Namespaces
namespace	internal

Typedefs
using	uint_t = unsigned int

using	ullint_t = unsigned long long int

using	llint_t = long long int

template<class T >
using	GCLIM = std::numeric_limits< T >

template<typename T >
using	return_t = typename std::conditional< std::is_integral< T >::value, double, T >::type

template<typename ... T>
using	common_t = typename std::common_type< T... >::type

template<typename ... T>
using	common_return_t = return_t< common_t< T... > >

Functions
template<typename T >
constexpr return_t< T >	acos (const T x) noexcept
	Compile-time arccosine function. More...

template<typename T1 , typename T2 >
constexpr common_t< T1, T2 >	binomial_coef (const T1 n, const T2 k) noexcept
	Compile-time binomial coefficient. More...

template<typename T1 , typename T2 , typename T3 >
constexpr common_return_t< T1, T2, T3 >	incomplete_beta (const T1 a, const T2 b, const T3 z) noexcept
	Compile-time regularized incomplete beta function. More...

template<typename T1 , typename T2 >
constexpr common_return_t< T1, T2 >	atan2 (const T1 y, const T2 x) noexcept
	Compile-time two-argument arctangent function. More...

template<typename T >
constexpr return_t< T >	expm1 (const T x) noexcept
	Compile-time exponential-minus-1 function. More...

template<typename T1 , typename T2 >
constexpr return_t< T1 >	lmgamma (const T1 a, const T2 p) noexcept
	Compile-time log multivariate gamma function. More...

template<typename T >
constexpr T	abs (const T x) noexcept
	Compile-time absolute value function. More...

template<typename T1 , typename T2 >
constexpr common_t< T1, T2 >	max (const T1 x, const T2 y) noexcept
	Compile-time pairwise maximum function. More...

template<typename T >
constexpr return_t< T >	trunc (const T x) noexcept
	Compile-time trunc function. More...

template<typename T1 , typename T2 >
constexpr common_t< T1, T2 >	lcm (const T1 a, const T2 b) noexcept
	Compile-time least common multiple (LCM) function. More...

template<typename T >
constexpr return_t< T >	erf (const T x) noexcept
	Compile-time Gaussian error function. More...

template<typename T >
constexpr return_t< T >	tanh (const T x) noexcept
	Compile-time hyperbolic tangent function. More...

template<typename T >
constexpr return_t< T >	atanh (const T x) noexcept
	Compile-time inverse hyperbolic tangent function. More...

template<typename T >
constexpr return_t< T >	inv_sqrt (const T x) noexcept
	Compile-time inverse-square-root function. More...

template<typename T >
constexpr return_t< T >	log1p (const T x) noexcept
	Compile-time natural-logarithm-plus-1 function. More...

template<typename T1 , typename T2 >
constexpr common_return_t< T1, T2 >	lbeta (const T1 a, const T2 b) noexcept
	Compile-time log-beta function. More...

template<typename T >
constexpr int	sgn (const T x) noexcept
	Compile-time sign function. More...

template<typename T1 , typename T2 >
constexpr common_return_t< T1, T2 >	log_binomial_coef (const T1 n, const T2 k) noexcept
	Compile-time log binomial coefficient. More...

template<typename T1 , typename T2 >
constexpr common_return_t< T1, T2 >	beta (const T1 a, const T2 b) noexcept
	Compile-time beta function. More...

template<typename T >
constexpr return_t< T >	round (const T x) noexcept
	Compile-time round function. More...

template<typename T >
constexpr return_t< T >	lgamma (const T x) noexcept
	Compile-time log-gamma function. More...

template<typename T >
constexpr return_t< T >	log (const T x) noexcept
	Compile-time natural logarithm function. More...

template<typename T1 , typename T2 >
constexpr common_t< T1, T2 >	pow (const T1 base, const T2 exp_term) noexcept
	Compile-time power function. More...

template<typename T1 , typename T2 >
constexpr common_t< T1, T2 >	min (const T1 x, const T2 y) noexcept
	Compile-time pairwise minimum function. More...

template<typename T >
constexpr return_t< T >	log2 (const T x) noexcept
	Compile-time binary logarithm function. More...

template<typename T >
constexpr return_t< T >	cos (const T x) noexcept
	Compile-time cosine function. More...

template<typename T >
constexpr return_t< T >	acosh (const T x) noexcept
	Compile-time inverse hyperbolic cosine function. More...

template<typename T >
constexpr bool	signbit (const T x) noexcept
	Compile-time sign bit detection function. More...

template<typename T >
constexpr return_t< T >	sinh (const T x) noexcept
	Compile-time hyperbolic sine function. More...

template<typename T >
constexpr return_t< T >	asinh (const T x) noexcept
	Compile-time inverse hyperbolic sine function. More...

