cpp-ch/local-engine/AggregateFunctions/AggregateFunctionUniqHyperLogLogPlusPlus.h (2,937 lines of code) (raw):
/*
* 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(static_cast<size_t>(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,
0.841399999999993,
0.879800000000003,
0.837599999999995,
0.469800000000006,
0.765600000000006,
0.331000000000003,
0.591399999999993,
0.601200000000006,
0.701599999999999,
0.558199999999999,
0.339399999999998,
0.354399999999998,
0.491200000000006,
0.308000000000007,
0.355199999999996,
-0.0254000000000048,
0.205200000000005,
-0.272999999999996,
0.132199999999997,
0.394400000000005,
-0.241200000000006,
0.242000000000004,
0.191400000000002,
0.253799999999998,
-0.122399999999999,
-0.370800000000003,
0.193200000000004,
-0.0848000000000013,
0.0867999999999967,
-0.327200000000005,
-0.285600000000002,
0.311400000000006,
-0.128399999999999,
-0.754999999999995,
-0.209199999999996,
-0.293599999999998,
-0.364000000000004,
-0.253600000000006,
-0.821200000000005,
-0.253600000000006,
-0.510400000000004,
-0.383399999999995,
-0.491799999999998,
-0.220200000000006,
-0.0972000000000008,
-0.557400000000001,
-0.114599999999996,
-0.295000000000002,
-0.534800000000004,
0.346399999999988,
-0.65379999999999,
0.0398000000000138,
0.0341999999999985,
-0.995800000000003,
-0.523400000000009,
-0.489000000000004,
-0.274799999999999,
-0.574999999999989,
-0.482799999999997,
0.0571999999999946,
-0.330600000000004,
-0.628800000000012,
-0.140199999999993,
-0.540600000000012,
-0.445999999999998,
-0.599400000000003,
-0.262599999999992,
0.163399999999996,
-0.100599999999986,
-0.39500000000001,
-1.06960000000001,
-0.836399999999998,
-0.753199999999993,
-0.412399999999991,
-0.790400000000005,
-0.29679999999999,
-0.28540000000001,
-0.193000000000012,
-0.0772000000000048,
-0.962799999999987,
-0.414800000000014},
// precision 6
{45,
44.1902,
43.271,
42.8358,
41.8142,
41.2854,
40.317,
39.354,
38.8924,
37.9436,
37.4596,
36.5262,
35.6248,
35.1574,
34.2822,
33.837,
32.9636,
32.074,
31.7042,
30.7976,
30.4772,
29.6564,
28.7942,
28.5004,
27.686,
27.291,
26.5672,
25.8556,
25.4982,
24.8204,
24.4252,
23.7744,
23.0786,
22.8344,
22.0294,
21.8098,
21.0794,
20.5732,
20.1878,
19.5648,
19.2902,
18.6784,
18.3352,
17.8946,
17.3712,
17.0852,
16.499,
16.2686,
15.6844,
15.2234,
14.9732,
14.3356,
14.2286,
13.7262,
13.3284,
13.1048,
12.5962,
12.3562,
12.1272,
11.4184,
11.4974,
11.0822,
10.856,
10.48,
10.2834,
10.0208,
9.637,
9.51739999999999,
9.05759999999999,
8.74760000000001,
8.42700000000001,
8.1326,
8.2372,
8.2788,
7.6776,
7.79259999999999,
7.1952,
6.9564,
6.6454,
6.87,
6.5428,
6.19999999999999,
6.02940000000001,
5.62780000000001,
5.6782,
5.792,
5.35159999999999,
5.28319999999999,
5.0394,
5.07480000000001,
4.49119999999999,
4.84899999999999,
4.696,
4.54040000000001,
4.07300000000001,
4.37139999999999,
3.7216,
3.7328,
3.42080000000001,
3.41839999999999,
3.94239999999999,
3.27719999999999,
3.411,
3.13079999999999,
2.76900000000001,
2.92580000000001,
2.68279999999999,
2.75020000000001,
2.70599999999999,
2.3886,
3.01859999999999,
2.45179999999999,
2.92699999999999,
2.41720000000001,
2.41139999999999,
2.03299999999999,
2.51240000000001,
2.5564,
2.60079999999999,
2.41720000000001,
1.80439999999999,
1.99700000000001,
2.45480000000001,
1.8948,
2.2346,
2.30860000000001,
2.15479999999999,
1.88419999999999,
1.6508,
0.677199999999999,
1.72540000000001,
1.4752,
1.72280000000001,
1.66139999999999,
1.16759999999999,
1.79300000000001,
1.00059999999999,
0.905200000000008,
0.659999999999997,
1.55879999999999,
1.1636,
0.688199999999995,
0.712600000000009,
0.450199999999995,
1.1978,
0.975599999999986,
0.165400000000005,
1.727,
1.19739999999999,
-0.252600000000001,
1.13460000000001,
1.3048,
1.19479999999999,
0.313400000000001,
0.878999999999991,
1.12039999999999,
0.853000000000009,
1.67920000000001,
0.856999999999999,
0.448599999999999,
1.2362,
0.953399999999988,
1.02859999999998,
0.563199999999995,
0.663000000000011,
0.723000000000013,
0.756599999999992,
0.256599999999992,
-0.837600000000009,
0.620000000000005,
0.821599999999989,
0.216600000000028,
0.205600000000004,
0.220199999999977,
0.372599999999977,
0.334400000000016,
0.928400000000011,
0.972800000000007,
0.192400000000021,
0.487199999999973,
-0.413000000000011,
0.807000000000016,
0.120600000000024,
0.769000000000005,
0.870799999999974,
0.66500000000002,
0.118200000000002,
0.401200000000017,
0.635199999999998,
0.135400000000004,
0.175599999999974,
1.16059999999999,
0.34620000000001,
0.521400000000028,
-0.586599999999976,
-1.16480000000001,
0.968399999999974,
0.836999999999989,
0.779600000000016,
0.985799999999983},
// precision 7
{91,
89.4934,
87.9758,
86.4574,
84.9718,
83.4954,
81.5302,
80.0756,
78.6374,
77.1782,
75.7888,
73.9522,
72.592,
71.2532,
69.9086,
68.5938,
66.9474,
65.6796,
64.4394,
63.2176,
61.9768,
60.4214,
59.2528,
58.0102,
56.8658,
55.7278,
54.3044,
53.1316,
52.093,
51.0032,
49.9092,
48.6306,
47.5294,
46.5756,
45.6508,
44.662,
43.552,
42.3724,
41.617,
40.5754,
