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new 3f3f223701 [GLUTEN-8528][CH]Support approx_count_distinct (#8550)
3f3f223701 is described below
commit 3f3f223701bc28f9573824303c7c802d06078897
Author: 李扬 <[email protected]>
AuthorDate: Thu Feb 13 15:02:43 2025 +0800
[GLUTEN-8528][CH]Support approx_count_distinct (#8550)
* first commit
* commit again
* finish dev
* finish debug
* commit again
* fix bugs
* commit again
* add spark uts
* fix building
* remove comments
* fix failed uts
---
.../execution/CHHashAggregateExecTransformer.scala | 20 +-
.../expression/CHExpressionTransformer.scala | 1 +
.../execution/GlutenFunctionValidateSuite.scala | 6 +
.../gluten/backendsapi/velox/VeloxBackend.scala | 5 +-
.../AggregateFunctionSparkAvg.cpp | 2 +-
.../AggregateFunctionUniqHyperLogLogPlusPlus.cpp | 46 +
.../AggregateFunctionUniqHyperLogLogPlusPlus.h | 3089 ++++++++++++++++++++
cpp-ch/local-engine/Common/CHUtil.cpp | 16 +-
.../ApproxCountDistinctFunctionParser.cpp | 148 +
cpp-ch/local-engine/tests/gtest_hllpp.cpp | 88 +
.../gluten/expression/ExpressionMappings.scala | 1 +
.../utils/clickhouse/ClickHouseTestSettings.scala | 1 -
.../utils/clickhouse/ClickHouseTestSettings.scala | 1 -
.../utils/clickhouse/ClickHouseTestSettings.scala | 1 -
.../apache/gluten/expression/ExpressionNames.scala | 1 +
15 files changed, 3413 insertions(+), 13 deletions(-)
diff --git
a/backends-clickhouse/src/main/scala/org/apache/gluten/execution/CHHashAggregateExecTransformer.scala
b/backends-clickhouse/src/main/scala/org/apache/gluten/execution/CHHashAggregateExecTransformer.scala
index a3f9749292..f58a41bbea 100644
---
a/backends-clickhouse/src/main/scala/org/apache/gluten/execution/CHHashAggregateExecTransformer.scala
+++
b/backends-clickhouse/src/main/scala/org/apache/gluten/execution/CHHashAggregateExecTransformer.scala
@@ -277,12 +277,24 @@ case class CHHashAggregateExecTransformer(
val childrenNodeList = new util.ArrayList[ExpressionNode]()
val childrenNodes = aggExpr.mode match {
case Partial | Complete =>
- aggregateFunc.children.toList.map(
+ val nodes = aggregateFunc.children.toList.map(
expr => {
ExpressionConverter
.replaceWithExpressionTransformer(expr, childOutput)
.doTransform(args)
})
+
+ val extraNodes = aggregateFunc match {
+ case hll: HyperLogLogPlusPlus =>
+ val relativeSDLiteral = Literal(hll.relativeSD)
+ Seq(
+ ExpressionConverter
+ .replaceWithExpressionTransformer(relativeSDLiteral,
child.output)
+ .doTransform(args))
+ case _ => Seq.empty
+ }
+
+ nodes ++ extraNodes
case PartialMerge if distinct_modes.contains(Partial) =>
// this is the case where PartialMerge co-exists with Partial
// so far, it only happens in a three-stage count distinct case
@@ -446,6 +458,12 @@ case class CHHashAggregateExecTransformer(
percentile.percentageExpression.dataType,
percentile.percentageExpression.nullable)
(makeStructType(fields), attr.nullable)
+ case hllpp: HyperLogLogPlusPlus =>
+ var fields = Seq[(DataType, Boolean)]()
+ fields = fields :+ (hllpp.child.dataType, hllpp.child.nullable)
+ val relativeSDLiteral = Literal(hllpp.relativeSD)
+ fields = fields :+ (relativeSDLiteral.dataType, false)
+ (makeStructType(fields), attr.nullable)
case _ =>
(makeStructTypeSingleOne(attr.dataType, attr.nullable),
attr.nullable)
}
diff --git
a/backends-clickhouse/src/main/scala/org/apache/gluten/expression/CHExpressionTransformer.scala
b/backends-clickhouse/src/main/scala/org/apache/gluten/expression/CHExpressionTransformer.scala
index 4f4b1bfba2..edc88f9ee5 100644
---
a/backends-clickhouse/src/main/scala/org/apache/gluten/expression/CHExpressionTransformer.scala
+++
b/backends-clickhouse/src/main/scala/org/apache/gluten/expression/CHExpressionTransformer.scala
@@ -260,6 +260,7 @@ case class GetArrayItemTransformer(
ConverterUtils.getTypeNode(getArrayItem.dataType, getArrayItem.nullable))
}
}
+
case class CHStringSplitTransformer(
substraitExprName: String,
children: Seq[ExpressionTransformer],
diff --git
a/backends-clickhouse/src/test/scala/org/apache/gluten/execution/GlutenFunctionValidateSuite.scala
b/backends-clickhouse/src/test/scala/org/apache/gluten/execution/GlutenFunctionValidateSuite.scala
index 0d6abd3fdc..75f9e9beb1 100644
---
a/backends-clickhouse/src/test/scala/org/apache/gluten/execution/GlutenFunctionValidateSuite.scala
+++
b/backends-clickhouse/src/test/scala/org/apache/gluten/execution/GlutenFunctionValidateSuite.scala
@@ -1023,4 +1023,10 @@ class GlutenFunctionValidateSuite extends
GlutenClickHouseWholeStageTransformerS
compareResultsAgainstVanillaSpark(sql, true, { _ => })
}
}
+
+ test("Test approx_count_distinct") {
+ val sql = "select approx_count_distinct(id, 0.001),
approx_count_distinct(id, 0.01), " +
+ "approx_count_distinct(id, 0.1) from range(1000)"
+ compareResultsAgainstVanillaSpark(sql, true, { _ => })
+ }
}
diff --git
a/backends-velox/src/main/scala/org/apache/gluten/backendsapi/velox/VeloxBackend.scala
b/backends-velox/src/main/scala/org/apache/gluten/backendsapi/velox/VeloxBackend.scala
index 0dabdfe31c..650d82dfed 100644
---
a/backends-velox/src/main/scala/org/apache/gluten/backendsapi/velox/VeloxBackend.scala
+++
b/backends-velox/src/main/scala/org/apache/gluten/backendsapi/velox/VeloxBackend.scala
@@ -35,7 +35,7 @@ import org.apache.gluten.utils._
import org.apache.spark.sql.catalyst.catalog.BucketSpec
import org.apache.spark.sql.catalyst.expressions.{Alias, CumeDist, DenseRank,
Descending, Expression, Lag, Lead, NamedExpression, NthValue, NTile,
PercentRank, RangeFrame, Rank, RowNumber, SortOrder, SpecialFrameBoundary,
SpecifiedWindowFrame}
-import
org.apache.spark.sql.catalyst.expressions.aggregate.{AggregateExpression,
ApproximatePercentile, Percentile}
+import
org.apache.spark.sql.catalyst.expressions.aggregate.{AggregateExpression,
ApproximatePercentile, HyperLogLogPlusPlus, Percentile}
import org.apache.spark.sql.catalyst.plans.{JoinType, LeftOuter, RightOuter}
import org.apache.spark.sql.catalyst.util.{CaseInsensitiveMap,
CharVarcharUtils}
import org.apache.spark.sql.connector.read.Scan
@@ -476,7 +476,8 @@ object VeloxBackendSettings extends BackendSettingsApi {
case l: Lead if !l.input.foldable =>
case aggrExpr: AggregateExpression
if
!aggrExpr.aggregateFunction.isInstanceOf[ApproximatePercentile]
- && !aggrExpr.aggregateFunction.isInstanceOf[Percentile] =>
+ && !aggrExpr.aggregateFunction.isInstanceOf[Percentile]
+ &&
!aggrExpr.aggregateFunction.isInstanceOf[HyperLogLogPlusPlus] =>
case _ =>
allSupported = false
}
diff --git
a/cpp-ch/local-engine/AggregateFunctions/AggregateFunctionSparkAvg.cpp
b/cpp-ch/local-engine/AggregateFunctions/AggregateFunctionSparkAvg.cpp
index 97aa561cb6..1cb34efdb1 100644
--- a/cpp-ch/local-engine/AggregateFunctions/AggregateFunctionSparkAvg.cpp
+++ b/cpp-ch/local-engine/AggregateFunctions/AggregateFunctionSparkAvg.cpp
@@ -29,7 +29,7 @@ struct Settings;
namespace ErrorCodes
{
-
+extern const int ILLEGAL_TYPE_OF_ARGUMENT;
}
}
diff --git
a/cpp-ch/local-engine/AggregateFunctions/AggregateFunctionUniqHyperLogLogPlusPlus.cpp
b/cpp-ch/local-engine/AggregateFunctions/AggregateFunctionUniqHyperLogLogPlusPlus.cpp
new file mode 100644
index 0000000000..3d715e457f
--- /dev/null
+++
b/cpp-ch/local-engine/AggregateFunctions/AggregateFunctionUniqHyperLogLogPlusPlus.cpp
@@ -0,0 +1,46 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements. See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License. You may obtain a copy of the License at
+ *
+ * http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+#include <AggregateFunctions/AggregateFunctionFactory.h>
+#include <AggregateFunctions/AggregateFunctionUniqHyperLogLogPlusPlus.h>
+#include <AggregateFunctions/Helpers.h>
+#include <DataTypes/DataTypeDate.h>
+#include <DataTypes/DataTypeDate32.h>
+#include <DataTypes/DataTypeDateTime.h>
+#include <DataTypes/DataTypeUUID.h>
+#include <DataTypes/DataTypeIPv4andIPv6.h>
+
+namespace DB
+{
+
+namespace
+{
+
+AggregateFunctionPtr createAggregateFunctionUniqHyperLogLogPlusPlus(
+ const std::string & , const DataTypes & argument_types, const Array &
params, const Settings *)
+{
+ return
std::make_shared<AggregateFunctionUniqHyperLogLogPlusPlus>(argument_types,
params);
+}
+
+}
+
+void registerAggregateFunctionUniqHyperLogLogPlusPlus(AggregateFunctionFactory
& factory)
+{
+ AggregateFunctionProperties properties = {.returns_default_when_only_null
= true, .is_order_dependent = false};
+ factory.registerFunction("uniqHLLPP",
{createAggregateFunctionUniqHyperLogLogPlusPlus, properties});
+}
+
+}
diff --git
a/cpp-ch/local-engine/AggregateFunctions/AggregateFunctionUniqHyperLogLogPlusPlus.h
b/cpp-ch/local-engine/AggregateFunctions/AggregateFunctionUniqHyperLogLogPlusPlus.h
new file mode 100644
index 0000000000..d785164b12
--- /dev/null
+++
b/cpp-ch/local-engine/AggregateFunctions/AggregateFunctionUniqHyperLogLogPlusPlus.h
@@ -0,0 +1,3089 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements. See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License. You may obtain a copy of the License at
+ *
+ * http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+#pragma once
+
+#include <AggregateFunctions/IAggregateFunction.h>
+#include <AggregateFunctions/AggregateFunctionFactory.h>
+#include <Columns/ColumnVector.h>
+#include <DataTypes/DataTypesNumber.h>
+#include <DataTypes/IDataType.h>
+#include <DataTypes/DataTypeAggregateFunction.h>
+#include <Common/FieldVisitorConvertToNumber.h>
+#include <Common/FieldVisitors.h>
+#include <Common/HashTable/Hash.h>
+#include <Common/HyperLogLogWithSmallSetOptimization.h>
+#include <Parsers/NullsAction.h>
+namespace DB
+{
+
+namespace ErrorCodes
+{
+extern const int PARAMETER_OUT_OF_BOUND;
+extern const int NUMBER_OF_ARGUMENTS_DOESNT_MATCH;
+extern const int ILLEGAL_TYPE_OF_ARGUMENT;
+extern const int INCORRECT_DATA;
+}
+
+
+/**
+ * @brief HyperLogLog++ algorithm for approximate distinct counting.
+ *
+ * This structure implements the HyperLogLog++ algorithm, which is an improved
version of the HyperLogLog algorithm.
+ * It provides an efficient way to estimate the number of distinct elements in
a multiset.
+ *
+ * The algorithm uses a fixed amount of memory and provides a trade-off
between memory usage and accuracy.
+ * The accuracy is controlled by the relative standard deviation (relative_sd)
parameter.
+ *
+ * The algorithm also includes bias correction and linear counting for
improved accuracy.