template<typename T1 , typename T2 >
constexpr T1	copysign (const T1 x, const T2 y) noexcept
	Compile-time copy sign function. More...

template<typename T >
constexpr return_t< T >	sin (const T x) noexcept
	Compile-time sine function. More...

template<typename T >
constexpr return_t< T >	atan (const T x) noexcept
	Compile-time arctangent function. More...

template<typename T >
constexpr return_t< T >	log10 (const T x) noexcept
	Compile-time common logarithm function. More...

template<typename T >
constexpr return_t< T >	tan (const T x) noexcept
	Compile-time tangent function. More...

template<typename T >
constexpr return_t< T >	exp (const T x) noexcept
	Compile-time exponential function. More...

template<typename T >
constexpr T	factorial (const T x) noexcept
	Compile-time factorial function. More...

template<typename T1 , typename T2 >
constexpr common_return_t< T1, T2 >	fmod (const T1 x, const T2 y) noexcept
	Compile-time remainder of division function. More...

template<typename T1 , typename T2 , typename T3 >
constexpr common_t< T1, T2, T3 >	incomplete_beta_inv (const T1 a, const T2 b, const T3 p) noexcept
	Compile-time inverse incomplete beta function. More...

template<typename T1 , typename T2 >
constexpr common_t< T1, T2 >	gcd (const T1 a, const T2 b) noexcept
	Compile-time greatest common divisor (GCD) function. More...

template<typename T >
constexpr return_t< T >	erf_inv (const T p) noexcept
	Compile-time inverse Gaussian error function. More...

template<typename T >
constexpr return_t< T >	sqrt (const T x) noexcept
	Compile-time square-root function. More...

template<typename T >
constexpr return_t< T >	cosh (const T x) noexcept
	Compile-time hyperbolic cosine function. More...

template<typename T >
constexpr return_t< T >	asin (const T x) noexcept
	Compile-time arcsine function. More...

template<typename T >
constexpr return_t< T >	ceil (const T x) noexcept
	Compile-time ceil function. More...

template<typename T1 , typename T2 >
constexpr common_return_t< T1, T2 >	incomplete_gamma_inv (const T1 a, const T2 p) noexcept
	Compile-time inverse incomplete gamma function. More...

template<typename T >
constexpr return_t< T >	tgamma (const T x) noexcept
	Compile-time gamma function. More...

template<typename T1 , typename T2 >
constexpr common_return_t< T1, T2 >	hypot (const T1 x, const T2 y) noexcept
	Compile-time Pythagorean addition function. More...

template<typename T1 , typename T2 , typename T3 >
constexpr common_return_t< T1, T2, T3 >	hypot (const T1 x, const T2 y, const T3 z) noexcept
	Compile-time Pythagorean addition function. More...

template<typename T >
constexpr return_t< T >	floor (const T x) noexcept
	Compile-time floor function. More...

template<typename T1 , typename T2 >
constexpr common_return_t< T1, T2 >	incomplete_gamma (const T1 a, const T2 x) noexcept
	Compile-time regularized lower incomplete gamma function. More...

Variables
static const long double	gauss_legendre_30_points [30]

static const long double	gauss_legendre_30_weights [30]

static const long double	gauss_legendre_50_points [50]

static const long double	gauss_legendre_50_weights [50]

Typedef Documentation

◆ common_return_t

template<typename ... T>

using gcem::common_return_t = typedef return_t<common_t<T...> >

◆ common_t

template<typename ... T>

using gcem::common_t = typedef typename std::common_type<T...>::type

◆ GCLIM

template<class T >

using gcem::GCLIM = typedef std::numeric_limits<T>

◆ llint_t

using gcem::llint_t = typedef long long int

◆ return_t

template<typename T >

using gcem::return_t = typedef typename std::conditional<std::is_integral<T>::value,double,T>::type

◆ uint_t

using gcem::uint_t = typedef unsigned int

◆ ullint_t

using gcem::ullint_t = typedef unsigned long long int

Function Documentation

◆ abs()

template<typename T >

constexpr T gcem::abs ( const T x )

constexprnoexcept

Compile-time absolute value function.

Parameters

x	a real-valued input.

Returns: the absolute value of x, \( |x| \).

◆ acos()

template<typename T >

constexpr return_t< T > gcem::acos ( const T x )

constexprnoexcept

Compile-time arccosine function.

Parameters

x	a real-valued input, where \( x \in [-1,1] \).

Returns: the inverse cosine function using
\[ \text{acos}(x) = \text{atan} \left( \frac{\sqrt{1-x^2}}{x} \right) \]

◆ acosh()

template<typename T >

constexpr return_t< T > gcem::acosh ( const T x )

constexprnoexcept

Compile-time inverse hyperbolic cosine function.

