Generalized Constant Expression Math

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GCE-Math (Generalized Constant Expression Math, or gcem) 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 floating-point precision against the C++ standard library.
  • Released under a permissive, non-GPL license.

Author: Keith O'Hara



Status and Documentation


  • basic C++ standard library functions:
    • abs, exp, log, max, min, pow, sqrt
    • , gcd, lcm, and more
  • trigonometric functions:
    • basic: cos, sin, tan
    • inverse: acos, asin, atan
  • hyperbolic (area) functions:
    • cosh, sinh, tanh, acosh, asinh, atanh
  • 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

Full documentation is available online:

Documentation Status

Installation and Tests

  • The source code is available on GitHub. Download the latest version with git:
  •     git clone ./gcem

GCE-Math is a header-only library and does not require any additional libraries (beyond a C++11 compatible compiler). Simply add the header files to your project using:

#include "gcem.hpp"


You can install GCE-Math using the conda package manager.

conda install gcem -c kthohr
Anaconda Builds


You can also install the library from source using CMake.

# clone gcem from GitHub
git clone ./gcem

# make a build directory
cd ./gcem
mkdir build
cd build

# generate Makefiles and install
cmake .. -DCMAKE_INSTALL_PREFIX=/gcem/install/location
make install

For example, /gcem/install/location could be /usr/local/.

Test Suite

There are two ways to build the test suite. On Unix-alike systems, a Makefile is available under tests/.

cd ./gcem/tests

With CMake, the option BUILD_TESTS=1 generates the necessary Makefiles to build the test suite.

cd ./gcem
mkdir build

cd build
cmake ../ -DBUILD_TESTS=1 -DCMAKE_INSTALL_PREFIX=/gcem/install/location
make gcem_tests

cd tests

Jupyter Notebook

You can test the library online using an interactive Jupyter notebook:


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. For example, the Gaussian error function (erf) is defined as:
template<typename T>
erf(const T x);
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.


To calculate 10!:

#include "gcem.hpp"

int main()
    constexpr int x = 10;
    constexpr int res = gcem::factorial(x);

    return 0;

Inspecting the assembly code generated by Clang:

    _main:                                  ## @main
## BB#0:
	push	rbp
	.cfi_def_cfa_offset 16
	.cfi_offset rbp, -16
	mov	rbp, rsp
	.cfi_def_cfa_register rbp
	xor	eax, eax
	mov	dword ptr [rbp - 4], 0
	mov	dword ptr [rbp - 8], 10
	mov	dword ptr [rbp - 12], 3628800
	pop	rbp

We see that a function call has been replaced by a numeric value (10! = 3628800).