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Update tools.md with more documentation about U-test Fixes https://github.com/google/benchmark/issues/1491
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docs/tools.md
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docs/tools.md
@ -186,6 +186,146 @@ Benchmark Time CPU Time Old
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This is a mix of the previous two modes, two (potentially different) benchmark binaries are run, and a different filter is applied to each one.
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As you can note, the values in `Time` and `CPU` columns are calculated as `(new - old) / |old|`.
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### Note: Interpreting the output
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Performance measurements are an art, and performance comparisons are doubly so.
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Results are often noisy and don't necessarily have large absolute differences to
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them, so just by visual inspection, it is not at all apparent if two
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measurements are actually showing a performance change or not. It is even more
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confusing with multiple benchmark repetitions.
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Thankfully, what we can do, is use statistical tests on the results to determine
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whether the performance has statistically-significantly changed. `compare.py`
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uses [Mann–Whitney U
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test](https://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test), with a null
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hypothesis being that there's no difference in performance.
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**The below output is a summary of a benchmark comparison with statistics
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provided for a multi-threaded process.**
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```
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Benchmark Time CPU Time Old Time New CPU Old CPU New
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-----------------------------------------------------------------------------------------------------------------------------
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benchmark/threads:1/process_time/real_time_pvalue 0.0000 0.0000 U Test, Repetitions: 27 vs 27
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benchmark/threads:1/process_time/real_time_mean -0.1442 -0.1442 90 77 90 77
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benchmark/threads:1/process_time/real_time_median -0.1444 -0.1444 90 77 90 77
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benchmark/threads:1/process_time/real_time_stddev +0.3974 +0.3933 0 0 0 0
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benchmark/threads:1/process_time/real_time_cv +0.6329 +0.6280 0 0 0 0
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OVERALL_GEOMEAN -0.1442 -0.1442 0 0 0 0
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```
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--------------------------------------------
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Here's a breakdown of each row:
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**benchmark/threads:1/process_time/real_time_pvalue**: This shows the _p-value_ for
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the statistical test comparing the performance of the process running with one
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thread. A value of 0.0000 suggests a statistically significant difference in
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performance. The comparison was conducted using the U Test (Mann-Whitney
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U Test) with 27 repetitions for each case.
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**benchmark/threads:1/process_time/real_time_mean**: This shows the relative
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difference in mean execution time between two different cases. The negative
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value (-0.1442) implies that the new process is faster by about 14.42%. The old
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time was 90 units, while the new time is 77 units.
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**benchmark/threads:1/process_time/real_time_median**: Similarly, this shows the
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relative difference in the median execution time. Again, the new process is
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faster by 14.44%.
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**benchmark/threads:1/process_time/real_time_stddev**: This is the relative
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difference in the standard deviation of the execution time, which is a measure
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of how much variation or dispersion there is from the mean. A positive value
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(+0.3974) implies there is more variance in the execution time in the new
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process.
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**benchmark/threads:1/process_time/real_time_cv**: CV stands for Coefficient of
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Variation. It is the ratio of the standard deviation to the mean. It provides a
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standardized measure of dispersion. An increase (+0.6329) indicates more
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relative variability in the new process.
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**OVERALL_GEOMEAN**: Geomean stands for geometric mean, a type of average that is
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less influenced by outliers. The negative value indicates a general improvement
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in the new process. However, given the values are all zero for the old and new
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times, this seems to be a mistake or placeholder in the output.
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-----------------------------------------
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Let's first try to see what the different columns represent in the above
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`compare.py` benchmarking output:
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1. **Benchmark:** The name of the function being benchmarked, along with the
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size of the input (after the slash).
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2. **Time:** The average time per operation, across all iterations.
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3. **CPU:** The average CPU time per operation, across all iterations.
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4. **Iterations:** The number of iterations the benchmark was run to get a
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stable estimate.
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5. **Time Old and Time New:** These represent the average time it takes for a
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function to run in two different scenarios or versions. For example, you
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might be comparing how fast a function runs before and after you make some
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changes to it.
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6. **CPU Old and CPU New:** These show the average amount of CPU time that the
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function uses in two different scenarios or versions. This is similar to
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Time Old and Time New, but focuses on CPU usage instead of overall time.
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In the comparison section, the relative differences in both time and CPU time
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are displayed for each input size.
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A statistically-significant difference is determined by a **p-value**, which is
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a measure of the probability that the observed difference could have occurred
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just by random chance. A smaller p-value indicates stronger evidence against the
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null hypothesis.
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**Therefore:**
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1. If the p-value is less than the chosen significance level (alpha), we
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reject the null hypothesis and conclude the benchmarks are significantly
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different.
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2. If the p-value is greater than or equal to alpha, we fail to reject the
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null hypothesis and treat the two benchmarks as similar.
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The result of said the statistical test is additionally communicated through color coding:
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```diff
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+ Green:
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```
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The benchmarks are _**statistically different**_. This could mean the
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performance has either **significantly improved** or **significantly
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deteriorated**. You should look at the actual performance numbers to see which
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is the case.
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```diff
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- Red:
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```
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The benchmarks are _**statistically similar**_. This means the performance
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**hasn't significantly changed**.
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In statistical terms, **'green'** means we reject the null hypothesis that
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there's no difference in performance, and **'red'** means we fail to reject the
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null hypothesis. This might seem counter-intuitive if you're expecting 'green'
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to mean 'improved performance' and 'red' to mean 'worsened performance'.
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```bash
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But remember, in this context:
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'Success' means 'successfully finding a difference'.
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'Failure' means 'failing to find a difference'.
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```
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Also, please note that **even if** we determine that there **is** a
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statistically-significant difference between the two measurements, it does not
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_necessarily_ mean that the actual benchmarks that were measured **are**
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different, or vice versa, even if we determine that there is **no**
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statistically-significant difference between the two measurements, it does not
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necessarily mean that the actual benchmarks that were measured **are not**
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different.
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### U test
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If there is a sufficient repetition count of the benchmarks, the tool can do
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