Parameters

x	a real-valued input.

Returns: the inverse hyperbolic cosine function using
\[ \text{acosh}(x) = \ln \left( x + \sqrt{x^2 - 1} \right) \]

◆ asin()

template<typename T >

constexpr return_t< T > gcem::asin ( const T x )

constexprnoexcept

Compile-time arcsine function.

Parameters

x	a real-valued input, where \( x \in [-1,1] \).

Returns: the inverse sine function using
\[ \text{asin}(x) = \text{atan} \left( \frac{x}{\sqrt{1-x^2}} \right) \]

◆ asinh()

template<typename T >

constexpr return_t< T > gcem::asinh ( const T x )

constexprnoexcept

Compile-time inverse hyperbolic sine function.

Parameters

x	a real-valued input.

Returns: the inverse hyperbolic sine function using
\[ \text{asinh}(x) = \ln \left( x + \sqrt{x^2 + 1} \right) \]

◆ atan()

template<typename T >

constexpr return_t< T > gcem::atan ( const T x )

constexprnoexcept

Compile-time arctangent function.

Parameters

x	a real-valued input.

Returns: the inverse tangent function using
\[ \text{atan}(x) = \dfrac{x}{1 + \dfrac{x^2}{3 + \dfrac{4x^2}{5 + \dfrac{9x^2}{7 + \ddots}}}} \]

◆ atan2()

template<typename T1 , typename T2 >

constexpr common_return_t< T1, T2 > gcem::atan2	(	const T1	y,
		const T2	x
	)

constexprnoexcept

Compile-time two-argument arctangent function.

Parameters

y	a real-valued input.
x	a real-valued input.

Returns

\[ \text{atan2}(y,x) = \begin{cases} \text{atan}(y/x) & \text{ if } x > 0 \\ \text{atan}(y/x) + \pi & \text{ if } x < 0 \text{ and } y \geq 0 \\ \text{atan}(y/x) - \pi & \text{ if } x < 0 \text{ and } y < 0 \\ + \pi/2 & \text{ if } x = 0 \text{ and } y > 0 \\ - \pi/2 & \text{ if } x = 0 \text{ and } y < 0 \end{cases} \]

The function is undefined at the origin, however the following conventions are used.

\[ \text{atan2}(y,x) = \begin{cases} +0 & \text{ if } x = +0 \text{ and } y = +0 \\ -0 & \text{ if } x = +0 \text{ and } y = -0 \\ +\pi & \text{ if } x = -0 \text{ and } y = +0 \\ - \pi & \text{ if } x = -0 \text{ and } y = -0 \end{cases} \]

◆ atanh()

template<typename T >

constexpr return_t< T > gcem::atanh ( const T x )

constexprnoexcept

Compile-time inverse hyperbolic tangent function.

Parameters

x	a real-valued input.

Returns: the inverse hyperbolic tangent function using
\[ \text{atanh}(x) = \frac{1}{2} \ln \left( \frac{1+x}{1-x} \right) \]

◆ beta()

template<typename T1 , typename T2 >

constexpr common_return_t< T1, T2 > gcem::beta	(	const T1	a,
		const T2	b
	)

constexprnoexcept

Compile-time beta function.

Parameters

a	a real-valued input.
b	a real-valued input.

Returns: the beta function using
\[ \text{B}(\alpha,\beta) := \int_0^1 t^{\alpha - 1} (1-t)^{\beta - 1} dt = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)} \]
where \( \Gamma \) denotes the gamma function.

◆ binomial_coef()

template<typename T1 , typename T2 >

constexpr common_t< T1, T2 > gcem::binomial_coef	(	const T1	n,
		const T2	k
	)

constexprnoexcept

Compile-time binomial coefficient.

Parameters

n	integral-valued input.
k	integral-valued input.

Returns: computes the Binomial coefficient
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
also known as 'n choose k '.

◆ ceil()

template<typename T >

constexpr return_t< T > gcem::ceil ( const T x )

constexprnoexcept

Compile-time ceil function.

Parameters

x	a real-valued input.

Returns: computes the ceiling-value of the input.

◆ copysign()

template<typename T1 , typename T2 >

constexpr T1 gcem::copysign	(	const T1	x,
		const T2	y
	)

constexprnoexcept

Compile-time copy sign function.

Parameters

x	a real-valued input
y	a real-valued input

Returns: replace the signbit of x with the signbit of y.

◆ cos()

template<typename T >

constexpr return_t< T > gcem::cos ( const T x )

constexprnoexcept

Compile-time cosine function.

Parameters

x	a real-valued input.