39.7872,
38.8444,
37.7988,
36.8606,
36.2118,
35.3566,
34.4476,
33.5882,
32.6816,
32.0824,
31.0258,
30.6048,
29.4436,
28.7274,
27.957,
27.147,
26.4364,
25.7592,
25.3386,
24.781,
23.8028,
23.656,
22.6544,
21.996,
21.4718,
21.1544,
20.6098,
19.5956,
19.0616,
18.5758,
18.4878,
17.5244,
17.2146,
16.724,
15.8722,
15.5198,
15.0414,
14.941,
14.9048,
13.87,
13.4304,
13.028,
12.4708,
12.37,
12.0624,
11.4668,
11.5532,
11.4352,
11.2564,
10.2744,
10.2118,
9.74720000000002,
10.1456,
9.2928,
8.75040000000001,
8.55279999999999,
8.97899999999998,
8.21019999999999,
8.18340000000001,
7.3494,
7.32499999999999,
7.66140000000001,
6.90300000000002,
7.25439999999998,
6.9042,
7.21499999999997,
6.28640000000001,
6.08139999999997,
6.6764,
6.30099999999999,
5.13900000000001,
5.65800000000002,
5.17320000000001,
4.59019999999998,
4.9538,
5.08280000000002,
4.92200000000003,
4.99020000000002,
4.7328,
5.4538,
4.11360000000002,
4.22340000000003,
4.08780000000002,
3.70800000000003,
4.15559999999999,
4.18520000000001,
3.63720000000001,
3.68220000000002,
3.77960000000002,
3.6078,
2.49160000000001,
3.13099999999997,
2.5376,
3.19880000000001,
3.21100000000001,
2.4502,
3.52820000000003,
2.91199999999998,
3.04480000000001,
2.7432,
2.85239999999999,
2.79880000000003,
2.78579999999999,
1.88679999999999,
2.98860000000002,
2.50639999999999,
1.91239999999999,
2.66160000000002,
2.46820000000002,
1.58199999999999,
1.30399999999997,
2.27379999999999,
2.68939999999998,
1.32900000000001,
3.10599999999999,
1.69080000000002,
2.13740000000001,
2.53219999999999,
1.88479999999998,
1.33240000000001,
1.45119999999997,
1.17899999999997,
2.44119999999998,
1.60659999999996,
2.16700000000003,
0.77940000000001,
2.37900000000002,
2.06700000000001,
1.46000000000004,
2.91160000000002,
1.69200000000001,
0.954600000000028,
2.49300000000005,
2.2722,
1.33500000000004,
2.44899999999996,
1.20140000000004,
3.07380000000001,
2.09739999999999,
2.85640000000001,
2.29960000000005,
2.40899999999999,
1.97040000000004,
0.809799999999996,
1.65279999999996,
2.59979999999996,
0.95799999999997,
2.06799999999998,
2.32780000000002,
4.20159999999998,
1.96320000000003,
1.86400000000003,
1.42999999999995,
3.77940000000001,
1.27200000000005,
1.86440000000005,
2.20600000000002,
3.21900000000005,
1.5154,
2.61019999999996},
// precision 8
{183.2152,
180.2454,
177.2096,
173.6652,
170.6312,
167.6822,
164.249,
161.3296,
158.0038,
155.2074,
152.4612,
149.27,
146.5178,
143.4412,
140.8032,
138.1634,
135.1688,
132.6074,
129.6946,
127.2664,
124.8228,
122.0432,
119.6824,
116.9464,
114.6268,
112.2626,
109.8376,
107.4034,
104.8956,
102.8522,
100.7638,
98.3552,
96.3556,
93.7526,
91.9292,
89.8954,
87.8198,
85.7668,
83.298,
81.6688,
79.9466,
77.9746,
76.1672,
74.3474,
72.3028,
70.8912,
69.114,
67.4646,
65.9744,
64.4092,
62.6022,
60.843,
59.5684,
58.1652,
56.5426,
55.4152,
53.5388,
52.3592,
51.1366,
49.486,
48.3918,
46.5076,
45.509,
44.3834,
43.3498,
42.0668,
40.7346,
40.1228,
38.4528,
37.7,
36.644,
36.0518,
34.5774,
33.9068,
32.432,
32.1666,
30.434,
29.6644,
28.4894,
27.6312,
26.3804,
26.292,
25.5496000000001,
25.0234,
24.8206,
22.6146,
22.4188,
22.117,
20.6762,
20.6576,
19.7864,
19.509,
18.5334,
17.9204,
17.772,
16.2924,
16.8654,
15.1836,
15.745,
15.1316,
15.0386,
14.0136,
13.6342,
12.6196,
12.1866,
12.4281999999999,
11.3324,
10.4794000000001,
11.5038,
10.129,
9.52800000000002,
10.3203999999999,
9.46299999999997,
9.79280000000006,
9.12300000000005,
8.74180000000001,
9.2192,
7.51020000000005,
7.60659999999996,
7.01840000000004,
7.22239999999999,
7.40139999999997,
6.76179999999999,
7.14359999999999,
5.65060000000005,
5.63779999999997,
5.76599999999996,
6.75139999999999,
5.57759999999996,
3.73220000000003,
5.8048,
5.63019999999995,
4.93359999999996,
3.47979999999995,
4.33879999999999,
3.98940000000005,
3.81960000000004,
3.31359999999995,
3.23080000000004,
3.4588,
3.08159999999998,
3.4076,
3.00639999999999,
2.38779999999997,
2.61900000000003,
1.99800000000005,
3.34820000000002,
2.95060000000001,
0.990999999999985,
2.11440000000005,
2.20299999999997,
2.82219999999995,
2.73239999999998,
2.7826,
3.76660000000004,
2.26480000000004,
2.31280000000004,
2.40819999999997,
2.75360000000001,
3.33759999999995,
2.71559999999999,
1.7478000000001,
1.42920000000004,
2.39300000000003,
2.22779999999989,
2.34339999999997,
0.87259999999992,
3.88400000000001,
1.80600000000004,
1.91759999999999,
1.16779999999994,
1.50320000000011,
2.52500000000009,
0.226400000000012,
2.31500000000005,
0.930000000000064,
1.25199999999995,
2.14959999999996,
0.0407999999999902,
2.5447999999999,
1.32960000000003,
0.197400000000016,
2.52620000000002,
3.33279999999991,
-1.34300000000007,
0.422199999999975,
0.917200000000093,
1.12920000000008,
1.46060000000011,
1.45779999999991,
2.8728000000001,
3.33359999999993,
-1.34079999999994,
1.57680000000005,
0.363000000000056,
1.40740000000005,
0.656600000000026,
0.801400000000058,
-0.454600000000028,
1.51919999999996},
// precision 9
{368,
361.8294,
355.2452,
348.6698,
342.1464,
336.2024,
329.8782,
