random: add benchmarks and drop unnecessary Shuffle function #30396

sipa · 2024-07-05T14:54:13Z

This adds benchmarks for various operations on FastRandomContext and InsecureRandomContext, and then removes the ad-hoc Shuffle functions, now that it appears that standard library std::shuffle has comparable performance. The other reason for keeping Shuffle, namely the fact that libstdc++ used self-move (which debug mode panics on) has been fixed as well (see #29625 (comment)).

DrahtBot · 2024-07-05T14:54:16Z

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sipa · 2024-07-05T15:02:45Z

I've also benchmarked Deniel Lemire's nearly divisionless random integer generation (which libstdc++'s std::shuffle switched to), for randrange itself, but opted not to include this in this PR.

For InsecureRandomContext it's unambiguously faster than the current code (but nothing performance-critical relies on InsecureRandomContext::randrange, so far). For FastRandomContext the advantage is there, but it's more tricky to achieve (it either needs fast 128-bit integer multiplication, or needs specialization for 32-bit randrange where a 64-bit multiply suffices).

src/bench/random.cpp

hodlinator

~~Is determinism of std::shuffle controlled/controllable? If we don't control it maybe it would be better to keep our custom one to ease reproducibility of test failures, despite it being slower.~~
I see now that you are passing the proper context. 👍

hodlinator · 2024-07-05T19:13:06Z

src/bench/random.cpp

251438 seems to be a fine hook for manhole covers. :)
Worth documenting the choice of seed initialization value?

It's the alphabetic position of the letters of "bench" (2-5-14-3-8) concatenated, but it's really just an arbitrary constant.

src/bench/random.cpp

Also rename the benchmark names to match the operation names

Benchmarks show it is no longer faster with modern standard C++ libraries, and the debug-mode failure due to self-move has been fixed as well.

hodlinator

Ran benchmarks on da28a26 with..

Clang, non-debug, Linux

$ src/bench/bench_bitcoin --filter=".*Random.*"

|           ns/number |            number/s |    err% |      ins/number |      cyc/number |    IPC |     bra/number |   miss% |     total | benchmark
|--------------------:|--------------------:|--------:|----------------:|----------------:|-------:|---------------:|--------:|----------:|:----------
|               12.84 |       77,883,564.55 |    0.2% |          103.12 |           47.37 |  2.177 |          15.66 |    4.2% |      0.01 | `FastRandom_Shuffle100`
|               10.35 |       96,651,871.86 |    0.1% |          116.75 |           38.15 |  3.060 |           9.38 |    0.0% |      0.01 | `FastRandom_rand32`
|               19.03 |       52,549,720.82 |    0.2% |          214.50 |           70.12 |  3.059 |          15.75 |    0.0% |      0.01 | `FastRandom_rand64`
|               14.56 |       68,674,993.41 |    0.2% |          149.17 |           53.67 |  2.779 |          14.49 |    1.2% |      0.01 | `FastRandom_randbits`
|                1.69 |      591,854,532.68 |    0.0% |           11.38 |            6.23 |  1.826 |           1.26 |    0.2% |      0.01 | `FastRandom_randbool`
|               12.11 |       82,569,289.00 |    0.1% |           95.33 |           44.68 |  2.133 |          13.70 |    4.4% |      0.01 | `FastRandom_randrange100`
|               13.55 |       73,808,837.05 |    0.6% |          108.46 |           49.99 |  2.170 |          14.63 |    4.4% |      0.01 | `FastRandom_randrange1000`
|               17.80 |       56,164,658.15 |    0.1% |          156.81 |           65.65 |  2.389 |          18.39 |    3.3% |      0.20 | `FastRandom_randrange1000000`
|               13.08 |       76,440,558.82 |    0.2% |          134.02 |           48.23 |  2.779 |          12.42 |    0.7% |      0.01 | `FastRandom_stdshuffle100`
|                7.55 |      132,409,503.40 |    0.9% |           52.83 |           27.85 |  1.897 |          10.39 |    6.3% |      0.01 | `InsecureRandom_Shuffle100`
|                1.70 |      589,139,570.89 |    0.2% |           19.50 |            6.25 |  3.118 |           1.50 |    0.0% |      0.01 | `InsecureRandom_rand32`
|                1.11 |      901,858,137.81 |    0.0% |            7.00 |            4.09 |  1.711 |           0.00 |    0.0% |      0.01 | `InsecureRandom_rand64`
|                2.10 |      475,829,327.28 |    0.4% |           20.89 |            7.75 |  2.697 |           2.48 |    0.0% |      0.01 | `InsecureRandom_randbits`
|                1.40 |      714,972,427.11 |    0.1% |           12.25 |            5.16 |  2.374 |           1.02 |    0.0% |      0.01 | `InsecureRandom_randbool`
|                6.46 |      154,870,239.99 |    0.3% |           39.00 |           23.80 |  1.639 |           8.14 |    7.7% |      0.01 | `InsecureRandom_randrange100`
|                7.08 |      141,237,665.66 |    0.4% |           39.80 |           26.10 |  1.525 |           7.97 |    7.9% |      0.01 | `InsecureRandom_randrange1000`
|                6.49 |      154,065,899.22 |    0.6% |           43.93 |           23.80 |  1.846 |           7.80 |    7.0% |      0.07 | `InsecureRandom_randrange1000000`
|                3.61 |      276,805,356.11 |    0.1% |           34.38 |           13.33 |  2.580 |           3.98 |    0.3% |      0.01 | `InsecureRandom_stdshuffle100`