+ */
+struct HyperLogLogPlusPlusData
+{
+ explicit HyperLogLogPlusPlusData(Float64 relative_sd_ = 0.05)
+ : relative_sd(relative_sd_)
+ , p(static_cast<UInt64>(std::ceil(2.0 * std::log(1.106 / relative_sd)
/ std::log(2.0))))
+ , idx_shift(64 - p)
+ , w_padding(1ULL << (p - 1))
+ , m(1ULL << p)
+ , num_words(m / REGISTERS_PER_WORD + 1)
+ , alpha_mm(computeAlphaMM())
+ {
+ if (p < 4)
+ throw Exception(
+ ErrorCodes::PARAMETER_OUT_OF_BOUND,
+ "HLL++ requires at least 4 bits for addressing instead of {}.
Use a lower error, at most 39%",
+ p);
+
+ if (p > 25)
+ throw Exception(
+ ErrorCodes::PARAMETER_OUT_OF_BOUND,
+ "HLL++ requires at most 25 bits for addressing instead of {}
to avoid allocating too much memory",
+ p);
+
+ registers = PaddedPODArray<UInt64>(num_words, 0); // Initialize
registers with zeros
+ }
+
+ void serialize(WriteBuffer & buf) const
+ {
+ writeBinaryLittleEndian(relative_sd, buf);
+
+ writeBinaryLittleEndian(registers.size(), buf);
+ for (const auto & word : registers)
+ writeBinaryLittleEndian(word, buf);
+ }
+
+ void deserialize(ReadBuffer & buf)
+ {
+ Float64 relative_rd_read;
+ readBinaryLittleEndian(relative_rd_read, buf);
+ if (relative_rd_read != relative_sd)
+ throw Exception(
+ ErrorCodes::INCORRECT_DATA, "The relative standard deviation
{} isn't the expected one {}", relative_rd_read, relative_sd);
+
+ size_t registers_size = 0;
+ readBinaryLittleEndian(registers_size, buf);
+ if (registers_size != registers.size())
+ throw Exception(
+ ErrorCodes::INCORRECT_DATA, "The number of registers {} isn't
the expected one {}", registers_size, registers.size());
+
+ for (size_t i = 0; i < registers_size; ++i)
+ readBinaryLittleEndian(registers[i], buf);
+ }
+
+ void add(UInt64 value)
+ {
+ UInt64 x = value;
+ UInt64 idx = x >> idx_shift;
+ UInt64 w = (x << p) | w_padding;
+ UInt64 pw = __builtin_clzll(w) + 1;
+
+ UInt64 word_offset = idx / REGISTERS_PER_WORD;
+ UInt64 word = registers[word_offset];
+
+ UInt64 shift = (idx - word_offset * REGISTERS_PER_WORD) *
REGISTER_SIZE;
+ UInt64 mask = REGISTER_WORD_MASK << shift;
+ UInt64 midx = (word & mask) >> shift;
+ if (pw > midx)
+ {
+ registers[word_offset] = (word & ~mask) | (pw << shift);
+ }
+ }
+
+ void merge(const HyperLogLogPlusPlusData & other)
+ {
+ size_t i = 0; // global register index
+ for (size_t word_i = 0; word_i < num_words; ++word_i)
+ {
+ UInt64 word1 = registers[word_i];
+ UInt64 word2 = other.registers[word_i];
+ UInt64 mask = REGISTER_WORD_MASK;
+ UInt64 merged_word = 0;
+ for (size_t register_i = 0; i < m && register_i <
REGISTERS_PER_WORD; ++register_i, ++i)
+ {
+ merged_word |= std::max((word1 & mask), (word2 & mask));
+ mask <<= REGISTER_SIZE;
+ }
+ registers[word_i] = merged_word;
+ }
+ }
+
+ UInt64 query() const
+ {
+ Float64 z_inverse = 0.0;
+ UInt64 v = 0;
+
+ size_t i = 0; // global register index
+ for (size_t word_i = 0; word_i < num_words; ++word_i)
+ {
+ UInt64 word = registers[word_i];
+ for (size_t register_i = 0; i < m && register_i <
REGISTERS_PER_WORD; ++register_i, ++i)
+ {
+ UInt64 midx = (word >> (register_i * REGISTER_SIZE)) &
REGISTER_WORD_MASK;
+ z_inverse += 1.0 / (1ULL << midx);
+
+ if (midx == 0)
+ ++v;
+ }
+ }
+
+ Float64 e = alpha_mm / z_inverse;
+
+ Float64 e_bias_corrected = e;
+ if (p < 19 && e < 5.0 * m)
+ e_bias_corrected = e - estimateBias(e);
+
+ if (v > 0)
+ {
+ Float64 h = m * std::log(static_cast<Float64>(m) / v);
+ if ((p < 19 && h <= THRESHOLDS[p - 4]) || e <= 2.5 * m)
+ {
+ return static_cast<UInt64>(std::round(h));
+ }
+ else
+ return static_cast<UInt64>(std::round(e_bias_corrected));
+ }
+ else
+ return static_cast<UInt64>(std::round(e_bias_corrected));
+ }
+
+private:
+ Float64 computeAlphaMM() const
+ {
+ // Compute alpha * m * m based on the value of m
+ if (p == 4)
+ return 0.673 * m * m;
+ else if (p == 5)
+ return 0.697 * m * m;
+ else if (p == 6)
+ return 0.709 * m * m;
+ else
+ return (0.7213 / (1 + 1.079 / m)) * m * m;
+ }
+
+
+ /// Estimate the bias using the raw estimates with their respective biases
from the HLL++
+ /// appendix. We currently use KNN interpolation to determine the bias (as
suggested in the
+ /// paper).
+ Float64 estimateBias(Float64 e) const
+ {
+ const auto & estimates = RAW_ESTIMATE_DATA[p - 4];
+ size_t num_estimates = estimates.size();
+
+ // Binary search to find the nearest estimate index
+ auto nearest_estimate_it = std::lower_bound(estimates.begin(),
estimates.end(), e);
+ size_t nearest_estimate_index = std::distance(estimates.begin(),
nearest_estimate_it);
+
+ // Distance metric: square of the difference
+ auto distance = [&](size_t i)
+ {
+ Float64 diff = e - estimates[i];
+ return diff * diff;
+ };
+
+ // Adjust bounds
+ size_t low = nearest_estimate_index + 1 > K ? nearest_estimate_index -
K + 1 : 0;
+ size_t high = std::min(low + K, num_estimates);
+ while (high < num_estimates && distance(high) < distance(low))
+ {
+ ++low;
+ ++high;
+ }
+
+ // Sum biases in the interval
+ const auto & biases = BIAS_DATA[p - 4];
+ Float64 bias_sum = 0.0;
+ for (size_t i = low; i < high; ++i)
+ bias_sum += biases[i];
+
+ // Calculate the bias
+ return bias_sum / (high - low);
+ }
+
+ /// Defines the maximum relative standard deviation allowed.
+ const Float64 relative_sd;
+
+ /// HLL++ uses 'p' bits for bucket addressing.
+ const UInt64 p;
+
+ /// Shift used to extract the index of the register from the hashed value.
+ /// This assumes the use of 64-bit hashcodes.
+ const UInt64 idx_shift;
+
+ /// Value to pad the 'w' value with before the number of leading zeros is
determined.
+ const UInt64 w_padding;
+
+ /// The number of registers used.
+ const UInt64 m;
+
+ /// The number of words used to store the registers. We use Longs for
storage because this is the
+ /// most compact way of storage; Spark aligns to 8-byte words or uses Long
wrappers.
+
+ /// We only store whole registers per word in order to prevent overly
complex bitwise operations.
+ /// In practice this means we only use 60 out of 64 bits.
+ const UInt64 num_words;
+
+ /// The pre-calculated combination of: alpha * m * m
+ /// 'alpha' corrects the raw cardinality estimate 'Z'. See the FlFuGaMe07
paper for its derivation.
+ const Float64 alpha_mm;
+
+ /// The number of bits that is required per register.
+ ///
+ /// This number is determined by the maximum number of leading binary
zeros a hashcode can
+ /// produce. This is equal to the number of bits the hashcode returns. The
current
+ /// implementation uses a 64-bit hashcode, this means 6-bits are (at most)
needed to store the
+ /// number of leading zeros.
+ static constexpr UInt64 REGISTER_SIZE = 6;
+ static constexpr UInt64 REGISTER_WORD_MASK = (1ULL << REGISTER_SIZE) - 1;
+
+ /// The size of a word used for storing registers: 64 Bits.
+ static constexpr UInt64 WORD_SIZE = 64;
+
+ static constexpr UInt64 REGISTERS_PER_WORD = WORD_SIZE / REGISTER_SIZE;
+
+ /// Number of points used for interpolating the bias value.
+ static constexpr UInt64 K = 6;
+
+ inline static const std::vector<std::vector<Float64>> BIAS_DATA = {
+ // precision 4
+ {10,
+ 9.717,
+ 9.207,
+ 8.7896,
+ 8.2882,
+ 7.8204,
+ 7.3772,
+ 6.9342,
+ 6.5202,
+ 6.161,
+ 5.7722,
+ 5.4636,
+ 5.0396,
+ 4.6766,
+ 4.3566,
+ 4.0454,
+ 3.7936,
+ 3.4856,
+ 3.2666,
+ 2.9946,
+ 2.766,
+ 2.4692,
+ 2.3638,
+ 2.0764,
+ 1.7864,
+ 1.7602,
+ 1.4814,
+ 1.433,
+ 1.2926,
+ 1.0664,
+ 0.999600000000001,
+ 0.7956,
+ 0.5366,
+ 0.589399999999998,
+ 0.573799999999999,
+ 0.269799999999996,
+ 0.368200000000002,
+ 0.0544000000000011,
+ 0.234200000000001,
+ 0.0108000000000033,
+ -0.203400000000002,
+ -0.0701999999999998,
+ -0.129600000000003,
+ -0.364199999999997,
+ -0.480600000000003,
+ -0.226999999999997,
+ -0.322800000000001,
+ -0.382599999999996,
+ -0.511200000000002,
+ -0.669600000000003,
+ -0.749400000000001,
+ -0.500399999999999,
+ -0.617600000000003,
+ -0.6922,
+ -0.601599999999998,
+ -0.416200000000003,
+ -0.338200000000001,
+ -0.782600000000002,
+ -0.648600000000002,
+ -0.919800000000002,
+ -0.851799999999997,
+ -0.962400000000002,
+ -0.6402,
+ -1.1922,
+ -1.0256,
+ -1.086,
+ -1.21899999999999,
+ -0.819400000000002,
+ -0.940600000000003,
+ -1.1554,
+ -1.2072,
+ -1.1752,
+ -1.16759999999999,
+ -1.14019999999999,
+ -1.3754,
+ -1.29859999999999,
+ -1.607,
+ -1.3292,
+ -1.7606},
+ // precision 5
+ {22,
+ 21.1194,
+ 20.8208,
+ 20.2318,
+ 19.77,
+ 19.2436,
+ 18.7774,
+ 18.2848,
+ 17.8224,
+ 17.3742,
+ 16.9336,
+ 16.503,
+ 16.0494,
+ 15.6292,
+ 15.2124,
+ 14.798,
+ 14.367,
+ 13.9728,
+ 13.5944,
+ 13.217,
+ 12.8438,
+ 12.3696,
+ 12.0956,
+ 11.7044,
+ 11.324,
+ 11.0668,
+ 10.6698,
+ 10.3644,
+ 10.049,
+ 9.6918,
+ 9.4146,
+ 9.082,
+ 8.687,
+ 8.5398,
+ 8.2462,
+ 7.857,
+ 7.6606,
+ 7.4168,
+ 7.1248,
+ 6.9222,
+ 6.6804,
+ 6.447,
+ 6.3454,
+ 5.9594,
+ 5.7636,
+ 5.5776,
+ 5.331,
+ 5.19,
+ 4.9676,
+ 4.7564,
+ 4.5314,
+ 4.4442,
+ 4.3708,
+ 3.9774,
+ 3.9624,
+ 3.8796,
+ 3.755,
+ 3.472,
+ 3.2076,
+ 3.1024,
+ 2.8908,
+ 2.7338,
+ 2.7728,
+ 2.629,
+ 2.413,
+ 2.3266,
+ 2.1524,
+ 2.2642,
+ 2.1806,
+ 2.0566,
+ 1.9192,
+ 1.7598,
+ 1.3516,
+ 1.5802,
+ 1.43859999999999,
+ 1.49160000000001,
+ 1.1524,
+ 1.1892,
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10984.1814, 10782.0086,
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28.2032000000036, 26.0078000000067, 17.0387999999948,
+ 9.22000000000116, 13.8387999999977, 8.07240000000456,
14.1549999999988, 15.3570000000036, 3.42660000000615,
+ 6.24820000000182, -2.96940000000177, -8.79940000000352,
-5.97860000000219, -14.4048000000039, -3.4143999999942,
+ -13.0148000000045, -11.6977999999945, -25.7878000000055,
-22.3185999999987, -24.409599999999, -31.9756000000052,
+ -18.9722000000038, -22.8678000000073, -30.8972000000067,
-32.3715999999986, -22.3907999999938, -43.6720000000059,
+ -35.9038, -39.7492000000057, -54.1641999999993,
-45.2749999999942, -42.2989999999991, -44.1089999999967,
+ -64.3564000000042, -49.9551999999967, -42.6116000000038},
+ // precision 15
+ {23634.0036, 23210.8034, 22792.4744, 22379.1524,
21969.7928, 21565.326,
+ 21165.3532, 20770.2806, 20379.9892, 19994.7098,
19613.318, 19236.799,