323.6598,
317.462,
311.2826,
305.7102,
299.7416,
293.9366,
288.1046,
282.285,
277.0668,
271.306,
265.8448,
260.301,
254.9886,
250.2422,
244.8138,
239.7074,
234.7428,
229.8402,
225.1664,
220.3534,
215.594,
210.6886,
205.7876,
201.65,
197.228,
192.8036,
188.1666,
184.0818,
180.0824,
176.2574,
172.302,
168.1644,
164.0056,
160.3802,
156.7192,
152.5234,
149.2084,
145.831,
142.485,
139.1112,
135.4764,
131.76,
129.3368,
126.5538,
122.5058,
119.2646,
116.5902,
113.3818,
110.8998,
107.9532,
105.2062,
102.2798,
99.4728,
96.9582,
94.3292,
92.171,
89.7809999999999,
87.5716,
84.7048,
82.5322,
79.875,
78.3972,
75.3464,
73.7274,
71.2834,
70.1444,
68.4263999999999,
66.0166,
64.018,
62.0437999999999,
60.3399999999999,
58.6856,
57.9836,
55.0311999999999,
54.6769999999999,
52.3188,
51.4846,
49.4423999999999,
47.739,
46.1487999999999,
44.9202,
43.4059999999999,
42.5342000000001,
41.2834,
38.8954000000001,
38.3286000000001,
36.2146,
36.6684,
35.9946,
33.123,
33.4338,
31.7378000000001,
29.076,
28.9692,
27.4964,
27.0998,
25.9864,
26.7754,
24.3208,
23.4838,
22.7388000000001,
24.0758000000001,
21.9097999999999,
20.9728,
19.9228000000001,
19.9292,
16.617,
17.05,
18.2996000000001,
15.6128000000001,
15.7392,
14.5174,
13.6322,
12.2583999999999,
13.3766000000001,
11.423,
13.1232,
9.51639999999998,
10.5938000000001,
9.59719999999993,
8.12220000000002,
9.76739999999995,
7.50440000000003,
7.56999999999994,
6.70440000000008,
6.41419999999994,
6.71019999999999,
5.60940000000005,
4.65219999999999,
6.84099999999989,
3.4072000000001,
3.97859999999991,
3.32760000000007,
5.52160000000003,
3.31860000000006,
2.06940000000009,
4.35400000000004,
1.57500000000005,
0.280799999999999,
2.12879999999996,
-0.214799999999968,
-0.0378000000000611,
-0.658200000000079,
0.654800000000023,
-0.0697999999999865,
0.858400000000074,
-2.52700000000004,
-2.1751999999999,
-3.35539999999992,
-1.04019999999991,
-0.651000000000067,
-2.14439999999991,
-1.96659999999997,
-3.97939999999994,
-0.604400000000169,
-3.08260000000018,
-3.39159999999993,
-5.29640000000018,
-5.38920000000007,
-5.08759999999984,
-4.69900000000007,
-5.23720000000003,
-3.15779999999995,
-4.97879999999986,
-4.89899999999989,
-7.48880000000008,
-5.94799999999987,
-5.68060000000014,
-6.67180000000008,
-4.70499999999993,
-7.27779999999984,
-4.6579999999999,
-4.4362000000001,
-4.32139999999981,
-5.18859999999995,
-6.66879999999992,
-6.48399999999992,
-5.1260000000002,
-4.4032000000002,
-6.13500000000022,
-5.80819999999994,
-4.16719999999987,
-4.15039999999999,
-7.45600000000013,
-7.24080000000004,
-9.83179999999993,
-5.80420000000004,
-8.6561999999999,
-6.99940000000015,
-10.5473999999999,
-7.34139999999979,
-6.80999999999995,
-6.29719999999998,
-6.23199999999997},
// precision 10
{737.1256,
724.4234,
711.1064,
698.4732,
685.4636,
673.0644,
660.488,
647.9654,
636.0832,
623.7864,
612.1992,
600.2176,
588.5228,
577.1716,
565.7752,
554.899,
543.6126,
532.6492,
521.9474,
511.5214,
501.1064,
490.6364,
480.2468,
470.4588,
460.3832,
451.0584,
440.8606,
431.3868,
422.5062,
413.1862,
404.463,
395.339,
386.1936,
378.1292,
369.1854,
361.2908,
353.3324,
344.8518,
337.5204,
329.4854,
321.9318,
314.552,
306.4658,
299.4256,
292.849,
286.152,
278.8956,
271.8792,
265.118,
258.62,
252.5132,
245.9322,
239.7726,
233.6086,
227.5332,
222.5918,
216.4294,
210.7662,
205.4106,
199.7338,
194.9012,
188.4486,
183.1556,
178.6338,
173.7312,
169.6264,
163.9526,
159.8742,
155.8326,
151.1966,
147.5594,
143.07,
140.037,
134.1804,
131.071,
127.4884,
124.0848,
120.2944,
117.333,
112.9626,
110.2902,
107.0814,
103.0334,
99.4832000000001,
96.3899999999999,
93.7202000000002,
90.1714000000002,
87.2357999999999,
85.9346,
82.8910000000001,
80.0264000000002,
78.3834000000002,
75.1543999999999,
73.8683999999998,
70.9895999999999,
69.4367999999999,
64.8701999999998,
65.0408000000002,
61.6738,
59.5207999999998,
57.0158000000001,
54.2302,
53.0962,
50.4985999999999,
52.2588000000001,
47.3914,
45.6244000000002,
42.8377999999998,
43.0072,
40.6516000000001,
40.2453999999998,
35.2136,
36.4546,
33.7849999999999,
33.2294000000002,
32.4679999999998,
30.8670000000002,
28.6507999999999,
28.9099999999999,
27.5983999999999,
26.1619999999998,
24.5563999999999,
23.2328000000002,
21.9484000000002,
21.5902000000001,
21.3346000000001,
17.7031999999999,
20.6111999999998,
19.5545999999999,
15.7375999999999,
17.0720000000001,
16.9517999999998,
15.326,
13.1817999999998,
14.6925999999999,
13.0859999999998,
13.2754,
10.8697999999999,
11.248,
7.3768,
4.72339999999986,
7.97899999999981,
8.7503999999999,
7.68119999999999,
9.7199999999998,
7.73919999999998,
5.6224000000002,
7.44560000000001,
6.6601999999998,
5.9058,
4.00199999999995,
4.51699999999983,
4.68240000000014,
3.86220000000003,
5.13639999999987,
5.98500000000013,
2.47719999999981,
2.61999999999989,
1.62800000000016,
4.65000000000009,
0.225599999999758,
0.831000000000131,
-0.359400000000278,
1.27599999999984,
-2.92559999999958,
-0.0303999999996449,
2.37079999999969,
-2.0033999999996,
0.804600000000391,