Returns: the cosine function using
\[ \cos(x) = \frac{1-\tan^2(x/2)}{1+\tan^2(x/2)} \]

◆ cosh()

template<typename T >

constexpr return_t< T > gcem::cosh ( const T x )

constexprnoexcept

Compile-time hyperbolic cosine function.

Parameters

x	a real-valued input.

Returns: the hyperbolic cosine function using
\[ \cosh(x) = \frac{\exp(x) + \exp(-x)}{2} \]

◆ erf()

template<typename T >

constexpr return_t< T > gcem::erf ( const T x )

constexprnoexcept

Compile-time Gaussian error function.

Parameters

x	a real-valued input.

Returns

computes the Gaussian error function

\[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp( - t^2) dt \]

using a continued fraction representation:

\[ \text{erf}(x) = \frac{2x}{\sqrt{\pi}} \exp(-x^2) \dfrac{1}{1 - 2x^2 + \dfrac{4x^2}{3 - 2x^2 + \dfrac{8x^2}{5 - 2x^2 + \dfrac{12x^2}{7 - 2x^2 + \ddots}}}} \]

◆ erf_inv()

template<typename T >

constexpr return_t< T > gcem::erf_inv ( const T p )

constexprnoexcept

Compile-time inverse Gaussian error function.

Parameters

p	a real-valued input with values in the unit-interval.

Returns

Computes the inverse Gaussian error function, a value \( x \) such that

\[ f(x) := \text{erf}(x) - p \]

is equal to zero, for a given p. GCE-Math finds this root using Halley's method:

\[ x_{n+1} = x_n - \frac{f(x_n)/f'(x_n)}{1 - 0.5 \frac{f(x_n)}{f'(x_n)} \frac{f''(x_n)}{f'(x_n)} } \]

where

\[ \frac{\partial}{\partial x} \text{erf}(x) = \exp(-x^2), \ \ \frac{\partial^2}{\partial x^2} \text{erf}(x) = -2x\exp(-x^2) \]

◆ exp()

template<typename T >

constexpr return_t< T > gcem::exp ( const T x )

constexprnoexcept

Compile-time exponential function.

Parameters

x	a real-valued input.

Returns: \( \exp(x) \) using
\[ \exp(x) = \dfrac{1}{1-\dfrac{x}{1+x-\dfrac{\frac{1}{2}x}{1 + \frac{1}{2}x - \dfrac{\frac{1}{3}x}{1 + \frac{1}{3}x - \ddots}}}} \]
The continued fraction argument is split into two parts: \( x = n + r \), where \( n \) is an integer and \( r \in [-0.5,0.5] \).

◆ expm1()

template<typename T >

constexpr return_t< T > gcem::expm1 ( const T x )

constexprnoexcept

Compile-time exponential-minus-1 function.

Parameters

x	a real-valued input.

Returns: \( \exp(x) - 1 \) using
\[ \exp(x) = \sum_{k=0}^\infty \dfrac{x^k}{k!} \]

◆ factorial()

template<typename T >

constexpr T gcem::factorial ( const T x )

constexprnoexcept

Compile-time factorial function.

Parameters

x	a real-valued input.

Returns: Computes the factorial value \( x! \). When x is an integral type (int, long int, etc.), a simple recursion method is used, along with table values. When x is real-valued, factorial(x) = tgamma(x+1).

◆ floor()

template<typename T >

constexpr return_t< T > gcem::floor ( const T x )

constexprnoexcept

Compile-time floor function.

Parameters

x	a real-valued input.

Returns: computes the floor-value of the input.

◆ fmod()

template<typename T1 , typename T2 >

constexpr common_return_t< T1, T2 > gcem::fmod	(	const T1	x,
		const T2	y
	)

constexprnoexcept

Compile-time remainder of division function.

Parameters

x	a real-valued input.
y	a real-valued input.

Returns: computes the floating-point remainder of \( x / y \) (rounded towards zero) using
\[ \text{fmod}(x,y) = x - \text{trunc}(x/y) \times y \]

◆ gcd()

template<typename T1 , typename T2 >

constexpr common_t< T1, T2 > gcem::gcd	(	const T1	a,
		const T2	b
	)

constexprnoexcept

Compile-time greatest common divisor (GCD) function.

Parameters

a	integral-valued input.
b	integral-valued input.

Returns: the greatest common divisor between integers a and b using a Euclidean algorithm.

◆ hypot() [1/2]

template<typename T1 , typename T2 >

constexpr common_return_t< T1, T2 > gcem::hypot	(	const T1	x,
		const T2	y
	)

constexprnoexcept

Compile-time Pythagorean addition function.

Parameters

x	a real-valued input.
y	a real-valued input.

Returns: Computes \( x \oplus y = \sqrt{x^2 + y^2} \).