+ 18865.4382, 18498.8244, 18136.5138, 17778.8668,
17426.2344, 17079.32,
+ 16734.778, 16397.2418, 16063.3324, 15734.0232,
15409.731, 15088.728,
+ 14772.9896, 14464.1402, 14157.5588, 13855.5958,
13559.3296, 13264.9096,
+ 12978.326, 12692.0826, 12413.8816, 12137.3192,
11870.2326, 11602.5554,
+ 11340.3142, 11079.613, 10829.5908, 10583.5466,
10334.0344, 10095.5072,
+ 9859.694, 9625.2822, 9395.7862, 9174.0586,
8957.3164, 8738.064,
+ 8524.155, 8313.7396, 8116.9168, 7913.542,
7718.4778, 7521.65,
+ 7335.5596, 7154.2906, 6968.7396, 6786.3996,
6613.236, 6437.406,
+ 6270.6598, 6107.7958, 5945.7174, 5787.6784,
5635.5784, 5482.308,
+ 5337.9784, 5190.0864, 5045.9158, 4919.1386,
4771.817, 4645.7742,
+ 4518.4774, 4385.5454, 4262.6622,
4142.74679999999, 4015.5318, 3897.9276,
+ 3790.7764, 3685.13800000001, 3573.6274, 3467.9706,
3368.61079999999, 3271.5202,
+ 3170.3848, 3076.4656, 2982.38400000001, 2888.4664,
2806.4868, 2711.9564,
+ 2634.1434, 2551.3204, 2469.7662,
2396.61139999999, 2318.9902, 2243.8658,
+ 2171.9246, 2105.01360000001, 2028.8536, 1960.9952,
1901.4096, 1841.86079999999,
+ 1777.54700000001, 1714.5802, 1654.65059999999, 1596.311,
1546.2016, 1492.3296,
+ 1433.8974, 1383.84600000001, 1339.4152, 1293.5518,
1245.8686, 1193.50659999999,
+ 1162.27959999999, 1107.19439999999, 1069.18060000001,
1035.09179999999, 999.679000000004, 957.679999999993,
+ 925.300199999998, 888.099400000006, 848.638600000006,
818.156400000007, 796.748399999997, 752.139200000005,
+ 725.271200000003, 692.216, 671.633600000001,
647.939799999993, 621.670599999998, 575.398799999995,
+ 561.226599999995, 532.237999999998, 521.787599999996,
483.095799999996, 467.049599999998, 465.286399999997,
+ 415.548599999995, 401.047399999996, 380.607999999993,
377.362599999993, 347.258799999996, 338.371599999999,
+ 310.096999999994, 301.409199999995, 276.280799999993,
265.586800000005, 258.994399999996, 223.915999999997,
+ 215.925399999993, 213.503800000006, 191.045400000003,
166.718200000003, 166.259000000005, 162.941200000001,
+ 148.829400000002, 141.645999999993, 123.535399999993,
122.329800000007, 89.473399999988, 80.1962000000058,
+ 77.5457999999926, 59.1056000000099, 83.3509999999951,
52.2906000000075, 36.3979999999865, 40.6558000000077,
+ 42.0003999999899, 19.6630000000005, 19.7153999999864,
-8.38539999999921, -0.692799999989802, 0.854800000000978,
+ 3.23219999999856, -3.89040000000386, -5.25880000001052,
-24.9052000000083, -22.6837999999989, -26.4286000000138,
+ -34.997000000003, -37.0216000000073, -43.430400000012,
-58.2390000000014, -68.8034000000043, -56.9245999999985,
+ -57.8583999999973, -77.3097999999882, -73.2793999999994,
-81.0738000000129, -87.4530000000086, -65.0254000000132,
+ -57.296399999992, -96.2746000000043, -103.25,
-96.081600000005, -91.5542000000132, -102.465200000006,
+ -107.688599999994, -101.458000000013, -109.715800000005},
+ // precision 16
+ {47270,
+ 46423.3584,
+ 45585.7074,
+ 44757.152,
+ 43938.8416,
+ 43130.9514,
+ 42330.03,
+ 41540.407,
+ 40759.6348,
+ 39988.206,
+ 39226.5144,
+ 38473.2096,
+ 37729.795,
+ 36997.268,
+ 36272.6448,
+ 35558.665,
+ 34853.0248,
+ 34157.4472,
+ 33470.5204,
+ 32793.5742,
+ 32127.0194,
+ 31469.4182,
+ 30817.6136,
+ 30178.6968,
+ 29546.8908,
+ 28922.8544,
+ 28312.271,
+ 27707.0924,
+ 27114.0326,
+ 26526.692,
+ 25948.6336,
+ 25383.7826,
+ 24823.5998,
+ 24272.2974,
+ 23732.2572,
+ 23201.4976,
+ 22674.2796,
+ 22163.6336,
+ 21656.515,
+ 21161.7362,
+ 20669.9368,
+ 20189.4424,
+ 19717.3358,
+ 19256.3744,
+ 18795.9638,
+ 18352.197,
+ 17908.5738,
+ 17474.391,
+ 17052.918,
+ 16637.2236,
+ 16228.4602,
+ 15823.3474,
+ 15428.6974,
+ 15043.0284,
+ 14667.6278,
+ 14297.4588,
+ 13935.2882,
+ 13578.5402,
+ 13234.6032,
+ 12882.1578,
+ 12548.0728,
+ 12219.231,
+ 11898.0072,
+ 11587.2626,
+ 11279.9072,
+ 10973.5048,
+ 10678.5186,
+ 10392.4876,
+ 10105.2556,
+ 9825.766,
+ 9562.5444,
+ 9294.2222,
+ 9038.2352,
+ 8784.848,
+ 8533.2644,
+ 8301.7776,
+ 8058.30859999999,
+ 7822.94579999999,
+ 7599.11319999999,
+ 7366.90779999999,
+ 7161.217,
+ 6957.53080000001,
+ 6736.212,
+ 6548.21220000001,
+ 6343.06839999999,
+ 6156.28719999999,
+ 5975.15419999999,
+ 5791.75719999999,
+ 5621.32019999999,
+ 5451.66,
+ 5287.61040000001,
+ 5118.09479999999,
+ 4957.288,
+ 4798.4246,
+ 4662.17559999999,
+ 4512.05900000001,
+ 4364.68539999999,
+ 4220.77720000001,
+ 4082.67259999999,
+ 3957.19519999999,
+ 3842.15779999999,
+ 3699.3328,
+ 3583.01180000001,
+ 3473.8964,
+ 3338.66639999999,
+ 3233.55559999999,
+ 3117.799,
+ 3008.111,
+ 2909.69140000001,
+ 2814.86499999999,
+ 2719.46119999999,
+ 2624.742,
+ 2532.46979999999,
+ 2444.7886,
+ 2370.1868,
+ 2272.45259999999,
+ 2196.19260000001,
+ 2117.90419999999,
+ 2023.2972,
+ 1969.76819999999,
+ 1885.58979999999,
+ 1833.2824,
+ 1733.91200000001,
+ 1682.54920000001,
+ 1604.57980000001,
+ 1556.11240000001,
+ 1491.3064,
+ 1421.71960000001,
+ 1371.22899999999,
+ 1322.1324,
+ 1264.7892,
+ 1196.23920000001,
+ 1143.8474,
+ 1088.67240000001,
+ 1073.60380000001,
+ 1023.11660000001,
+ 959.036400000012,
+ 927.433199999999,
+ 906.792799999996,
+ 853.433599999989,
+ 841.873800000001,
+ 791.1054,
+ 756.899999999994,
+ 704.343200000003,
+ 672.495599999995,
+ 622.790399999998,
+ 611.254799999995,
+ 567.283200000005,
+ 519.406599999988,
+ 519.188400000014,
+ 495.312800000014,
+ 451.350799999986,
+ 443.973399999988,
+ 431.882199999993,
+ 392.027000000002,
+ 380.924200000009,
+ 345.128999999986,
+ 298.901400000002,
+ 287.771999999997,
+ 272.625,
+ 247.253000000026,
+ 222.490600000019,
+ 223.590000000026,
+ 196.407599999977,
+ 176.425999999978,
+ 134.725199999986,
+ 132.4804,
+ 110.445599999977,
+ 86.7939999999944,
+ 56.7038000000175,
+ 64.915399999998,
+ 38.3726000000024,
+ 37.1606000000029,
+ 46.170999999973,
+ 49.1716000000015,
+ 15.3362000000197,
+ 6.71639999997569,
+ -34.8185999999987,
+ -39.4476000000141,
+ 12.6830000000191,
+ -12.3331999999937,
+ -50.6565999999875,
+ -59.9538000000175,
+ -65.1054000000004,
+ -70.7576000000117,
+ -106.325200000021,
+ -126.852200000023,
+ -110.227599999984,
+ -132.885999999999,
+ -113.897200000007,
+ -142.713800000027,
+ -151.145399999979,
+ -150.799200000009,
+ -177.756200000003,
+ -156.036399999983,
+ -182.735199999996,
+ -177.259399999981,
+ -198.663600000029,
+ -174.577600000019,
+ -193.84580000001},
+ // precision 17
+ {94541,
+ 92848.811,
+ 91174.019,
+ 89517.558,
+ 87879.9705,
+ 86262.7565,
+ 84663.5125,
+ 83083.7435,
+ 81521.7865,
+ 79977.272,
+ 78455.9465,
+ 76950.219,
+ 75465.432,
+ 73994.152,
+ 72546.71,
+ 71115.2345,
+ 69705.6765,
+ 68314.937,
+ 66944.2705,
+ 65591.255,
+ 64252.9485,
+ 62938.016,
+ 61636.8225,
+ 60355.592,
+ 59092.789,
+ 57850.568,
+ 56624.518,
+ 55417.343,
+ 54231.1415,
+ 53067.387,
+ 51903.526,
+ 50774.649,
+ 49657.6415,
+ 48561.05,
+ 47475.7575,
+ 46410.159,
+ 45364.852,
+ 44327.053,
+ 43318.4005,
+ 42325.6165,
+ 41348.4595,
+ 40383.6265,
+ 39436.77,
+ 38509.502,
+ 37594.035,
+ 36695.939,
+ 35818.6895,
+ 34955.691,
+ 34115.8095,
+ 33293.949,
+ 32465.0775,
+ 31657.6715,
+ 30877.2585,
+ 30093.78,
+ 29351.3695,
+ 28594.1365,
+ 27872.115,
+ 27168.7465,
+ 26477.076,
+ 25774.541,
+ 25106.5375,
+ 24452.5135,
+ 23815.5125,
+ 23174.0655,
+ 22555.2685,
+ 21960.2065,
+ 21376.3555,
+ 20785.1925,
+ 20211.517,
+ 19657.0725,
+ 19141.6865,
+ 18579.737,
+ 18081.3955,
+ 17578.995,
+ 17073.44,
+ 16608.335,
+ 16119.911,
+ 15651.266,
+ 15194.583,
+ 14749.0495,
+ 14343.4835,
+ 13925.639,
+ 13504.509,
+ 13099.3885,
+ 12691.2855,
+ 12328.018,
+ 11969.0345,
+ 11596.5145,
+ 11245.6355,
+ 10917.6575,
+ 10580.9785,
+ 10277.8605,
+ 9926.58100000001,
+ 9605.538,
+ 9300.42950000003,
+ 8989.97850000003,
+ 8728.73249999998,
+ 8448.3235,
+ 8175.31050000002,
+ 7898.98700000002,
+ 7629.79100000003,
+ 7413.76199999999,
+ 7149.92300000001,
+ 6921.12650000001,
+ 6677.1545,
+ 6443.28000000003,
+ 6278.23450000002,
+ 6014.20049999998,
+ 5791.20299999998,
+ 5605.78450000001,
+ 5438.48800000001,
+ 5234.2255,
+ 5059.6825,
+ 4887.43349999998,
+ 4682.935,
+ 4496.31099999999,
+ 4322.52250000002,
+ 4191.42499999999,
+ 4021.24200000003,
+ 3900.64799999999,
+ 3762.84250000003,
+ 3609.98050000001,
+ 3502.29599999997,
+ 3363.84250000003,
+ 3206.54849999998,
+ 3079.70000000001,
+ 2971.42300000001,
+ 2867.80349999998,
+ 2727.08100000001,
+ 2630.74900000001,
+ 2496.6165,
+ 2440.902,
+ 2356.19150000002,
+ 2235.58199999999,
+ 2120.54149999999,
+ 2012.25449999998,
+ 1933.35600000003,
+ 1820.93099999998,
+ 1761.54800000001,
+ 1663.09350000002,
+ 1578.84600000002,
+ 1509.48149999999,
+ 1427.3345,
+ 1379.56150000001,
+ 1306.68099999998,
+ 1212.63449999999,
+ 1084.17300000001,
+ 1124.16450000001,
+ 1060.69949999999,
+ 1007.48849999998,
+ 941.194499999983,
+ 879.880500000028,
+ 836.007500000007,
+ 782.802000000025,
+ 748.385499999975,
+ 647.991500000004,
+ 626.730500000005,
+ 570.776000000013,
+ 484.000500000024,
+ 513.98550000001,
+ 418.985499999952,
+ 386.996999999974,
+ 370.026500000036,
+ 355.496999999974,
+ 356.731499999994,
+ 255.92200000002,
+ 259.094000000041,
+ 205.434499999974,
+ 165.374500000034,
+ 197.347500000033,
+ 95.718499999959,
+ 67.6165000000037,
+ 54.6970000000438,
+ 31.7395000000251,
+ -15.8784999999916,
+ 8.42500000004657,
+ -26.3754999999655,
+ -118.425500000012,
+ -66.6629999999423,
+ -42.9745000000112,
+ -107.364999999991,
+ -189.839000000036,
+ -162.611499999999,
+ -164.964999999967,
+ -189.079999999958,
+ -223.931499999948,
+ -235.329999999958,
+ -269.639500000048,
+ -249.087999999989,
+ -206.475499999942,
+ -283.04449999996,
+ -290.667000000016,
+ -304.561499999953,
+ -336.784499999951,
+ -380.386500000022,
+ -283.280499999993,
+ -364.533000000054,
+ -389.059499999974,
+ -364.454000000027,
+ -415.748000000021,
+ -417.155000000028},
+ // precision 18
+ {189083,
+ 185696.913,
+ 182348.774,
+ 179035.946,
+ 175762.762,
+ 172526.444,
+ 169329.754,
+ 166166.099,
+ 163043.269,
+ 159958.91,
+ 156907.912,
+ 153906.845,
+ 150924.199,
+ 147996.568,
+ 145093.457,
+ 142239.233,
+ 139421.475,
+ 136632.27,
+ 133889.588,
+ 131174.2,
+ 128511.619,
+ 125868.621,
+ 123265.385,
+ 120721.061,
+ 118181.769,
+ 115709.456,
+ 113252.446,
+ 110840.198,
+ 108465.099,
+ 106126.164,