FastRandom - std::shuffle slower by ~2%.
InsecureRandom - std::shuffle is ~2.1x faster, the case for migrating is strong here.

GCC equivalent

$ src/bench/bench_bitcoin --filter=".*Random.*"

|           ns/number |            number/s |    err% |      ins/number |      cyc/number |    IPC |     bra/number |   miss% |     total | benchmark
|--------------------:|--------------------:|--------:|----------------:|----------------:|-------:|---------------:|--------:|----------:|:----------
|               10.90 |       91,706,416.37 |    0.4% |           73.17 |           40.22 |  1.819 |           8.91 |    7.6% |      0.01 | `FastRandom_Shuffle100`
|               10.29 |       97,154,304.49 |    0.1% |          115.31 |           37.96 |  3.037 |           8.94 |    0.0% |      0.01 | `FastRandom_rand32`
|               18.97 |       52,727,508.94 |    0.4% |          210.63 |           69.79 |  3.018 |          15.88 |    0.0% |      0.01 | `FastRandom_rand64`
|               11.56 |       86,507,143.42 |    0.5% |          122.79 |           42.62 |  2.881 |          10.55 |    0.4% |      0.01 | `FastRandom_randbits`
|                1.68 |      594,607,147.44 |    0.1% |           12.32 |            6.20 |  1.987 |           1.25 |    0.2% |      0.01 | `FastRandom_randbool`
|                9.99 |      100,126,193.36 |    0.4% |           61.84 |           36.83 |  1.679 |           6.91 |    9.1% |      0.01 | `FastRandom_randrange100`
|               11.50 |       86,963,003.45 |    0.3% |           75.50 |           42.40 |  1.781 |           7.89 |    8.1% |      0.01 | `FastRandom_randrange1000`
|               15.36 |       65,104,420.98 |    0.4% |          122.93 |           56.09 |  2.192 |          11.41 |    5.2% |      0.17 | `FastRandom_randrange1000000`
|               14.70 |       68,050,253.74 |    0.0% |          138.64 |           54.21 |  2.558 |          11.02 |    0.1% |      0.01 | `FastRandom_stdshuffle100`
|                8.32 |      120,167,896.55 |    0.4% |           51.58 |           30.66 |  1.682 |           7.84 |    8.3% |      0.01 | `InsecureRandom_Shuffle100`
|                1.09 |      920,716,470.48 |    0.0% |           13.00 |            4.01 |  3.244 |           1.00 |    0.0% |      0.01 | `InsecureRandom_rand32`
|                0.88 |    1,133,795,846.45 |    0.1% |            7.00 |            3.25 |  2.153 |           0.00 |   57.0% |      0.01 | `InsecureRandom_rand64`
|                1.66 |      601,775,499.66 |    1.2% |           19.86 |            6.11 |  3.251 |           1.98 |    0.0% |      0.01 | `InsecureRandom_randbits`
|                0.56 |    1,771,130,575.63 |    0.1% |            4.28 |            2.08 |  2.057 |           1.00 |    0.0% |      0.01 | `InsecureRandom_randbool`
|                6.97 |      143,428,585.12 |    0.2% |           30.93 |           25.70 |  1.203 |           5.84 |   10.5% |      0.01 | `InsecureRandom_randrange100`
|                7.25 |      137,895,833.08 |    0.2% |           32.18 |           26.75 |  1.203 |           5.78 |   10.6% |      0.01 | `InsecureRandom_randrange1000`
|                7.37 |      135,732,504.89 |    0.4% |           37.62 |           26.87 |  1.400 |           5.81 |    9.2% |      0.08 | `InsecureRandom_randrange1000000`
|                5.81 |      172,220,492.06 |    0.1% |           33.82 |           21.42 |  1.579 |           3.09 |    0.3% |      0.01 | `InsecureRandom_stdshuffle100`

