Basic functions

template<typename T>
constexpr T abs(const T x) noexcept

Compile-time absolute value function.

Parameters

x – a real-valued input.

Returns

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

template<typename T>
constexpr return_t<T> ceil(const T x) noexcept

Compile-time ceil function.

Parameters

x – a real-valued input.

Returns

computes the ceiling-value of the input.

template<typename T1, typename T2>
constexpr T1 copysign(const T1 x, const T2 y) noexcept

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.

template<typename T>
constexpr return_t<T> exp(const T x) noexcept

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 \( r \in [-0.5,0.5] \).

template<typename T>
constexpr return_t<T> expm1(const T x) noexcept

Compile-time exponential-minus-1 function.

Parameters

x – a real-valued input.

Returns

\( \exp(x) - 1 \) using

\[ \exp(x) = \sum_{k=0}^\infty \dfrac{x^k}{k!} \]

template<typename T>
constexpr T factorial(const T x) noexcept

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).

template<typename T>
constexpr return_t<T> floor(const T x) noexcept

Compile-time floor function.

Parameters

x – a real-valued input.

Returns

computes the floor-value of the input.

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.

Parameters
  • x – a real-valued input.

  • y – a real-valued input.

Returns

computes the floating-point remainder of \( x / y \) (rounded towards zero) using

\[ \text{fmod}(x,y) = x - \text{trunc}(x/y) \times y \]

template<typename T>
constexpr return_t<T> log(const T x) noexcept

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.

template<typename T>
constexpr return_t<T> log1p(const T x) noexcept

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

Parameters

x – a real-valued input.

Returns

\( \log_e(x+1) \) using

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

template<typename T>
constexpr return_t<T> log2(const T x) noexcept

Compile-time binary logarithm function.

Parameters

x – a real-valued input.

Returns

\( \log_2(x) \) using

\[ \log_{2}(x) = \frac{\log_e(x)}{\log_e(2)} \]

template<typename T>
constexpr return_t<T> log10(const T x) noexcept

Compile-time common logarithm function.

Parameters

x – a real-valued input.

Returns

\( \log_{10}(x) \) using

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

template<typename T1, typename T2>
constexpr common_t<T1, T2> max(const T1 x, const T2 y) noexcept

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.)

template<typename T1, typename T2>
constexpr common_t<T1, T2> min(const T1 x, const T2 y) noexcept

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.)

template<typename T1, typename T2>
constexpr common_t<T1, T2> pow(const T1 base, const T2 exp_term) noexcept

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 \( \text{base}^{\text{exp\_term}} = e^{\text{exp\_term} \log(\text{base})} \)

template<typename T>
constexpr return_t<T> round(const T x) noexcept

Compile-time round function.

Parameters

x – a real-valued input.

Returns

computes the rounding value of the input.

template<typename T>
constexpr bool signbit(const T x) noexcept

Compile-time sign bit detection function.

Parameters

x – a real-valued input

Returns

return true if x is negative, otherwise return false.

template<typename T>
constexpr int sgn(const T x) noexcept

Compile-time sign function.

Parameters

x – a real-valued input

Returns

a value \( y \) such that

\[ y = \begin{cases} 1 \ &\text{ if } x > 0 \\ 0 \ &\text{ if } x = 0 \\ -1 \ &\text{ if } x < 0 \end{cases} \]

template<typename T>
constexpr return_t<T> sqrt(const T x) noexcept

Compile-time square-root function.

Parameters

x – a real-valued input.

Returns

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

template<typename T>
constexpr return_t<T> inv_sqrt(const T x) noexcept

Compile-time inverse-square-root function.

Parameters

x – a real-valued input.

Returns

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

template<typename T>
constexpr return_t<T> trunc(const T x) noexcept

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.