/*
 * 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;
};

}