FastRandom - std::shuffle shows a performance loss of 26%. :/
InsecureRandom - std::shuffle wins with ~1.4x speed multiplier.
(Side note: std::shuffle seems much better than Shuffle when it comes to branch misses in this configuration (already significantly better in Clang)).

GCC in debug mode

$ src/bench/bench_bitcoin --filter=".*Random.*"
Warning, results might be unstable:
* DEBUG defined

Recommendations
* Make sure you compile for Release

|           ns/number |            number/s |    err% |      ins/number |      cyc/number |    IPC |     bra/number |   miss% |     total | benchmark
|--------------------:|--------------------:|--------:|----------------:|----------------:|-------:|---------------:|--------:|----------:|:----------
|               96.61 |       10,350,962.32 |    0.4% |          844.51 |          356.24 |  2.371 |         124.47 |    0.5% |      0.01 | `FastRandom_Shuffle100`
|              182.54 |        5,478,307.47 |    0.0% |        1,703.98 |          673.49 |  2.530 |         204.73 |    0.1% |      0.01 | `FastRandom_rand32`
|              342.44 |        2,920,180.80 |    0.0% |        3,197.03 |        1,263.38 |  2.531 |         370.47 |    0.1% |      0.01 | `FastRandom_rand64`
|              192.05 |        5,207,050.93 |    0.1% |        1,756.26 |          708.80 |  2.478 |         213.54 |    0.3% |      0.01 | `FastRandom_randbits`
|               16.01 |       62,471,637.02 |    0.1% |          148.23 |           59.00 |  2.513 |          23.84 |    0.1% |      0.01 | `FastRandom_randbool`
|               87.24 |       11,461,986.84 |    0.6% |          758.66 |          321.66 |  2.359 |         108.08 |    0.6% |      0.01 | `FastRandom_randrange100`
|              110.16 |        9,077,576.97 |    0.5% |          965.24 |          405.83 |  2.378 |         132.06 |    0.5% |      0.01 | `FastRandom_randrange1000`
|              186.39 |        5,365,223.88 |    0.1% |        1,676.54 |          687.82 |  2.437 |         215.45 |    0.3% |      2.05 | `FastRandom_randrange1000000`
|              215.54 |        4,639,483.16 |    0.2% |        1,906.59 |          794.16 |  2.401 |         229.95 |    0.1% |      0.01 | `FastRandom_stdshuffle100`
|               53.91 |       18,549,333.04 |    0.3% |          457.13 |          198.59 |  2.302 |          79.37 |    0.8% |      0.01 | `InsecureRandom_Shuffle100`
|               22.03 |       45,393,295.79 |    0.1% |          215.50 |           81.27 |  2.652 |          31.50 |    0.0% |      0.01 | `InsecureRandom_rand32`
|               22.72 |       44,019,773.42 |    0.0% |          220.00 |           83.84 |  2.624 |          24.00 |    0.0% |      0.01 | `InsecureRandom_rand64`
|               27.54 |       36,307,742.75 |    0.6% |          244.77 |          101.11 |  2.421 |          37.63 |    1.0% |      0.01 | `InsecureRandom_randbits`
|               10.95 |       91,358,394.49 |    0.1% |          101.70 |           40.40 |  2.518 |          18.42 |    0.1% |      0.01 | `InsecureRandom_randbool`
|               44.06 |       22,693,987.66 |    0.3% |          371.40 |          162.65 |  2.283 |          63.01 |    0.9% |      0.01 | `InsecureRandom_randrange100`
|               45.33 |       22,061,595.98 |    0.2% |          382.82 |          167.30 |  2.288 |          64.26 |    0.9% |      0.01 | `InsecureRandom_randrange1000`
|               50.17 |       19,933,844.35 |    0.2% |          438.63 |          184.96 |  2.372 |          71.38 |    0.8% |      0.55 | `InsecureRandom_randrange1000000`
|               52.96 |       18,880,998.91 |    0.2% |          418.53 |          195.25 |  2.144 |          56.76 |    0.0% |      0.01 | `InsecureRandom_stdshuffle100`

