This improves accuracy for the bessel function at large arguments, and this in turn
should improve the quality of the Kaiser window. It also improves the
performance of the bessel function and hence build_filter by ~ 20%.
Details are given below.
Algorithm: taken from the Boost project, who have done a detailed
investigation of the accuracy of their method, as compared with e.g the
GNU Scientific Library (GSL):
http://www.boost.org/doc/libs/1_52_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/mbessel.html.
Boost source code (also cited and licensed in the code):
https://searchcode.com/codesearch/view/14918379/.
Accuracy: sample values may be obtained as follows. i0 denotes the old bessel code,
i0_boost the approach here, and i0_real an arbitrary precision result (truncated) from Wolfram Alpha:
type "bessel i0(6.0)" to reproduce. These are evaluation points that occur for
the default kaiser_beta = 9.
Some illustrations:
bessel(8.0)
i0 (8.000000) = 427.564115721804739678191254
i0_boost(8.000000) = 427.564115721804796521610115
i0_real (8.000000) = 427.564115721804785177396791
bessel(6.0)
i0 (6.000000) = 67.234406976477956163762428
i0_boost(6.000000) = 67.234406976477970374617144
i0_real (6.000000) = 67.234406976477975326188025
Reason for accuracy: Main accuracy benefits come at larger bessel arguments, where the
Taylor-Maclaurin method is not that good: 23+ iterations
(at large arguments, since the series is about 0) can cause
significant floating point error accumulation.
Benchmarks: Obtained on x86-64, Haswell, GNU/Linux via a loop calling
build_filter 1000 times:
test: fate-swr-resample-dblp-44100-2626
new:
995894468 decicycles in build_filter(loop 1000), 256 runs, 0 skips
1029719302 decicycles in build_filter(loop 1000), 512 runs, 0 skips
984101131 decicycles in build_filter(loop 1000), 1024 runs, 0 skips
old:
1250020763 decicycles in build_filter(loop 1000), 256 runs, 0 skips
1246353282 decicycles in build_filter(loop 1000), 512 runs, 0 skips
1220017565 decicycles in build_filter(loop 1000), 1024 runs, 0 skips
A further ~ 5% may be squeezed by enabling -ftree-vectorize. However,
this is a separate issue from this patch.
Reviewed-by: Michael Niedermayer <michael@niedermayer.cc>
Signed-off-by: Ganesh Ajjanagadde <gajjanagadde@gmail.com>
Ganesh Ajjanagadde
a5202bc968