// File: solidity/contracts/converter/interfaces/IBancorFormula.sol

pragma solidity 0.4.26;

/*
    Bancor Formula interface
*/
contract IBancorFormula {
    function purchaseTargetAmount(uint256 _supply,
                                  uint256 _reserveBalance,
                                  uint32 _reserveWeight,
                                  uint256 _amount)
                                  public view returns (uint256);

    function saleTargetAmount(uint256 _supply,
                              uint256 _reserveBalance,
                              uint32 _reserveWeight,
                              uint256 _amount)
                              public view returns (uint256);

    function crossReserveTargetAmount(uint256 _sourceReserveBalance,
                                      uint32 _sourceReserveWeight,
                                      uint256 _targetReserveBalance,
                                      uint32 _targetReserveWeight,
                                      uint256 _amount)
                                      public view returns (uint256);

    function fundCost(uint256 _supply,
                      uint256 _reserveBalance,
                      uint32 _reserveRatio,
                      uint256 _amount)
                      public view returns (uint256);

    function fundSupplyAmount(uint256 _supply,
                              uint256 _reserveBalance,
                              uint32 _reserveRatio,
                              uint256 _amount)
                              public view returns (uint256);

    function liquidateReserveAmount(uint256 _supply,
                                    uint256 _reserveBalance,
                                    uint32 _reserveRatio,
                                    uint256 _amount)
                                    public view returns (uint256);

    function balancedWeights(uint256 _primaryReserveStakedBalance,
                             uint256 _primaryReserveBalance,
                             uint256 _secondaryReserveBalance,
                             uint256 _reserveRateNumerator,
                             uint256 _reserveRateDenominator)
                             public view returns (uint32, uint32);
}

// File: solidity/contracts/utility/SafeMath.sol

pragma solidity 0.4.26;

/**
  * @dev Library for basic math operations with overflow/underflow protection
*/
library SafeMath {
    /**
      * @dev returns the sum of _x and _y, reverts if the calculation overflows
      *
      * @param _x   value 1
      * @param _y   value 2
      *
      * @return sum
    */
    function add(uint256 _x, uint256 _y) internal pure returns (uint256) {
        uint256 z = _x + _y;
        require(z >= _x, "ERR_OVERFLOW");
        return z;
    }

    /**
      * @dev returns the difference of _x minus _y, reverts if the calculation underflows
      *
      * @param _x   minuend
      * @param _y   subtrahend
      *
      * @return difference
    */
    function sub(uint256 _x, uint256 _y) internal pure returns (uint256) {
        require(_x >= _y, "ERR_UNDERFLOW");
        return _x - _y;
    }

    /**
      * @dev returns the product of multiplying _x by _y, reverts if the calculation overflows
      *
      * @param _x   factor 1
      * @param _y   factor 2
      *
      * @return product
    */
    function mul(uint256 _x, uint256 _y) internal pure returns (uint256) {
        // gas optimization
        if (_x == 0)
            return 0;

        uint256 z = _x * _y;
        require(z / _x == _y, "ERR_OVERFLOW");
        return z;
    }

    /**
      * @dev Integer division of two numbers truncating the quotient, reverts on division by zero.
      *
      * @param _x   dividend
      * @param _y   divisor
      *
      * @return quotient
    */
    function div(uint256 _x, uint256 _y) internal pure returns (uint256) {
        require(_y > 0, "ERR_DIVIDE_BY_ZERO");
        uint256 c = _x / _y;
        return c;
    }
}

// File: solidity/contracts/converter/BancorFormula.sol

pragma solidity 0.4.26;

contract BancorFormula is IBancorFormula {
    using SafeMath for uint256;

    uint16 public constant version = 8;

    uint256 private constant ONE = 1;
    uint32 private constant MAX_WEIGHT = 1000000;
    uint8 private constant MIN_PRECISION = 32;
    uint8 private constant MAX_PRECISION = 127;

    /**
      * Auto-generated via 'PrintIntScalingFactors.py'
    */
    uint256 private constant FIXED_1 = 0x080000000000000000000000000000000;
    uint256 private constant FIXED_2 = 0x100000000000000000000000000000000;
    uint256 private constant MAX_NUM = 0x200000000000000000000000000000000;

    /**
      * Auto-generated via 'PrintLn2ScalingFactors.py'
    */
    uint256 private constant LN2_NUMERATOR   = 0x3f80fe03f80fe03f80fe03f80fe03f8;
    uint256 private constant LN2_DENOMINATOR = 0x5b9de1d10bf4103d647b0955897ba80;

    /**
      * Auto-generated via 'PrintFunctionOptimalLog.py' and 'PrintFunctionOptimalExp.py'
    */
    uint256 private constant OPT_LOG_MAX_VAL = 0x15bf0a8b1457695355fb8ac404e7a79e3;
    uint256 private constant OPT_EXP_MAX_VAL = 0x800000000000000000000000000000000;

    /**
      * Auto-generated via 'PrintLambertFactors.py'
    */
    uint256 private constant LAMBERT_CONV_RADIUS = 0x002f16ac6c59de6f8d5d6f63c1482a7c86;
    uint256 private constant LAMBERT_POS2_SAMPLE = 0x0003060c183060c183060c183060c18306;
    uint256 private constant LAMBERT_POS2_MAXVAL = 0x01af16ac6c59de6f8d5d6f63c1482a7c80;
    uint256 private constant LAMBERT_POS3_MAXVAL = 0x6b22d43e72c326539cceeef8bb48f255ff;

    /**
      * Auto-generated via 'PrintWeightFactors.py'
    */
    uint256 private constant MAX_UNF_WEIGHT = 0x10c6f7a0b5ed8d36b4c7f34938583621fafc8b0079a2834d26fa3fcc9ea9;