0.30199999999968,
1.1247999999996,
-2.6880000000001,
0.0321999999996478,
-1.18099999999959,
-3.9402,
-1.47940000000017,
-0.188400000000001,
-2.10720000000038,
-2.04159999999956,
-3.12880000000041,
-4.16160000000036,
-0.612799999999879,
-3.48719999999958,
-8.17900000000009,
-5.37780000000021,
-4.01379999999972,
-5.58259999999973,
-5.73719999999958,
-7.66799999999967,
-5.69520000000011,
-1.1247999999996,
-5.58520000000044,
-8.04560000000038,
-4.64840000000004,
-11.6468000000004,
-7.97519999999986,
-5.78300000000036,
-7.67420000000038,
-10.6328000000003,
-9.81720000000041},
// precision 11
{1476,
1449.6014,
1423.5802,
1397.7942,
1372.3042,
1347.2062,
1321.8402,
1297.2292,
1272.9462,
1248.9926,
1225.3026,
1201.4252,
1178.0578,
1155.6092,
1132.626,
1110.5568,
1088.527,
1066.5154,
1045.1874,
1024.3878,
1003.37,
982.1972,
962.5728,
942.1012,
922.9668,
903.292,
884.0772,
864.8578,
846.6562,
828.041,
809.714,
792.3112,
775.1806,
757.9854,
740.656,
724.346,
707.5154,
691.8378,
675.7448,
659.6722,
645.5722,
630.1462,
614.4124,
600.8728,
585.898,
572.408,
558.4926,
544.4938,
531.6776,
517.282,
505.7704,
493.1012,
480.7388,
467.6876,
456.1872,
445.5048,
433.0214,
420.806,
411.409,
400.4144,
389.4294,
379.2286,
369.651,
360.6156,
350.337,
342.083,
332.1538,
322.5094,
315.01,
305.6686,
298.1678,
287.8116,
280.9978,
271.9204,
265.3286,
257.5706,
249.6014,
242.544,
235.5976,
229.583,
220.9438,
214.672,
208.2786,
201.8628,
195.1834,
191.505,
186.1816,
178.5188,
172.2294,
167.8908,
161.0194,
158.052,
151.4588,
148.1596,
143.4344,
138.5238,
133.13,
127.6374,
124.8162,
118.7894,
117.3984,
114.6078,
109.0858,
105.1036,
103.6258,
98.6018000000004,
95.7618000000002,
93.5821999999998,
88.5900000000001,
86.9992000000002,
82.8800000000001,
80.4539999999997,
74.6981999999998,
74.3644000000004,
73.2914000000001,
65.5709999999999,
66.9232000000002,
65.1913999999997,
62.5882000000001,
61.5702000000001,
55.7035999999998,
56.1764000000003,
52.7596000000003,
53.0302000000001,
49.0609999999997,
48.4694,
44.933,
46.0474000000004,
44.7165999999997,
41.9416000000001,
39.9207999999999,
35.6328000000003,
35.5276000000003,
33.1934000000001,
33.2371999999996,
33.3864000000003,
33.9228000000003,
30.2371999999996,
29.1373999999996,
25.2272000000003,
24.2942000000003,
19.8338000000003,
18.9005999999999,
23.0907999999999,
21.8544000000002,
19.5176000000001,
15.4147999999996,
16.9314000000004,
18.6737999999996,
12.9877999999999,
14.3688000000002,
12.0447999999997,
15.5219999999999,
12.5299999999997,
14.5940000000001,
14.3131999999996,
9.45499999999993,
12.9441999999999,
3.91139999999996,
13.1373999999996,
5.44720000000052,
9.82779999999912,
7.87279999999919,
3.67760000000089,
5.46980000000076,
5.55099999999948,
5.65979999999945,
3.89439999999922,
3.1275999999998,
5.65140000000065,
6.3062000000009,
3.90799999999945,
1.87060000000019,
5.17020000000048,
2.46680000000015,
0.770000000000437,
-3.72340000000077,
1.16400000000067,
8.05340000000069,
0.135399999999208,
2.15940000000046,
0.766999999999825,
1.0594000000001,
3.15500000000065,
-0.287399999999252,
2.37219999999979,
-2.86620000000039,
-1.63199999999961,
-2.22979999999916,
-0.15519999999924,
-1.46039999999994,
-0.262199999999211,
-2.34460000000036,
-2.8078000000005,
-3.22179999999935,
-5.60159999999996,
-8.42200000000048,
-9.43740000000071,
0.161799999999857,
-10.4755999999998,
-10.0823999999993},
// precision 12
{2953,
2900.4782,
2848.3568,
2796.3666,
2745.324,
2694.9598,
2644.648,
2595.539,
2546.1474,
2498.2576,
2450.8376,
2403.6076,
2357.451,
2311.38,
2266.4104,
2221.5638,
2176.9676,
2134.193,
2090.838,
2048.8548,
2007.018,
1966.1742,
1925.4482,
1885.1294,
1846.4776,
1807.4044,
1768.8724,
1731.3732,
1693.4304,
1657.5326,
1621.949,
1586.5532,
1551.7256,
1517.6182,
1483.5186,
1450.4528,
1417.865,
1385.7164,
1352.6828,
1322.6708,
1291.8312,
1260.9036,
1231.476,
1201.8652,
1173.6718,
1145.757,
1119.2072,
1092.2828,
1065.0434,
1038.6264,
1014.3192,
988.5746,
965.0816,
940.1176,
917.9796,
894.5576,
871.1858,
849.9144,
827.1142,
805.0818,
783.9664,
763.9096,
742.0816,
724.3962,
706.3454,
688.018,
667.4214,
650.3106,
633.0686,
613.8094,
597.818,
581.4248,
563.834,
547.363,
531.5066,
520.455400000001,
505.583199999999,
488.366,
476.480799999999,
459.7682,
450.0522,
434.328799999999,
423.952799999999,
408.727000000001,
399.079400000001,
387.252200000001,
373.987999999999,
360.852000000001,
351.6394,
339.642,
330.902400000001,
322.661599999999,
311.662200000001,
301.3254,
291.7484,
279.939200000001,
276.7508,
263.215200000001,
254.811400000001,
245.5494,
242.306399999999,
234.8734,
223.787200000001,
217.7156,
212.0196,
200.793,
195.9748,
189.0702,
182.449199999999,
177.2772,
170.2336,
164.741,
158.613600000001,
155.311,
147.5964,
142.837,
137.3724,
132.0162,
130.0424,
121.9804,
120.451800000001,
114.8968,
111.585999999999,
105.933199999999,
101.705,
98.5141999999996,
95.0488000000005,
89.7880000000005,
91.4750000000004,
83.7764000000006,
80.9698000000008,
72.8574000000008,
73.1615999999995,
