https://github.com/expander/fast_clog
Fast implementation of complex logarithm
https://github.com/expander/fast_clog
Last synced: 4 months ago
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Fast implementation of complex logarithm
- Host: GitHub
- URL: https://github.com/expander/fast_clog
- Owner: Expander
- Created: 2020-06-03T10:46:49.000Z (almost 6 years ago)
- Default Branch: master
- Last Pushed: 2020-06-06T21:50:22.000Z (almost 6 years ago)
- Last Synced: 2025-01-12T09:22:31.975Z (about 1 year ago)
- Language: C++
- Size: 13.7 KB
- Stars: 1
- Watchers: 3
- Forks: 0
- Open Issues: 0
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Metadata Files:
- Readme: README.md
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README
# Fast implementation of complex logarithms
Real and complex logarithms frequently appear in high-energy physics
calculations. In numerical applications, these logarithms have to be
calculated hundreds to millions of times, depending on the considered
example. For this reason, a fast numerical evaluation is of high
importance, in particular in studies of models with a parameter space
with a large dimensionality.
The usual implementation of the complex logarithm in mathematical
libraries is of the form
log(z) = log(|z|) + i*arg(z)
where `|z|` is the magnitude and `arg(z)` is the polar angle of the
complex number `z`. The evaluation of this expression requires the
calculation of a square root (sometimes implemented by calling the
`hypot()` function) and sometimes a small extra overhead when
calculating the polar angle with `arg(z)`.
The complex logarithm can be calculated in a more performant way by
writing it as
log(z) = 0.5*log(re*re + im*im) + i*atan2(im, re)
where `re = Re(z)` and `im = Im(z)`. This expression avoids the
calculation of the square root and a potential overhead from the
`arg(z)` function. Note, however, that the more performant form is
restricted to complex numbers with `|z|^2 < inf`, while the slower
form is in principle valid for `|z| < inf`.
The following table compares the average run-time of the `std::log()`
function from the C++ STL with more performant implementations
contained in this package (compiler: clang++ 11, compiler flags:
`-Ofast -ffast-math`, CPU: i7-5600U):
| Function | run-time in ms |
|-----------------------------------------------|----------------|
| `std::log(const std::complex&)` (C++) | 5.6e-05 |
| `fast_clog` (C++) | 3.7e-05 |
| `fast_clog` (C) | 3.9e-05 |
| | |
| `log(double complex)` (FORTRAN) | 6.0e-05 |
| `fast_clog` (FORTRAN) | 4.2e-05 |
License: CC0