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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++11/14/17/20 compatible.
  • Continued fraction expansions and series expansions are implemented using recursive templates.
  • The gcem:: syntax is identical to that of 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

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

  • Basic library functions:
    • abs, max, min, pow, sqrt, inv_sqrt,
    • ceil, floor, round, trunc, fmod,
    • exp, expm1, log, log1p, log2, log10, 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

Full documentation is available online:

Documentation Status

A PDF version of the documentation is available here.


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

#include "gcem.hpp"


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

conda install -c conda-forge gcem


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 GCEM_BUILD_TESTS=1 generates the necessary Makefiles to build the test suite.

cd ./gcem
mkdir build

cd build
cmake ../ -DGCEM_BUILD_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 appear 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) noexcept;

A set of internal templated constexpr functions will implement a continued fraction expansion and return a value of type return_t<T>. The output type (‘return_t<T>’) is generally determined by the input type, e.g., int, float, double, long double, etc.; when T is an integral 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 7.0.0:

        push    rbp
        mov     rbp, rsp
        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).

Similarly, to compute the log Gamma function at a point:

#include "gcem.hpp"

int main()
    constexpr long double x = 1.5;
    constexpr long double res = gcem::lgamma(x);

    return 0;

Assembly code:

        .long   1069547520              # float 1.5
        .quad   -622431863250842976     # x86_fp80 -0.120782237635245222719
        .short  49147
        .zero   6
main:                                   # @main
        push    rbp
        mov     rbp, rsp
        xor     eax, eax
        mov     dword ptr [rbp - 4], 0
        fld     dword ptr [rip + .LCPI0_0]
        fstp    tbyte ptr [rbp - 32]
        fld     tbyte ptr [rip + .LCPI0_1]
        fstp    tbyte ptr [rbp - 48]
        pop     rbp

Related libraries

  • StatsLib is built on GCEM’s compile-time functionality.

A C++ compile-time math library using generalized constant expressions

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