε (Mean = 0.461ε)

Max = 1.54ε (Mean = 0.439ε)

tgamma(a, z) large values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.02e+03ε (Mean = 105ε))
(Rmath 3.2.3: Max = 1.11e+03ε (Mean = 67.5ε))

Max = 3.08e+04ε (Mean = 1.86e+03ε)

Max = 3.02e+04ε (Mean = 1.91e+03ε)

Max = 243ε (Mean = 20.2ε)

tgamma(a, z) integer and half integer values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 128ε (Mean = 22.6ε))
(Rmath 3.2.3: Max = 66.2ε (Mean = 12.2ε))

Max = 11.8ε (Mean = 2.66ε)

Max = 71.6ε (Mean = 9.47ε)

Max = 13ε (Mean = 2.97ε)


Table 23.48. Error rates for gamma_p_inv

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

incomplete gamma inverse(a, z) medium values

Max = 0.993ε (Mean = 0.15ε)

(Rmath 3.2.3: Max = 4.88ε (Mean = 0.868ε))

Max = 1.8ε (Mean = 0.406ε)

Max = 1.89ε (Mean = 0.466ε)

Max = 1.71ε (Mean = 0.34ε)

incomplete gamma inverse(a, z) large values

Max = 0ε (Mean = 0ε)

(Rmath 3.2.3: Max = 0.816ε (Mean = 0.0874ε))

Max = 0.509ε (Mean = 0.0447ε)

Max = 0.509ε (Mean = 0.0447ε)

Max = 0.924ε (Mean = 0.108ε)

incomplete gamma inverse(a, z) small values

Max = 441ε (Mean = 53.9ε)

(Rmath 3.2.3: Max = 547ε (Mean = 61.6ε))

Max = 9.17e+03ε (Mean = 1.45e+03ε)

Max = 1.09e+04ε (Mean = 1.3e+03ε)

Max = 1.1e+03ε (Mean = 131ε)


Table 23.49. Error rates for gamma_p_inva

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Incomplete gamma inverses.

Max = 0ε (Mean = 0ε)

Max = 7.87ε (Mean = 1.15ε)

Max = 4.08ε (Mean = 1.12ε)

Max = 4.92ε (Mean = 1.03ε)


Table 23.50. Error rates for gamma_q

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

tgamma(a, z) medium values

Max = 0.927ε (Mean = 0.035ε)

(GSL 2.1: Max = 201ε (Mean = 13.5ε))
(Rmath 3.2.3: Max = 131ε (Mean = 12.7ε))

Max = 32.3ε (Mean = 6.61ε)

Max = 199ε (Mean = 26.6ε)

Max = 23.7ε (Mean = 4ε)

tgamma(a, z) small values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.38e+10ε (Mean = 1.05e+09ε))
(Rmath 3.2.3: Max = 65.6ε (Mean = 11ε))

Max = 2.45ε (Mean = 0.885ε)

Max = 2.45ε (Mean = 0.819ε)

Max = 2.26ε (Mean = 0.74ε)

tgamma(a, z) large values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.71e+04ε (Mean = 2.16e+03ε))
(Rmath 3.2.3: Max = 1.02e+03ε (Mean = 62.7ε))

Max = 6.82e+03ε (Mean = 414ε)

Max = 1.15e+04ε (Mean = 733ε)

Max = 469ε (Mean = 31.5ε)

tgamma(a, z) integer and half integer values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 118ε (Mean = 12.5ε))
(Rmath 3.2.3: Max = 138ε (Mean = 16.9ε))

Max = 11.1ε (Mean = 2.07ε)

Max = 54.7ε (Mean = 6.16ε)

Max = 8.72ε (Mean = 1.48ε)


Table 23.51. Error rates for gamma_q_inv

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

incomplete gamma inverse(a, z) medium values

Max = 0.912ε (Mean = 0.154ε)

(Rmath 3.2.3: Max = 4.66ε (Mean = 0.792ε))

Max = 6.2ε (Mean = 0.627ε)

Max = 6.2ε (Mean = 0.683ε)

Max = 2.88ε (Mean = 0.469ε)

incomplete gamma inverse(a, z) large values

Max = 0.894ε (Mean = 0.0915ε)

(Rmath 3.2.3: Max = 0.894ε (Mean = 0.106ε))

Max = 0ε (Mean = 0ε)

Max = 0ε (Mean = 0ε)

Max = 0.814ε (Mean = 0.0856ε)

incomplete gamma inverse(a, z) small values

Max = 292ε (Mean = 36.4ε)

(Rmath 3.2.3: Max = 415ε (Mean = 48.7ε))

Max = 8.28e+03ε (Mean = 1.09e+03ε)

Max = 8.98e+03ε (Mean = 877ε)

Max = 451ε (Mean = 64.7ε)


Table 23.52. Error rates for gamma_q_inva

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Incomplete gamma inverses.