+ 103823.469,
+ 101556.618,
+ 99308.004,
+ 97124.508,
+ 94937.803,
+ 92833.731,
+ 90745.061,
+ 88677.627,
+ 86617.47,
+ 84650.442,
+ 82697.833,
+ 80769.132,
+ 78879.629,
+ 77014.432,
+ 75215.626,
+ 73384.587,
+ 71652.482,
+ 69895.93,
+ 68209.301,
+ 66553.669,
+ 64921.981,
+ 63310.323,
+ 61742.115,
+ 60205.018,
+ 58698.658,
+ 57190.657,
+ 55760.865,
+ 54331.169,
+ 52908.167,
+ 51550.273,
+ 50225.254,
+ 48922.421,
+ 47614.533,
+ 46362.049,
+ 45098.569,
+ 43926.083,
+ 42736.03,
+ 41593.473,
+ 40425.26,
+ 39316.237,
+ 38243.651,
+ 37170.617,
+ 36114.609,
+ 35084.19,
+ 34117.233,
+ 33206.509,
+ 32231.505,
+ 31318.728,
+ 30403.404,
+ 29540.0550000001,
+ 28679.236,
+ 27825.862,
+ 26965.216,
+ 26179.148,
+ 25462.08,
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+ 23922.523,
+ 23198.144,
+ 22529.128,
+ 21762.4179999999,
+ 21134.779,
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+ 17982.831,
+ 17439.7389999999,
+ 16874.547,
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+ 15835.684,
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+ 13342.874,
+ 12880.882,
+ 12491.648,
+ 12021.254,
+ 11625.392,
+ 11293.7610000001,
+ 10813.697,
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+ 8033,
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+ 7258.47699999996,
+ 6924.58200000005,
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+ 3341.67200000002,
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+ 1022.71900000004,
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+ 959.03600000008,
+ 876.095999999903,
+ 791.183999999892,
+ 703.337000000058,
+ 731.949999999953,
+ 586.86400000006,
+ 526.024999999907,
+ 323.004999999888,
+ 320.448000000091,
+ 340.672999999952,
+ 309.638999999966,
+ 216.601999999955,
+ 102.922999999952,
+ 19.2399999999907,
+ -0.114000000059605,
+ -32.6240000000689,
+ -89.3179999999702,
+ -153.497999999905,
+ -64.2970000000205,
+ -143.695999999996,
+ -259.497999999905,
+ -253.017999999924,
+ -213.948000000091,
+ -397.590000000084,
+ -434.006000000052,
+ -403.475000000093,
+ -297.958000000101,
+ -404.317000000039,
+ -528.898999999976,
+ -506.621000000043,
+ -513.205000000075,
+ -479.351000000024,
+ -596.139999999898,
+ -527.016999999993,
+ -664.681000000099,
+ -680.306000000099,
+ -704.050000000047,
+ -850.486000000034,
+ -757.43200000003,
+ -713.308999999892}};
+
+
+ /// Thresholds which decide if the linear counting or the regular
algorithm is used.
+ static constexpr Float64 THRESHOLDS[15] = {10, 20, 40, 80, 220, 400, 900,
1800, 3100, 6500, 15500, 20000, 50000, 120000, 350000};
+
+ /// Lookup table used to find the (index of the) bias correction for a
given precision (exact)
+ /// and estimate (nearest).
+ inline static const std::vector<std::vector<Float64>> RAW_ESTIMATE_DATA = {
+ {// precision 4
+ {11, 11.717, 12.207, 12.7896, 13.2882, 13.8204, 14.3772,
14.9342, 15.5202, 16.161, 16.7722, 17.4636, 18.0396, 18.6766,
+ 19.3566, 20.0454, 20.7936, 21.4856, 22.2666, 22.9946, 23.766,
24.4692, 25.3638, 26.0764, 26.7864, 27.7602, 28.4814, 29.433,
+ 30.2926, 31.0664, 31.9996, 32.7956, 33.5366, 34.5894, 35.5738,
36.2698, 37.3682, 38.0544, 39.2342, 40.0108, 40.7966, 41.9298,
+ 42.8704, 43.6358, 44.5194, 45.773, 46.6772, 47.6174, 48.4888,
49.3304, 50.2506, 51.4996, 52.3824, 53.3078, 54.3984, 55.5838,
+ 56.6618, 57.2174, 58.3514, 59.0802, 60.1482, 61.0376, 62.3598,
62.8078, 63.9744, 64.914, 65.781, 67.1806, 68.0594, 68.8446,
+ 69.7928, 70.8248, 71.8324, 72.8598, 73.6246, 74.7014, 75.393,
76.6708, 77.2394},
+ // precision 5
+ {23, 23.1194, 23.8208, 24.2318, 24.77, 25.2436, 25.7774,
26.2848, 26.8224, 27.3742, 27.9336, 28.503, 29.0494,
+ 29.6292, 30.2124, 30.798, 31.367, 31.9728, 32.5944, 33.217,
33.8438, 34.3696, 35.0956, 35.7044, 36.324, 37.0668,
+ 37.6698, 38.3644, 39.049, 39.6918, 40.4146, 41.082, 41.687,
42.5398, 43.2462, 43.857, 44.6606, 45.4168, 46.1248,
+ 46.9222, 47.6804, 48.447, 49.3454, 49.9594, 50.7636, 51.5776,
52.331, 53.19, 53.9676, 54.7564, 55.5314, 56.4442,
+ 57.3708, 57.9774, 58.9624, 59.8796, 60.755, 61.472, 62.2076,
63.1024, 63.8908, 64.7338, 65.7728, 66.629, 67.413,
+ 68.3266, 69.1524, 70.2642, 71.1806, 72.0566, 72.9192, 73.7598,
74.3516, 75.5802, 76.4386, 77.4916, 78.1524, 79.1892,
+ 79.8414, 80.8798, 81.8376, 82.4698, 83.7656, 84.331, 85.5914,
86.6012, 87.7016, 88.5582, 89.3394, 90.3544, 91.4912,
+ 92.308, 93.3552, 93.9746, 95.2052, 95.727, 97.1322, 98.3944,
98.7588, 100.242, 101.1914, 102.2538, 102.8776, 103.6292,
+ 105.1932, 105.9152, 107.0868, 107.6728, 108.7144, 110.3114,
110.8716, 111.245, 112.7908, 113.7064, 114.636, 115.7464, 116.1788,
+ 117.7464, 118.4896, 119.6166, 120.5082, 121.7798, 122.9028,
123.4426, 124.8854, 125.705, 126.4652, 128.3464, 128.3462, 130.0398,
+ 131.0342, 131.0042, 132.4766, 133.511, 134.7252, 135.425,
136.5172, 138.0572, 138.6694, 139.3712, 140.8598, 141.4594, 142.554,
+ 143.4006, 144.7374, 146.1634, 146.8994, 147.605, 147.9304,
149.1636, 150.2468, 151.5876, 152.2096, 153.7032, 154.7146, 155.807,
+ 156.9228, 157.0372, 158.5852},
+ // precision 6
+ {46, 46.1902, 47.271, 47.8358, 48.8142, 49.2854, 50.317,
51.354, 51.8924, 52.9436, 53.4596, 54.5262, 55.6248,
+ 56.1574, 57.2822, 57.837, 58.9636, 60.074, 60.7042, 61.7976,
62.4772, 63.6564, 64.7942, 65.5004, 66.686, 67.291,
+ 68.5672, 69.8556, 70.4982, 71.8204, 72.4252, 73.7744, 75.0786,
75.8344, 77.0294, 77.8098, 79.0794, 80.5732, 81.1878,
+ 82.5648, 83.2902, 84.6784, 85.3352, 86.8946, 88.3712, 89.0852,
90.499, 91.2686, 92.6844, 94.2234, 94.9732, 96.3356,
+ 97.2286, 98.7262, 100.3284, 101.1048, 102.5962, 103.3562,
105.1272, 106.4184, 107.4974, 109.0822, 109.856, 111.48, 113.2834,
+ 114.0208, 115.637, 116.5174, 118.0576, 119.7476, 120.427,
122.1326, 123.2372, 125.2788, 126.6776, 127.7926, 129.1952, 129.9564,
+ 131.6454, 133.87, 134.5428, 136.2, 137.0294, 138.6278,
139.6782, 141.792, 143.3516, 144.2832, 146.0394, 147.0748, 148.4912,
+ 150.849, 151.696, 153.5404, 154.073, 156.3714, 157.7216,
158.7328, 160.4208, 161.4184, 163.9424, 165.2772, 166.411, 168.1308,
+ 168.769, 170.9258, 172.6828, 173.7502, 175.706, 176.3886,
179.0186, 180.4518, 181.927, 183.4172, 184.4114, 186.033, 188.5124,
+ 189.5564, 191.6008, 192.4172, 193.8044, 194.997, 197.4548,
198.8948, 200.2346, 202.3086, 203.1548, 204.8842, 206.6508, 206.6772,
+ 209.7254, 210.4752, 212.7228, 214.6614, 215.1676, 217.793,
218.0006, 219.9052, 221.66, 223.5588, 225.1636, 225.6882, 227.7126,
+ 229.4502, 231.1978, 232.9756, 233.1654, 236.727, 238.1974,
237.7474, 241.1346, 242.3048, 244.1948, 245.3134, 246.879, 249.1204,
+ 249.853, 252.6792, 253.857, 254.4486, 257.2362, 257.9534,
260.0286, 260.5632, 262.663, 264.723, 265.7566, 267.2566, 267.1624,
+ 270.62, 272.8216, 273.2166, 275.2056, 276.2202, 278.3726,
280.3344, 281.9284, 283.9728, 284.1924, 286.4872, 287.587, 289.807,
+ 291.1206, 292.769, 294.8708, 296.665, 297.1182, 299.4012,
300.6352, 302.1354, 304.1756, 306.1606, 307.3462, 308.5214, 309.4134,
+ 310.8352, 313.9684, 315.837, 316.7796, 318.9858},
+ // precision 7
+ {92, 93.4934, 94.9758, 96.4574, 97.9718, 99.4954,
101.5302, 103.0756, 104.6374, 106.1782, 107.7888, 109.9522, 111.592,
+ 113.2532, 114.9086, 116.5938, 118.9474, 120.6796, 122.4394,
124.2176, 125.9768, 128.4214, 130.2528, 132.0102, 133.8658, 135.7278,
+ 138.3044, 140.1316, 142.093, 144.0032, 145.9092, 148.6306,
150.5294, 152.5756, 154.6508, 156.662, 159.552, 161.3724, 163.617,
+ 165.5754, 167.7872, 169.8444, 172.7988, 174.8606, 177.2118,
179.3566, 181.4476, 184.5882, 186.6816, 189.0824, 191.0258, 193.6048,
+ 196.4436, 198.7274, 200.957, 203.147, 205.4364, 208.7592,
211.3386, 213.781, 215.8028, 218.656, 221.6544, 223.996, 226.4718,
+ 229.1544, 231.6098, 234.5956, 237.0616, 239.5758, 242.4878,
244.5244, 248.2146, 250.724, 252.8722, 255.5198, 258.0414, 261.941,
+ 264.9048, 266.87, 269.4304, 272.028, 274.4708, 278.37,
281.0624, 283.4668, 286.5532, 289.4352, 293.2564, 295.2744, 298.2118,
+ 300.7472, 304.1456, 307.2928, 309.7504, 312.5528, 315.979,
318.2102, 322.1834, 324.3494, 327.325, 330.6614, 332.903, 337.2544,
+ 339.9042, 343.215, 345.2864, 348.0814, 352.6764, 355.301, 357.139,
360.658, 363.1732, 366.5902, 369.9538, 373.0828, 375.922,
+ 378.9902, 382.7328, 386.4538, 388.1136, 391.2234, 394.0878, 396.708,
401.1556, 404.1852, 406.6372, 409.6822, 412.7796, 416.6078,
+ 418.4916, 422.131, 424.5376, 428.1988, 432.211, 434.4502,
438.5282, 440.912, 444.0448, 447.7432, 450.8524, 453.7988, 456.7858,
+ 458.8868, 463.9886, 466.5064, 468.9124, 472.6616, 475.4682, 478.582,
481.304, 485.2738, 488.6894, 490.329, 496.106, 497.6908,
+ 501.1374, 504.5322, 506.8848, 510.3324, 513.4512, 516.179,
520.4412, 522.6066, 526.167, 528.7794, 533.379, 536.067, 538.46,
+ 542.9116, 545.692, 547.9546, 552.493, 555.2722, 557.335, 562.449,
564.2014, 569.0738, 571.0974, 574.8564, 578.2996, 581.409,
+ 583.9704, 585.8098, 589.6528, 594.5998, 595.958, 600.068,
603.3278, 608.2016, 609.9632, 612.864, 615.43, 620.7794, 621.272,
+ 625.8644, 629.206, 633.219, 634.5154, 638.6102},
+ // precision 8
+ {184.2152, 187.2454, 190.2096, 193.6652, 196.6312, 199.6822,
203.249, 206.3296, 210.0038, 213.2074, 216.4612,
+ 220.27, 223.5178, 227.4412, 230.8032, 234.1634, 238.1688,
241.6074, 245.6946, 249.2664, 252.8228, 257.0432,
+ 260.6824, 264.9464, 268.6268, 272.2626, 276.8376, 280.4034,
284.8956, 288.8522, 292.7638, 297.3552, 301.3556,
+ 305.7526, 309.9292, 313.8954, 318.8198, 322.7668, 327.298,
331.6688, 335.9466, 340.9746, 345.1672, 349.3474,
+ 354.3028, 358.8912, 364.114, 368.4646, 372.9744, 378.4092,
382.6022, 387.843, 392.5684, 397.1652, 402.5426,
+ 407.4152, 412.5388, 417.3592, 422.1366, 427.486, 432.3918,
437.5076, 442.509, 447.3834, 453.3498, 458.0668,
+ 463.7346, 469.1228, 473.4528, 479.7, 484.644, 491.0518,
495.5774, 500.9068, 506.432, 512.1666, 517.434,
+ 522.6644, 527.4894, 533.6312, 538.3804, 544.292, 550.5496,
556.0234, 562.8206, 566.6146, 572.4188, 579.117,
+ 583.6762, 590.6576, 595.7864, 601.509, 607.5334, 612.9204,
619.772, 624.2924, 630.8654, 636.1836, 642.745,
+ 649.1316, 655.0386, 660.0136, 666.6342, 671.6196, 678.1866,
684.4282, 689.3324, 695.4794, 702.5038, 708.129,