67.5838000000003,
62.6263999999992,
63.2638000000006,
66.0977999999996,
52.0843999999997,
58.9956000000002,
47.0912000000008,
46.4956000000002,
48.4383999999991,
47.1082000000006,
43.2392,
37.2759999999998,
40.0283999999992,
35.1864000000005,
35.8595999999998,
32.0998,
28.027,
23.6694000000007,
33.8266000000003,
26.3736000000008,
27.2008000000005,
21.3245999999999,
26.4115999999995,
23.4521999999997,
19.5013999999992,
19.8513999999996,
10.7492000000002,
18.6424000000006,
13.1265999999996,
18.2436000000016,
6.71860000000015,
3.39459999999963,
6.33759999999893,
7.76719999999841,
0.813999999998487,
3.82819999999992,
0.826199999999517,
8.07440000000133,
-1.59080000000176,
5.01780000000144,
0.455399999998917,
-0.24199999999837,
0.174800000000687,
-9.07640000000174,
-4.20160000000033,
-3.77520000000004,
-4.75179999999818,
-5.3724000000002,
-8.90680000000066,
-6.10239999999976,
-5.74120000000039,
-9.95339999999851,
-3.86339999999836,
-13.7304000000004,
-16.2710000000006,
-7.51359999999841,
-3.30679999999847,
-13.1339999999982,
-10.0551999999989,
-6.72019999999975,
-8.59660000000076,
-10.9307999999983,
-1.8775999999998,
-4.82259999999951,
-13.7788,
-21.6470000000008,
-10.6735999999983,
-15.7799999999988},
// precision 13
{5907.5052, 5802.2672,
5697.347, 5593.5794,
5491.2622, 5390.5514,
5290.3376, 5191.6952,
5093.5988, 4997.3552,
4902.5972, 4808.3082,
4715.5646, 4624.109,
4533.8216, 4444.4344,
4356.3802, 4269.2962,
4183.3784, 4098.292,
4014.79, 3932.4574,
3850.6036, 3771.2712,
3691.7708, 3615.099,
3538.1858, 3463.4746,
3388.8496, 3315.6794,
3244.5448, 3173.7516,
3103.3106, 3033.6094,
2966.5642, 2900.794,
2833.7256, 2769.81,
2707.3196, 2644.0778,
2583.9916, 2523.4662,
2464.124, 2406.073,
2347.0362, 2292.1006,
2238.1716, 2182.7514,
2128.4884, 2077.1314,
2025.037, 1975.3756,
1928.933, 1879.311,
1831.0006, 1783.2144,
1738.3096, 1694.5144,
1649.024, 1606.847,
1564.7528, 1525.3168,
1482.5372, 1443.9668,
1406.5074, 1365.867,
1329.2186, 1295.4186,
1257.9716, 1225.339,
1193.2972, 1156.3578,
1125.8686, 1091.187,
1061.4094, 1029.4188,
1000.9126, 972.3272,
944.004199999999, 915.7592,
889.965, 862.834200000001,
840.4254, 812.598399999999,
785.924200000001, 763.050999999999,
741.793799999999, 721.466,
699.040799999999, 677.997200000002,
649.866999999998, 634.911800000002,
609.8694, 591.981599999999,
570.2922, 557.129199999999,
538.3858, 521.872599999999,
502.951400000002, 495.776399999999,
475.171399999999, 459.751,
439.995200000001, 426.708999999999,
413.7016, 402.3868,
387.262599999998, 372.0524,
357.050999999999, 342.5098,
334.849200000001, 322.529399999999,
311.613799999999, 295.848000000002,
289.273000000001, 274.093000000001,
263.329600000001, 251.389599999999,
245.7392, 231.9614,
229.7952, 217.155200000001,
208.9588, 199.016599999999,
190.839199999999, 180.6976,
176.272799999999, 166.976999999999,
162.5252, 151.196400000001,
149.386999999999, 133.981199999998,
130.0586, 130.164000000001,
122.053400000001, 110.7428,
108.1276, 106.232400000001,
100.381600000001, 98.7668000000012,
86.6440000000002, 79.9768000000004,
82.4722000000002, 68.7026000000005,
70.1186000000016, 71.9948000000004,
58.998599999999, 59.0492000000013,
56.9818000000014, 47.5338000000011,
42.9928, 51.1591999999982,
37.2740000000013, 42.7220000000016,
31.3734000000004, 26.8090000000011,
25.8934000000008, 26.5286000000015,
29.5442000000003, 19.3503999999994,
26.0760000000009, 17.9527999999991,
14.8419999999969, 10.4683999999979,
8.65899999999965, 9.86720000000059,
4.34139999999752, -0.907800000000861,
-3.32080000000133, -0.936199999996461,
-11.9916000000012, -8.87000000000262,
-6.33099999999831, -11.3366000000024,
-15.9207999999999, -9.34659999999712,
-15.5034000000014, -19.2097999999969,
-15.357799999998, -28.2235999999975,
-30.6898000000001, -19.3271999999997,
-25.6083999999973, -24.409599999999,
-13.6385999999984, -33.4473999999973,
-32.6949999999997, -28.9063999999998,
-31.7483999999968, -32.2935999999972,
-35.8329999999987, -47.620600000002,
-39.0855999999985, -33.1434000000008,
-46.1371999999974, -37.5892000000022,
-46.8164000000033, -47.3142000000007,
-60.2914000000019, -37.7575999999972},
// precision 14
{11816.475, 11605.0046, 11395.3792, 11188.7504, 10984.1814, 10782.0086,
10582.0072, 10384.503, 10189.178, 9996.2738, 9806.0344, 9617.9798,
9431.394, 9248.7784, 9067.6894, 8889.6824, 8712.9134, 8538.8624,
8368.4944, 8197.7956, 8031.8916, 7866.6316, 7703.733, 7544.5726,
7386.204, 7230.666, 7077.8516, 6926.7886, 6778.6902, 6631.9632,
6487.304, 6346.7486, 6206.4408, 6070.202, 5935.2576, 5799.924,
5671.0324, 5541.9788, 5414.6112, 5290.0274, 5166.723, 5047.6906,
4929.162, 4815.1406, 4699.127, 4588.5606, 4477.7394, 4369.4014,
4264.2728, 4155.9224, 4055.581, 3955.505, 3856.9618, 3761.3828,
3666.9702, 3575.7764, 3482.4132, 3395.0186, 3305.8852, 3221.415,
3138.6024, 3056.296, 2970.4494, 2896.1526, 2816.8008, 2740.2156,
2670.497, 2594.1458, 2527.111, 2460.8168, 2387.5114, 2322.9498,
2260.6752, 2194.2686, 2133.7792, 2074.767, 2015.204, 1959.4226,
