## Overview

The special package of Hipparchus gathers several useful special functions not provided by java.lang.Math.

## Erf functions

Erf contains several useful functions involving the Error Function, Erf.

Function Method Reference
Error Function erf See MathWorld

## Gamma functions

Class Gamma contains several useful functions involving the Gamma Function.

### Gamma

Gamma.gamma(x) computes the Gamma function, $$\Gamma(x)$$ (see MathWorld, DLMF). The accuracy of the Hipparchus implementation is assessed by comparison with high precision values computed with the Maxima Computer Algebra System.

Interval Values tested Average error Standard deviation Maximum error
$$-5 < x < -4$$ x[i] = i / 1024, i = -5119, ..., -4097 0.49 ulps 0.57 ulps 3.0 ulps
$$-4 < x < -3$$ x[i] = i / 1024, i = -4095, ..., -3073 0.36 ulps 0.51 ulps 2.0 ulps
$$-3 < x < -2$$ x[i] = i / 1024, i = -3071, ..., -2049 0.41 ulps 0.53 ulps 2.0 ulps
$$-2 < x < -1$$ x[i] = i / 1024, i = -2047, ..., -1025 0.37 ulps 0.50 ulps 2.0 ulps
$$-1 < x < 0$$ x[i] = i / 1024, i = -1023, ..., -1 0.46 ulps 0.54 ulps 2.0 ulps
$$0 < x ≤ 8$$ x[i] = i / 1024, i = 1, ..., 8192 0.33 ulps 0.48 ulps 2.0 ulps
$$8 < x ≤ 141$$ x[i] = i / 64, i = 513, ..., 9024 1.32 ulps 1.19 ulps 7.0 ulps

### Log Gamma

Gamma.logGamma(x) computes the natural logarithm of the Gamma function, $$\log \Gamma(x)$$, for $$x > 0$$ (see MathWorld, DLMF). The accuracy of the Hipparchus implementation is assessed by comparison with high precision values computed with the Maxima Computer Algebra System.

Interval Values tested Average error Standard deviation Maximum error
$$0 < x \le 8$$ x[i] = i / 1024, i = 1, ..., 8192 0.32 ulps 0.50 ulps 4.0 ulps
$$8 < x \le 1024$$ x[i] = i / 8, i = 65, ..., 8192 0.43 ulps 0.53 ulps 3.0 ulps
$$1024 < x \le 8192$$ x[i], i = 1025, ..., 8192 0.53 ulps 0.56 ulps 3.0 ulps
$$8933.439345993791 \le x \le 1.75555970201398e+305$$ x[i] = 2**(i / 8), i = 105, ..., 8112 0.35 ulps 0.49 ulps 2.0 ulps

### Regularized Gamma

Gamma.regularizedGammaP(a, x) computes the value of the regularized Gamma function, P(a, x) (see MathWorld).

## Beta functions

Beta contains several useful functions involving the Beta Function.

### Log Beta

Beta.logBeta(a, b) computes the value of the natural logarithm of the Beta function, log B(a, b). (see MathWorld, DLMF). The accuracy of the Hipparchus implementation is assessed by comparison with high precision values computed with the Maxima Computer Algebra System.

Interval Values tested Average error Standard deviation Maximum error
$$0 < x \le 8$$
$$0 < y \le 8$$
x[i] = i / 32, i = 1, ..., 256
y[j] = j / 32, j = 1, ..., 256
1.80 ulps 81.08 ulps 14031.0 ulps
$$0 < x \le 8$$
$$8 < y \le 16$$
x[i] = i / 32, i = 1, ..., 256
y[j] = j / 32, j = 257, ..., 512
0.50 ulps 3.64 ulps 694.0 ulps
$$0 < x \le 8$$
$$16 < y \le 256$$
x[i] = i / 32, i = 1, ..., 256
y[j] = j, j = 17, ..., 256
1.04 ulps 139.32 ulps 34509.0 ulps
$$8 < x \le 16$$
$$8 < y \le 16$$
x[i] = i / 32, i = 257, ..., 512
y[j] = j / 32, j = 257, ..., 512
0.35 ulps 0.48 ulps 2.0 ulps
$$8 < x \le 16$$
$$16 < y \le 256$$
x[i] = i / 32, i = 257, ..., 512
y[j] = j, j = 17, ..., 256
0.31 ulps 0.47 ulps 2.0 ulps
$$16 < x \le 256$$
$$16 < y \le 256$$
x[i] = i, i = 17, ..., 256
y[j] = j, j = 17, ..., 256
0.35 ulps 0.49 ulps 2.0 ulps

(see MathWorld)