◆ hypot() [2/2]

template<typename T1 , typename T2 , typename T3 >

constexpr common_return_t< T1, T2, T3 > gcem::hypot	(	const T1	x,
		const T2	y,
		const T3	z
	)

constexprnoexcept

Compile-time Pythagorean addition function.

Parameters

x	a real-valued input.
y	a real-valued input.
z	a real-valued input.

Returns: Computes \( x \oplus y \oplus z = \sqrt{x^2 + y^2 + z^2} \).

◆ incomplete_beta()

template<typename T1 , typename T2 , typename T3 >

constexpr common_return_t< T1, T2, T3 > gcem::incomplete_beta	(	const T1	a,
		const T2	b,
		const T3	z
	)

constexprnoexcept

Compile-time regularized incomplete beta function.

Parameters

a	a real-valued, non-negative input.
b	a real-valued, non-negative input.
z	a real-valued, non-negative input.

Returns

computes the regularized incomplete beta function,

\[ \frac{\text{B}(z;\alpha,\beta)}{\text{B}(\alpha,\beta)} = \frac{1}{\text{B}(\alpha,\beta)}\int_0^z t^{a-1} (1-t)^{\beta-1} dt \]

using a continued fraction representation, found in the Handbook of Continued Fractions for Special Functions, and a modified Lentz method.

\[ \frac{\text{B}(z;\alpha,\beta)}{\text{B}(\alpha,\beta)} = \frac{z^{\alpha} (1-t)^{\beta}}{\alpha \text{B}(\alpha,\beta)} \dfrac{a_1}{1 + \dfrac{a_2}{1 + \dfrac{a_3}{1 + \dfrac{a_4}{1 + \ddots}}}} \]

where \( a_1 = 1 \) and

\[ a_{2m+2} = - \frac{(\alpha + m)(\alpha + \beta + m)}{(\alpha + 2m)(\alpha + 2m + 1)}, \ m \geq 0 \]

\[ a_{2m+1} = \frac{m(\beta - m)}{(\alpha + 2m - 1)(\alpha + 2m)}, \ m \geq 1 \]

The Lentz method works as follows: let \( f_j \) denote the value of the continued fraction up to the first \( j \) terms; \( f_j \) is updated as follows:

\[ c_j = 1 + a_j / c_{j-1}, \ \ d_j = 1 / (1 + a_j d_{j-1}) \]

\[ f_j = c_j d_j f_{j-1} \]

◆ incomplete_beta_inv()

template<typename T1 , typename T2 , typename T3 >

constexpr common_t< T1, T2, T3 > gcem::incomplete_beta_inv	(	const T1	a,
		const T2	b,
		const T3	p
	)

constexprnoexcept

Compile-time inverse incomplete beta function.

Parameters

a	a real-valued, non-negative input.
b	a real-valued, non-negative input.
p	a real-valued input with values in the unit-interval.

Returns

Computes the inverse incomplete beta function, a value \( x \) such that

\[ f(x) := \frac{\text{B}(x;\alpha,\beta)}{\text{B}(\alpha,\beta)} - p \]

equal to zero, for a given p. GCE-Math finds this root using Halley's method:

\[ x_{n+1} = x_n - \frac{f(x_n)/f'(x_n)}{1 - 0.5 \frac{f(x_n)}{f'(x_n)} \frac{f''(x_n)}{f'(x_n)} } \]

where

\[ \frac{\partial}{\partial x} \left(\frac{\text{B}(x;\alpha,\beta)}{\text{B}(\alpha,\beta)}\right) = \frac{1}{\text{B}(\alpha,\beta)} x^{\alpha-1} (1-x)^{\beta-1} \]

\[ \frac{\partial^2}{\partial x^2} \left(\frac{\text{B}(x;\alpha,\beta)}{\text{B}(\alpha,\beta)}\right) = \frac{1}{\text{B}(\alpha,\beta)} x^{\alpha-1} (1-x)^{\beta-1} \left( \frac{\alpha-1}{x} - \frac{\beta-1}{1 - x} \right) \]

◆ incomplete_gamma()

template<typename T1 , typename T2 >

constexpr common_return_t< T1, T2 > gcem::incomplete_gamma	(	const T1	a,
		const T2	x
	)

constexprnoexcept

Compile-time regularized lower incomplete gamma function.

Parameters

a	a real-valued, non-negative input.
x	a real-valued, non-negative input.