1898.6502, 1850.006, 1792.849, 1741.4838, 1687.9778, 1638.1322,
1589.3266, 1543.1394, 1496.8266, 1447.8516, 1402.7354, 1361.9606,
1327.0692, 1285.4106, 1241.8112, 1201.6726, 1161.973, 1130.261,
1094.2036, 1048.2036, 1020.6436, 990.901400000002, 961.199800000002, 924.769800000002,
899.526400000002, 872.346400000002, 834.375, 810.432000000001, 780.659800000001, 756.013800000001,
733.479399999997, 707.923999999999, 673.858, 652.222399999999, 636.572399999997, 615.738599999997,
586.696400000001, 564.147199999999, 541.679600000003, 523.943599999999, 505.714599999999, 475.729599999999,
461.779600000002, 449.750800000002, 439.020799999998, 412.7886, 400.245600000002, 383.188199999997,
362.079599999997, 357.533799999997, 334.319000000003, 327.553399999997, 308.559399999998, 291.270199999999,
279.351999999999, 271.791400000002, 252.576999999997, 247.482400000001, 236.174800000001, 218.774599999997,
220.155200000001, 208.794399999999, 201.223599999998, 182.995600000002, 185.5268, 164.547400000003,
176.5962, 150.689599999998, 157.8004, 138.378799999999, 134.021200000003, 117.614399999999,
108.194000000003, 97.0696000000025, 89.6042000000016, 95.6030000000028, 84.7810000000027, 72.635000000002,
77.3482000000004, 59.4907999999996, 55.5875999999989, 50.7346000000034, 61.3916000000027, 50.9149999999936,
39.0384000000049, 58.9395999999979, 29.633600000001, 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,
24645.952,
23922.523,
23198.144,
22529.128,
21762.4179999999,
21134.779,
20459.117,
19840.818,
19187.04,
18636.3689999999,
17982.831,
17439.7389999999,
16874.547,
16358.2169999999,
15835.684,
15352.914,
14823.681,
14329.313,
13816.897,
13342.874,
12880.882,
12491.648,
12021.254,
11625.392,
11293.7610000001,
10813.697,
10456.209,
10099.074,
9755.39000000001,
9393.18500000006,
9047.57900000003,
8657.98499999999,
8395.85900000005,
8033,
7736.95900000003,
7430.59699999995,
7258.47699999996,
6924.58200000005,
6691.29399999999,
6357.92500000005,
6202.05700000003,
5921.19700000004,
5628.28399999999,
5404.96799999999,
5226.71100000001,
4990.75600000005,
4799.77399999998,
4622.93099999998,
4472.478,
4171.78700000001,
3957.46299999999,
3868.95200000005,
3691.14300000004,
3474.63100000005,
3341.67200000002,
3109.14000000001,
3071.97400000005,
2796.40399999998,
2756.17799999996,
2611.46999999997,
2471.93000000005,
2382.26399999997,
2209.22400000005,
2142.28399999999,
2013.96100000001,
1911.18999999994,
1818.27099999995,
1668.47900000005,
1519.65800000005,
1469.67599999998,
1367.13800000004,
1248.52899999998,
1181.23600000003,
1022.71900000004,
1088.20700000005,
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,
6927.9664, 7010.9096, 7091.0816, 7175.3962, 7260.3454, 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,
8642.0522, 8728.3288, 8820.9528, 8907.727, 9001.0794, 9091.2522, 9179.988, 9269.852, 9362.6394, 9453.642,
9546.9024, 9640.6616, 9732.6622, 9824.3254, 9917.7484, 10007.9392, 10106.7508, 10196.2152, 10289.8114, 10383.5494,
10482.3064, 10576.8734, 10668.7872, 10764.7156, 10862.0196, 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,
12408.4518, 12504.8968, 12604.586, 12700.9332, 12798.705, 12898.5142, 12997.0488, 13094.788, 13198.475, 13292.7764,
13392.9698, 13486.8574, 13590.1616, 13686.5838, 13783.6264, 13887.2638, 13992.0978, 14081.0844, 14189.9956, 14280.0912,
14382.4956, 14486.4384, 14588.1082, 14686.2392, 14782.276, 14888.0284, 14985.1864, 15088.8596, 15187.0998, 15285.027,
15383.6694, 15495.8266, 15591.3736, 15694.2008, 15790.3246, 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
{5908.5052, 6007.2672, 6107.347, 6208.5794, 6311.2622, 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,
10775.9916, 10920.4662, 11066.124, 11213.073, 11358.0362, 11508.1006, 11659.1716, 11808.7514, 11959.4884, 12112.1314,
12265.037, 12420.3756, 12578.933, 12734.311, 12890.0006, 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,
15529.2972, 15697.3578, 15871.8686, 16042.187, 16216.4094, 16389.4188, 16565.9126, 16742.3272, 16919.0042, 17094.7592,
17273.965, 17451.8342, 17634.4254, 17810.5984, 17988.9242, 18171.051, 18354.7938, 18539.466, 18721.0408, 18904.9972,
19081.867, 19271.9118, 19451.8694, 19637.9816, 19821.2922, 20013.1292, 20199.3858, 20387.8726, 20572.9514, 20770.7764,
20955.1714, 21144.751, 21329.9952, 21520.709, 21712.7016, 21906.3868, 22096.2626, 22286.0524, 22475.051, 22665.5098,
22862.8492, 23055.5294, 23249.6138, 23437.848, 23636.273, 23826.093, 24020.3296, 24213.3896, 24411.7392, 24602.9614,
24805.7952, 24998.1552, 25193.9588, 25389.0166, 25585.8392, 25780.6976, 25981.2728, 26175.977, 26376.5252, 26570.1964,
26773.387, 26962.9812, 27163.0586, 27368.164, 27565.0534, 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,
30762.9928, 30976.1592, 31166.274, 31376.722, 31570.3734, 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,
34804.0084, 35011.13, 35218.669, 35418.6634, 35619.0792, 35830.6534, 36028.4966, 36229.7902, 36438.6422, 36630.7764,