Max = 0ε (Mean = 0ε)

Max = 8.42ε (Mean = 1.3ε)

Max = 7.86ε (Mean = 1.24ε)

Max = 5.05ε (Mean = 1.08ε)


Table 23.53. Error rates for hermite

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Hermite Polynomials

Max = 0ε (Mean = 0ε)

Max = 6.24ε (Mean = 2.07ε)

Max = 6.24ε (Mean = 2.07ε)

Max = 4.46ε (Mean = 1.41ε)


Table 23.54. Error rates for heuman_lambda

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Elliptic Integral Jacobi Zeta: Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 1.89ε (Mean = 0.887ε)

Max = 1.89ε (Mean = 0.887ε)

Max = 1.08ε (Mean = 0.734ε)

Elliptic Integral Heuman Lambda: Random Data

Max = 0ε (Mean = 0ε)

Max = 3.82ε (Mean = 0.609ε)

Max = 3.82ε (Mean = 0.608ε)

Max = 2.12ε (Mean = 0.588ε)


Table 23.55. Error rates for ibeta

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Incomplete Beta Function: Small Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 682ε (Mean = 32.6ε))
(Rmath 3.2.3: Max = 22.9ε (Mean = 3.35ε))

Max = 8.97ε (Mean = 2.09ε)

Max = 21.3ε (Mean = 2.75ε)

Max = 8.4ε (Mean = 1.93ε)

Incomplete Beta Function: Medium Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 690ε (Mean = 151ε))
(Rmath 3.2.3: Max = 232ε (Mean = 27.9ε))

Max = 50ε (Mean = 12.1ε)

Max = 124ε (Mean = 18.4ε)

Max = 106ε (Mean = 16.3ε)

Incomplete Beta Function: Large and Diverse Values

Max = 1.26ε (Mean = 0.063ε)

(GSL 2.1: Max = 3.9e+05ε (Mean = 1.82e+04ε) And other failures.)
(Rmath 3.2.3: Max = 574ε (Mean = 49.4ε))

Max = 1.96e+04ε (Mean = 997ε)

Max = 4.98e+04ε (Mean = 2.07e+03ε)

Max = 1.32e+03ε (Mean = 68.5ε)

Incomplete Beta Function: Small Integer Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 254ε (Mean = 50.9ε))
(Rmath 3.2.3: Max = 62.2ε (Mean = 8.95ε))

Max = 4.45ε (Mean = 0.814ε)

Max = 44.5ε (Mean = 10.1ε)

Max = 3.85ε (Mean = 0.791ε)


Table 23.56. Error rates for ibeta_inv

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Inverse incomplete beta

Max = 11ε (Mean = 0.345ε)

(Rmath 3.2.3: Max = 1.14e+121ε (Mean = 3.28e+119ε) And other failures.)

Max = 3.8e+04ε (Mean = 2.66e+03ε)

Max = 4.07e+04ε (Mean = 2.86e+03ε)

Max = 8.59e+03ε (Mean = 277ε)


Table 23.57. Error rates for ibeta_inva

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Inverse incomplete beta

Max = 0.602ε (Mean = 0.0239ε)

Max = 377ε (Mean = 24.4ε)

Max = 438ε (Mean = 31.3ε)

Max = 242ε (Mean = 22.9ε)


Table 23.58. Error rates for ibeta_invb

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Inverse incomplete beta

Max = 0.765ε (Mean = 0.0422ε)

Max = 407ε (Mean = 27.2ε)

Max = 407ε (Mean = 24.4ε)

Max = 409ε (Mean = 19.3ε)


Table 23.59. Error rates for ibetac

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Incomplete Beta Function: Small Values

Max = 0ε (Mean = 0ε)

(Rmath 3.2.3: Max = 22.4ε (Mean = 3.67ε))

Max = 10.6ε (Mean = 2.22ε)

Max = 13.8ε (Mean = 2.68ε)

Max = 6.94ε (Mean = 1.71ε)

Incomplete Beta Function: Medium Values

Max = 0ε (Mean = 0ε)

(Rmath 3.2.3: Max = 204ε (Mean = 25.8ε))

Max = 73.9ε (Mean = 11.2ε)

Max = 132ε (Mean = 19.8ε)

Max = 56.7ε (Mean = 14.3ε)

Incomplete Beta Function: Large and Diverse Values

Max = 0.981ε (Mean = 0.0573ε)

(Rmath 3.2.3: Max = 889ε (Mean = 68.4ε))

Max = 3.45e+04ε (Mean = 1.32e+03ε)

Max = 6.31e+04ε (Mean = 2.04e+03ε)

Max = 1.88e+03ε (Mean = 82.7ε)

Incomplete Beta Function: Small Integer Values

Max = 0ε (Mean = 0ε)

(Rmath 3.2.3: Max = 84.6ε (Mean = 18ε))

Max = 5.34ε (Mean = 1.11ε)

Max = 107ε (Mean = 17.1ε)

Max = 6.37ε (Mean = 1.03ε)


Table 23.60. Error rates for ibetac_inv

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Inverse incomplete beta

Max = 0.977ε (Mean = 0.0976ε)

(Rmath 3.2.3: Max = 3.01e+132ε (Mean = 8.65e+130ε) And other failures.)

Max = 4.88e+04ε (Mean = 3.16e+03ε)

Max = 5.05e+04ε (Mean = 3.33e+03ε)

Max = 2.93e+03ε (Mean = 198ε)


Table 23.61. Error rates for ibetac_inva

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Inverse incomplete beta

Max = 0.683ε (Mean = 0.0314ε)

Max = 382ε (Mean = 22.2ε)

Max = 315ε (Mean = 23.7ε)

Max = 408ε (Mean = 26.7ε)


Table 23.62. Error rates for ibetac_invb

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Inverse incomplete beta

Max = 0.724ε (Mean = 0.0303ε)

Max = 317ε (Mean = 19.8ε)

Max = 369ε (Mean = 22.6ε)

Max = 271ε (Mean = 16.4ε)


Table 23.63. Error rates for jacobi_cn

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Jacobi Elliptic: Mathworld Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 17.3ε (Mean = 4.29ε) And other failures.)

