Update formula for PG-HyperLogLog small cardinality estimates

Based on the original HLL paper and referenced linear probabilistic
counting paper, this should be m*log(m/V) instead of using the
alpha(m) constant for small cardinality, as linear counting is
used instead.

lib/gitlab/database/postgres_hll/buckets.rb

@@ -65,8 +65,7 @@ module Gitlab
           ).to_i
 
           if num_zero_buckets > 0 && num_uniques < 2.5 * TOTAL_BUCKETS
-            ((0.7213 / (1 + 1.079 / TOTAL_BUCKETS)) * (TOTAL_BUCKETS *
-              Math.log2(TOTAL_BUCKETS.to_f / num_zero_buckets)))
+            TOTAL_BUCKETS * Math.log(TOTAL_BUCKETS.to_f / num_zero_buckets)
           else
             num_uniques
           end

spec/lib/gitlab/usage/metrics/aggregates/sources/postgres_hll_spec.rb

@@ -11,6 +11,7 @@ RSpec.describe Gitlab::Usage::Metrics::Aggregates::Sources::PostgresHll, :clean_
   let(:metric_1) { 'metric_1' }
   let(:metric_2) { 'metric_2' }
   let(:metric_names) { [metric_1, metric_2] }
+  let(:error_rate) { Gitlab::Database::PostgresHll::BatchDistinctCounter::ERROR_RATE }
 
   describe 'metric calculations' do
     before do
@@ -38,7 +39,7 @@ RSpec.describe Gitlab::Usage::Metrics::Aggregates::Sources::PostgresHll, :clean_
       end
 
       it 'returns the number of unique events in the union of all metrics' do
-        expect(calculate_metrics_union.round(2)).to eq(3.12)
+        expect(calculate_metrics_union.round(2)).to be_within(error_rate).percent_of(3)
       end
 
       context 'when there is no aggregated data saved' do
@@ -53,7 +54,7 @@ RSpec.describe Gitlab::Usage::Metrics::Aggregates::Sources::PostgresHll, :clean_
         let(:metric_names) { [metric_1] }
 
         it 'returns the number of unique events for that metric' do
-          expect(calculate_metrics_union.round(2)).to eq(2.08)
+          expect(calculate_metrics_union.round(2)).to be_within(error_rate).percent_of(2)
         end
       end
     end
@@ -64,7 +65,7 @@ RSpec.describe Gitlab::Usage::Metrics::Aggregates::Sources::PostgresHll, :clean_
       end
 
       it 'returns the number of common events in the intersection of all metrics' do
-        expect(calculate_metrics_intersections.round(2)).to eq(1.04)
+        expect(calculate_metrics_intersections.round(2)).to be_within(error_rate).percent_of(1)
       end
 
       context 'when there is no aggregated data saved' do
@@ -79,7 +80,7 @@ RSpec.describe Gitlab::Usage::Metrics::Aggregates::Sources::PostgresHll, :clean_
         let(:metric_names) { [metric_1] }
 
         it 'returns the number of common/unique events for the intersection of that metric' do
-          expect(calculate_metrics_intersections.round(2)).to eq(2.08)
+          expect(calculate_metrics_intersections.round(2)).to be_within(error_rate).percent_of(2)
         end
       end
     end

spec/lib/gitlab/usage/metrics/instrumentations/database_metric_spec.rb

@@ -82,7 +82,7 @@ RSpec.describe Gitlab::Usage::Metrics::Instrumentations::DatabaseMetric do
         end.new(time_frame: 'all')
       end
 
-      it 'calculates a correct result', quarantine: 'https://gitlab.com/gitlab-org/gitlab/-/issues/348139' do
+      it 'calculates a correct result' do
         expect(subject.value).to be_within(Gitlab::Database::PostgresHll::BatchDistinctCounter::ERROR_RATE).percent_of(3)
       end
 

spec/lib/gitlab/utils/usage_data_spec.rb

@@ -118,7 +118,7 @@ RSpec.describe Gitlab::Utils::UsageData do
       # build_needs set: ['1', '2', '3', '4', '5']
       # ci_build set ['a', 'b']
       # with them, current implementation is expected to consistently report
-      # 5.217656147118495 and 2.0809220082170614 values
+      # the same static values
       # This test suite is expected to assure, that HyperLogLog implementation
       # behaves consistently between changes made to other parts of codebase.
       # In case of fine tuning or changes to HyperLogLog algorithm implementation
@@ -130,8 +130,8 @@ RSpec.describe Gitlab::Utils::UsageData do
 
       let(:model) { Ci::BuildNeed }
       let(:column) { :name }
-      let(:build_needs_estimated_cardinality) { 5.217656147118495 }
-      let(:ci_builds_estimated_cardinality) { 2.0809220082170614 }
+      let(:build_needs_estimated_cardinality) { 5.024574181542231 }
+      let(:ci_builds_estimated_cardinality) { 2.003916452421793 }
 
       before do
         allow(model.connection).to receive(:transaction_open?).and_return(false)