FastRandom - std::shuffle is less than half the speed. :/ A lot more instructions & branches.
InsecureRandom - Negligible difference.

Conclusion so far

InsecureRandom is only used in tests, so this would probably be a win for developers/testers/fuzzing but not really for end users. Maybe other systems/configurations show a clearer win.

Also ran make check & test/functional/test_runner.py on 6ecda04 in GCC debug without any failures. Confirming that the self-assign issue with std::shuffle doesn't appear to be an issue. (Haven't tried to create a configuration that triggers the panic though).

sipa · 2024-07-06T20:54:02Z

@hodlinator FWIW, my motivation here is just FastRandomContext non-debug performance. I did notice that std::shuffle is a bit slower than the custom Shuffle on my system too (AMD 5950X, GCC 13.2) , but not the huge (5x-10x) slowdown it once was. With 2% or 26% or even 50% slowdown, I don't think it's worth keeping a custom function around (I don't think any of the Shuffle uses are that performance critical). Things like randbool and randrange are probably more relevant overall.

I do expect InsecureRandomContext will get used for non-testing purposes eventually, but that's not a huge motivation now.

To explain what we're seeing here, worked out for a shuffle of 100 elements:

Shuffle would use randrange 100 times, which for the ranges involved would invoke rand64 on average ~13.005 times (~195.433 times for 1000, ~416003.423 times for 1000000).
std::shuffle would invoke rand64 ~100 times directly.

In FastRandomContext, the cost of a rand64 is relevatively high, while for InsecureRandomContext it is very low. On the other hand, randrange itself has some overhead too. For FastRandomContext this randrange overhead is worth it as it is compensated by less of the expensive rand64 calls, while for InsecureRandomContext the reduction in rand64 cost is negligible.

If we really wanted, it would be possible to use the technique that std::shuffle uses for converting rand64s directly into integers of arbitrary range, in InsecureRandomContext::randrange, but it making it efficient would need a few different variants for different compilers/architectures. As there is no use case for efficient InsecureRandomContext::randrange right now that doesn't seem worth it, but that can change.

For posterity, this is sufficient to make 32-bit InsecureRandomContext::randrange work significantly faster:

+
+    template<std::integral I>
+    I randrange(I range) noexcept
+    {
+        if constexpr (std::numeric_limits<I>::digits <= 32) {
+            uint32_t x = rand32();
+            uint64_t m = uint64_t{x} * uint32_t(range);
+            uint32_t l = uint32_t(m);
+            if (l < uint32_t(range)) {
+                uint32_t t = (0 - range) % range;
+                while (l < t) {
+                    x = rand32();
+                    m = uint64_t{x} * uint32_t(range);
+                    l = uint32_t(m);
+                }
+            }
+            return m >> 32;
+        }
+        return RandomMixin<InsecureRandomContext>::randrange(range);
+    }

hodlinator

ACK 6ecda04

Thanks for taking the time to add valuable context!!
(I read up a bit, understand you implemented a 32-bit version of Daniel Lemire's nearly divisionless algo. Nice).

I did notice that std::shuffle is a bit slower than the custom Shuffle on my system too (AMD 5950X, GCC 13.2)

Maybe worth adding some nuance to the PR summary which currently says "now that it appears that standard library std::shuffle beats it.".

# `rand64()` invocations

Shuffle would use randrange 100 times, which for the ranges involved would invoke rand64 on average ~13.005 times

Agreed, verified with added instrumentation (rough reasoning through: range of 100 > 0b1100100 > 7 binary digits wide. 64 bits / 7 bits => ~9 randbits() calls per rand64()... with some extra randbits() because of if (ret <= maxval) in randrange() failing => 100 / 9 + extra = ~13).

std::shuffle would invoke rand64 ~100 times directly.

nit: With the added instrumentation my recent testing shows it to be =50 times - as the ranges are small enough std::shuffle() will perform 2 swaps for each rand64(). This goes for both Clang & GCC.