    /**
      * Auto-generated via 'PrintMaxExpArray.py'
    */
    uint256[128] private maxExpArray;
    function initMaxExpArray() private {
    //  maxExpArray[  0] = 0x6bffffffffffffffffffffffffffffffff;
    //  maxExpArray[  1] = 0x67ffffffffffffffffffffffffffffffff;
    //  maxExpArray[  2] = 0x637fffffffffffffffffffffffffffffff;
    //  maxExpArray[  3] = 0x5f6fffffffffffffffffffffffffffffff;
    //  maxExpArray[  4] = 0x5b77ffffffffffffffffffffffffffffff;
    //  maxExpArray[  5] = 0x57b3ffffffffffffffffffffffffffffff;
    //  maxExpArray[  6] = 0x5419ffffffffffffffffffffffffffffff;
    //  maxExpArray[  7] = 0x50a2ffffffffffffffffffffffffffffff;
    //  maxExpArray[  8] = 0x4d517fffffffffffffffffffffffffffff;
    //  maxExpArray[  9] = 0x4a233fffffffffffffffffffffffffffff;
    //  maxExpArray[ 10] = 0x47165fffffffffffffffffffffffffffff;
    //  maxExpArray[ 11] = 0x4429afffffffffffffffffffffffffffff;
    //  maxExpArray[ 12] = 0x415bc7ffffffffffffffffffffffffffff;
    //  maxExpArray[ 13] = 0x3eab73ffffffffffffffffffffffffffff;
    //  maxExpArray[ 14] = 0x3c1771ffffffffffffffffffffffffffff;
    //  maxExpArray[ 15] = 0x399e96ffffffffffffffffffffffffffff;
    //  maxExpArray[ 16] = 0x373fc47fffffffffffffffffffffffffff;
    //  maxExpArray[ 17] = 0x34f9e8ffffffffffffffffffffffffffff;
    //  maxExpArray[ 18] = 0x32cbfd5fffffffffffffffffffffffffff;
    //  maxExpArray[ 19] = 0x30b5057fffffffffffffffffffffffffff;
    //  maxExpArray[ 20] = 0x2eb40f9fffffffffffffffffffffffffff;
    //  maxExpArray[ 21] = 0x2cc8340fffffffffffffffffffffffffff;
    //  maxExpArray[ 22] = 0x2af09481ffffffffffffffffffffffffff;
    //  maxExpArray[ 23] = 0x292c5bddffffffffffffffffffffffffff;
    //  maxExpArray[ 24] = 0x277abdcdffffffffffffffffffffffffff;
    //  maxExpArray[ 25] = 0x25daf6657fffffffffffffffffffffffff;
    //  maxExpArray[ 26] = 0x244c49c65fffffffffffffffffffffffff;
    //  maxExpArray[ 27] = 0x22ce03cd5fffffffffffffffffffffffff;
    //  maxExpArray[ 28] = 0x215f77c047ffffffffffffffffffffffff;
    //  maxExpArray[ 29] = 0x1fffffffffffffffffffffffffffffffff;
    //  maxExpArray[ 30] = 0x1eaefdbdabffffffffffffffffffffffff;
    //  maxExpArray[ 31] = 0x1d6bd8b2ebffffffffffffffffffffffff;
        maxExpArray[ 32] = 0x1c35fedd14ffffffffffffffffffffffff;
        maxExpArray[ 33] = 0x1b0ce43b323fffffffffffffffffffffff;
        maxExpArray[ 34] = 0x19f0028ec1ffffffffffffffffffffffff;
        maxExpArray[ 35] = 0x18ded91f0e7fffffffffffffffffffffff;
        maxExpArray[ 36] = 0x17d8ec7f0417ffffffffffffffffffffff;
        maxExpArray[ 37] = 0x16ddc6556cdbffffffffffffffffffffff;
        maxExpArray[ 38] = 0x15ecf52776a1ffffffffffffffffffffff;
        maxExpArray[ 39] = 0x15060c256cb2ffffffffffffffffffffff;
        maxExpArray[ 40] = 0x1428a2f98d72ffffffffffffffffffffff;
        maxExpArray[ 41] = 0x13545598e5c23fffffffffffffffffffff;
        maxExpArray[ 42] = 0x1288c4161ce1dfffffffffffffffffffff;
        maxExpArray[ 43] = 0x11c592761c666fffffffffffffffffffff;
        maxExpArray[ 44] = 0x110a688680a757ffffffffffffffffffff;
        maxExpArray[ 45] = 0x1056f1b5bedf77ffffffffffffffffffff;
        maxExpArray[ 46] = 0x0faadceceeff8bffffffffffffffffffff;
        maxExpArray[ 47] = 0x0f05dc6b27edadffffffffffffffffffff;
        maxExpArray[ 48] = 0x0e67a5a25da4107fffffffffffffffffff;
        maxExpArray[ 49] = 0x0dcff115b14eedffffffffffffffffffff;
        maxExpArray[ 50] = 0x0d3e7a392431239fffffffffffffffffff;
        maxExpArray[ 51] = 0x0cb2ff529eb71e4fffffffffffffffffff;
        maxExpArray[ 52] = 0x0c2d415c3db974afffffffffffffffffff;
        maxExpArray[ 53] = 0x0bad03e7d883f69bffffffffffffffffff;
        maxExpArray[ 54] = 0x0b320d03b2c343d5ffffffffffffffffff;
        maxExpArray[ 55] = 0x0abc25204e02828dffffffffffffffffff;
        maxExpArray[ 56] = 0x0a4b16f74ee4bb207fffffffffffffffff;
        maxExpArray[ 57] = 0x09deaf736ac1f569ffffffffffffffffff;
        maxExpArray[ 58] = 0x0976bd9952c7aa957fffffffffffffffff;
        maxExpArray[ 59] = 0x09131271922eaa606fffffffffffffffff;
        maxExpArray[ 60] = 0x08b380f3558668c46fffffffffffffffff;
        maxExpArray[ 61] = 0x0857ddf0117efa215bffffffffffffffff;
        maxExpArray[ 62] = 0x07ffffffffffffffffffffffffffffffff;
        maxExpArray[ 63] = 0x07abbf6f6abb9d087fffffffffffffffff;
        maxExpArray[ 64] = 0x075af62cbac95f7dfa7fffffffffffffff;
        maxExpArray[ 65] = 0x070d7fb7452e187ac13fffffffffffffff;
        maxExpArray[ 66] = 0x06c3390ecc8af379295fffffffffffffff;
        maxExpArray[ 67] = 0x067c00a3b07ffc01fd6fffffffffffffff;
        maxExpArray[ 68] = 0x0637b647c39cbb9d3d27ffffffffffffff;
        maxExpArray[ 69] = 0x05f63b1fc104dbd39587ffffffffffffff;
        maxExpArray[ 70] = 0x05b771955b36e12f7235ffffffffffffff;
        maxExpArray[ 71] = 0x057b3d49dda84556d6f6ffffffffffffff;
        maxExpArray[ 72] = 0x054183095b2c8ececf30ffffffffffffff;
        maxExpArray[ 73] = 0x050a28be635ca2b888f77fffffffffffff;
        maxExpArray[ 74] = 0x04d5156639708c9db33c3fffffffffffff;
        maxExpArray[ 75] = 0x04a23105873875bd52dfdfffffffffffff;
        maxExpArray[ 76] = 0x0471649d87199aa990756fffffffffffff;
        maxExpArray[ 77] = 0x04429a21a029d4c1457cfbffffffffffff;
        maxExpArray[ 78] = 0x0415bc6d6fb7dd71af2cb3ffffffffffff;
        maxExpArray[ 79] = 0x03eab73b3bbfe282243ce1ffffffffffff;
        maxExpArray[ 80] = 0x03c1771ac9fb6b4c18e229ffffffffffff;
        maxExpArray[ 81] = 0x0399e96897690418f785257fffffffffff;
        maxExpArray[ 82] = 0x0373fc456c53bb779bf0ea9fffffffffff;
        maxExpArray[ 83] = 0x034f9e8e490c48e67e6ab8bfffffffffff;
        maxExpArray[ 84] = 0x032cbfd4a7adc790560b3337ffffffffff;
        maxExpArray[ 85] = 0x030b50570f6e5d2acca94613ffffffffff;
        maxExpArray[ 86] = 0x02eb40f9f620fda6b56c2861ffffffffff;
        maxExpArray[ 87] = 0x02cc8340ecb0d0f520a6af58ffffffffff;
        maxExpArray[ 88] = 0x02af09481380a0a35cf1ba02ffffffffff;
        maxExpArray[ 89] = 0x0292c5bdd3b92ec810287b1b3fffffffff;
        maxExpArray[ 90] = 0x0277abdcdab07d5a77ac6d6b9fffffffff;
        maxExpArray[ 91] = 0x025daf6654b1eaa55fd64df5efffffffff;
        maxExpArray[ 92] = 0x0244c49c648baa98192dce88b7ffffffff;
        maxExpArray[ 93] = 0x022ce03cd5619a311b2471268bffffffff;
        maxExpArray[ 94] = 0x0215f77c045fbe885654a44a0fffffffff;
        maxExpArray[ 95] = 0x01ffffffffffffffffffffffffffffffff;
        maxExpArray[ 96] = 0x01eaefdbdaaee7421fc4d3ede5ffffffff;
        maxExpArray[ 97] = 0x01d6bd8b2eb257df7e8ca57b09bfffffff;
        maxExpArray[ 98] = 0x01c35fedd14b861eb0443f7f133fffffff;
        maxExpArray[ 99] = 0x01b0ce43b322bcde4a56e8ada5afffffff;
        maxExpArray[100] = 0x019f0028ec1fff007f5a195a39dfffffff;
        maxExpArray[101] = 0x018ded91f0e72ee74f49b15ba527ffffff;
        maxExpArray[102] = 0x017d8ec7f04136f4e5615fd41a63ffffff;
        maxExpArray[103] = 0x016ddc6556cdb84bdc8d12d22e6fffffff;
        maxExpArray[104] = 0x015ecf52776a1155b5bd8395814f7fffff;
        maxExpArray[105] = 0x015060c256cb23b3b3cc3754cf40ffffff;
        maxExpArray[106] = 0x01428a2f98d728ae223ddab715be3fffff;
        maxExpArray[107] = 0x013545598e5c23276ccf0ede68034fffff;
        maxExpArray[108] = 0x01288c4161ce1d6f54b7f61081194fffff;
        maxExpArray[109] = 0x011c592761c666aa641d5a01a40f17ffff;
        maxExpArray[110] = 0x0110a688680a7530515f3e6e6cfdcdffff;
        maxExpArray[111] = 0x01056f1b5bedf75c6bcb2ce8aed428ffff;
        maxExpArray[112] = 0x00faadceceeff8a0890f3875f008277fff;
        maxExpArray[113] = 0x00f05dc6b27edad306388a600f6ba0bfff;
        maxExpArray[114] = 0x00e67a5a25da41063de1495d5b18cdbfff;
        maxExpArray[115] = 0x00dcff115b14eedde6fc3aa5353f2e4fff;
        maxExpArray[116] = 0x00d3e7a3924312399f9aae2e0f868f8fff;
        maxExpArray[117] = 0x00cb2ff529eb71e41582cccd5a1ee26fff;
        maxExpArray[118] = 0x00c2d415c3db974ab32a51840c0b67edff;
        maxExpArray[119] = 0x00bad03e7d883f69ad5b0a186184e06bff;
        maxExpArray[120] = 0x00b320d03b2c343d4829abd6075f0cc5ff;
        maxExpArray[121] = 0x00abc25204e02828d73c6e80bcdb1a95bf;
        maxExpArray[122] = 0x00a4b16f74ee4bb2040a1ec6c15fbbf2df;
        maxExpArray[123] = 0x009deaf736ac1f569deb1b5ae3f36c130f;
        maxExpArray[124] = 0x00976bd9952c7aa957f5937d790ef65037;
        maxExpArray[125] = 0x009131271922eaa6064b73a22d0bd4f2bf;
        maxExpArray[126] = 0x008b380f3558668c46c91c49a2f8e967b9;
        maxExpArray[127] = 0x00857ddf0117efa215952912839f6473e6;
    }

