math_approx is a C++ library for math approximations.
Currently supported:
- sin/cos/tan
- arcsin/arccos/arctan
- exp/exp2/exp10/expm1
- log/log2/log10/log1p
- sinh/cosh/tanh
- arcsinh/arccosh/arctanh
- Sigmoid function
- Wright-Omega function
- Dilogarithm function
At the moment, most of these implementations have been "good enough" for my own use cases (both in terms of performance and accuracy). That said, I definitely believe that it's possible to achieve better results for many of these functions. If you have ideas for improving these approximations, either by:
- Modifying an approximation to achieve better accuracy with the same (or similar) performance
- Modifying an approximation to achieve better performance with the same (or similar) accuracy
then please get in touch with a GitHub issue or pull request!
math_approx is set up as a CMake INTERFACE library. To use it as
such, you'll need to add the following to your CMakeLists.txt file:
add_subdirectory(math_approx)
target_link_libraries(<your_target> math_approx)And then in your C++ code, you can use the approximations like so:
#include <math_approx/math_approx.hpp>
constexpr auto sin_half = math_approx::sin<5> (0.5f);To use math_approx without CMake, you'll need to add
/path/to/repo/include to your include path. If you're
compiling your program with MSVC, you may also need to
add the pre-processor definition _USE_MATH_DEFINES.
Most of the methods in this library are provided with template arguments which control the "order" of the approximation. The "order" typically refers to the order of a polynomial used in the approximation. In general, higher-order approximations will achieve greater accuracy, while taking longer to compute.
Since the approximations in this library are primarily based on polynomial approximations, I've tried to provide the details for how those polynomials were derived, by providing a zipped folder containing the Mathematica notebooks that were used to derive the polynomials. Since not everyone has access to Mathematica, the folder also contains a PDF version of each notebook. At the moment, I'm planning to upload an updated copy of the zipped folder with each release of the library, but if I can think of a better method of distribution, that doesn't involve adding the notebook files to the repository directly, I'll do that instead.
This library uses three approaches for measuring accuracy:
- Absolute error (
Error = |actual - approx|) - Relative error (
Error = |(actual - approx) / actual|) - ULP Distance
At the moment, the approximations in this library have been primarily tested with single-precision floating-point numbers. It is possible (maybe even likely) that most of the approximations do not achieve sufficient accuracy for double-precision computations.
The library has been mostly developed and tested with C++20, with a little bit of effort to provide compatibility with C++17. Personally, I would rather not extend support to C++14 or earlier.
These approximations are intended to work for both scalar floating-point data types, as well as SIMD floating-point data types. At the moment, the library is set up to be compatible with the XSIMD library. That said, I would like to make it as easy as possible to use this library with other SIMD libraries (or matrix math libraries), so if anyone has some suggestions, please let me know!
The majority of the approximations in this library are implemented
so as to be constexpr-compatible. That said, there are some
approximations that are only constexpr if the compiler supports
std::bit_cast (typically C++20 and later), and some that cannot
be made constexpr because they depend on std::sqrt. If someone
knows of any portable constexpr-compatible implementations of these
methods, I would be happy to add them to the library!
math_approx is open source, and is licensed under the
BSD 3-clause license.
Enjoy!