Max = 71.6ε (Mean = 19.3ε)

Max = 71.6ε (Mean = 19.4ε)

Max = 45.8ε (Mean = 11.4ε)

Jacobi Elliptic: Random Data

Max = 0.816ε (Mean = 0.0563ε)

(GSL 2.1: Max = 2.43ε (Mean = 0.803ε))

Max = 1.68ε (Mean = 0.443ε)

Max = 1.68ε (Mean = 0.454ε)

Max = 1.83ε (Mean = 0.455ε)

Jacobi Elliptic: Random Small Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 55.2ε (Mean = 1.64ε) And other failures.)

Max = 10.4ε (Mean = 0.594ε)

Max = 10.4ε (Mean = 0.602ε)

Max = 26.2ε (Mean = 1.17ε)

Jacobi Elliptic: Modulus near 1

Max = 0.919ε (Mean = 0.127ε)

(GSL 2.1: Max = 0ε (Mean = 0ε) And other failures.)

Max = 675ε (Mean = 87.1ε)

Max = 675ε (Mean = 86.8ε)

Max = 513ε (Mean = 126ε)

Jacobi Elliptic: Large Phi

Max = 14.2ε (Mean = 0.927ε)

(GSL 2.1: Max = 5.92e+03ε (Mean = 477ε))

Max = 2.97e+04ε (Mean = 1.9e+03ε)

Max = 2.97e+04ε (Mean = 1.9e+03ε)

Max = 3.27e+04ε (Mean = 1.93e+03ε)


Table 23.64. Error rates for jacobi_dn

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Jacobi Elliptic: Mathworld Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.82ε (Mean = 1.18ε) And other failures.)

Max = 49ε (Mean = 14ε)

Max = 49ε (Mean = 14ε)

Max = 34.3ε (Mean = 8.71ε)

Jacobi Elliptic: Random Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 3ε (Mean = 0.61ε))

Max = 1.53ε (Mean = 0.473ε)

Max = 1.53ε (Mean = 0.481ε)

Max = 1.52ε (Mean = 0.466ε)

Jacobi Elliptic: Random Small Values

Max = 0.5ε (Mean = 0.0122ε)

(GSL 2.1: Max = 1.5ε (Mean = 0.391ε) And other failures.)

Max = 22.4ε (Mean = 0.777ε)

Max = 22.4ε (Mean = 0.763ε)

Max = 16.1ε (Mean = 0.685ε)

Jacobi Elliptic: Modulus near 1

Max = 2.28ε (Mean = 0.194ε)

(GSL 2.1: Max = 0ε (Mean = 0ε) And other failures.)

Max = 3.75e+03ε (Mean = 293ε)

Max = 3.75e+03ε (Mean = 293ε)

Max = 6.24e+03ε (Mean = 482ε)

Jacobi Elliptic: Large Phi

Max = 14.1ε (Mean = 0.897ε)

(GSL 2.1: Max = 121ε (Mean = 22ε))

Max = 2.82e+04ε (Mean = 1.79e+03ε)

Max = 2.82e+04ε (Mean = 1.79e+03ε)

Max = 1.67e+04ε (Mean = 1e+03ε)


Table 23.65. Error rates for jacobi_sn

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Jacobi Elliptic: Mathworld Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 588ε (Mean = 146ε) And other failures.)

Max = 341ε (Mean = 80.7ε)

Max = 341ε (Mean = 80.7ε)

Max = 481ε (Mean = 113ε)

Jacobi Elliptic: Random Data

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 4.02ε (Mean = 1.07ε))

Max = 2.01ε (Mean = 0.584ε)

Max = 2.01ε (Mean = 0.593ε)

Max = 1.92ε (Mean = 0.567ε)

Jacobi Elliptic: Random Small Values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 11.7ε (Mean = 1.65ε) And other failures.)

Max = 1.99ε (Mean = 0.347ε)

Max = 1.99ε (Mean = 0.347ε)

Max = 2.11ε (Mean = 0.385ε)

Jacobi Elliptic: Modulus near 1

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0ε (Mean = 0ε) And other failures.)

Max = 109ε (Mean = 7.35ε)

Max = 109ε (Mean = 7.38ε)

Max = 23.2ε (Mean = 1.85ε)

Jacobi Elliptic: Large Phi

Max = 12ε (Mean = 0.771ε)

(GSL 2.1: Max = 4.54e+04ε (Mean = 2.63e+03ε))

Max = 2.45e+04ε (Mean = 1.51e+03ε)

Max = 2.45e+04ε (Mean = 1.51e+03ε)

Max = 4.36e+04ε (Mean = 2.54e+03ε)


Table 23.66. Error rates for jacobi_zeta

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Elliptic Integral Jacobi Zeta: Mathworld Data

Max = 0ε (Mean = 0ε)

Max = 1.66ε (Mean = 0.48ε)

Max = 1.66ε (Mean = 0.48ε)

Max = 1.52ε (Mean = 0.357ε)

Elliptic Integral Jacobi Zeta: Random Data

Max = 0ε (Mean = 0ε)