As there is no use case for efficient InsecureRandomContext::randrange right now that doesn't seem worth it, but that can change.

Seems like it could make fuzz-tests more efficient even now and hence free up resources for more test coverage. But agree it's probably minor.

(Bigger ranges converge)

I'm also on GCC 13.2. CPU: Intel 9th gen Coffee Lake. Did some test with bigger ranges and performance seems to converge between Shuffle & std::shuffle for FastRandom.

|           ns/number |            number/s |    err% |      ins/number |      cyc/number |    IPC |     bra/number |   miss% |     total | benchmark
|--------------------:|--------------------:|--------:|----------------:|----------------:|-------:|---------------:|--------:|----------:|:----------
|               12.01 |       83,247,932.44 |    0.4% |           86.95 |           44.32 |  1.962 |           9.88 |    6.8% |      0.01 | `FastRandom_Shuffle1000`
|               14.61 |       68,427,974.21 |    0.1% |          138.35 |           53.91 |  2.566 |          10.95 |    0.0% |      0.01 | `FastRandom_stdshuffle1000`
|                9.65 |      103,602,289.99 |    0.3% |           52.55 |           35.61 |  1.475 |           7.78 |    7.9% |      0.01 | `InsecureRandom_Shuffle1000`
|                5.67 |      176,274,334.46 |    0.0% |           33.53 |           20.92 |  1.603 |           3.01 |    0.0% |      0.01 | `InsecureRandom_stdshuffle1000`
|--------------------:|--------------------:|--------:|----------------:|----------------:|-------:|---------------:|--------:|----------:|:----------
|               13.99 |       71,471,645.86 |    0.4% |          107.03 |           51.56 |  2.076 |          11.44 |    6.2% |      0.01 | `FastRandom_Shuffle10000`
|               14.68 |       68,116,488.28 |    0.1% |          138.32 |           54.15 |  2.554 |          10.94 |    0.0% |      0.01 | `FastRandom_stdshuffle10000`
|                8.60 |      116,299,489.79 |    0.1% |           55.08 |           31.70 |  1.738 |           7.92 |    8.3% |      0.01 | `InsecureRandom_Shuffle10000`
|                5.66 |      176,764,575.63 |    0.1% |           33.50 |           20.87 |  1.606 |           3.00 |    0.0% |      0.01 | `InsecureRandom_stdshuffle10000`
|--------------------:|--------------------:|--------:|----------------:|----------------:|-------:|---------------:|--------:|----------:|:----------
|               15.24 |       65,617,271.52 |    0.3% |          123.32 |           56.05 |  2.200 |          12.64 |    5.0% |      0.02 | `FastRandom_Shuffle100000`
|               15.44 |       64,762,393.26 |    0.1% |          138.31 |           56.89 |  2.431 |          10.94 |    0.0% |      0.02 | `FastRandom_stdshuffle100000`
|                8.41 |      118,884,436.01 |    0.3% |           56.50 |           30.95 |  1.826 |           7.93 |    7.4% |      0.01 | `InsecureRandom_Shuffle100000`
|                5.79 |      172,767,668.95 |    0.0% |           33.50 |           21.35 |  1.569 |           3.00 |    0.0% |      0.01 | `InsecureRandom_stdshuffle100000`
|--------------------:|--------------------:|--------:|----------------:|----------------:|-------:|---------------:|--------:|----------:|:----------
|               16.44 |       60,831,272.72 |    0.4% |          134.57 |           60.60 |  2.220 |          13.41 |    4.4% |      0.18 | `FastRandom_Shuffle1000000`
|               16.49 |       60,636,239.08 |    0.0% |          138.31 |           60.82 |  2.274 |          10.94 |    0.0% |      0.18 | `FastRandom_stdshuffle1000000`
|                8.41 |      118,931,462.65 |    0.2% |           56.74 |           31.00 |  1.830 |           7.81 |    6.9% |      0.09 | `InsecureRandom_Shuffle1000000`
|                6.23 |      160,482,706.30 |    0.6% |           33.50 |           22.98 |  1.458 |           3.00 |    0.0% |      0.07 | `InsecureRandom_stdshuffle1000000`

dergoegge

Code review ACK 6ecda04

achow101 · 2024-07-09T21:41:24Z

ACK 6ecda04

maflcko · 2024-07-10T08:30:42Z

For fuzzing, just to keep it in mind, there is (obviously) no guarantee by the standard as to how std::shuffle behaves. See also https://en.cppreference.com/w/cpp/algorithm/random_shuffle#Notes

Note that the implementation is not dictated by the standard, so even if you use exactly the same RandomFunc or URBG (Uniform Random Number Generator) you may get different results with different standard library implementations.