36833.3102, 37048.6728, 37247.3916, 37453.5904, 37669.3614, 37854.5526, 38059.305, 38268.0936, 38470.2516, 38674.7064,
38876.167, 39068.3794, 39281.9144, 39492.8566, 39684.8628, 39898.4108, 40093.1836, 40297.6858, 40489.7086, 40717.2424},
// precision 14
{11817.475, 12015.0046, 12215.3792, 12417.7504, 12623.1814, 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,
18775.304, 19044.7486, 19314.4408, 19587.202, 19862.2576, 20135.924, 20417.0324, 20697.9788, 20979.6112, 21265.0274,
21550.723, 21841.6906, 22132.162, 22428.1406, 22722.127, 23020.5606, 23319.7394, 23620.4014, 23925.2728, 24226.9224,
24535.581, 24845.505, 25155.9618, 25470.3828, 25785.9702, 26103.7764, 26420.4132, 26742.0186, 27062.8852, 27388.415,
27714.6024, 28042.296, 28365.4494, 28701.1526, 29031.8008, 29364.2156, 29704.497, 30037.1458, 30380.111, 30723.8168,
31059.5114, 31404.9498, 31751.6752, 32095.2686, 32444.7792, 32794.767, 33145.204, 33498.4226, 33847.6502, 34209.006,
34560.849, 34919.4838, 35274.9778, 35635.1322, 35996.3266, 36359.1394, 36722.8266, 37082.8516, 37447.7354, 37815.9606,
38191.0692, 38559.4106, 38924.8112, 39294.6726, 39663.973, 40042.261, 40416.2036, 40779.2036, 41161.6436, 41540.9014,
41921.1998, 42294.7698, 42678.5264, 43061.3464, 43432.375, 43818.432, 44198.6598, 44583.0138, 44970.4794, 45353.924,
45729.858, 46118.2224, 46511.5724, 46900.7386, 47280.6964, 47668.1472, 48055.6796, 48446.9436, 48838.7146, 49217.7296,
49613.7796, 50010.7508, 50410.0208, 50793.7886, 51190.2456, 51583.1882, 51971.0796, 52376.5338, 52763.319, 53165.5534,
53556.5594, 53948.2702, 54346.352, 54748.7914, 55138.577, 55543.4824, 55941.1748, 56333.7746, 56745.1552, 57142.7944,
57545.2236, 57935.9956, 58348.5268, 58737.5474, 59158.5962, 59542.6896, 59958.8004, 60349.3788, 60755.0212, 61147.6144,
61548.194, 61946.0696, 62348.6042, 62763.603, 63162.781, 63560.635, 63974.3482, 64366.4908, 64771.5876, 65176.7346,
65597.3916, 65995.915, 66394.0384, 66822.9396, 67203.6336, 67612.2032, 68019.0078, 68420.0388, 68821.22, 69235.8388,
69640.0724, 70055.155, 70466.357, 70863.4266, 71276.2482, 71677.0306, 72080.2006, 72493.0214, 72893.5952, 73314.5856,
73714.9852, 74125.3022, 74521.2122, 74933.6814, 75341.5904, 75743.0244, 76166.0278, 76572.1322, 76973.1028, 77381.6284,
77800.6092, 78189.328, 78607.0962, 79012.2508, 79407.8358, 79825.725, 80238.701, 80646.891, 81035.6436, 81460.0448,
81876.3884},
// precision 15
{23635.0036, 24030.8034, 24431.4744, 24837.1524, 25246.7928, 25661.326, 26081.3532, 26505.2806, 26933.9892, 27367.7098,
27805.318, 28248.799, 28696.4382, 29148.8244, 29605.5138, 30066.8668, 30534.2344, 31006.32, 31480.778, 31962.2418,
32447.3324, 32938.0232, 33432.731, 33930.728, 34433.9896, 34944.1402, 35457.5588, 35974.5958, 36497.3296, 37021.9096,
37554.326, 38088.0826, 38628.8816, 39171.3192, 39723.2326, 40274.5554, 40832.3142, 41390.613, 41959.5908, 42532.5466,
43102.0344, 43683.5072, 44266.694, 44851.2822, 45440.7862, 46038.0586, 46640.3164, 47241.064, 47846.155, 48454.7396,
49076.9168, 49692.542, 50317.4778, 50939.65, 51572.5596, 52210.2906, 52843.7396, 53481.3996, 54127.236, 54770.406,
55422.6598, 56078.7958, 56736.7174, 57397.6784, 58064.5784, 58730.308, 59404.9784, 60077.0864, 60751.9158, 61444.1386,
62115.817, 62808.7742, 63501.4774, 64187.5454, 64883.6622, 65582.7468, 66274.5318, 66976.9276, 67688.7764, 68402.138,
69109.6274, 69822.9706, 70543.6108, 71265.5202, 71983.3848, 72708.4656, 73433.384, 74158.4664, 74896.4868, 75620.9564,
76362.1434, 77098.3204, 77835.7662, 78582.6114, 79323.9902, 80067.8658, 80814.9246, 81567.0136, 82310.8536, 83061.9952,
83821.4096, 84580.8608, 85335.547, 86092.5802, 86851.6506, 87612.311, 88381.2016, 89146.3296, 89907.8974, 90676.846,
91451.4152, 92224.5518, 92995.8686, 93763.5066, 94551.2796, 95315.1944, 96096.1806, 96881.0918, 97665.679, 98442.68,
99229.3002, 100011.0994, 100790.6386, 101580.1564, 102377.7484, 103152.1392, 103944.2712, 104730.216, 105528.6336, 106324.9398,
107117.6706, 107890.3988, 108695.2266, 109485.238, 110294.7876, 111075.0958, 111878.0496, 112695.2864, 113464.5486, 114270.0474,
115068.608, 115884.3626, 116673.2588, 117483.3716, 118275.097, 119085.4092, 119879.2808, 120687.5868, 121499.9944, 122284.916,
123095.9254, 123912.5038, 124709.0454, 125503.7182, 126323.259, 127138.9412, 127943.8294, 128755.646, 129556.5354, 130375.3298,
131161.4734, 131971.1962, 132787.5458, 133588.1056, 134431.351, 135220.2906, 136023.398, 136846.6558, 137667.0004, 138463.663,
139283.7154, 140074.6146, 140901.3072, 141721.8548, 142543.2322, 143356.1096, 144173.7412, 144973.0948, 145794.3162, 146609.5714,
147420.003, 148237.9784, 149050.5696, 149854.761, 150663.1966, 151494.0754, 152313.1416, 153112.6902, 153935.7206, 154746.9262,
155559.547, 156401.9746, 157228.7036, 158008.7254, 158820.75, 159646.9184, 160470.4458, 161279.5348, 162093.3114, 162918.542,
163729.2842},
// precision 16
{47271, 48062.3584, 48862.7074, 49673.152, 50492.8416, 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,