Returns

the regularized lower incomplete gamma function evaluated at (a, x),

\[ \frac{\gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} \exp(-t) dt \]

When a is not too large, the value is computed using the continued fraction representation of the upper incomplete gamma function, \( \Gamma(a,x) \), using

\[ \Gamma(a,x) = \Gamma(a) - \dfrac{x^a\exp(-x)}{a - \dfrac{ax}{a + 1 + \dfrac{x}{a + 2 - \dfrac{(a+1)x}{a + 3 + \dfrac{2x}{a + 4 - \ddots}}}}} \]

where \( \gamma(a,x) \) and \( \Gamma(a,x) \) are connected via

\[ \frac{\gamma(a,x)}{\Gamma(a)} + \frac{\Gamma(a,x)}{\Gamma(a)} = 1 \]

When \( a > 10 \), a 50-point Gauss-Legendre quadrature scheme is employed.

◆ incomplete_gamma_inv()

template<typename T1 , typename T2 >

constexpr common_return_t< T1, T2 > gcem::incomplete_gamma_inv	(	const T1	a,
		const T2	p
	)

constexprnoexcept

Compile-time inverse incomplete gamma function.

Parameters

a	a real-valued, non-negative input.
p	a real-valued input with values in the unit-interval.

Returns

Computes the inverse incomplete gamma function, a value \( x \) such that

\[ f(x) := \frac{\gamma(a,x)}{\Gamma(a)} - p \]

equal to zero, for a given p. GCE-Math finds this root using Halley's method:

\[ x_{n+1} = x_n - \frac{f(x_n)/f'(x_n)}{1 - 0.5 \frac{f(x_n)}{f'(x_n)} \frac{f''(x_n)}{f'(x_n)} } \]

where

\[ \frac{\partial}{\partial x} \left(\frac{\gamma(a,x)}{\Gamma(a)}\right) = \frac{1}{\Gamma(a)} x^{a-1} \exp(-x) \]

\[ \frac{\partial^2}{\partial x^2} \left(\frac{\gamma(a,x)}{\Gamma(a)}\right) = \frac{1}{\Gamma(a)} x^{a-1} \exp(-x) \left( \frac{a-1}{x} - 1 \right) \]

◆ inv_sqrt()

template<typename T >

constexpr return_t< T > gcem::inv_sqrt ( const T x )

constexprnoexcept

Compile-time inverse-square-root function.

Parameters

x	a real-valued input.

Returns: Computes \( 1 / \sqrt{x} \) using a Newton-Raphson approach.

◆ lbeta()

template<typename T1 , typename T2 >

constexpr common_return_t< T1, T2 > gcem::lbeta	(	const T1	a,
		const T2	b
	)

constexprnoexcept

Compile-time log-beta function.

Parameters

a	a real-valued input.
b	a real-valued input.

Returns: the log-beta function using
\[ \ln \text{B}(\alpha,\beta) := \ln \int_0^1 t^{\alpha - 1} (1-t)^{\beta - 1} dt = \ln \Gamma(\alpha) + \ln \Gamma(\beta) - \ln \Gamma(\alpha + \beta) \]
where \( \Gamma \) denotes the gamma function.

◆ lcm()

template<typename T1 , typename T2 >

constexpr common_t< T1, T2 > gcem::lcm	(	const T1	a,
		const T2	b
	)

constexprnoexcept

Compile-time least common multiple (LCM) function.

Parameters

a	integral-valued input.
b	integral-valued input.

Returns: the least common multiple between integers a and b using the representation
\[ \text{lcm}(a,b) = \dfrac{| a b |}{\text{gcd}(a,b)} \]
where \( \text{gcd}(a,b) \) denotes the greatest common divisor between \( a \) and \( b \).

◆ lgamma()

template<typename T >

constexpr return_t< T > gcem::lgamma ( const T x )

constexprnoexcept

Compile-time log-gamma function.

Parameters

x	a real-valued input.

Returns

computes the log-gamma function

\[ \ln \Gamma(x) = \ln \int_0^\infty y^{x-1} \exp(-y) dy \]

using a polynomial form:

\[ \Gamma(x+1) \approx (x+g+0.5)^{x+0.5} \exp(-x-g-0.5) \sqrt{2 \pi} \left[ c_0 + \frac{c_1}{x+1} + \frac{c_2}{x+2} + \cdots + \frac{c_n}{x+n} \right] \]

where the value \( g \) and the coefficients \( (c_0, c_1, \ldots, c_n) \) are taken from Paul Godfrey, whose note can be found here: http://my.fit.edu/~gabdo/gamma.txt

◆ lmgamma()

template<typename T1 , typename T2 >

constexpr return_t< T1 > gcem::lmgamma	(	const T1	a,
		const T2	p
	)

constexprnoexcept

Compile-time log multivariate gamma function.

Parameters

a	a real-valued input.
p	integral-valued input.

Returns: computes log-multivariate gamma function via recursion
\[ \Gamma_p(a) = \pi^{(p-1)/2} \Gamma(a) \Gamma_{p-1}(a-0.5) \]
where \( \Gamma_1(a) = \Gamma(a) \).