Max = 2.99ε (Mean = 0.824ε)

Max = 3.96ε (Mean = 1.06ε)

Max = 3.89ε (Mean = 0.824ε)

Elliptic Integral Jacobi Zeta: Large Phi Values

Max = 0ε (Mean = 0ε)

Max = 2.92ε (Mean = 0.951ε)

Max = 3.05ε (Mean = 1.13ε)

Max = 2.52ε (Mean = 0.977ε)


Table 23.67. Error rates for laguerre(n, m, x)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Associated Laguerre Polynomials

Max = 0.84ε (Mean = 0.0358ε)

(GSL 2.1: Max = 434ε (Mean = 10.7ε))

Max = 167ε (Mean = 6.38ε)

(<cmath>: Max = 206ε (Mean = 6.86ε))

Max = 167ε (Mean = 6.38ε)

Max = 434ε (Mean = 11.1ε)


Table 23.68. Error rates for laguerre(n, x)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Laguerre Polynomials

Max = 6.82ε (Mean = 0.408ε)

(GSL 2.1: Max = 3.1e+03ε (Mean = 185ε))

Max = 1.39e+04ε (Mean = 828ε)

(<cmath>: Max = 4.2e+03ε (Mean = 251ε))

Max = 1.39e+04ε (Mean = 828ε)

Max = 3.1e+03ε (Mean = 185ε)


Table 23.69. Error rates for legendre_p

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Legendre Polynomials: Small Values

Max = 0.732ε (Mean = 0.0619ε)

(GSL 2.1: Max = 211ε (Mean = 20.4ε))

Max = 69.2ε (Mean = 9.58ε)

(<cmath>: Max = 124ε (Mean = 13.2ε))

Max = 69.2ε (Mean = 9.58ε)

Max = 211ε (Mean = 20.4ε)

Legendre Polynomials: Large Values

Max = 0.632ε (Mean = 0.0693ε)

(GSL 2.1: Max = 300ε (Mean = 33.2ε))

Max = 699ε (Mean = 59.6ε)

(<cmath>: Max = 343ε (Mean = 32.1ε))

Max = 699ε (Mean = 59.6ε)

Max = 300ε (Mean = 33.2ε)


Table 23.70. Error rates for legendre_p (associated)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Associated Legendre Polynomials: Small Values

Max = 0.999ε (Mean = 0.05ε)

(GSL 2.1: Max = 121ε (Mean = 6.75ε) And other failures.)

Max = 175ε (Mean = 9.88ε)

(<cmath>: Max = 175ε (Mean = 9.36ε) And other failures.)

Max = 77.7ε (Mean = 5.59ε)

Max = 121ε (Mean = 7.14ε)


Table 23.71. Error rates for legendre_q

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Legendre Polynomials: Small Values

Max = 0.612ε (Mean = 0.0517ε)

(GSL 2.1: Max = 46.4ε (Mean = 7.46ε))

Max = 50.9ε (Mean = 9ε)

Max = 50.9ε (Mean = 8.98ε)

Max = 46.4ε (Mean = 7.32ε)

Legendre Polynomials: Large Values

Max = 2.49ε (Mean = 0.202ε)

(GSL 2.1: Max = 4.6e+03ε (Mean = 366ε))

Max = 5.98e+03ε (Mean = 478ε)

Max = 5.98e+03ε (Mean = 478ε)

Max = 4.6e+03ε (Mean = 366ε)


Table 23.72. Error rates for lgamma

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

factorials

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 33.6ε (Mean = 2.78ε))
(Rmath 3.2.3: Max = 1.55ε (Mean = 0.592ε))

Max = 0.991ε (Mean = 0.308ε)

(<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))

Max = 0.914ε (Mean = 0.175ε)

(<math.h>: Max = 0.958ε (Mean = 0.38ε))

near 0

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 5.21ε (Mean = 1.57ε))
(Rmath 3.2.3: Max = 0ε (Mean = 0ε))

Max = 1.42ε (Mean = 0.566ε)

(<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))

Max = 0.964ε (Mean = 0.462ε)

(<math.h>: Max = 0.962ε (Mean = 0.372ε))

near 1

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 442ε (Mean = 88.8ε))
(Rmath 3.2.3: Max = 7.99e+04ε (Mean = 1.68e+04ε))

Max = 0.948ε (Mean = 0.36ε)

(<cmath>: Max = 0.615ε (Mean = 0.096ε))
(<math.h>: Max = 0.615ε (Mean = 0.096ε))

Max = 0.948ε (Mean = 0.36ε)

(<math.h>: Max = 1.71ε (Mean = 0.581ε))

Max = 0.867ε (Mean = 0.468ε)

(<math.h>: Max = 0.906ε (Mean = 0.565ε))

near 2

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.17e+03ε (Mean = 274ε))
(Rmath 3.2.3: Max = 2.63e+05ε (Mean = 5.84e+04ε))

Max = 0.878ε (Mean = 0.242ε)

(<cmath>: Max = 0.741ε (Mean = 0.263ε))
(<math.h>: Max = 0.741ε (Mean = 0.263ε))

Max = 0.878ε (Mean = 0.242ε)

(<math.h>: Max = 0.598ε (Mean = 0.235ε))

Max = 0.591ε (Mean = 0.159ε)