    /**
      * Auto-generated via 'PrintLambertArray.py'
    */
    uint256[128] private lambertArray;
    function initLambertArray() private {
        lambertArray[  0] = 0x60e393c68d20b1bd09deaabc0373b9c5;
        lambertArray[  1] = 0x5f8f46e4854120989ed94719fb4c2011;
        lambertArray[  2] = 0x5e479ebb9129fb1b7e72a648f992b606;
        lambertArray[  3] = 0x5d0bd23fe42dfedde2e9586be12b85fe;
        lambertArray[  4] = 0x5bdb29ddee979308ddfca81aeeb8095a;
        lambertArray[  5] = 0x5ab4fd8a260d2c7e2c0d2afcf0009dad;
        lambertArray[  6] = 0x5998b31359a55d48724c65cf09001221;
        lambertArray[  7] = 0x5885bcad2b322dfc43e8860f9c018cf5;
        lambertArray[  8] = 0x577b97aa1fe222bb452fdf111b1f0be2;
        lambertArray[  9] = 0x5679cb5e3575632e5baa27e2b949f704;
        lambertArray[ 10] = 0x557fe8241b3a31c83c732f1cdff4a1c5;
        lambertArray[ 11] = 0x548d868026504875d6e59bbe95fc2a6b;
        lambertArray[ 12] = 0x53a2465ce347cf34d05a867c17dd3088;
        lambertArray[ 13] = 0x52bdce5dcd4faed59c7f5511cf8f8acc;
        lambertArray[ 14] = 0x51dfcb453c07f8da817606e7885f7c3e;
        lambertArray[ 15] = 0x5107ef6b0a5a2be8f8ff15590daa3cce;
        lambertArray[ 16] = 0x5035f241d6eae0cd7bacba119993de7b;
        lambertArray[ 17] = 0x4f698fe90d5b53d532171e1210164c66;
        lambertArray[ 18] = 0x4ea288ca297a0e6a09a0eee240e16c85;
        lambertArray[ 19] = 0x4de0a13fdcf5d4213fc398ba6e3becde;
        lambertArray[ 20] = 0x4d23a145eef91fec06b06140804c4808;
        lambertArray[ 21] = 0x4c6b5430d4c1ee5526473db4ae0f11de;
        lambertArray[ 22] = 0x4bb7886c240562eba11f4963a53b4240;
        lambertArray[ 23] = 0x4b080f3f1cb491d2d521e0ea4583521e;
        lambertArray[ 24] = 0x4a5cbc96a05589cb4d86be1db3168364;
        lambertArray[ 25] = 0x49b566d40243517658d78c33162d6ece;
        lambertArray[ 26] = 0x4911e6a02e5507a30f947383fd9a3276;
        lambertArray[ 27] = 0x487216c2b31be4adc41db8a8d5cc0c88;
        lambertArray[ 28] = 0x47d5d3fc4a7a1b188cd3d788b5c5e9fc;
        lambertArray[ 29] = 0x473cfce4871a2c40bc4f9e1c32b955d0;
        lambertArray[ 30] = 0x46a771ca578ab878485810e285e31c67;
        lambertArray[ 31] = 0x4615149718aed4c258c373dc676aa72d;
        lambertArray[ 32] = 0x4585c8b3f8fe489c6e1833ca47871384;
        lambertArray[ 33] = 0x44f972f174e41e5efb7e9d63c29ce735;
        lambertArray[ 34] = 0x446ff970ba86d8b00beb05ecebf3c4dc;
        lambertArray[ 35] = 0x43e9438ec88971812d6f198b5ccaad96;
        lambertArray[ 36] = 0x436539d11ff7bea657aeddb394e809ef;
        lambertArray[ 37] = 0x42e3c5d3e5a913401d86f66db5d81c2c;
        lambertArray[ 38] = 0x4264d2395303070ea726cbe98df62174;
        lambertArray[ 39] = 0x41e84a9a593bb7194c3a6349ecae4eea;
        lambertArray[ 40] = 0x416e1b785d13eba07a08f3f18876a5ab;
        lambertArray[ 41] = 0x40f6322ff389d423ba9dd7e7e7b7e809;
        lambertArray[ 42] = 0x40807cec8a466880ecf4184545d240a4;
        lambertArray[ 43] = 0x400cea9ce88a8d3ae668e8ea0d9bf07f;
        lambertArray[ 44] = 0x3f9b6ae8772d4c55091e0ed7dfea0ac1;
        lambertArray[ 45] = 0x3f2bee253fd84594f54bcaafac383a13;
        lambertArray[ 46] = 0x3ebe654e95208bb9210c575c081c5958;
        lambertArray[ 47] = 0x3e52c1fc5665635b78ce1f05ad53c086;
        lambertArray[ 48] = 0x3de8f65ac388101ddf718a6f5c1eff65;
        lambertArray[ 49] = 0x3d80f522d59bd0b328ca012df4cd2d49;
        lambertArray[ 50] = 0x3d1ab193129ea72b23648a161163a85a;
        lambertArray[ 51] = 0x3cb61f68d32576c135b95cfb53f76d75;
        lambertArray[ 52] = 0x3c5332d9f1aae851a3619e77e4cc8473;
        lambertArray[ 53] = 0x3bf1e08edbe2aa109e1525f65759ef73;
        lambertArray[ 54] = 0x3b921d9cff13fa2c197746a3dfc4918f;
        lambertArray[ 55] = 0x3b33df818910bfc1a5aefb8f63ae2ac4;
        lambertArray[ 56] = 0x3ad71c1c77e34fa32a9f184967eccbf6;
        lambertArray[ 57] = 0x3a7bc9abf2c5bb53e2f7384a8a16521a;
        lambertArray[ 58] = 0x3a21dec7e76369783a68a0c6385a1c57;
        lambertArray[ 59] = 0x39c9525de6c9cdf7c1c157ca4a7a6ee3;
        lambertArray[ 60] = 0x39721bad3dc85d1240ff0190e0adaac3;
        lambertArray[ 61] = 0x391c324344d3248f0469eb28dd3d77e0;
        lambertArray[ 62] = 0x38c78df7e3c796279fb4ff84394ab3da;
        lambertArray[ 63] = 0x387426ea4638ae9aae08049d3554c20a;
        lambertArray[ 64] = 0x3821f57dbd2763256c1a99bbd2051378;
        lambertArray[ 65] = 0x37d0f256cb46a8c92ff62fbbef289698;
        lambertArray[ 66] = 0x37811658591ffc7abdd1feaf3cef9b73;
        lambertArray[ 67] = 0x37325aa10e9e82f7df0f380f7997154b;
        lambertArray[ 68] = 0x36e4b888cfb408d873b9a80d439311c6;
        lambertArray[ 69] = 0x3698299e59f4bb9de645fc9b08c64cca;
        lambertArray[ 70] = 0x364ca7a5012cb603023b57dd3ebfd50d;
        lambertArray[ 71] = 0x36022c928915b778ab1b06aaee7e61d4;
        lambertArray[ 72] = 0x35b8b28d1a73dc27500ffe35559cc028;
        lambertArray[ 73] = 0x357033e951fe250ec5eb4e60955132d7;
        lambertArray[ 74] = 0x3528ab2867934e3a21b5412e4c4f8881;
        lambertArray[ 75] = 0x34e212f66c55057f9676c80094a61d59;
        lambertArray[ 76] = 0x349c66289e5b3c4b540c24f42fa4b9bb;
        lambertArray[ 77] = 0x34579fbbd0c733a9c8d6af6b0f7d00f7;
        lambertArray[ 78] = 0x3413bad2e712288b924b5882b5b369bf;
        lambertArray[ 79] = 0x33d0b2b56286510ef730e213f71f12e9;
        lambertArray[ 80] = 0x338e82ce00e2496262c64457535ba1a1;
        lambertArray[ 81] = 0x334d26a96b373bb7c2f8ea1827f27a92;
        lambertArray[ 82] = 0x330c99f4f4211469e00b3e18c31475ea;
        lambertArray[ 83] = 0x32ccd87d6486094999c7d5e6f33237d8;
        lambertArray[ 84] = 0x328dde2dd617b6665a2e8556f250c1af;
        lambertArray[ 85] = 0x324fa70e9adc270f8262755af5a99af9;
        lambertArray[ 86] = 0x32122f443110611ca51040f41fa6e1e3;
        lambertArray[ 87] = 0x31d5730e42c0831482f0f1485c4263d8;
        lambertArray[ 88] = 0x31996ec6b07b4a83421b5ebc4ab4e1f1;
        lambertArray[ 89] = 0x315e1ee0a68ff46bb43ec2b85032e876;
        lambertArray[ 90] = 0x31237fe7bc4deacf6775b9efa1a145f8;
        lambertArray[ 91] = 0x30e98e7f1cc5a356e44627a6972ea2ff;
        lambertArray[ 92] = 0x30b04760b8917ec74205a3002650ec05;
        lambertArray[ 93] = 0x3077a75c803468e9132ce0cf3224241d;
        lambertArray[ 94] = 0x303fab57a6a275c36f19cda9bace667a;
        lambertArray[ 95] = 0x3008504beb8dcbd2cf3bc1f6d5a064f0;
        lambertArray[ 96] = 0x2fd19346ed17dac61219ce0c2c5ac4b0;
        lambertArray[ 97] = 0x2f9b7169808c324b5852fd3d54ba9714;
        lambertArray[ 98] = 0x2f65e7e711cf4b064eea9c08cbdad574;
        lambertArray[ 99] = 0x2f30f405093042ddff8a251b6bf6d103;
        lambertArray[100] = 0x2efc931a3750f2e8bfe323edfe037574;
        lambertArray[101] = 0x2ec8c28e46dbe56d98685278339400cb;
        lambertArray[102] = 0x2e957fd933c3926d8a599b602379b851;
        lambertArray[103] = 0x2e62c882c7c9ed4473412702f08ba0e5;
        lambertArray[104] = 0x2e309a221c12ba361e3ed695167feee2;
        lambertArray[105] = 0x2dfef25d1f865ae18dd07cfea4bcea10;
        lambertArray[106] = 0x2dcdcee821cdc80decc02c44344aeb31;
        lambertArray[107] = 0x2d9d2d8562b34944d0b201bb87260c83;
        lambertArray[108] = 0x2d6d0c04a5b62a2c42636308669b729a;
        lambertArray[109] = 0x2d3d6842c9a235517fc5a0332691528f;
        lambertArray[110] = 0x2d0e402963fe1ea2834abc408c437c10;
        lambertArray[111] = 0x2cdf91ae602647908aff975e4d6a2a8c;
        lambertArray[112] = 0x2cb15ad3a1eb65f6d74a75da09a1b6c5;
        lambertArray[113] = 0x2c8399a6ab8e9774d6fcff373d210727;
        lambertArray[114] = 0x2c564c4046f64edba6883ca06bbc4535;
        lambertArray[115] = 0x2c2970c431f952641e05cb493e23eed3;
        lambertArray[116] = 0x2bfd0560cd9eb14563bc7c0732856c18;
        lambertArray[117] = 0x2bd1084ed0332f7ff4150f9d0ef41a2c;
        lambertArray[118] = 0x2ba577d0fa1628b76d040b12a82492fb;
        lambertArray[119] = 0x2b7a5233cd21581e855e89dc2f1e8a92;
        lambertArray[120] = 0x2b4f95cd46904d05d72bdcde337d9cc7;
        lambertArray[121] = 0x2b2540fc9b4d9abba3faca6691914675;
        lambertArray[122] = 0x2afb5229f68d0830d8be8adb0a0db70f;
        lambertArray[123] = 0x2ad1c7c63a9b294c5bc73a3ba3ab7a2b;
        lambertArray[124] = 0x2aa8a04ac3cbe1ee1c9c86361465dbb8;
        lambertArray[125] = 0x2a7fda392d725a44a2c8aeb9ab35430d;
        lambertArray[126] = 0x2a57741b18cde618717792b4faa216db;
        lambertArray[127] = 0x2a2f6c81f5d84dd950a35626d6d5503a;
    }