◆ log()

template<typename T >

constexpr return_t< T > gcem::log ( const T x )

constexprnoexcept

Compile-time natural logarithm function.

Parameters

x	a real-valued input.

Returns: \( \log_e(x) \) using
\[ \log\left(\frac{1+x}{1-x}\right) = \dfrac{2x}{1-\dfrac{x^2}{3-\dfrac{4x^2}{5 - \dfrac{9x^3}{7 - \ddots}}}}, \ \ x \in [-1,1] \]
The continued fraction argument is split into two parts: \( x = a \times 10^c \), where \( c \) is an integer.

◆ log10()

template<typename T >

constexpr return_t< T > gcem::log10 ( const T x )

constexprnoexcept

Compile-time common logarithm function.

Parameters

x	a real-valued input.

Returns: \( \log_{10}(x) \) using
\[ \log_{10}(x) = \frac{\log_e(x)}{\log_e(10)} \]

◆ log1p()

template<typename T >

constexpr return_t< T > gcem::log1p ( const T x )

constexprnoexcept

Compile-time natural-logarithm-plus-1 function.

Parameters

x	a real-valued input.

Returns: \( \log_e(x+1) \) using
\[ \log(x+1) = \sum_{k=1}^\infty \dfrac{(-1)^{k-1}x^k}{k}, \ \ |x| < 1 \]

◆ log2()

template<typename T >

constexpr return_t< T > gcem::log2 ( const T x )

constexprnoexcept

Compile-time binary logarithm function.

Parameters

x	a real-valued input.

Returns: \( \log_2(x) \) using
\[ \log_{2}(x) = \frac{\log_e(x)}{\log_e(2)} \]

◆ log_binomial_coef()

template<typename T1 , typename T2 >

constexpr common_return_t< T1, T2 > gcem::log_binomial_coef	(	const T1	n,
		const T2	k
	)

constexprnoexcept

Compile-time log binomial coefficient.

Parameters

n	integral-valued input.
k	integral-valued input.

Returns: computes the log Binomial coefficient
\[ \ln \frac{n!}{k!(n-k)!} = \ln \Gamma(n+1) - [ \ln \Gamma(k+1) + \ln \Gamma(n-k+1) ] \]

◆ max()

template<typename T1 , typename T2 >

constexpr common_t< T1, T2 > gcem::max	(	const T1	x,
		const T2	y
	)

constexprnoexcept

Compile-time pairwise maximum function.

Parameters

x	a real-valued input.
y	a real-valued input.

Returns: Computes the maximum between x and y, where x and y have the same type (e.g., int, double, etc.)

◆ min()

template<typename T1 , typename T2 >

constexpr common_t< T1, T2 > gcem::min	(	const T1	x,
		const T2	y
	)

constexprnoexcept

Compile-time pairwise minimum function.

Parameters

x	a real-valued input.
y	a real-valued input.

Returns: Computes the minimum between x and y, where x and y have the same type (e.g., int, double, etc.)

◆ pow()

template<typename T1 , typename T2 >

constexpr common_t< T1, T2 > gcem::pow	(	const T1	base,
		const T2	exp_term
	)

constexprnoexcept

Compile-time power function.

Parameters

base	a real-valued input.
exp_term	a real-valued input.

Returns: Computes base raised to the power exp_term. In the case where exp_term is integral-valued, recursion by squaring is used, otherwise \( \text{base}^{\text{exp\_term}} = e^{\text{exp\_term} \log(\text{base})} \)

◆ round()

template<typename T >

constexpr return_t< T > gcem::round ( const T x )

constexprnoexcept

Compile-time round function.

Parameters

x	a real-valued input.

Returns: computes the rounding value of the input.

◆ sgn()

template<typename T >

constexpr int gcem::sgn ( const T x )

constexprnoexcept

Compile-time sign function.

Parameters

x	a real-valued input

Returns: a value \( y \) such that
\[ y = \begin{cases} 1 \ &\text{ if } x > 0 \\ 0 \ &\text{ if } x = 0 \\ -1 \ &\text{ if } x < 0 \end{cases} \]

◆ signbit()

template<typename T >

constexpr bool gcem::signbit ( const T x )

constexprnoexcept

Compile-time sign bit detection function.

Parameters

x	a real-valued input

Returns: return true if x is negative, otherwise return false.

◆ sin()

template<typename T >

constexpr return_t< T > gcem::sin ( const T x )

constexprnoexcept

Compile-time sine function.

Parameters

x	a real-valued input.

Returns: the sine function using
\[ \sin(x) = \frac{2\tan(x/2)}{1+\tan^2(x/2)} \]

◆ sinh()

template<typename T >

constexpr return_t< T > gcem::sinh ( const T x )

constexprnoexcept

Compile-time hyperbolic sine function.