(<math.h>: Max = 0.741ε (Mean = 0.473ε))

near -10

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 24.9ε (Mean = 4.6ε))
(Rmath 3.2.3: Max = 4.22ε (Mean = 1.26ε))

Max = 3.81ε (Mean = 1.01ε)

(<cmath>: Max = 0.997ε (Mean = 0.412ε))
(<math.h>: Max = 0.997ε (Mean = 0.412ε))

Max = 3.81ε (Mean = 1.01ε)

(<math.h>: Max = 3.04ε (Mean = 1.01ε))

Max = 4.22ε (Mean = 1.33ε)

(<math.h>: Max = 0.997ε (Mean = 0.444ε))

near -55

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 7.02ε (Mean = 1.47ε))
(Rmath 3.2.3: Max = 250ε (Mean = 60.9ε))

Max = 0.821ε (Mean = 0.513ε)

(<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))

Max = 0.821ε (Mean = 0.419ε)

(<math.h>: Max = 249ε (Mean = 43.1ε))


Table 23.73. Error rates for log1p

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Random test data

Max = 0.818ε (Mean = 0.227ε)

(<cmath>: Max = 0.818ε (Mean = 0.227ε))
(<math.h>: Max = 0.818ε (Mean = 0.227ε))

Max = 0.846ε (Mean = 0.153ε)

(Rmath 3.2.3: Max = 0.846ε (Mean = 0.153ε))

Max = 2.3ε (Mean = 0.66ε)

(<math.h>: Max = 0.818ε (Mean = 0.249ε))

Max = 0.509ε (Mean = 0.057ε)

(<math.h>: Max = 0.509ε (Mean = 0.057ε))


Table 23.74. Error rates for non central beta CDF

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Non Central Beta, medium parameters

Max = 0.998ε (Mean = 0.0649ε)

(Rmath 3.2.3: Max = 1.46e+26ε (Mean = 3.5e+24ε) And other failures.)

Max = 824ε (Mean = 27.4ε)

Max = 832ε (Mean = 38.1ε)

Max = 242ε (Mean = 31ε)

Non Central Beta, large parameters

Max = 1.18ε (Mean = 0.175ε)

(Rmath 3.2.3: Max = 1.01e+36ε (Mean = 1.19e+35ε) And other failures.)

Max = 2.5e+04ε (Mean = 3.78e+03ε)

Max = 2.57e+04ε (Mean = 4.45e+03ε)

Max = 3.66e+03ε (Mean = 500ε)


Table 23.75. Error rates for non central beta CDF complement

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Non Central Beta, medium parameters

Max = 0.998ε (Mean = 0.0936ε)

(Rmath 3.2.3: Max = 7.5e+97ε (Mean = 1.37e+96ε) And other failures.)

Max = 396ε (Mean = 50.7ε)

Max = 554ε (Mean = 57.2ε)

Max = 624ε (Mean = 62.7ε)

Non Central Beta, large parameters

Max = 0.986ε (Mean = 0.188ε)

(Rmath 3.2.3: Max = +INFε (Mean = +INFε) And other failures.)

Max = 6.83e+03ε (Mean = 993ε)

Max = 3.56e+03ε (Mean = 707ε)

Max = 1.25e+04ε (Mean = 1.49e+03ε)


Table 23.76. Error rates for non central chi squared CDF

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Non Central Chi Squared, medium parameters

Max = 0.99ε (Mean = 0.0544ε)

(Rmath 3.2.3: Max = 727ε (Mean = 121ε))

Max = 46.5ε (Mean = 10.3ε)

Max = 115ε (Mean = 13.9ε)

Max = 48.9ε (Mean = 10ε)

Non Central Chi Squared, large parameters

Max = 1.07ε (Mean = 0.102ε)

(Rmath 3.2.3: Max = 3.27e+08ε (Mean = 2.23e+07ε))

Max = 3.07e+03ε (Mean = 336ε)

Max = 6.17e+03ε (Mean = 677ε)

Max = 9.79e+03ε (Mean = 723ε)


Table 23.77. Error rates for non central chi squared CDF complement

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Non Central Chi Squared, medium parameters

Max = 0.96ε (Mean = 0.0635ε)

(Rmath 3.2.3: Max = +INFε (Mean = +INFε) And other failures.)

Max = 107ε (Mean = 17.2ε)

Max = 171ε (Mean = 22.8ε)

Max = 98.6ε (Mean = 15.8ε)

Non Central Chi Squared, large parameters

Max = 2.11ε (Mean = 0.278ε)

(Rmath 3.2.3: Max = +INFε (Mean = +INFε) And other failures.)

Max = 5.02e+03ε (Mean = 630ε)

Max = 5.1e+03ε (Mean = 577ε)

Max = 5.43e+03ε (Mean = 705ε)


Table 23.78. Error rates for non central t CDF

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Non Central T

Max = 0.796ε (Mean = 0.0691ε)

(Rmath 3.2.3: Max = 5.28e+15ε (Mean = 8.49e+14ε) And other failures.)