    /**
      * @dev should be executed after construction (too large for the constructor)
    */
    function init() public {
        initMaxExpArray();
        initLambertArray();
    }

    /**
      * @dev given a token supply, reserve balance, weight and a deposit amount (in the reserve token),
      * calculates the target amount for a given conversion (in the main token)
      *
      * Formula:
      * return = _supply * ((1 + _amount / _reserveBalance) ^ (_reserveWeight / 1000000) - 1)
      *
      * @param _supply          smart token supply
      * @param _reserveBalance  reserve balance
      * @param _reserveWeight   reserve weight, represented in ppm (1-1000000)
      * @param _amount          amount of reserve tokens to get the target amount for
      *
      * @return smart token amount
    */
    function purchaseTargetAmount(uint256 _supply,
                                  uint256 _reserveBalance,
                                  uint32 _reserveWeight,
                                  uint256 _amount)
                                  public view returns (uint256)
    {
        // validate input
        require(_supply > 0, "ERR_INVALID_SUPPLY");
        require(_reserveBalance > 0, "ERR_INVALID_RESERVE_BALANCE");
        require(_reserveWeight > 0 && _reserveWeight <= MAX_WEIGHT, "ERR_INVALID_RESERVE_WEIGHT");

        // special case for 0 deposit amount
        if (_amount == 0)
            return 0;

        // special case if the weight = 100%
        if (_reserveWeight == MAX_WEIGHT)
            return _supply.mul(_amount) / _reserveBalance;

        uint256 result;
        uint8 precision;
        uint256 baseN = _amount.add(_reserveBalance);
        (result, precision) = power(baseN, _reserveBalance, _reserveWeight, MAX_WEIGHT);
        uint256 temp = _supply.mul(result) >> precision;
        return temp - _supply;
    }

    /**
      * @dev given a token supply, reserve balance, weight and a sell amount (in the main token),
      * calculates the target amount for a given conversion (in the reserve token)
      *
      * Formula:
      * return = _reserveBalance * (1 - (1 - _amount / _supply) ^ (1000000 / _reserveWeight))
      *
      * @param _supply          smart token supply
      * @param _reserveBalance  reserve balance
      * @param _reserveWeight   reserve weight, represented in ppm (1-1000000)
      * @param _amount          amount of smart tokens to get the target amount for
      *
      * @return reserve token amount
    */
    function saleTargetAmount(uint256 _supply,
                              uint256 _reserveBalance,
                              uint32 _reserveWeight,
                              uint256 _amount)
                              public view returns (uint256)
    {
        // validate input
        require(_supply > 0, "ERR_INVALID_SUPPLY");
        require(_reserveBalance > 0, "ERR_INVALID_RESERVE_BALANCE");
        require(_reserveWeight > 0 && _reserveWeight <= MAX_WEIGHT, "ERR_INVALID_RESERVE_WEIGHT");
        require(_amount <= _supply, "ERR_INVALID_AMOUNT");

        // special case for 0 sell amount
        if (_amount == 0)
            return 0;

        // special case for selling the entire supply
        if (_amount == _supply)
            return _reserveBalance;

        // special case if the weight = 100%
        if (_reserveWeight == MAX_WEIGHT)
            return _reserveBalance.mul(_amount) / _supply;

        uint256 result;
        uint8 precision;
        uint256 baseD = _supply - _amount;
        (result, precision) = power(_supply, baseD, MAX_WEIGHT, _reserveWeight);
        uint256 temp1 = _reserveBalance.mul(result);
        uint256 temp2 = _reserveBalance << precision;
        return (temp1 - temp2) / result;
    }

    /**
      * @dev given two reserve balances/weights and a sell amount (in the first reserve token),
      * calculates the target amount for a conversion from the source reserve token to the target reserve token
      *
      * Formula:
      * return = _targetReserveBalance * (1 - (_sourceReserveBalance / (_sourceReserveBalance + _amount)) ^ (_sourceReserveWeight / _targetReserveWeight))
      *
      * @param _sourceReserveBalance    source reserve balance
      * @param _sourceReserveWeight     source reserve weight, represented in ppm (1-1000000)
      * @param _targetReserveBalance    target reserve balance
      * @param _targetReserveWeight     target reserve weight, represented in ppm (1-1000000)
      * @param _amount                  source reserve amount
      *
      * @return target reserve amount
    */
    function crossReserveTargetAmount(uint256 _sourceReserveBalance,
                                      uint32 _sourceReserveWeight,
                                      uint256 _targetReserveBalance,
                                      uint32 _targetReserveWeight,
                                      uint256 _amount)
                                      public view returns (uint256)
    {
        // validate input
        require(_sourceReserveBalance > 0 && _targetReserveBalance > 0, "ERR_INVALID_RESERVE_BALANCE");
        require(_sourceReserveWeight > 0 && _sourceReserveWeight <= MAX_WEIGHT &&
                _targetReserveWeight > 0 && _targetReserveWeight <= MAX_WEIGHT, "ERR_INVALID_RESERVE_WEIGHT");

