GCE-Math (Generalized Constant Expression Math) is a templated C++ library enabling compile-time computation of mathematical functions.

  • The library is written in C++11 constexpr format, and is C++11/14/17 compatible.
  • Continued fraction and series expansions are implemented using recursive templates.
  • The gcem:: syntax is identical to the C++ standard library (std::).
  • Tested and accurate to machine precision against the C++ standard library.
  • Released under a permissive, non-GPL license.

Author: Keith O’Hara

License: Apache 2.0


The library is actively maintained, and is still being extended. A list of features includes:

  • basic library functions:
    • abs, max, min, pow, sqrt
    • exp, expm1, log, log1p, and more
  • trigonometric functions:
    • basic: cos, sin, tan
    • inverse: acos, asin, atan, atan2
  • hyperbolic (area) functions:
    • cosh, sinh, tanh, acosh, asinh, atanh
  • algorithms:
    • gcd, lcm
  • special functions:
    • factorials and the binomial coefficient: factorial, binomial_coef
    • beta, gamma, and multivariate gamma functions: beta, lbeta, lgamma, tgamma, lmgamma
    • the Gaussian error function and inverse error function: erf, erf_inv
    • (regularized) incomplete beta and incomplete gamma functions: incomplete_beta, incomplete_gamma
    • inverse incomplete beta and incomplete gamma functions: incomplete_beta_inv, incomplete_gamma_inv

General Syntax

GCE-Math functions are written as C++ templates with constexpr specifiers, the format of which might be confusing to users unfamiliar with template-based programming. As an example, the Gaussian error function (erf) is defined as:

template<typename T>
erf(const T x) noexcept;

where a set of internal templated constexpr functions will implement a continued fraction expansion to return a value of type return_t<T>. This output type (‘return_t<T>’) is generally determined by the input type, e.g., int, float, double, long double, etc. When T is an intergral type, the output will be upgraded to return_t<T> = double, otherwise return_t<T> = T. For types not covered by std::is_integral, recasts should be used.