Max = 139ε (Mean = 31ε)

Max = 145ε (Mean = 30.9ε)

Max = 135ε (Mean = 32.1ε)

Non Central T (small non-centrality)

Max = 0ε (Mean = 0ε)

(Rmath 3.2.3: Max = 2.09e+03ε (Mean = 244ε))

Max = 3.86ε (Mean = 1.4ε)

Max = 9.15ε (Mean = 2.13ε)

Max = 6.17ε (Mean = 1.45ε)

Non Central T (large parameters)

Max = 257ε (Mean = 72.1ε)

(Rmath 3.2.3: Max = 2.46ε (Mean = 0.657ε))

Max = 5.26e+05ε (Mean = 1.48e+05ε)

Max = 5.24e+05ε (Mean = 1.47e+05ε)

Max = 286ε (Mean = 62.8ε)


Table 23.79. Error rates for non central t CDF complement

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Non Central T

Max = 0.707ε (Mean = 0.0497ε)

(Rmath 3.2.3: Max = 6.19e+15ε (Mean = 6.72e+14ε) And other failures.)

Max = 201ε (Mean = 31.7ε)

Max = 340ε (Mean = 43.2ε)

Max = 154ε (Mean = 32.1ε)

Non Central T (small non-centrality)

Max = 0ε (Mean = 0ε)

(Rmath 3.2.3: Max = 1.87e+03ε (Mean = 263ε))

Max = 10.5ε (Mean = 2.13ε)

Max = 10.5ε (Mean = 2.39ε)

Max = 4.6ε (Mean = 1.63ε)

Non Central T (large parameters)

Max = 478ε (Mean = 96.3ε)

(Rmath 3.2.3: Max = 2.24ε (Mean = 0.945ε))

Max = 9.79e+05ε (Mean = 1.97e+05ε)

Max = 9.79e+05ε (Mean = 1.97e+05ε)

Max = 227ε (Mean = 50.4ε)


Table 23.80. Error rates for owens_t

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Owens T (medium small values)

Max = 0ε (Mean = 0ε)

Max = 3.34ε (Mean = 0.944ε)

Max = 3.34ε (Mean = 0.911ε)

Max = 4.37ε (Mean = 0.98ε)

Owens T (large and diverse values)

Max = 0ε (Mean = 0ε)

Max = 49ε (Mean = 2.16ε)

Max = 24.5ε (Mean = 1.39ε)

Max = 3.78ε (Mean = 0.621ε)


Table 23.81. Error rates for polygamma

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Mathematica Data

Max = 0.824ε (Mean = 0.0574ε)

(GSL 2.1: Max = 62.9ε (Mean = 12.8ε))
(Rmath 3.2.3: Max = 108ε (Mean = 15.2ε))

Max = 7.38ε (Mean = 1.84ε)

Max = 34.3ε (Mean = 7.65ε)

Max = 9.32ε (Mean = 1.95ε)

Mathematica Data - large arguments

Max = 0.998ε (Mean = 0.0592ε)

(GSL 2.1: Max = 244ε (Mean = 32.8ε) And other failures.)
(Rmath 3.2.3: Max = 1.71e+56ε (Mean = 1.01e+55ε) And other failures.)

Max = 2.23ε (Mean = 0.323ε)

Max = 11.1ε (Mean = 0.848ε)

Max = 150ε (Mean = 13.9ε)

Mathematica Data - negative arguments

Max = 0.516ε (Mean = 0.022ε)

(GSL 2.1: Max = 36.6ε (Mean = 3.04ε) And other failures.)
(Rmath 3.2.3: Max = 0ε (Mean = 0ε) And other failures.)

Max = 269ε (Mean = 87.7ε)

Max = 269ε (Mean = 88.4ε)

Max = 497ε (Mean = 129ε)

Mathematica Data - large negative arguments

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.79ε (Mean = 0.197ε) And other failures.)
(Rmath 3.2.3: Max = 0ε (Mean = 0ε) And other failures.)

Max = 155ε (Mean = 96.4ε)

Max = 155ε (Mean = 96.4ε)

Max = 162ε (Mean = 101ε)

Mathematica Data - small arguments

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 15.2ε (Mean = 5.03ε))
(Rmath 3.2.3: Max = 106ε (Mean = 20ε))

Max = 3.33ε (Mean = 0.75ε)

Max = 3.33ε (Mean = 0.75ε)

Max = 3ε (Mean = 0.496ε)

Mathematica Data - Large orders and other bug cases

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 151ε (Mean = 39.3ε) And other failures.)
(Rmath 3.2.3: Max = +INFε (Mean = +INFε) And other failures.)

Max = 54.5ε (Mean = 13.3ε)

Max = 145ε (Mean = 55.9ε)

Max = 200ε (Mean = 57.2ε)


Table 23.82. Error rates for powm1

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

powm1

Max = 1.06ε (Mean = 0.425ε)

Max = 2.04ε (Mean = 0.493ε)

Max = 1.88ε (Mean = 0.49ε)

Max = 1.84ε (Mean = 0.486ε)


Table 23.83. Error rates for sin_pi

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

sin_pi and cos_pi

Max = 0ε (Mean = 0ε)

Max = 0.996ε (Mean = 0.335ε)

Max = 0.996ε (Mean = 0.336ε)

Max = 0.99ε (Mean = 0.328ε)

sin_pi and cos_pi near integers and half integers

Max = 0ε (Mean = 0ε)

Max = 0.976ε (Mean = 0.293ε)

Max = 0.976ε (Mean = 0.293ε)

Max = 0.996ε (Mean = 0.343ε)


Table 23.84. Error rates for sph_bessel

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel j: Random Data

Max = 243ε (Mean = 13.3ε)

(<cmath>: Max = 1.91e+06ε (Mean = 1.09e+05ε))