+ 713.528, 720.3204, 726.463, 732.7928, 739.123, 744.7418,
751.2192, 756.5102, 762.6066, 769.0184, 775.2224,
+ 781.4014, 787.7618, 794.1436, 798.6506, 805.6378, 811.766,
819.7514, 824.5776, 828.7322, 837.8048, 843.6302,
+ 849.9336, 854.4798, 861.3388, 867.9894, 873.8196, 880.3136,
886.2308, 892.4588, 899.0816, 905.4076, 912.0064,
+ 917.3878, 923.619, 929.998, 937.3482, 943.9506, 947.991,
955.1144, 962.203, 968.8222, 975.7324, 981.7826,
+ 988.7666, 994.2648, 1000.3128, 1007.4082, 1013.7536, 1020.3376,
1026.7156, 1031.7478, 1037.4292, 1045.393, 1051.2278,
+ 1058.3434, 1062.8726, 1071.884, 1076.806, 1082.9176, 1089.1678,
1095.5032, 1102.525, 1107.2264, 1115.315, 1120.93,
+ 1127.252, 1134.1496, 1139.0408, 1147.5448, 1153.3296, 1158.1974,
1166.5262, 1174.3328, 1175.657, 1184.4222, 1190.9172,
+ 1197.1292, 1204.4606, 1210.4578, 1218.8728, 1225.3336, 1226.6592,
1236.5768, 1241.363, 1249.4074, 1254.6566, 1260.8014,
+ 1266.5454, 1274.5192},
+ // precision 9
+ {369, 374.8294, 381.2452, 387.6698, 394.1464, 400.2024,
406.8782, 413.6598, 420.462, 427.2826, 433.7102,
+ 440.7416, 447.9366, 455.1046, 462.285, 469.0668, 476.306,
483.8448, 491.301, 498.9886, 506.2422, 513.8138,
+ 521.7074, 529.7428, 537.8402, 545.1664, 553.3534, 561.594,
569.6886, 577.7876, 585.65, 594.228, 602.8036,
+ 611.1666, 620.0818, 628.0824, 637.2574, 646.302, 655.1644,
664.0056, 672.3802, 681.7192, 690.5234, 700.2084,
+ 708.831, 718.485, 728.1112, 737.4764, 746.76, 756.3368,
766.5538, 775.5058, 785.2646, 795.5902, 804.3818,
+ 814.8998, 824.9532, 835.2062, 845.2798, 854.4728, 864.9582,
875.3292, 886.171, 896.781, 906.5716, 916.7048,
+ 927.5322, 937.875, 949.3972, 958.3464, 969.7274, 980.2834,
992.1444, 1003.4264, 1013.0166, 1024.018, 1035.0438,
+ 1046.34, 1057.6856, 1068.9836, 1079.0312, 1091.677, 1102.3188,
1113.4846, 1124.4424, 1135.739, 1147.1488, 1158.9202,
+ 1169.406, 1181.5342, 1193.2834, 1203.8954, 1216.3286, 1226.2146,
1239.6684, 1251.9946, 1262.123, 1275.4338, 1285.7378,
+ 1296.076, 1308.9692, 1320.4964, 1333.0998, 1343.9864, 1357.7754,
1368.3208, 1380.4838, 1392.7388, 1406.0758, 1416.9098,
+ 1428.9728, 1440.9228, 1453.9292, 1462.617, 1476.05, 1490.2996,
1500.6128, 1513.7392, 1524.5174, 1536.6322, 1548.2584,
+ 1562.3766, 1572.423, 1587.1232, 1596.5164, 1610.5938, 1622.5972,
1633.1222, 1647.7674, 1658.5044, 1671.57, 1683.7044,
+ 1695.4142, 1708.7102, 1720.6094, 1732.6522, 1747.841, 1756.4072,
1769.9786, 1782.3276, 1797.5216, 1808.3186, 1819.0694,
+ 1834.354, 1844.575, 1856.2808, 1871.1288, 1880.7852, 1893.9622,
1906.3418, 1920.6548, 1932.9302, 1945.8584, 1955.473,
+ 1968.8248, 1980.6446, 1995.9598, 2008.349, 2019.8556, 2033.0334,
2044.0206, 2059.3956, 2069.9174, 2082.6084, 2093.7036,
+ 2106.6108, 2118.9124, 2132.301, 2144.7628, 2159.8422, 2171.0212,
2183.101, 2193.5112, 2208.052, 2221.3194, 2233.3282,
+ 2247.295, 2257.7222, 2273.342, 2286.5638, 2299.6786, 2310.8114,
2322.3312, 2335.516, 2349.874, 2363.5968, 2373.865,
+ 2387.1918, 2401.8328, 2414.8496, 2424.544, 2436.7592, 2447.1682,
2464.1958, 2474.3438, 2489.0006, 2497.4526, 2513.6586,
+ 2527.19, 2540.7028, 2553.768},
+ // precision 10
+ {738.1256, 750.4234, 763.1064, 775.4732, 788.4636, 801.0644,
814.488, 827.9654, 841.0832, 854.7864, 868.1992,
+ 882.2176, 896.5228, 910.1716, 924.7752, 938.899, 953.6126,
968.6492, 982.9474, 998.5214, 1013.1064, 1028.6364,
+ 1044.2468, 1059.4588, 1075.3832, 1091.0584, 1106.8606, 1123.3868,
1139.5062, 1156.1862, 1172.463, 1189.339, 1206.1936,
+ 1223.1292, 1240.1854, 1257.2908, 1275.3324, 1292.8518, 1310.5204,
1328.4854, 1345.9318, 1364.552, 1381.4658, 1400.4256,
+ 1419.849, 1438.152, 1456.8956, 1474.8792, 1494.118, 1513.62,
1532.5132, 1551.9322, 1570.7726, 1590.6086, 1610.5332,
+ 1630.5918, 1650.4294, 1669.7662, 1690.4106, 1710.7338, 1730.9012,
1750.4486, 1770.1556, 1791.6338, 1812.7312, 1833.6264,
+ 1853.9526, 1874.8742, 1896.8326, 1918.1966, 1939.5594, 1961.07,
1983.037, 2003.1804, 2026.071, 2047.4884, 2070.0848,
+ 2091.2944, 2114.333, 2135.9626, 2158.2902, 2181.0814, 2202.0334,
2224.4832, 2246.39, 2269.7202, 2292.1714, 2314.2358,
+ 2338.9346, 2360.891, 2384.0264, 2408.3834, 2430.1544, 2454.8684,
2476.9896, 2501.4368, 2522.8702, 2548.0408, 2570.6738,
+ 2593.5208, 2617.0158, 2640.2302, 2664.0962, 2687.4986, 2714.2588,
2735.3914, 2759.6244, 2781.8378, 2808.0072, 2830.6516,
+ 2856.2454, 2877.2136, 2903.4546, 2926.785, 2951.2294, 2976.468,
3000.867, 3023.6508, 3049.91, 3073.5984, 3098.162,
+ 3121.5564, 3146.2328, 3170.9484, 3195.5902, 3221.3346, 3242.7032,
3271.6112, 3296.5546, 3317.7376, 3345.072, 3369.9518,
+ 3394.326, 3418.1818, 3444.6926, 3469.086, 3494.2754, 3517.8698,
3544.248, 3565.3768, 3588.7234, 3616.979, 3643.7504,
+ 3668.6812, 3695.72, 3719.7392, 3742.6224, 3770.4456, 3795.6602,
3819.9058, 3844.002, 3869.517, 3895.6824, 3920.8622,
+ 3947.1364, 3973.985, 3995.4772, 4021.62, 4046.628, 4074.65,
4096.2256, 4121.831, 4146.6406, 4173.276, 4195.0744,
+ 4223.9696, 4251.3708, 4272.9966, 4300.8046, 4326.302, 4353.1248,
4374.312, 4403.0322, 4426.819, 4450.0598, 4478.5206,
+ 4504.8116, 4528.8928, 4553.9584, 4578.8712, 4603.8384, 4632.3872,
4655.5128, 4675.821, 4704.6222, 4731.9862, 4755.4174,
+ 4781.2628, 4804.332, 4832.3048, 4862.8752, 4883.4148, 4906.9544,
4935.3516, 4954.3532, 4984.0248, 5011.217, 5035.3258,
+ 5057.3672, 5084.1828},
+ // precision 11
+ {1477, 1501.6014, 1526.5802, 1551.7942, 1577.3042, 1603.2062,
1629.8402, 1656.2292, 1682.9462, 1709.9926, 1737.3026,
+ 1765.4252, 1793.0578, 1821.6092, 1849.626, 1878.5568, 1908.527,
1937.5154, 1967.1874, 1997.3878, 2027.37, 2058.1972,
+ 2089.5728, 2120.1012, 2151.9668, 2183.292, 2216.0772, 2247.8578,
2280.6562, 2313.041, 2345.714, 2380.3112, 2414.1806,
+ 2447.9854, 2481.656, 2516.346, 2551.5154, 2586.8378, 2621.7448,
2656.6722, 2693.5722, 2729.1462, 2765.4124, 2802.8728,
+ 2838.898, 2876.408, 2913.4926, 2951.4938, 2989.6776, 3026.282,
3065.7704, 3104.1012, 3143.7388, 3181.6876, 3221.1872,
+ 3261.5048, 3300.0214, 3339.806, 3381.409, 3421.4144, 3461.4294,
3502.2286, 3544.651, 3586.6156, 3627.337, 3670.083,
+ 3711.1538, 3753.5094, 3797.01, 3838.6686, 3882.1678, 3922.8116,
3967.9978, 4009.9204, 4054.3286, 4097.5706, 4140.6014,
+ 4185.544, 4229.5976, 4274.583, 4316.9438, 4361.672, 4406.2786,
4451.8628, 4496.1834, 4543.505, 4589.1816, 4632.5188,
+ 4678.2294, 4724.8908, 4769.0194, 4817.052, 4861.4588, 4910.1596,
4956.4344, 5002.5238, 5048.13, 5093.6374, 5142.8162,
+ 5187.7894, 5237.3984, 5285.6078, 5331.0858, 5379.1036, 5428.6258,
5474.6018, 5522.7618, 5571.5822, 5618.59, 5667.9992,
+ 5714.88, 5763.454, 5808.6982, 5860.3644, 5910.2914, 5953.571,
6005.9232, 6055.1914, 6104.5882, 6154.5702, 6199.7036,
+ 6251.1764, 6298.7596, 6350.0302, 6398.061, 6448.4694, 6495.933,
6548.0474, 6597.7166, 6646.9416, 6695.9208, 6742.6328,
+ 6793.5276, 6842.1934, 6894.2372, 6945.3864, 6996.9228, 7044.2372,
7094.1374, 7142.2272, 7192.2942, 7238.8338, 7288.9006,
+ 7344.0908, 7394.8544, 7443.5176, 7490.4148, 7542.9314, 7595.6738,
7641.9878, 7694.3688, 7743.0448, 7797.522, 7845.53,
+ 7899.594, 7950.3132, 7996.455, 8050.9442, 8092.9114, 8153.1374,
8197.4472, 8252.8278, 8301.8728, 8348.6776, 8401.4698,
+ 8453.551, 8504.6598, 8553.8944, 8604.1276, 8657.6514, 8710.3062,
8758.908, 8807.8706, 8862.1702, 8910.4668, 8960.77,
+ 9007.2766, 9063.164, 9121.0534, 9164.1354, 9218.1594, 9267.767,
9319.0594, 9372.155, 9419.7126, 9474.3722, 9520.1338,
+ 9572.368, 9622.7702, 9675.8448, 9726.5396, 9778.7378, 9827.6554,
9878.1922, 9928.7782, 9978.3984, 10026.578, 10076.5626,
+ 10137.1618, 10177.5244, 10229.9176},
+ // precision 12
+ {2954, 3003.4782, 3053.3568, 3104.3666, 3155.324,
3206.9598, 3259.648, 3312.539, 3366.1474, 3420.2576,
+ 3474.8376, 3530.6076, 3586.451, 3643.38, 3700.4104,
3757.5638, 3815.9676, 3875.193, 3934.838, 3994.8548,
+ 4055.018, 4117.1742, 4178.4482, 4241.1294, 4304.4776,
4367.4044, 4431.8724, 4496.3732, 4561.4304, 4627.5326,
+ 4693.949, 4761.5532, 4828.7256, 4897.6182, 4965.5186,
5034.4528, 5104.865, 5174.7164, 5244.6828, 5316.6708,
+ 5387.8312, 5459.9036, 5532.476, 5604.8652, 5679.6718,
5753.757, 5830.2072, 5905.2828, 5980.0434, 6056.6264,
+ 6134.3192, 6211.5746, 6290.0816, 6367.1176, 6447.9796,
6526.5576, 6606.1858, 6686.9144, 6766.1142, 6847.0818,
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7344.018, 7426.4214, 7511.3106, 7596.0686, 7679.8094,
+ 7765.818, 7852.4248, 7936.834, 8022.363, 8109.5066,
8200.4554, 8288.5832, 8373.366, 8463.4808, 8549.7682,
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9091.2522, 9179.988, 9269.852, 9362.6394, 9453.642,
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10007.9392, 10106.7508, 10196.2152, 10289.8114, 10383.5494,
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10952.793, 11049.9748, 11146.0702, 11241.4492, 11339.2772,
+ 11434.2336, 11530.741, 11627.6136, 11726.311, 11821.5964,
11918.837, 12015.3724, 12113.0162, 12213.0424, 12306.9804,
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12898.5142, 12997.0488, 13094.788, 13198.475, 13292.7764,
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13887.2638, 13992.0978, 14081.0844, 14189.9956, 14280.0912,
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14888.0284, 14985.1864, 15088.8596, 15187.0998, 15285.027,
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15898.4116, 15997.4522, 16095.5014, 16198.8514, 16291.7492,
+ 16402.6424, 16499.1266, 16606.2436, 16697.7186, 16796.3946,
16902.3376, 17005.7672, 17100.814, 17206.8282, 17305.8262,
+ 17416.0744, 17508.4092, 17617.0178, 17715.4554, 17816.758,
17920.1748, 18012.9236, 18119.7984, 18223.2248, 18324.2482,
+ 18426.6276, 18525.0932, 18629.8976, 18733.2588, 18831.0466,
18940.1366, 19032.2696, 19131.729, 19243.4864, 19349.6932,
+ 19442.866, 19547.9448, 19653.2798, 19754.4034, 19854.0692,
19965.1224, 20065.1774, 20158.2212, 20253.353, 20366.3264,
+ 20463.22},
+ // precision 13
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6414.5514, 6519.3376, 6625.6952, 6732.5988, 6841.3552,
+ 6950.5972, 7061.3082, 7173.5646, 7287.109, 7401.8216,
7516.4344, 7633.3802, 7751.2962, 7870.3784, 7990.292,
