perf: Try fast pow for EigenStepper step size scaling

[andiwand]

Currently the step scaling calculation has a big impact on the EigenStepper performance. In this PR the pow(x, 0.25) is approximated with fastPow (similar to fast inverse square root) which relies on the bit representation of the floating point number.

The assumption is that we do not really care about the precision of this value but rather that it gives a good approximation for the step size scaling.

Edit: After measuring the performance it seems like there is no measurable improvement. I made this approximation compile time optional for now. See this #3153 (comment) for more details

References

• https://martin.ankerl.com/2012/01/25/optimized-approximative-pow-in-c-and-cpp

[AJPfleger]

Since we are clamping anyway, we could look at the representation of double x and the clamping limits 0.25 and 4.

so we want to end up with a number in the range $(1 \cdot 2^{-2},1 \cdot 2^{2})$ for this our initial x needs to be in $(1 \cdot 2^{-2 \cdot 4},1 \cdot 2^{2 \cdot 4})$

we could then construct the function using std::frexp, maybe like this:

int exponent;
double significant = std::frexp(x, &i);

// -7 instead of -8, because significant is in the interval [0.5, 1)
if ( i < -7) {
    return 0.25;
} else if ( i > 8) {
    return 4;
} else {
    int remainder = exponent % 4;
    // TODO extra-step on `significant` using remainder and pow-estimate
    return ldexp(significant, exponent / 4);
}

Even with the other technique, it might be useful to do the clamping before.

[github-actions]

📊: Physics performance monitoring for 441aa99

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[andiwand]

Apparently std::sqrt(std::sqrt(x)) is pretty fast.

Benchmarking a=247.842, b=0.25...
- void: 20000 runs of 10000 iteration(s), 119.2ms total, 5.8750+/-0.0005µs per run, 0.588+/-0.005ns per iteration
- std::pow: 20000 runs of 10000 iteration(s), 1134.6ms total, 56.2090+/-0.0032µs per run, 5.621+/-0.032ns per iteration
- std::exp: 20000 runs of 10000 iteration(s), 901.2ms total, 44.8340+/-0.0016µs per run, 4.483+/-0.016ns per iteration
- std::sqrt: 20000 runs of 10000 iteration(s), 197.8ms total, 9.8750+/-0.0005µs per run, 0.988+/-0.005ns per iteration
- fastPow: 20000 runs of 10000 iteration(s), 147.6ms total, 7.4590+/-0.0011µs per run, 0.746+/-0.011ns per iteration
- fastPowMorePrecise: 20000 runs of 10000 iteration(s), 209.8ms total, 10.5830+/-0.0005µs per run, 1.058+/-0.005ns per iteration
- fastPowAnother: 20000 runs of 10000 iteration(s), 178.2ms total, 9.0000+/-0.0080µs per run, 0.900+/-0.080ns per iteration

fastPow is still faster but it does not have an impact on the overall stepper performance.

I suspect the reason for higher performance when removing the second scaling is that our current scaling strategy increases and decreases the step size periodically which results in a lot of attempted steps and possibly mispredicted branches.

I will make the function selection configurable for now.

[acts-project-service]

✅ Athena integration test results [620069e]

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status	job	report
🟢	run_unit_tests
🟢	test_ActsEFTrackFit
🟢	test_ActsPersistifySeeds
🟢	test_ActsBenchmarkWithSpot
🟢	test_ActsAnalogueClustering
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🟢	test_ActsValidateActsCoreSpacePoints
🟢	test_ActsValidateActsSpacePoints
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🟢	test_ActsPersistifyEDM
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🟢	test_ActsKfRefitting
🟢	test_ActsExtrapolationAlgTest
🟢	test_ActsITkTest
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🟢	run_workflow_tests_run3_data
🟢	run_art_test: test_data18_13TeV_1000evt
🟢	run_art_test: test_ttbarPU40_reco