Max = 0.978ε (Mean = 0.0445ε)

(GSL 2.1: Max = 1.79e+03ε (Mean = 107ε))

Max = 243ε (Mean = 33.7ε)

Max = 245ε (Mean = 16.3ε)


Table 23.85. Error rates for sph_bessel_prime

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Bessel j': Random Data

Max = 0.753ε (Mean = 0.0343ε)

Max = 167ε (Mean = 12ε)

Max = 167ε (Mean = 33.2ε)

Max = 307ε (Mean = 25.2ε)


Table 23.86. Error rates for sph_neumann

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

y: Random Data

Max = 234ε (Mean = 19.5ε)

(<cmath>: Max = 1.6e+06ε (Mean = 1.4e+05ε))

Max = 0.995ε (Mean = 0.0665ε)

(GSL 2.1: Max = 8.5e+04ε (Mean = 5.33e+03ε))

Max = 234ε (Mean = 19.8ε)

Max = 281ε (Mean = 31.1ε)


Table 23.87. Error rates for sph_neumann_prime

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

y': Random Data

Max = 0.988ε (Mean = 0.0869ε)

Max = 158ε (Mean = 18.8ε)

Max = 158ε (Mean = 20.2ε)

Max = 296ε (Mean = 25.6ε)


Table 23.88. Error rates for spherical_harmonic_i

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Spherical Harmonics

Max = 1.36ε (Mean = 0.0765ε)

Max = 2.89e+03ε (Mean = 108ε)

Max = 1.03e+04ε (Mean = 327ε)

Max = 2.27e+04ε (Mean = 725ε)


Table 23.89. Error rates for spherical_harmonic_r

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Spherical Harmonics

Max = 1.58ε (Mean = 0.0707ε)

Max = 2.89e+03ε (Mean = 108ε)

Max = 1.03e+04ε (Mean = 327ε)

Max = 2.27e+04ε (Mean = 725ε)


Table 23.90. Error rates for sqrt1pm1

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

sqrt1pm1

Max = 1.3ε (Mean = 0.404ε)

Max = 1.33ε (Mean = 0.404ε)

Max = 1.54ε (Mean = 0.563ε)

Max = 1.35ε (Mean = 0.497ε)


Table 23.91. Error rates for tgamma

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

factorials

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 3.95ε (Mean = 0.783ε))
(Rmath 3.2.3: Max = 314ε (Mean = 93.4ε))

Max = 2.67ε (Mean = 0.617ε)

(<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))

Max = 1.85ε (Mean = 0.566ε)

(<math.h>: Max = 3.17ε (Mean = 0.928ε))

near 0

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 4.51ε (Mean = 1.92ε))
(Rmath 3.2.3: Max = 1ε (Mean = 0.335ε))

Max = 2ε (Mean = 0.608ε)

(<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))

Max = 1.5ε (Mean = 0.635ε)

(<math.h>: Max = 1ε (Mean = 0.405ε))

near 1

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 4.41ε (Mean = 1.81ε))
(Rmath 3.2.3: Max = 1ε (Mean = 0.32ε))

Max = 2.51ε (Mean = 1.02ε)

(<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))

Max = 1.1ε (Mean = 0.59ε)

(<math.h>: Max = 1ε (Mean = 0.4ε))

near 2

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 7.95ε (Mean = 3.12ε))
(Rmath 3.2.3: Max = 1ε (Mean = 0.191ε))

Max = 4.1ε (Mean = 1.55ε)

(<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))

Max = 2ε (Mean = 0.733ε)

(<math.h>: Max = 0ε (Mean = 0ε))

near -10

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 2.6ε (Mean = 1.05ε))
(Rmath 3.2.3: Max = 34.9ε (Mean = 9.2ε))

Max = 1.75ε (Mean = 0.895ε)

(<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))

Max = 1.86ε (Mean = 0.881ε)

(<math.h>: Max = 0.866ε (Mean = 0.445ε))

near -55

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 1.8ε (Mean = 0.782ε))
(Rmath 3.2.3: Max = 3.89e+04ε (Mean = 9.52e+03ε))

Max = 2.69ε (Mean = 1.09ε)

(<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))

Max = 2.7ε (Mean = 1.35ε)

(<math.h>: Max = 3.87e+04ε (Mean = 6.71e+03ε))


Table 23.92. Error rates for tgamma1pm1

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

tgamma1pm1(dz)

Max = 0ε (Mean = 0ε)

Max = 1.12ε (Mean = 0.49ε)

Max = 6.61ε (Mean = 0.84ε)

Max = 3.31ε (Mean = 0.517ε)


Table 23.93. Error rates for tgamma_delta_ratio

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

tgamma + small delta ratios

Max = 0ε (Mean = 0ε)

Max = 5.83ε (Mean = 1.3ε)

Max = 15.4ε (Mean = 2.09ε)

Max = 7.56ε (Mean = 1.31ε)

tgamma + small delta ratios (negative delta)

Max = 0ε (Mean = 0ε)

Max = 7.94ε (Mean = 1.4ε)

Max = 18.3ε (Mean = 2.03ε)

Max = 7.43ε (Mean = 1.42ε)

tgamma + small integer ratios

Max = 0ε (Mean = 0ε)

Max = 1.96ε (Mean = 0.677ε)

Max = 1.96ε (Mean = 0.677ε)