+ 8110.79, 8233.4574, 8356.6036, 8482.2712, 8607.7708,
8735.099, 8863.1858, 8993.4746, 9123.8496, 9255.6794,
+ 9388.5448, 9522.7516, 9657.3106, 9792.6094, 9930.5642,
10068.794, 10206.7256, 10347.81, 10490.3196, 10632.0778,
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11508.1006, 11659.1716, 11808.7514, 11959.4884, 12112.1314,
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13047.2144, 13207.3096, 13368.5144, 13528.024, 13689.847,
+ 13852.7528, 14018.3168, 14180.5372, 14346.9668, 14513.5074,
14677.867, 14846.2186, 15017.4186, 15184.9716, 15356.339,
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16389.4188, 16565.9126, 16742.3272, 16919.0042, 17094.7592,
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18171.051, 18354.7938, 18539.466, 18721.0408, 18904.9972,
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20013.1292, 20199.3858, 20387.8726, 20572.9514, 20770.7764,
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21906.3868, 22096.2626, 22286.0524, 22475.051, 22665.5098,
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23826.093, 24020.3296, 24213.3896, 24411.7392, 24602.9614,
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25780.6976, 25981.2728, 26175.977, 26376.5252, 26570.1964,
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27758.7428, 27961.1276, 28163.2324, 28362.3816, 28565.7668,
+ 28758.644, 28956.9768, 29163.4722, 29354.7026, 29561.1186,
29767.9948, 29959.9986, 30164.0492, 30366.9818, 30562.5338,
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31770.809, 31974.8934, 32179.5286, 32387.5442, 32582.3504,
+ 32794.076, 32989.9528, 33191.842, 33392.4684, 33595.659,
33801.8672, 34000.3414, 34200.0922, 34402.6792, 34610.0638,
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35830.6534, 36028.4966, 36229.7902, 36438.6422, 36630.7764,
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37854.5526, 38059.305, 38268.0936, 38470.2516, 38674.7064,
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39898.4108, 40093.1836, 40297.6858, 40489.7086, 40717.2424},
+ // precision 14
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12830.0086, 13040.0072, 13252.503, 13466.178, 13683.2738,
+ 13902.0344, 14123.9798, 14347.394, 14573.7784, 14802.6894,
15033.6824, 15266.9134, 15502.8624, 15741.4944, 15980.7956,
+ 16223.8916, 16468.6316, 16715.733, 16965.5726, 17217.204,
17470.666, 17727.8516, 17986.7886, 18247.6902, 18510.9632,
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20135.924, 20417.0324, 20697.9788, 20979.6112, 21265.0274,
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23020.5606, 23319.7394, 23620.4014, 23925.2728, 24226.9224,
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26103.7764, 26420.4132, 26742.0186, 27062.8852, 27388.415,
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29364.2156, 29704.497, 30037.1458, 30380.111, 30723.8168,
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32794.767, 33145.204, 33498.4226, 33847.6502, 34209.006,
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36359.1394, 36722.8266, 37082.8516, 37447.7354, 37815.9606,
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40042.261, 40416.2036, 40779.2036, 41161.6436, 41540.9014,
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43818.432, 44198.6598, 44583.0138, 44970.4794, 45353.924,
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47668.1472, 48055.6796, 48446.9436, 48838.7146, 49217.7296,
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51583.1882, 51971.0796, 52376.5338, 52763.319, 53165.5534,
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55543.4824, 55941.1748, 56333.7746, 56745.1552, 57142.7944,
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59542.6896, 59958.8004, 60349.3788, 60755.0212, 61147.6144,
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63560.635, 63974.3482, 64366.4908, 64771.5876, 65176.7346,
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67612.2032, 68019.0078, 68420.0388, 68821.22, 69235.8388,
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79825.725, 80238.701, 80646.891, 81035.6436, 81460.0448,
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+ // precision 15
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25661.326, 26081.3532, 26505.2806, 26933.9892, 27367.7098,
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30066.8668, 30534.2344, 31006.32, 31480.778, 31962.2418,
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34944.1402, 35457.5588, 35974.5958, 36497.3296, 37021.9096,
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40274.5554, 40832.3142, 41390.613, 41959.5908, 42532.5466,
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46038.0586, 46640.3164, 47241.064, 47846.155, 48454.7396,
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52210.2906, 52843.7396, 53481.3996, 54127.236, 54770.406,
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58730.308, 59404.9784, 60077.0864, 60751.9158, 61444.1386,
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65582.7468, 66274.5318, 66976.9276, 67688.7764, 68402.138,
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72708.4656, 73433.384, 74158.4664, 74896.4868, 75620.9564,
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80067.8658, 80814.9246, 81567.0136, 82310.8536, 83061.9952,
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103152.1392, 103944.2712, 104730.216, 105528.6336, 106324.9398,
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111075.0958, 111878.0496, 112695.2864, 113464.5486, 114270.0474,
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119085.4092, 119879.2808, 120687.5868, 121499.9944, 122284.916,
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127138.9412, 127943.8294, 128755.646, 129556.5354, 130375.3298,
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135220.2906, 136023.398, 136846.6558, 137667.0004, 138463.663,
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143356.1096, 144173.7412, 144973.0948, 145794.3162, 146609.5714,
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151494.0754, 152313.1416, 153112.6902, 153935.7206, 154746.9262,
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159646.9184, 160470.4458, 161279.5348, 162093.3114, 162918.542,
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+ // precision 16
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51322.9514, 52161.03, 53009.407, 53867.6348, 54734.206,
+ 55610.5144, 56496.2096, 57390.795, 58297.268, 59210.6448,
60134.665, 61068.0248, 62010.4472, 62962.5204, 63923.5742,
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69882.8544, 70911.271, 71944.0924, 72990.0326, 74040.692,
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80545.4976, 81657.2796, 82784.6336, 83915.515, 85059.7362,
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92080.197, 93275.5738, 94479.391, 95695.918, 96919.2236,
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104409.4588, 105686.2882, 106967.5402, 108261.6032, 109548.1578,
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131181.7776, 132577.3086, 133979.9458, 135394.1132, 136800.9078,
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145420.2872, 146878.1542, 148332.7572, 149800.3202, 151269.66,
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160160.059, 161650.6854, 163145.7772, 164645.6726, 166159.1952,
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175265.5556, 176787.799, 178317.111, 179856.6914, 181400.865,
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190688.4526, 192250.1926, 193810.9042, 195354.2972, 196938.7682,
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222207.1166, 223781.0364, 225387.4332, 227005.7928, 228590.4336,
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238190.7904, 239817.2548, 241411.2832, 243002.4066, 244640.1884,
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254332.9242, 255935.129, 257526.9014, 259154.772, 260777.625,
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270470.7252, 272106.4804, 273722.4456, 275337.794, 276945.7038,
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286735.3362, 288364.7164, 289961.1814, 291595.5524, 293285.683,
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302997.6748, 304615.1478, 306269.7724, 307886.114, 309543.1028,
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319305.2648, 320948.7406, 322566.3364, 324228.4224, 325847.1542},
+ // precision 17
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102646.7565, 104324.5125, 106021.7435, 107736.7865, 109469.272,
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120267.2345, 122134.6765, 124020.937, 125927.2705, 127851.255,
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139770.568, 141821.518, 143891.343, 145982.1415, 148095.387,
+ 150207.526, 152355.649, 154515.6415, 156696.05, 158887.7575,
161098.159, 163329.852, 165569.053, 167837.4005, 170121.6165,
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184151.939, 186551.6895, 188965.691, 191402.8095, 193857.949,
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208818.1365, 211373.115, 213946.7465, 216532.076, 219105.541,
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234952.2065, 237645.3555, 240331.1925, 243034.517, 245756.0725,
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262368.335, 265156.911, 267965.266, 270785.583, 273616.0495,
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290856.018, 293774.0345, 296678.5145, 299603.6355, 302552.6575,
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320285.9785, 323301.7325, 326298.3235, 329301.3105, 332301.987,
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350507.28, 353619.2345, 356631.2005, 359685.203, 362776.7845,
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381328.311, 384430.5225, 387576.425, 390683.242, 393839.648,
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412678.7, 415847.423, 419020.8035, 422157.081, 425337.749,
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444379.2545, 447577.356, 450741.931, 453959.548, 457137.0935,
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476347.6345, 479496.173, 482813.1645, 486025.6995, 489249.4885,