        // special case for equal weights
        if (_sourceReserveWeight == _targetReserveWeight)
            return _targetReserveBalance.mul(_amount) / _sourceReserveBalance.add(_amount);

        uint256 result;
        uint8 precision;
        uint256 baseN = _sourceReserveBalance.add(_amount);
        (result, precision) = power(baseN, _sourceReserveBalance, _sourceReserveWeight, _targetReserveWeight);
        uint256 temp1 = _targetReserveBalance.mul(result);
        uint256 temp2 = _targetReserveBalance << precision;
        return (temp1 - temp2) / result;
    }

    /**
      * @dev given a smart token supply, reserve balance, reserve ratio and an amount of requested smart tokens,
      * calculates the amount of reserve tokens required for purchasing the given amount of smart tokens
      *
      * Formula:
      * return = _reserveBalance * (((_supply + _amount) / _supply) ^ (MAX_WEIGHT / _reserveRatio) - 1)
      *
      * @param _supply          smart token supply
      * @param _reserveBalance  reserve balance
      * @param _reserveRatio    reserve ratio, represented in ppm (2-2000000)
      * @param _amount          requested amount of smart tokens
      *
      * @return reserve token amount
    */
    function fundCost(uint256 _supply,
                      uint256 _reserveBalance,
                      uint32 _reserveRatio,
                      uint256 _amount)
                      public view returns (uint256)
    {
        // validate input
        require(_supply > 0, "ERR_INVALID_SUPPLY");
        require(_reserveBalance > 0, "ERR_INVALID_RESERVE_BALANCE");
        require(_reserveRatio > 1 && _reserveRatio <= MAX_WEIGHT * 2, "ERR_INVALID_RESERVE_RATIO");

        // special case for 0 amount
        if (_amount == 0)
            return 0;

        // special case if the reserve ratio = 100%
        if (_reserveRatio == MAX_WEIGHT)
            return (_amount.mul(_reserveBalance) - 1) / _supply + 1;

        uint256 result;
        uint8 precision;
        uint256 baseN = _supply.add(_amount);
        (result, precision) = power(baseN, _supply, MAX_WEIGHT, _reserveRatio);
        uint256 temp = ((_reserveBalance.mul(result) - 1) >> precision) + 1;
        return temp - _reserveBalance;
    }

    /**
      * @dev given a smart token supply, reserve balance, reserve ratio and an amount of reserve tokens to fund with,
      * calculates the amount of smart tokens received for purchasing with the given amount of reserve tokens
      *
      * Formula:
      * return = _supply * ((_amount / _reserveBalance + 1) ^ (_reserveRatio / MAX_WEIGHT) - 1)
      *
      * @param _supply          smart token supply
      * @param _reserveBalance  reserve balance
      * @param _reserveRatio    reserve ratio, represented in ppm (2-2000000)
      * @param _amount          amount of reserve tokens to fund with
      *
      * @return smart token amount
    */
    function fundSupplyAmount(uint256 _supply,
                              uint256 _reserveBalance,
                              uint32 _reserveRatio,
                              uint256 _amount)
                              public view returns (uint256)
    {
        // validate input
        require(_supply > 0, "ERR_INVALID_SUPPLY");
        require(_reserveBalance > 0, "ERR_INVALID_RESERVE_BALANCE");
        require(_reserveRatio > 1 && _reserveRatio <= MAX_WEIGHT * 2, "ERR_INVALID_RESERVE_RATIO");

        // special case for 0 amount
        if (_amount == 0)
            return 0;

        // special case if the reserve ratio = 100%
        if (_reserveRatio == MAX_WEIGHT)
            return _amount.mul(_supply) / _reserveBalance;

        uint256 result;
        uint8 precision;
        uint256 baseN = _reserveBalance.add(_amount);
        (result, precision) = power(baseN, _reserveBalance, _reserveRatio, MAX_WEIGHT);
        uint256 temp = _supply.mul(result) >> precision;
        return temp - _supply;
    }

    /**
      * @dev given a smart token supply, reserve balance, reserve ratio and an amount of smart tokens to liquidate,
      * calculates the amount of reserve tokens received for selling the given amount of smart tokens
      *
      * Formula:
      * return = _reserveBalance * (1 - ((_supply - _amount) / _supply) ^ (MAX_WEIGHT / _reserveRatio))
      *
      * @param _supply          smart token supply
      * @param _reserveBalance  reserve balance
      * @param _reserveRatio    reserve ratio, represented in ppm (2-2000000)
      * @param _amount          amount of smart tokens to liquidate
      *
      * @return reserve token amount
    */
    function liquidateReserveAmount(uint256 _supply,
                                    uint256 _reserveBalance,
                                    uint32 _reserveRatio,
                                    uint256 _amount)
                                    public view returns (uint256)
    {
        // validate input
        require(_supply > 0, "ERR_INVALID_SUPPLY");
        require(_reserveBalance > 0, "ERR_INVALID_RESERVE_BALANCE");
        require(_reserveRatio > 1 && _reserveRatio <= MAX_WEIGHT * 2, "ERR_INVALID_RESERVE_RATIO");
        require(_amount <= _supply, "ERR_INVALID_AMOUNT");

        // special case for 0 amount
        if (_amount == 0)
            return 0;

        // special case for liquidating the entire supply
        if (_amount == _supply)
            return _reserveBalance;

        // special case if the reserve ratio = 100%
        if (_reserveRatio == MAX_WEIGHT)
            return _amount.mul(_reserveBalance) / _supply;

        uint256 result;
        uint8 precision;
        uint256 baseD = _supply - _amount;
        (result, precision) = power(_supply, baseD, MAX_WEIGHT, _reserveRatio);
        uint256 temp1 = _reserveBalance.mul(result);
        uint256 temp2 = _reserveBalance << precision;
        return (temp1 - temp2) / result;
    }

    /**
      * @dev The arbitrage incentive is to convert to the point where the on-chain price is equal to the off-chain price.
      * We want this operation to also impact the primary reserve balance becoming equal to the primary reserve staked balance.
      * In other words, we want the arbitrager to convert the difference between the reserve balance and the reserve staked balance.
      *
      * Formula input:
      * - let t denote the primary reserve token staked balance
      * - let s denote the primary reserve token balance
      * - let r denote the secondary reserve token balance
      * - let q denote the numerator of the rate between the tokens
      * - let p denote the denominator of the rate between the tokens
      * Where p primary tokens are equal to q secondary tokens
      *
      * Formula output:
      * - compute x = W(t / r * q / p * log(s / t)) / log(s / t)
      * - return x / (1 + x) as the weight of the primary reserve token
      * - return 1 / (1 + x) as the weight of the secondary reserve token
      * Where W is the Lambert W Function
      *
      * If the rate-provider provides the rates for a common unit, for example:
      * - P = 2 ==> 2 primary reserve tokens = 1 ether
      * - Q = 3 ==> 3 secondary reserve tokens = 1 ether
      * Then you can simply use p = P and q = Q
      *
      * If the rate-provider provides the rates for a single unit, for example:
      * - P = 2 ==> 1 primary reserve token = 2 ethers
      * - Q = 3 ==> 1 secondary reserve token = 3 ethers
      * Then you can simply use p = Q and q = P
      *
      * @param _primaryReserveStakedBalance the primary reserve token staked balance
      * @param _primaryReserveBalance       the primary reserve token balance
      * @param _secondaryReserveBalance     the secondary reserve token balance
      * @param _reserveRateNumerator        the numerator of the rate between the tokens
      * @param _reserveRateDenominator      the denominator of the rate between the tokens
      *
      * Note that `numerator / denominator` should represent the amount of secondary tokens equal to one primary token
      *
      * @return the weight of the primary reserve token and the weight of the secondary reserve token, both in ppm (0-1000000)
    */
    function balancedWeights(uint256 _primaryReserveStakedBalance,
                             uint256 _primaryReserveBalance,
                             uint256 _secondaryReserveBalance,
                             uint256 _reserveRateNumerator,
                             uint256 _reserveRateDenominator)
                             public view returns (uint32, uint32)
    {
        if (_primaryReserveStakedBalance == _primaryReserveBalance)
            require(_primaryReserveStakedBalance > 0 || _secondaryReserveBalance > 0, "ERR_INVALID_RESERVE_BALANCE");
        else
            require(_primaryReserveStakedBalance > 0 && _primaryReserveBalance > 0 && _secondaryReserveBalance > 0, "ERR_INVALID_RESERVE_BALANCE");
        require(_reserveRateNumerator > 0 && _reserveRateDenominator > 0, "ERR_INVALID_RESERVE_RATE");