Max = 2.74ε (Mean = 0.736ε)

tgamma + small integer ratios (negative delta)

Max = 0ε (Mean = 0ε)

Max = 1.62ε (Mean = 0.451ε)

Max = 1.62ε (Mean = 0.451ε)

Max = 2.15ε (Mean = 0.685ε)

integer tgamma ratios

Max = 0ε (Mean = 0ε)

Max = 0.997ε (Mean = 0.4ε)

Max = 0.997ε (Mean = 0.4ε)

Max = 0.968ε (Mean = 0.386ε)

integer tgamma ratios (negative delta)

Max = 0ε (Mean = 0ε)

Max = 0.853ε (Mean = 0.176ε)

Max = 0.853ε (Mean = 0.176ε)

Max = 0.974ε (Mean = 0.175ε)


Table 23.94. Error rates for tgamma (incomplete)

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

tgamma(a, z) medium values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 200ε (Mean = 13.3ε))

Max = 8.47ε (Mean = 1.9ε)

Max = 412ε (Mean = 95.5ε)

Max = 8.14ε (Mean = 1.76ε)

tgamma(a, z) small values

Max = 0.753ε (Mean = 0.0474ε)

(GSL 2.1: Max = 1.38e+10ε (Mean = 1.05e+09ε))

Max = 2.31ε (Mean = 0.775ε)

Max = 2.13ε (Mean = 0.717ε)

Max = 2.53ε (Mean = 0.66ε)

tgamma(a, z) integer and half integer values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 117ε (Mean = 12.5ε))

Max = 5.52ε (Mean = 1.48ε)

Max = 79.6ε (Mean = 20.9ε)

Max = 5.16ε (Mean = 1.33ε)


Table 23.95. Error rates for tgamma_lower

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

tgamma(a, z) medium values

Max = 0.833ε (Mean = 0.0315ε)

(GSL 2.1: Max = 0.833ε (Mean = 0.0315ε))

Max = 6.79ε (Mean = 1.46ε)

Max = 363ε (Mean = 63.8ε)

Max = 5.62ε (Mean = 1.49ε)

tgamma(a, z) small values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0ε (Mean = 0ε))

Max = 1.97ε (Mean = 0.555ε)

Max = 1.97ε (Mean = 0.558ε)

Max = 1.57ε (Mean = 0.525ε)

tgamma(a, z) integer and half integer values

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0ε (Mean = 0ε))

Max = 4.83ε (Mean = 1.15ε)

Max = 84.7ε (Mean = 17.5ε)

Max = 2.69ε (Mean = 0.849ε)


Table 23.96. Error rates for tgamma_ratio

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

tgamma ratios

Max = 0.694ε (Mean = 0.0347ε)

Max = 2.99ε (Mean = 1.15ε)

Max = 174ε (Mean = 61.2ε)

Max = 3.28ε (Mean = 1.12ε)


Table 23.97. Error rates for trigamma

GNU C++ version 7.1.0
linux
double

GNU C++ version 7.1.0
linux
long double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Mathematica Data

Max = 0.998ε (Mean = 0.105ε)

(GSL 2.1: Max = 1.34e+04ε (Mean = 1.49e+03ε))
(Rmath 3.2.3: Max = 1.34e+04ε (Mean = 1.51e+03ε))

Max = 1.28ε (Mean = 0.449ε)

Max = 1.28ε (Mean = 0.449ε)

Max = 1ε (Mean = 0.382ε)


Table 23.98. Error rates for zeta

GNU C++ version 7.1.0
linux
long double

GNU C++ version 7.1.0
linux
double

Sun compiler version 0x5150
Sun Solaris
long double

Microsoft Visual C++ version 14.1
Win32
double

Zeta: Random values greater than 1

Max = 0.846ε (Mean = 0.0833ε)

(<cmath>: Max = 5.45ε (Mean = 1ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 8.69ε (Mean = 1.03ε))

Max = 0.846ε (Mean = 0.0833ε)

Max = 0.836ε (Mean = 0.093ε)

Zeta: Random values less than 1

Max = 7.03ε (Mean = 2.93ε)

(<cmath>: Max = 538ε (Mean = 59.3ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 137ε (Mean = 13.8ε))

Max = 70.1ε (Mean = 17.1ε)

Max = 6.84ε (Mean = 3.12ε)

Zeta: Values close to and greater than 1

Max = 0.995ε (Mean = 0.5ε)

(<cmath>: Max = 1.9e+06ε (Mean = 5.11e+05ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 7.73ε (Mean = 4.07ε))

Max = 0.995ε (Mean = 0.5ε)

Max = 0.994ε (Mean = 0.421ε)

Zeta: Values close to and less than 1

Max = 0.998ε (Mean = 0.508ε)

(<cmath>: Max = 8.53e+06ε (Mean = 1.87e+06ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 0.991ε (Mean = 0.28ε))

Max = 0.998ε (Mean = 0.508ε)

Max = 0.991ε (Mean = 0.375ε)

Zeta: Integer arguments

Max = 9ε (Mean = 3.06ε)

(<cmath>: Max = 70.3ε (Mean = 17.4ε))

Max = 0ε (Mean = 0ε)

(GSL 2.1: Max = 3.75ε (Mean = 1.1ε))

Max = 28ε (Mean = 5.62ε)

Max = 9ε (Mean = 3ε)



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