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508550.9915, 511806.7305, 515026.776, 518217.0005, 521523.9855,
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540926.922, 544207.094, 547429.4345, 550666.3745, 553975.3475,
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573447.425, 576689.6245, 579874.5745, 583202.337, 586503.0255,
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638691.7195, 641887.467, 645139.9405, 648441.546, 651666.252,
+ 654941.845},
+ // precision 18
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205294.444, 208651.754, 212042.099, 215472.269,
+ 218941.91, 222443.912, 225996.845, 229568.199, 233193.568,
236844.457, 240543.233, 244279.475, 248044.27,
+ 251854.588, 255693.2, 259583.619, 263494.621, 267445.385,
271454.061, 275468.769, 279549.456, 283646.446,
+ 287788.198, 291966.099, 296181.164, 300431.469, 304718.618,
309024.004, 313393.508, 317760.803, 322209.731,
+ 326675.061, 331160.627, 335654.47, 340241.442, 344841.833,
349467.132, 354130.629, 358819.432, 363574.626,
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392601.981, 397544.323, 402529.115, 407546.018,
+ 412593.658, 417638.657, 422762.865, 427886.169, 433017.167,
438213.273, 443441.254, 448692.421, 453937.533,
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486070.26, 491515.237, 496995.651, 502476.617,
+ 507973.609, 513497.19, 519083.233, 524726.509, 530305.505,
535945.728, 541584.404, 547274.055, 552967.236,
+ 558667.862, 564360.216, 570128.148, 575965.08, 581701.952,
587532.523, 593361.144, 599246.128, 605033.418,
+ 610958.779, 616837.117, 622772.818, 628672.04, 634675.369,
640574.831, 646585.739, 652574.547, 658611.217,
+ 664642.684, 670713.914, 676737.681, 682797.313, 688837.897,
694917.874, 701009.882, 707173.648, 713257.254,
+ 719415.392, 725636.761, 731710.697, 737906.209, 744103.074,
750313.39, 756504.185, 762712.579, 768876.985,
+ 775167.859, 781359, 787615.959, 793863.597, 800245.477,
806464.582, 812785.294, 819005.925, 825403.057,
+ 831676.197, 837936.284, 844266.968, 850642.711, 856959.756,
863322.774, 869699.931, 876102.478, 882355.787,
+ 888694.463, 895159.952, 901536.143, 907872.631, 914293.672,
920615.14, 927130.974, 933409.404, 939922.178,
+ 946331.47, 952745.93, 959209.264, 965590.224, 972077.284,
978501.961, 984953.19, 991413.271, 997817.479,
+ 1004222.658, 1010725.676, 1017177.138, 1023612.529, 1030098.236,
1036493.719, 1043112.207, 1049537.036, 1056008.096,
+ 1062476.184, 1068942.337, 1075524.95, 1081932.864, 1088426.025,
1094776.005, 1101327.448, 1107901.673, 1114423.639,
+ 1120884.602, 1127324.923, 1133794.24, 1140328.886, 1146849.376,
1153346.682, 1159836.502, 1166478.703, 1172953.304,
+ 1179391.502, 1185950.982, 1192544.052, 1198913.41, 1205430.994,
1212015.525, 1218674.042, 1225121.683, 1231551.101,
+ 1238126.379, 1244673.795, 1251260.649, 1257697.86, 1264320.983,
1270736.319, 1277274.694, 1283804.95, 1290211.514,
+ 1296858.568, 1303455.691}}};
+
+ PaddedPODArray<UInt64> registers; // Declaration of registers
+};
+
+class AggregateFunctionUniqHyperLogLogPlusPlus final
+ : public IAggregateFunctionDataHelper<HyperLogLogPlusPlusData,
AggregateFunctionUniqHyperLogLogPlusPlus>
+{
+public:
+ explicit AggregateFunctionUniqHyperLogLogPlusPlus(const DataTypes &
argument_types_, const Array & params)
+ : IAggregateFunctionDataHelper(argument_types_, {}, createResultType())
+ {
+ if (argument_types.size() != 1)
+ throw Exception(
+ ErrorCodes::NUMBER_OF_ARGUMENTS_DOESNT_MATCH,
+ "Aggregate function '{}' requires exactly one argument: got
{}",
+ getName(),
+ argument_types.size());
+
+ WhichDataType which(argument_types[0]);
+ if (!which.isUInt64())
+ throw Exception(
+ ErrorCodes::ILLEGAL_TYPE_OF_ARGUMENT,
+ "Aggregate function '{}' requires an argument of type UInt64,
got {}",
+ getName(),
+ argument_types[0]->getName());
+
+ if (params.empty())
+ relative_sd = 0.05;
+ else if (params.size() == 1)
+ {
+ relative_sd = applyVisitor(FieldVisitorConvertToNumber<Float64>(),
params[0]);
+ if (isNaN(relative_sd) || relative_sd <= 0 || relative_sd > 1)
+ throw Exception(ErrorCodes::PARAMETER_OUT_OF_BOUND, "Relative
standard deviation must be in the range (0, 1]");
+ }
+ else
+ throw Exception(
+ ErrorCodes::NUMBER_OF_ARGUMENTS_DOESNT_MATCH, "Aggregate
function {} requires one parameter: relative_sd", getName());
+ }
+
+ String getName() const override { return "uniqHLLPP"; }
+
+ static DataTypePtr createResultType() { return
std::make_shared<DataTypeUInt64>(); }
+
+ bool allocatesMemoryInArena() const override { return false; }
+
+ void create(AggregateDataPtr __restrict place) const override /// NOLINT
+ {
+ new (place) Data(relative_sd);
+ }
+
+ void add(AggregateDataPtr __restrict place, const IColumn ** columns,
size_t row_num, Arena *) const override
+ {
+ auto value = static_cast<const ColumnUInt64
&>(*columns[0]).getData()[row_num];
+ data(place).add(value);
+ }
+
+ void merge(AggregateDataPtr __restrict place, ConstAggregateDataPtr rhs,
Arena *) const override { data(place).merge(data(rhs)); }
+
+ void serialize(ConstAggregateDataPtr place, WriteBuffer & buf,
std::optional<size_t>) const override { data(place).serialize(buf); }
+
+ void deserialize(AggregateDataPtr place, ReadBuffer & buf,
std::optional<size_t>, Arena *) const override
+ {
+ data(place).deserialize(buf);
+ }
+
+ DataTypePtr getNormalizedStateType() const override
+ {
+ /// Return normalized state type: quantiles*(1)(...)
+ Array params{0.05};
+ AggregateFunctionProperties properties =
{.returns_default_when_only_null = true, .is_order_dependent = false};
+ return std::make_shared<DataTypeAggregateFunction>(
+ AggregateFunctionFactory::instance().get(getName(),
NullsAction::EMPTY, this->argument_types, params, properties),
+ this->argument_types,
+ params);
+ }
+
+ void insertResultInto(AggregateDataPtr __restrict place, IColumn & to,
Arena *) const override
+ {
+ static_cast<ColumnUInt64
&>(to).getData().push_back(data(place).query());
+ }
+
+private:
+ /// Defines the maximum relative standard deviation allowed.
+ Float64 relative_sd;
+};
+
+}
diff --git a/cpp-ch/local-engine/Common/CHUtil.cpp
b/cpp-ch/local-engine/Common/CHUtil.cpp
index 5d7a5c803b..29f4013ee5 100644
--- a/cpp-ch/local-engine/Common/CHUtil.cpp
+++ b/cpp-ch/local-engine/Common/CHUtil.cpp
@@ -109,6 +109,8 @@ extern const ServerSettingsUInt64 thread_pool_queue_size;
extern const ServerSettingsUInt64 max_io_thread_pool_size;
extern const ServerSettingsUInt64 io_thread_pool_queue_size;
}
+
+extern void
registerAggregateFunctionUniqHyperLogLogPlusPlus(AggregateFunctionFactory &);
}
namespace local_engine
@@ -855,22 +857,24 @@ extern void
registerAggregateFunctionCombinatorPartialMerge(AggregateFunctionCom
extern void registerAggregateFunctionsBloomFilter(AggregateFunctionFactory &);
extern void registerAggregateFunctionSparkAvg(AggregateFunctionFactory &);
extern void registerAggregateFunctionRowNumGroup(AggregateFunctionFactory &);
+
+
extern void registerFunctions(FunctionFactory &);
void registerAllFunctions()
{
DB::registerFunctions();
-
DB::registerAggregateFunctions();
+
auto & agg_factory = AggregateFunctionFactory::instance();
registerAggregateFunctionsBloomFilter(agg_factory);
registerAggregateFunctionSparkAvg(agg_factory);
registerAggregateFunctionRowNumGroup(agg_factory);
- {
- /// register aggregate function combinators from local_engine
- auto & factory = AggregateFunctionCombinatorFactory::instance();
- registerAggregateFunctionCombinatorPartialMerge(factory);
- }
+ DB::registerAggregateFunctionUniqHyperLogLogPlusPlus(agg_factory);
+
+ /// register aggregate function combinators from local_engine
+ auto & combinator_factory = AggregateFunctionCombinatorFactory::instance();
+ registerAggregateFunctionCombinatorPartialMerge(combinator_factory);
}
void registerGlutenDisks()
diff --git
a/cpp-ch/local-engine/Parser/aggregate_function_parser/ApproxCountDistinctFunctionParser.cpp
b/cpp-ch/local-engine/Parser/aggregate_function_parser/ApproxCountDistinctFunctionParser.cpp
new file mode 100644
index 0000000000..d214858365
--- /dev/null
+++
b/cpp-ch/local-engine/Parser/aggregate_function_parser/ApproxCountDistinctFunctionParser.cpp
@@ -0,0 +1,148 @@
+
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements. See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License. You may obtain a copy of the License at
+ *
+ * http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+#include <DataTypes/DataTypeAggregateFunction.h>
+#include <DataTypes/DataTypeNullable.h>
+#include <DataTypes/DataTypesNumber.h>
+#include <Functions/FunctionHelpers.h>
+#include <Parser/AggregateFunctionParser.h>
+#include <Poco/Logger.h>
+#include <Common/logger_useful.h>
+
+namespace DB
+{
+namespace ErrorCodes
+{
+ extern const int NUMBER_OF_ARGUMENTS_DOESNT_MATCH;
+ extern const int BAD_ARGUMENTS;
+}
+}
+
+namespace local_engine
+{
+
+using namespace DB;
+
+struct ApproxCountDistinctNameStruct
+{
+ static constexpr auto spark_name = "approx_count_distinct";
+ static constexpr auto ch_name = "uniqHLLPP";
+};
+
+/// Spark approx_count_distinct(expr, relative_sd) = CH
uniqHLLPP(relative_sd)(if(isNull(expr), null, sparkXxHash64(expr)))
+/// Spark approx_count_distinct(expr) = CH
uniqHLLPP(0.05)(if(isNull(expr), null, sparkXxHash64(expr)))
+template <typename NameStruct>
+class AggregateFunctionParserApproxCountDistinct final : public
AggregateFunctionParser
+{
+public:
+ static constexpr auto name = NameStruct::spark_name;
+
+ AggregateFunctionParserApproxCountDistinct(ParserContextPtr
parser_context_) : AggregateFunctionParser(parser_context_) { }
+ ~AggregateFunctionParserApproxCountDistinct() override = default;
+
+ String getName() const override { return NameStruct::spark_name; }
+
+ String getCHFunctionName(const CommonFunctionInfo &) const override {
return NameStruct::ch_name; }
+
+ String getCHFunctionName(DataTypes & types) const override
+ {
+ /// Always invoked during second stage, the first argument is expr,
the second argument is relative_sd.