        uint256 tq = _primaryReserveStakedBalance.mul(_reserveRateNumerator);
        uint256 rp = _secondaryReserveBalance.mul(_reserveRateDenominator);

        if (_primaryReserveStakedBalance < _primaryReserveBalance)
            return balancedWeightsByStake(_primaryReserveBalance, _primaryReserveStakedBalance, tq, rp, true);

        if (_primaryReserveStakedBalance > _primaryReserveBalance)
            return balancedWeightsByStake(_primaryReserveStakedBalance, _primaryReserveBalance, tq, rp, false);

        return normalizedWeights(tq, rp);
    }

    /**
      * @dev General Description:
      *     Determine a value of precision.
      *     Calculate an integer approximation of (_baseN / _baseD) ^ (_expN / _expD) * 2 ^ precision.
      *     Return the result along with the precision used.
      *
      * Detailed Description:
      *     Instead of calculating "base ^ exp", we calculate "e ^ (log(base) * exp)".
      *     The value of "log(base)" is represented with an integer slightly smaller than "log(base) * 2 ^ precision".
      *     The larger "precision" is, the more accurately this value represents the real value.
      *     However, the larger "precision" is, the more bits are required in order to store this value.
      *     And the exponentiation function, which takes "x" and calculates "e ^ x", is limited to a maximum exponent (maximum value of "x").
      *     This maximum exponent depends on the "precision" used, and it is given by "maxExpArray[precision] >> (MAX_PRECISION - precision)".
      *     Hence we need to determine the highest precision which can be used for the given input, before calling the exponentiation function.
      *     This allows us to compute "base ^ exp" with maximum accuracy and without exceeding 256 bits in any of the intermediate computations.
      *     This functions assumes that "_expN < 2 ^ 256 / log(MAX_NUM - 1)", otherwise the multiplication should be replaced with a "safeMul".
      *     Since we rely on unsigned-integer arithmetic and "base < 1" ==> "log(base) < 0", this function does not support "_baseN < _baseD".
    */
    function power(uint256 _baseN, uint256 _baseD, uint32 _expN, uint32 _expD) internal view returns (uint256, uint8) {
        require(_baseN < MAX_NUM);

        uint256 baseLog;
        uint256 base = _baseN * FIXED_1 / _baseD;
        if (base < OPT_LOG_MAX_VAL) {
            baseLog = optimalLog(base);
        }
        else {
            baseLog = generalLog(base);
        }

        uint256 baseLogTimesExp = baseLog * _expN / _expD;
        if (baseLogTimesExp < OPT_EXP_MAX_VAL) {
            return (optimalExp(baseLogTimesExp), MAX_PRECISION);
        }
        else {
            uint8 precision = findPositionInMaxExpArray(baseLogTimesExp);
            return (generalExp(baseLogTimesExp >> (MAX_PRECISION - precision), precision), precision);
        }
    }

    /**
      * @dev computes log(x / FIXED_1) * FIXED_1.
      * This functions assumes that "x >= FIXED_1", because the output would be negative otherwise.
    */
    function generalLog(uint256 x) internal pure returns (uint256) {
        uint256 res = 0;

        // If x >= 2, then we compute the integer part of log2(x), which is larger than 0.
        if (x >= FIXED_2) {
            uint8 count = floorLog2(x / FIXED_1);
            x >>= count; // now x < 2
            res = count * FIXED_1;
        }

        // If x > 1, then we compute the fraction part of log2(x), which is larger than 0.
        if (x > FIXED_1) {
            for (uint8 i = MAX_PRECISION; i > 0; --i) {
                x = (x * x) / FIXED_1; // now 1 < x < 4
                if (x >= FIXED_2) {
                    x >>= 1; // now 1 < x < 2
                    res += ONE << (i - 1);
                }
            }
        }

        return res * LN2_NUMERATOR / LN2_DENOMINATOR;
    }

    /**
      * @dev computes the largest integer smaller than or equal to the binary logarithm of the input.
    */
    function floorLog2(uint256 _n) internal pure returns (uint8) {
        uint8 res = 0;

        if (_n < 256) {
            // At most 8 iterations
            while (_n > 1) {
                _n >>= 1;
                res += 1;
            }
        }
        else {
            // Exactly 8 iterations
            for (uint8 s = 128; s > 0; s >>= 1) {
                if (_n >= (ONE << s)) {
                    _n >>= s;
                    res |= s;
                }
            }
        }

        return res;
    }

    /**
      * @dev the global "maxExpArray" is sorted in descending order, and therefore the following statements are equivalent:
      * - This function finds the position of [the smallest value in "maxExpArray" larger than or equal to "x"]
      * - This function finds the highest position of [a value in "maxExpArray" larger than or equal to "x"]
    */
    function findPositionInMaxExpArray(uint256 _x) internal view returns (uint8) {
        uint8 lo = MIN_PRECISION;
        uint8 hi = MAX_PRECISION;

        while (lo + 1 < hi) {
            uint8 mid = (lo + hi) / 2;
            if (maxExpArray[mid] >= _x)
                lo = mid;
            else
                hi = mid;
        }

        if (maxExpArray[hi] >= _x)
            return hi;
        if (maxExpArray[lo] >= _x)
            return lo;

        require(false);
    }

    /**
      * @dev this function can be auto-generated by the script 'PrintFunctionGeneralExp.py'.
      * it approximates "e ^ x" via maclaurin summation: "(x^0)/0! + (x^1)/1! + ... + (x^n)/n!".
      * it returns "e ^ (x / 2 ^ precision) * 2 ^ precision", that is, the result is upshifted for accuracy.
      * the global "maxExpArray" maps each "precision" to "((maximumExponent + 1) << (MAX_PRECISION - precision)) - 1".
      * the maximum permitted value for "x" is therefore given by "maxExpArray[precision] >> (MAX_PRECISION - precision)".
    */
    function generalExp(uint256 _x, uint8 _precision) internal pure returns (uint256) {
        uint256 xi = _x;
        uint256 res = 0;

        xi = (xi * _x) >> _precision; res += xi * 0x3442c4e6074a82f1797f72ac0000000; // add x^02 * (33! / 02!)
        xi = (xi * _x) >> _precision; res += xi * 0x116b96f757c380fb287fd0e40000000; // add x^03 * (33! / 03!)
        xi = (xi * _x) >> _precision; res += xi * 0x045ae5bdd5f0e03eca1ff4390000000; // add x^04 * (33! / 04!)
        xi = (xi * _x) >> _precision; res += xi * 0x00defabf91302cd95b9ffda50000000; // add x^05 * (33! / 05!)
        xi = (xi * _x) >> _precision; res += xi * 0x002529ca9832b22439efff9b8000000; // add x^06 * (33! / 06!)
        xi = (xi * _x) >> _precision; res += xi * 0x00054f1cf12bd04e516b6da88000000; // add x^07 * (33! / 07!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000a9e39e257a09ca2d6db51000000; // add x^08 * (33! / 08!)
        xi = (xi * _x) >> _precision; res += xi * 0x000012e066e7b839fa050c309000000; // add x^09 * (33! / 09!)
        xi = (xi * _x) >> _precision; res += xi * 0x000001e33d7d926c329a1ad1a800000; // add x^10 * (33! / 10!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000002bee513bdb4a6b19b5f800000; // add x^11 * (33! / 11!)
        xi = (xi * _x) >> _precision; res += xi * 0x00000003a9316fa79b88eccf2a00000; // add x^12 * (33! / 12!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000000048177ebe1fa812375200000; // add x^13 * (33! / 13!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000000005263fe90242dcbacf00000; // add x^14 * (33! / 14!)
        xi = (xi * _x) >> _precision; res += xi * 0x000000000057e22099c030d94100000; // add x^15 * (33! / 15!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000000000057e22099c030d9410000; // add x^16 * (33! / 16!)
        xi = (xi * _x) >> _precision; res += xi * 0x00000000000052b6b54569976310000; // add x^17 * (33! / 17!)
        xi = (xi * _x) >> _precision; res += xi * 0x00000000000004985f67696bf748000; // add x^18 * (33! / 18!)
        xi = (xi * _x) >> _precision; res += xi * 0x000000000000003dea12ea99e498000; // add x^19 * (33! / 19!)
        xi = (xi * _x) >> _precision; res += xi * 0x00000000000000031880f2214b6e000; // add x^20 * (33! / 20!)
        xi = (xi * _x) >> _precision; res += xi * 0x000000000000000025bcff56eb36000; // add x^21 * (33! / 21!)
        xi = (xi * _x) >> _precision; res += xi * 0x000000000000000001b722e10ab1000; // add x^22 * (33! / 22!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000001317c70077000; // add x^23 * (33! / 23!)
        xi = (xi * _x) >> _precision; res += xi * 0x00000000000000000000cba84aafa00; // add x^24 * (33! / 24!)
        xi = (xi * _x) >> _precision; res += xi * 0x00000000000000000000082573a0a00; // add x^25 * (33! / 25!)
        xi = (xi * _x) >> _precision; res += xi * 0x00000000000000000000005035ad900; // add x^26 * (33! / 26!)
        xi = (xi * _x) >> _precision; res += xi * 0x000000000000000000000002f881b00; // add x^27 * (33! / 27!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000000000001b29340; // add x^28 * (33! / 28!)
        xi = (xi * _x) >> _precision; res += xi * 0x00000000000000000000000000efc40; // add x^29 * (33! / 29!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000000000000007fe0; // add x^30 * (33! / 30!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000000000000000420; // add x^31 * (33! / 31!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000000000000000021; // add x^32 * (33! / 32!)
        xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000000000000000001; // add x^33 * (33! / 33!)

