GCE-Math

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

Features:

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

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

basic library functions:
- abs, exp, log, max, min, pow, sqrt, gcd, lcm, and more
trigonometric functions:
- basic: cos, sin, tan
- inverse: acos, asin, atan, atan2
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:

Installation and Tests

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"

Conda

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

conda install gcem -c kthohr

CMake

You can also install the library from source using CMake.

# clone gcem from GitHub
git clone https://github.com/kthohr/gcem ./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
make
./run_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
./exp.test

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. As an example, the Gaussian error function (erf) is defined as:

template<typename T>
constexpr
return_t<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.

Examples

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
	.cfi_startproc
## BB#0:
	push	rbp
Lcfi0:
	.cfi_def_cfa_offset 16
Lcfi1:
	.cfi_offset rbp, -16
	mov	rbp, rsp
Lcfi2:
	.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
	ret
	.cfi_endproc

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:

LCPI0_0:
	.long	1069547520              ## float 1.5
	.section	__TEXT,__literal16,16byte_literals
	.p2align	4
LCPI0_1:
	.quad	-622431863250851168     ## x86_fp80 -0.120782237635245167208
	.short	49147
	.space	6
	.section	__TEXT,__text,regular,pure_instructions
	.globl	_main
	.p2align	4, 0x90
_main:                                  ## @main
	.cfi_startproc
## BB#0:
	push	rbp
Lcfi0:
	.cfi_def_cfa_offset 16
Lcfi1:
	.cfi_offset rbp, -16
	mov	rbp, rsp
Lcfi2:
	.cfi_def_cfa_register rbp
	xor	eax, eax
	mov	dword ptr [rbp - 4], 0
	fld	dword ptr [rip + LCPI0_0]
	fstp	xword ptr [rbp - 32]
	fld	xword ptr [rip + LCPI0_1]
	fstp	xword ptr [rbp - 48]
	pop	rbp
	ret
	.cfi_endproc

Related libraries

For a library built on the GCEM compile-time functionality, see StatsLib.

maksimus1210/gcem