Parameters

x	a real-valued input.

Returns: the hyperbolic sine function using
\[ \sinh(x) = \frac{\exp(x) - \exp(-x)}{2} \]

◆ sqrt()

template<typename T >

constexpr return_t< T > gcem::sqrt ( const T x )

constexprnoexcept

Compile-time square-root function.

Parameters

x	a real-valued input.

Returns: Computes \( \sqrt{x} \) using a Newton-Raphson approach.

◆ tan()

template<typename T >

constexpr return_t< T > gcem::tan ( const T x )

constexprnoexcept

Compile-time tangent function.

Parameters

x	a real-valued input.

Returns

the tangent function using

\[ \tan(x) = \dfrac{x}{1 - \dfrac{x^2}{3 - \dfrac{x^2}{5 - \ddots}}} \]

To deal with a singularity at \( \pi / 2 \), the following expansion is employed:

\[ \tan(x) = - \frac{1}{x-\pi/2} - \sum_{k=1}^\infty \frac{(-1)^k 2^{2k} B_{2k}}{(2k)!} (x - \pi/2)^{2k - 1} \]

where \( B_n \) is the n-th Bernoulli number.

◆ tanh()

template<typename T >

constexpr return_t< T > gcem::tanh ( const T x )

constexprnoexcept

Compile-time hyperbolic tangent function.

Parameters

x	a real-valued input.

Returns: the hyperbolic tangent function using
\[ \tanh(x) = \dfrac{x}{1 + \dfrac{x^2}{3 + \dfrac{x^2}{5 + \dfrac{x^2}{7 + \ddots}}}} \]

◆ tgamma()

template<typename T >

constexpr return_t< T > gcem::tgamma ( const T x )

constexprnoexcept

Compile-time gamma function.

Parameters

x	a real-valued input.

Returns

computes the ‘true’ gamma function

\[ \Gamma(x) = \int_0^\infty y^{x-1} \exp(-y) dy \]

using a polynomial form:

\[ \Gamma(x+1) \approx (x+g+0.5)^{x+0.5} \exp(-x-g-0.5) \sqrt{2 \pi} \left[ c_0 + \frac{c_1}{x+1} + \frac{c_2}{x+2} + \cdots + \frac{c_n}{x+n} \right] \]

where the value \( g \) and the coefficients \( (c_0, c_1, \ldots, c_n) \) are taken from Paul Godfrey, whose note can be found here: http://my.fit.edu/~gabdo/gamma.txt

◆ trunc()

template<typename T >

constexpr return_t< T > gcem::trunc ( const T x )

constexprnoexcept

Compile-time trunc function.

Parameters

x	a real-valued input.

Returns: computes the trunc-value of the input, essentially returning the integer part of the input.

Variable Documentation

◆ gauss_legendre_30_points

const long double gcem::gauss_legendre_30_points[30]

static

◆ gauss_legendre_30_weights

const long double gcem::gauss_legendre_30_weights[30]

static

◆ gauss_legendre_50_points

const long double gcem::gauss_legendre_50_points[50]

static

◆ gauss_legendre_50_weights

const long double gcem::gauss_legendre_50_weights[50]

static

Namespaces

Typedefs

Functions

Variables

Typedef Documentation

◆ common_return_t

◆ common_t

◆ GCLIM

◆ llint_t

◆ return_t

◆ uint_t

◆ ullint_t

Function Documentation

◆ abs()

◆ acos()

◆ acosh()

◆ asin()

◆ asinh()

◆ atan()

◆ atan2()

◆ atanh()

◆ beta()

◆ binomial_coef()

◆ ceil()

◆ copysign()

◆ cos()

◆ cosh()

◆ erf()

◆ erf_inv()

◆ exp()

◆ expm1()

◆ factorial()

◆ floor()

◆ fmod()

◆ gcd()

◆ hypot() [1/2]

◆ hypot() [2/2]

◆ incomplete_beta()

◆ incomplete_beta_inv()

◆ incomplete_gamma()

◆ incomplete_gamma_inv()

◆ inv_sqrt()

◆ lbeta()

◆ lcm()

◆ lgamma()

◆ lmgamma()

◆ log()

◆ log10()

◆ log1p()

◆ log2()

◆ log_binomial_coef()

◆ max()

◆ min()

◆ pow()

◆ round()

◆ sgn()

◆ signbit()

◆ sin()

◆ sinh()

◆ sqrt()

◆ tan()

◆ tanh()

◆ tgamma()

◆ trunc()

Variable Documentation

◆ gauss_legendre_30_points

◆ gauss_legendre_30_weights

◆ gauss_legendre_50_points

◆ gauss_legendre_50_weights