        return res / 0x688589cc0e9505e2f2fee5580000000 + _x + (ONE << _precision); // divide by 33! and then add x^1 / 1! + x^0 / 0!
    }

    /**
      * @dev computes log(x / FIXED_1) * FIXED_1
      * Input range: FIXED_1 <= x <= OPT_LOG_MAX_VAL - 1
      * Auto-generated via 'PrintFunctionOptimalLog.py'
      * Detailed description:
      * - Rewrite the input as a product of natural exponents and a single residual r, such that 1 < r < 2
      * - The natural logarithm of each (pre-calculated) exponent is the degree of the exponent
      * - The natural logarithm of r is calculated via Taylor series for log(1 + x), where x = r - 1
      * - The natural logarithm of the input is calculated by summing up the intermediate results above
      * - For example: log(250) = log(e^4 * e^1 * e^0.5 * 1.021692859) = 4 + 1 + 0.5 + log(1 + 0.021692859)
    */
    function optimalLog(uint256 x) internal pure returns (uint256) {
        uint256 res = 0;

        uint256 y;
        uint256 z;
        uint256 w;

        if (x >= 0xd3094c70f034de4b96ff7d5b6f99fcd8) {res += 0x40000000000000000000000000000000; x = x * FIXED_1 / 0xd3094c70f034de4b96ff7d5b6f99fcd8;} // add 1 / 2^1
        if (x >= 0xa45af1e1f40c333b3de1db4dd55f29a7) {res += 0x20000000000000000000000000000000; x = x * FIXED_1 / 0xa45af1e1f40c333b3de1db4dd55f29a7;} // add 1 / 2^2
        if (x >= 0x910b022db7ae67ce76b441c27035c6a1) {res += 0x10000000000000000000000000000000; x = x * FIXED_1 / 0x910b022db7ae67ce76b441c27035c6a1;} // add 1 / 2^3
        if (x >= 0x88415abbe9a76bead8d00cf112e4d4a8) {res += 0x08000000000000000000000000000000; x = x * FIXED_1 / 0x88415abbe9a76bead8d00cf112e4d4a8;} // add 1 / 2^4
        if (x >= 0x84102b00893f64c705e841d5d4064bd3) {res += 0x04000000000000000000000000000000; x = x * FIXED_1 / 0x84102b00893f64c705e841d5d4064bd3;} // add 1 / 2^5
        if (x >= 0x8204055aaef1c8bd5c3259f4822735a2) {res += 0x02000000000000000000000000000000; x = x * FIXED_1 / 0x8204055aaef1c8bd5c3259f4822735a2;} // add 1 / 2^6
        if (x >= 0x810100ab00222d861931c15e39b44e99) {res += 0x01000000000000000000000000000000; x = x * FIXED_1 / 0x810100ab00222d861931c15e39b44e99;} // add 1 / 2^7
        if (x >= 0x808040155aabbbe9451521693554f733) {res += 0x00800000000000000000000000000000; x = x * FIXED_1 / 0x808040155aabbbe9451521693554f733;} // add 1 / 2^8

        z = y = x - FIXED_1;
        w = y * y / FIXED_1;
        res += z * (0x100000000000000000000000000000000 - y) / 0x100000000000000000000000000000000; z = z * w / FIXED_1; // add y^01 / 01 - y^02 / 02
        res += z * (0x0aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa - y) / 0x200000000000000000000000000000000; z = z * w / FIXED_1; // add y^03 / 03 - y^04 / 04
        res += z * (0x099999999999999999999999999999999 - y) / 0x300000000000000000000000000000000; z = z * w / FIXED_1; // add y^05 / 05 - y^06 / 06
        res += z * (0x092492492492492492492492492492492 - y) / 0x400000000000000000000000000000000; z = z * w / FIXED_1; // add y^07 / 07 - y^08 / 08
        res += z * (0x08e38e38e38e38e38e38e38e38e38e38e - y) / 0x500000000000000000000000000000000; z = z * w / FIXED_1; // add y^09 / 09 - y^10 / 10
        res += z * (0x08ba2e8ba2e8ba2e8ba2e8ba2e8ba2e8b - y) / 0x600000000000000000000000000000000; z = z * w / FIXED_1; // add y^11 / 11 - y^12 / 12
        res += z * (0x089d89d89d89d89d89d89d89d89d89d89 - y) / 0x700000000000000000000000000000000; z = z * w / FIXED_1; // add y^13 / 13 - y^14 / 14
        res += z * (0x088888888888888888888888888888888 - y) / 0x800000000000000000000000000000000;                      // add y^15 / 15 - y^16 / 16

        return res;
    }

    /**
      * @dev computes e ^ (x / FIXED_1) * FIXED_1
      * input range: 0 <= x <= OPT_EXP_MAX_VAL - 1
      * auto-generated via 'PrintFunctionOptimalExp.py'
      * Detailed description:
      * - Rewrite the input as a sum of binary exponents and a single residual r, as small as possible
      * - The exponentiation of each binary exponent is given (pre-calculated)
      * - The exponentiation of r is calculated via Taylor series for e^x, where x = r
      * - The exponentiation of the input is calculated by multiplying the intermediate results above
      * - For example: e^5.521692859 = e^(4 + 1 + 0.5 + 0.021692859) = e^4 * e^1 * e^0.5 * e^0.021692859
    */
    function optimalExp(uint256 x) internal pure returns (uint256) {
        uint256 res = 0;

        uint256 y;
        uint256 z;

        z = y = x % 0x10000000000000000000000000000000; // get the input modulo 2^(-3)
        z = z * y / FIXED_1; res += z * 0x10e1b3be415a0000; // add y^02 * (20! / 02!)
        z = z * y / FIXED_1; res += z * 0x05a0913f6b1e0000; // add y^03 * (20! / 03!)
        z = z * y / FIXED_1; res += z * 0x0168244fdac78000; // add y^04 * (20! / 04!)
        z = z * y / FIXED_1; res += z * 0x004807432bc18000; // add y^05 * (20! / 05!)
        z = z * y / FIXED_1; res += z * 0x000c0135dca04000; // add y^06 * (20! / 06!)
        z = z * y / FIXED_1; res += z * 0x0001b707b1cdc000; // add y^07 * (20! / 07!)
        z = z * y / FIXED_1; res += z * 0x000036e0f639b800; // add y^08 * (20! / 08!)
        z = z * y / FIXED_1; res += z * 0x00000618fee9f800; // add y^09 * (20! / 09!)
        z = z * y / FIXED_1; res += z * 0x0000009c197dcc00; // add y^10 * (20! / 10!)
        z = z * y / FIXED_1; res += z * 0x0000000e30dce400; // add y^11 * (20! / 11!)
        z = z * y / FIXED_1; res += z * 0x000000012ebd1300; // add y^12 * (20! / 12!)
        z = z * y / FIXED_1; res += z * 0x0000000017499f00; // add y^13 * (20! / 13!)
        z = z * y / FIXED_1; res += z * 0x0000000001a9d480; // add y^14 * (20! / 14!)
        z = z * y / FIXED_1; res += z * 0x00000000001c6380; // add y^15 * (20! / 15!)
        z = z * y / FIXED_1; res += z * 0x000000000001c638; // add y^16 * (20! / 16!)
        z = z * y / FIXED_1; res += z * 0x0000000000001ab8; // add y^17 * (20! / 17!)
        z = z * y / FIXED_1; res += z * 0x000000000000017c; // add y^18 * (20! / 18!)
        z = z * y / FIXED_1; res += z * 0x0000000000000014; // add y^19 * (20! / 19!)
        z = z * y / FIXED_1; res += z * 0x0000000000000001; // add y^20 * (20! / 20!)
        res = res / 0x21c3677c82b40000 + y + FIXED_1; // divide by 20! and then add y^1 / 1! + y^0 / 0!