So going forward, I guess one can only assume to surely reproduce a fuzz crash by using the exact same stdlib (and compiler). For context there is also a compiler behavior mismatch discussed in #29043.

glozow · 2024-07-10T11:06:57Z

post merge ACK. My local bench results had comparable performance as well

hodlinator · 2024-07-10T15:58:02Z

So going forward, I guess one can only assume to surely reproduce a fuzz crash by using the exact same stdlib (and compiler). For context there is also a compiler behavior mismatch discussed in #29043.

This is a strong counter-point. I guess other stdlib-differences like std::unordered_map implementations already made this true though.

sipa · 2024-07-10T16:01:30Z

I don't think there are any std::shuffle in the fuzz tests, though (apart from one in the random module's tests itself).

It may make sense to add something like the old Shuffle back if we do need it, though.

maflcko · 2024-07-10T16:08:40Z

I don't think there are any std::shuffle in the fuzz tests, though?

I hope all code touched in this pull request is covered by fuzz tests. If not, then that should be fixed ;)

For example, joinpsbts is an RPC (not in the fuzz dir), but it is fuzzed, and it calls std::shuffle: https://maflcko.github.io/b-c-cov/fuzz.coverage/src/rpc/rawtransaction.cpp.gcov.html#L1793

I don't have a strong feeling either way, because if an effort is made to have uniform fuzz behavior across compilers and stdlibs, it will probably be too hard.

hebasto · 2024-07-14T06:57:58Z

Ported to the CMake-based build system in hebasto#264.

sipa · 2024-07-16T18:06:50Z

I hope all code touched in this pull request is covered by fuzz tests. If not, then that should be fixed ;)

Hmm, for some reason I only considered calls to Shuffle / std::shuffle in the fuzz tests themselves. But you're right, shuffles in the code being tested would also result in fuzz inconsistency between platforms.

random bench refactor: move to new bench/random.cpp

0a9bbc6

sipa mentioned this pull request Jul 5, 2024

refactor: use existing RNG object in ProcessGetBlockData #30393

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maflcko added Utils/log/libs Needs CMake port labels Jul 5, 2024

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sipa force-pushed the 202407_rng_opt branch 3 times, most recently from d37b7a3 to e8985f4 Compare July 5, 2024 15:31

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sipa added 2 commits July 6, 2024 09:06

bench random: benchmark more functions, and add InsecureRandomContext

da28a26

Also rename the benchmark names to match the operation names

random: drop ad-hoc Shuffle in favor of std::shuffle

6ecda04

Benchmarks show it is no longer faster with modern standard C++ libraries, and the debug-mode failure due to self-move has been fixed as well.

sipa force-pushed the 202407_rng_opt branch from e8985f4 to 6ecda04 Compare July 6, 2024 13:07

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dergoegge approved these changes Jul 9, 2024

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achow101 merged commit 1067771 into bitcoin:master Jul 9, 2024

hebasto mentioned this pull request Jul 14, 2024

cmake: Regular rebasing of the cmake-staging branch hebasto/bitcoin#264

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hebasto removed the Needs CMake port label Jul 14, 2024

maflcko mentioned this pull request Jul 26, 2024

fuzz: make FuzzedDataProvider usage deterministic #29043

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bitcoin locked and limited conversation to collaborators Jul 16, 2025

random: add benchmarks and drop unnecessary Shuffle function #30396

random: add benchmarks and drop unnecessary Shuffle function #30396

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Clang, non-debug, Linux

GCC equivalent

GCC in debug mode

Conclusion so far

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sipa commented Jul 6, 2024 • edited Loading Uh oh! There was an error while loading. Please reload this page.

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# rand64() invocations

(Bigger ranges converge)

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# `rand64()` invocations

sipa commented Jul 10, 2024 •

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