64895.0194, 65876.4182, 66862.6136, 67862.6968, 68868.8908, 69882.8544, 70911.271, 71944.0924, 72990.0326, 74040.692,
75100.6336, 76174.7826, 77252.5998, 78340.2974, 79438.2572, 80545.4976, 81657.2796, 82784.6336, 83915.515, 85059.7362,
86205.9368, 87364.4424, 88530.3358, 89707.3744, 90885.9638, 92080.197, 93275.5738, 94479.391, 95695.918, 96919.2236,
98148.4602, 99382.3474, 100625.6974, 101878.0284, 103141.6278, 104409.4588, 105686.2882, 106967.5402, 108261.6032, 109548.1578,
110852.0728, 112162.231, 113479.0072, 114806.2626, 116137.9072, 117469.5048, 118813.5186, 120165.4876, 121516.2556, 122875.766,
124250.5444, 125621.2222, 127003.2352, 128387.848, 129775.2644, 131181.7776, 132577.3086, 133979.9458, 135394.1132, 136800.9078,
138233.217, 139668.5308, 141085.212, 142535.2122, 143969.0684, 145420.2872, 146878.1542, 148332.7572, 149800.3202, 151269.66,
152743.6104, 154213.0948, 155690.288, 157169.4246, 158672.1756, 160160.059, 161650.6854, 163145.7772, 164645.6726, 166159.1952,
167682.1578, 169177.3328, 170700.0118, 172228.8964, 173732.6664, 175265.5556, 176787.799, 178317.111, 179856.6914, 181400.865,
182943.4612, 184486.742, 186033.4698, 187583.7886, 189148.1868, 190688.4526, 192250.1926, 193810.9042, 195354.2972, 196938.7682,
198493.5898, 200079.2824, 201618.912, 203205.5492, 204765.5798, 206356.1124, 207929.3064, 209498.7196, 211086.229, 212675.1324,
214256.7892, 215826.2392, 217412.8474, 218995.6724, 220618.6038, 222207.1166, 223781.0364, 225387.4332, 227005.7928, 228590.4336,
230217.8738, 231805.1054, 233408.9, 234995.3432, 236601.4956, 238190.7904, 239817.2548, 241411.2832, 243002.4066, 244640.1884,
246255.3128, 247849.3508, 249479.9734, 251106.8822, 252705.027, 254332.9242, 255935.129, 257526.9014, 259154.772, 260777.625,
262390.253, 264004.4906, 265643.59, 267255.4076, 268873.426, 270470.7252, 272106.4804, 273722.4456, 275337.794, 276945.7038,
278592.9154, 280204.3726, 281841.1606, 283489.171, 285130.1716, 286735.3362, 288364.7164, 289961.1814, 291595.5524, 293285.683,
294899.6668, 296499.3434, 298128.0462, 299761.8946, 301394.2424, 302997.6748, 304615.1478, 306269.7724, 307886.114, 309543.1028,
311153.2862, 312782.8546, 314421.2008, 316033.2438, 317692.9636, 319305.2648, 320948.7406, 322566.3364, 324228.4224, 325847.1542},
// precision 17
{94542, 96125.811, 97728.019, 99348.558, 100987.9705, 102646.7565, 104324.5125, 106021.7435, 107736.7865, 109469.272,
111223.9465, 112995.219, 114787.432, 116593.152, 118422.71, 120267.2345, 122134.6765, 124020.937, 125927.2705, 127851.255,
129788.9485, 131751.016, 133726.8225, 135722.592, 137736.789, 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,
172420.4595, 174732.6265, 177062.77, 179412.502, 181774.035, 184151.939, 186551.6895, 188965.691, 191402.8095, 193857.949,
196305.0775, 198774.6715, 201271.2585, 203764.78, 206299.3695, 208818.1365, 211373.115, 213946.7465, 216532.076, 219105.541,
221714.5375, 224337.5135, 226977.5125, 229613.0655, 232270.2685, 234952.2065, 237645.3555, 240331.1925, 243034.517, 245756.0725,
248517.6865, 251232.737, 254011.3955, 256785.995, 259556.44, 262368.335, 265156.911, 267965.266, 270785.583, 273616.0495,
276487.4835, 279346.639, 282202.509, 285074.3885, 287942.2855, 290856.018, 293774.0345, 296678.5145, 299603.6355, 302552.6575,
305492.9785, 308466.8605, 311392.581, 314347.538, 317319.4295, 320285.9785, 323301.7325, 326298.3235, 329301.3105, 332301.987,
335309.791, 338370.762, 341382.923, 344431.1265, 347464.1545, 350507.28, 353619.2345, 356631.2005, 359685.203, 362776.7845,
365886.488, 368958.2255, 372060.6825, 375165.4335, 378237.935, 381328.311, 384430.5225, 387576.425, 390683.242, 393839.648,
396977.8425, 400101.9805, 403271.296, 406409.8425, 409529.5485, 412678.7, 415847.423, 419020.8035, 422157.081, 425337.749,
428479.6165, 431700.902, 434893.1915, 438049.582, 441210.5415, 444379.2545, 447577.356, 450741.931, 453959.548, 457137.0935,
460329.846, 463537.4815, 466732.3345, 469960.5615, 473164.681, 476347.6345, 479496.173, 482813.1645, 486025.6995, 489249.4885,
492460.1945, 495675.8805, 498908.0075, 502131.802, 505374.3855, 508550.9915, 511806.7305, 515026.776, 518217.0005, 521523.9855,
524705.9855, 527950.997, 531210.0265, 534472.497, 537750.7315, 540926.922, 544207.094, 547429.4345, 550666.3745, 553975.3475,
557150.7185, 560399.6165, 563662.697, 566916.7395, 570146.1215, 573447.425, 576689.6245, 579874.5745, 583202.337, 586503.0255,
589715.635, 592910.161, 596214.3885, 599488.035, 602740.92, 605983.0685, 609248.67, 612491.3605, 615787.912, 619107.5245,
622307.9555, 625577.333, 628840.4385, 632085.2155, 635317.6135, 638691.7195, 641887.467, 645139.9405, 648441.546, 651666.252,
654941.845},
// precision 18
{189084, 192250.913, 195456.774, 198696.946, 201977.762, 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,
368296.587, 373118.482, 377914.93, 382782.301, 387680.669, 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,
459239.049, 464529.569, 469910.083, 475274.03, 480684.473, 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;
};
}