+ /// 1. Remove the second argument because types are used to create the
aggregate function.
+ /// 2. Replace the first argument type with UInt64 or Nullable(UInt64)
because uniqHLLPP requres it.
+ types.resize(1);
+ const auto old_type = types[0];
+ types[0] = std::make_shared<DataTypeUInt64>();
+ if (old_type->isNullable())
+ types[0] = std::make_shared<DataTypeNullable>(types[0]);
+
+ return NameStruct::ch_name;
+ }
+
+ Array parseFunctionParameters(
+ const CommonFunctionInfo & func_info, ActionsDAG::NodeRawConstPtrs &
arg_nodes, ActionsDAG & actions_dag) const override
+ {
+ if (func_info.phase ==
substrait::AGGREGATION_PHASE_INITIAL_TO_INTERMEDIATE
+ || func_info.phase ==
substrait::AGGREGATION_PHASE_INITIAL_TO_RESULT
+ || func_info.phase == substrait::AGGREGATION_PHASE_UNSPECIFIED)
+ {
+ const auto & arguments = func_info.arguments;
+ const size_t num_args = arguments.size();
+ const size_t num_nodes = arg_nodes.size();
+ if (num_args != num_nodes || num_args > 2 || num_args < 1 ||
num_nodes > 2 || num_nodes < 1)
+ throw Exception(
+ ErrorCodes::NUMBER_OF_ARGUMENTS_DOESNT_MATCH,
+ "Function {} takes 1 or 2 arguments in phase {}",
+ getName(),
+ magic_enum::enum_name(func_info.phase));
+
+ Array params;
+ if (num_args == 2)
+ {
+ const auto & relative_sd_arg = arguments[1].value();
+ if (relative_sd_arg.has_literal())
+ {
+ auto [_, field] = parseLiteral(relative_sd_arg.literal());
+ params.push_back(std::move(field));
+ }
+ else
+ throw Exception(ErrorCodes::BAD_ARGUMENTS, "Second
argument of function {} must be literal", getName());
+ }
+ else
+ {
+ params.push_back(0.05);
+ }
+
+ const auto & expr_arg = arg_nodes[0];
+ const auto * is_null_node = toFunctionNode(actions_dag, "isNull",
{expr_arg});
+ const auto * hash_node = toFunctionNode(actions_dag,
"sparkXxHash64", {expr_arg});
+ const auto * null_node
+ = addColumnToActionsDAG(actions_dag,
std::make_shared<DataTypeNullable>(std::make_shared<DataTypeUInt64>()), {});
+ const auto * if_node = toFunctionNode(actions_dag, "if",
{is_null_node, null_node, hash_node});
+ /// Replace the first argument expr with if(isNull(expr), null,
sparkXxHash64(expr))
+ arg_nodes[0] = if_node;
+ arg_nodes.resize(1);
+
+ return params;
+ }
+ else
+ {
+ if (arg_nodes.size() != 1)
+ throw Exception(
+ ErrorCodes::NUMBER_OF_ARGUMENTS_DOESNT_MATCH,
+ "Function {} takes 1 argument in phase {}",
+ getName(),
+ magic_enum::enum_name(func_info.phase));
+
+ const auto & result_type = arg_nodes[0]->result_type;
+ const auto * aggregate_function_type =
checkAndGetDataType<DataTypeAggregateFunction>(result_type.get());
+ if (!aggregate_function_type)
+ throw Exception(
+ ErrorCodes::BAD_ARGUMENTS,
+ "The first argument type of function {} in phase {} must
be AggregateFunction, but is {}",
+ getName(),
+ magic_enum::enum_name(func_info.phase),
+ result_type->getName());
+
+ return aggregate_function_type->getParameters();
+ }
+ }
+
+ Array getDefaultFunctionParameters() const override { return {0.05}; }
+};
+
+static const
AggregateFunctionParserRegister<AggregateFunctionParserApproxCountDistinct<ApproxCountDistinctNameStruct>>
registerer_approx_count_distinct;
+}
diff --git a/cpp-ch/local-engine/tests/gtest_hllpp.cpp
b/cpp-ch/local-engine/tests/gtest_hllpp.cpp
new file mode 100644
index 0000000000..8ca36bbbb4
--- /dev/null
+++ b/cpp-ch/local-engine/tests/gtest_hllpp.cpp
@@ -0,0 +1,88 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements. See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License. You may obtain a copy of the License at
+ *
+ * http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+#include <gtest/gtest.h>
+#include <AggregateFunctions/AggregateFunctionUniqHyperLogLogPlusPlus.h>
+#include <IO/ReadBufferFromString.h>
+
+using namespace DB;
+
+static std::vector<UInt64> random_uint64s
+ = {17956993516945311251ULL,
+ 4306050051188505054ULL,
+ 14289061765075743502ULL,
+ 16763375724458316157ULL,
+ 6144297519955185930ULL,
+ 18446472757487308114ULL,
+ 16923578592198257123ULL,
+ 13557354668567515845ULL,
+ 15328387702200001967ULL,
+ 15878166530370497646ULL};
+
+static void initSmallHLL(HyperLogLogPlusPlusData & hll)
+{
+ for (auto x : random_uint64s)
+ hll.add(x);
+}
+
+static void initLargeHLL(HyperLogLogPlusPlusData & hll)
+{
+ for (auto x : random_uint64s)
+ {
+ for (size_t i = 0; i < 100; ++i)
+ hll.add(x * (i+1));
+ }
+}
+
+TEST(HyperLogLogPlusPlusDataTest, Small)
+{
+ HyperLogLogPlusPlusData hll;
+ initSmallHLL(hll);
+ EXPECT_EQ(hll.query(), 10);
+}
+
+TEST(HyperLogLogPlusPlusDataTest, Large)
+{
+ HyperLogLogPlusPlusData hll;
+ initLargeHLL(hll);
+ EXPECT_EQ(hll.query(), 806);
+}
+
+TEST(HyperLogLogPlusPlusDataTest, Merge) {
+ HyperLogLogPlusPlusData hll1;
+ initSmallHLL(hll1);
+
+ HyperLogLogPlusPlusData hll2;
+ initLargeHLL(hll2);
+
+ hll1.merge(hll2);
+ EXPECT_EQ(hll1.query(), 806);
+}
+
+TEST(HyperLogLogPlusPlusDataTest, SerializeAndDeserialize) {
+ HyperLogLogPlusPlusData hll1;
+ initLargeHLL(hll1);
+
+ WriteBufferFromOwnString write_buffer;
+ hll1.serialize(write_buffer);
+
+ ReadBufferFromString read_buffer(write_buffer.str());
+ HyperLogLogPlusPlusData hll2;
+ hll2.deserialize(read_buffer);
+
+ EXPECT_EQ(hll2.query(), 806);
+}
diff --git
a/gluten-substrait/src/main/scala/org/apache/gluten/expression/ExpressionMappings.scala
b/gluten-substrait/src/main/scala/org/apache/gluten/expression/ExpressionMappings.scala
index 14f912cb47..354ad9ea79 100644
---
a/gluten-substrait/src/main/scala/org/apache/gluten/expression/ExpressionMappings.scala
+++
b/gluten-substrait/src/main/scala/org/apache/gluten/expression/ExpressionMappings.scala
@@ -323,6 +323,7 @@ object ExpressionMappings {
Sig[Skewness](SKEWNESS),
Sig[Kurtosis](KURTOSIS),
Sig[ApproximatePercentile](APPROX_PERCENTILE),
+ Sig[HyperLogLogPlusPlus](APPROX_COUNT_DISTINCT),
Sig[Percentile](PERCENTILE)
) ++ SparkShimLoader.getSparkShims.aggregateExpressionMappings
diff --git
a/gluten-ut/spark32/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
b/gluten-ut/spark32/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
index a8eaab5486..a27e56fc32 100644
---
a/gluten-ut/spark32/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
+++
b/gluten-ut/spark32/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
@@ -221,7 +221,6 @@ class ClickHouseTestSettings extends BackendTestSettings {
.exclude("SPARK-28067: Aggregate sum should not return wrong results for
decimal overflow")
.exclude("SPARK-35955: Aggregate avg should not return wrong results for
decimal overflow")
.exclude("describe")
- .exclude("SPARK-34165: Add count_distinct to summary")
.exclude("getRows: array")
.exclude("showString: array")
.exclude("showString: array, vertical = true")
diff --git
a/gluten-ut/spark33/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
b/gluten-ut/spark33/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
index b979651ae2..c0f80e4f53 100644
---
a/gluten-ut/spark33/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
+++
b/gluten-ut/spark33/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
@@ -243,7 +243,6 @@ class ClickHouseTestSettings extends BackendTestSettings {
.exclude("SPARK-28067: Aggregate sum should not return wrong results for
decimal overflow")
.exclude("SPARK-35955: Aggregate avg should not return wrong results for
decimal overflow")
.exclude("describe")
- .exclude("SPARK-34165: Add count_distinct to summary")
.exclude("getRows: array")
.exclude("showString: array")
.exclude("showString: array, vertical = true")
diff --git
a/gluten-ut/spark34/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
b/gluten-ut/spark34/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
index 2c315d8aae..495e764e1d 100644
---
a/gluten-ut/spark34/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
+++
b/gluten-ut/spark34/src/test/scala/org/apache/gluten/utils/clickhouse/ClickHouseTestSettings.scala
@@ -241,7 +241,6 @@ class ClickHouseTestSettings extends BackendTestSettings {
.exclude("SPARK-28067: Aggregate sum should not return wrong results for
decimal overflow")
.exclude("SPARK-35955: Aggregate avg should not return wrong results for
decimal overflow")
.exclude("describe")
- .exclude("SPARK-34165: Add count_distinct to summary")
.exclude("getRows: array")
.exclude("showString: array")
.exclude("showString: array, vertical = true")
diff --git
a/shims/common/src/main/scala/org/apache/gluten/expression/ExpressionNames.scala
b/shims/common/src/main/scala/org/apache/gluten/expression/ExpressionNames.scala
index 9e974120fa..2c919cebe9 100644
---
a/shims/common/src/main/scala/org/apache/gluten/expression/ExpressionNames.scala
+++
b/shims/common/src/main/scala/org/apache/gluten/expression/ExpressionNames.scala
@@ -47,6 +47,7 @@ object ExpressionNames {
final val FIRST_IGNORE_NULL = "first_ignore_null"
final val APPROX_DISTINCT = "approx_distinct"
final val APPROX_PERCENTILE = "approx_percentile"
+ final val APPROX_COUNT_DISTINCT = "approx_count_distinct"
final val PERCENTILE = "percentile"
final val SKEWNESS = "skewness"
final val KURTOSIS = "kurtosis"
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