        if ((x & 0x010000000000000000000000000000000) != 0) res = res * 0x1c3d6a24ed82218787d624d3e5eba95f9 / 0x18ebef9eac820ae8682b9793ac6d1e776; // multiply by e^2^(-3)
        if ((x & 0x020000000000000000000000000000000) != 0) res = res * 0x18ebef9eac820ae8682b9793ac6d1e778 / 0x1368b2fc6f9609fe7aceb46aa619baed4; // multiply by e^2^(-2)
        if ((x & 0x040000000000000000000000000000000) != 0) res = res * 0x1368b2fc6f9609fe7aceb46aa619baed5 / 0x0bc5ab1b16779be3575bd8f0520a9f21f; // multiply by e^2^(-1)
        if ((x & 0x080000000000000000000000000000000) != 0) res = res * 0x0bc5ab1b16779be3575bd8f0520a9f21e / 0x0454aaa8efe072e7f6ddbab84b40a55c9; // multiply by e^2^(+0)
        if ((x & 0x100000000000000000000000000000000) != 0) res = res * 0x0454aaa8efe072e7f6ddbab84b40a55c5 / 0x00960aadc109e7a3bf4578099615711ea; // multiply by e^2^(+1)
        if ((x & 0x200000000000000000000000000000000) != 0) res = res * 0x00960aadc109e7a3bf4578099615711d7 / 0x0002bf84208204f5977f9a8cf01fdce3d; // multiply by e^2^(+2)
        if ((x & 0x400000000000000000000000000000000) != 0) res = res * 0x0002bf84208204f5977f9a8cf01fdc307 / 0x0000003c6ab775dd0b95b4cbee7e65d11; // multiply by e^2^(+3)

        return res;
    }

    /**
      * @dev computes W(x / FIXED_1) / (x / FIXED_1) * FIXED_1
    */
    function lowerStake(uint256 _x) internal view returns (uint256) {
        if (_x <= LAMBERT_CONV_RADIUS)
            return lambertPos1(_x);
        if (_x <= LAMBERT_POS2_MAXVAL)
            return lambertPos2(_x);
        if (_x <= LAMBERT_POS3_MAXVAL)
            return lambertPos3(_x);
        require(false);
    }

    /**
      * @dev computes W(-x / FIXED_1) / (-x / FIXED_1) * FIXED_1
    */
    function higherStake(uint256 _x) internal pure returns (uint256) {
        if (_x <= LAMBERT_CONV_RADIUS)
            return lambertNeg1(_x);
        return FIXED_1 * FIXED_1 / _x;
    }

    /**
      * @dev computes W(x / FIXED_1) / (x / FIXED_1) * FIXED_1
      * input range: 1 <= x <= 1 / e * FIXED_1
      * auto-generated via 'PrintFunctionLambertPos1.py'
    */
    function lambertPos1(uint256 _x) internal pure returns (uint256) {
        uint256 xi = _x;
        uint256 res = (FIXED_1 - _x) * 0xde1bc4d19efcac82445da75b00000000; // x^(1-1) * (34! * 1^(1-1) / 1!) - x^(2-1) * (34! * 2^(2-1) / 2!)

        xi = (xi * _x) / FIXED_1; res += xi * 0x00000000014d29a73a6e7b02c3668c7b0880000000; // add x^(03-1) * (34! * 03^(03-1) / 03!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x0000000002504a0cd9a7f7215b60f9be4800000000; // sub x^(04-1) * (34! * 04^(04-1) / 04!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000000000484d0a1191c0ead267967c7a4a0000000; // add x^(05-1) * (34! * 05^(05-1) / 05!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x00000000095ec580d7e8427a4baf26a90a00000000; // sub x^(06-1) * (34! * 06^(06-1) / 06!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000000001440b0be1615a47dba6e5b3b1f10000000; // add x^(07-1) * (34! * 07^(07-1) / 07!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x000000002d207601f46a99b4112418400000000000; // sub x^(08-1) * (34! * 08^(08-1) / 08!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000000066ebaac4c37c622dd8288a7eb1b2000000; // add x^(09-1) * (34! * 09^(09-1) / 09!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x00000000ef17240135f7dbd43a1ba10cf200000000; // sub x^(10-1) * (34! * 10^(10-1) / 10!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000000233c33c676a5eb2416094a87b3657000000; // add x^(11-1) * (34! * 11^(11-1) / 11!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x0000000541cde48bc0254bed49a9f8700000000000; // sub x^(12-1) * (34! * 12^(12-1) / 12!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000000cae1fad2cdd4d4cb8d73abca0d19a400000; // add x^(13-1) * (34! * 13^(13-1) / 13!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x0000001edb2aa2f760d15c41ceedba956400000000; // sub x^(14-1) * (34! * 14^(14-1) / 14!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000004ba8d20d2dabd386c9529659841a2e200000; // add x^(15-1) * (34! * 15^(15-1) / 15!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x000000bac08546b867cdaa20000000000000000000; // sub x^(16-1) * (34! * 16^(16-1) / 16!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000001cfa8e70c03625b9db76c8ebf5bbf24820000; // add x^(17-1) * (34! * 17^(17-1) / 17!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x000004851d99f82060df265f3309b26f8200000000; // sub x^(18-1) * (34! * 18^(18-1) / 18!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x00000b550d19b129d270c44f6f55f027723cbb0000; // add x^(19-1) * (34! * 19^(19-1) / 19!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x00001c877dadc761dc272deb65d4b0000000000000; // sub x^(20-1) * (34! * 20^(20-1) / 20!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000048178ece97479f33a77f2ad22a81b64406c000; // add x^(21-1) * (34! * 21^(21-1) / 21!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x0000b6ca8268b9d810fedf6695ef2f8a6c00000000; // sub x^(22-1) * (34! * 22^(22-1) / 22!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0001d0e76631a5b05d007b8cb72a7c7f11ec36e000; // add x^(23-1) * (34! * 23^(23-1) / 23!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x0004a1c37bd9f85fd9c6c780000000000000000000; // sub x^(24-1) * (34! * 24^(24-1) / 24!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000bd8369f1b702bf491e2ebfcee08250313b65400; // add x^(25-1) * (34! * 25^(25-1) / 25!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x001e5c7c32a9f6c70ab2cb59d9225764d400000000; // sub x^(26-1) * (34! * 26^(26-1) / 26!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x004dff5820e165e910f95120a708e742496221e600; // add x^(27-1) * (34! * 27^(27-1) / 27!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x00c8c8f66db1fced378ee50e536000000000000000; // sub x^(28-1) * (34! * 28^(28-1) / 28!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0205db8dffff45bfa2938f128f599dbf16eb11d880; // add x^(29-1) * (34! * 29^(29-1) / 29!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x053a044ebd984351493e1786af38d39a0800000000; // sub x^(30-1) * (34! * 30^(30-1) / 30!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0d86dae2a4cc0f47633a544479735869b487b59c40; // add x^(31-1) * (34! * 31^(31-1) / 31!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0x231000000000000000000000000000000000000000; // sub x^(32-1) * (34! * 32^(32-1) / 32!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x5b0485a76f6646c2039db1507cdd51b08649680822; // add x^(33-1) * (34! * 33^(33-1) / 33!)
        xi = (xi * _x) / FIXED_1; res -= xi * 0xec983c46c49545bc17efa6b5b0055e242200000000; // sub x^(34-1) * (34! * 34^(34-1) / 34!)

        return res / 0xde1bc4d19efcac82445da75b00000000; // divide by 34!
    }

    /**
      * @dev computes W(x / FIXED_1) / (x / FIXED_1) * FIXED_1
      * input range: LAMBERT_CONV_RADIUS + 1 <= x <= LAMBERT_POS2_MAXVAL
    */
    function lambertPos2(uint256 _x) internal view returns (uint256) {
        uint256 x = _x - LAMBERT_CONV_RADIUS - 1;
        uint256 i = x / LAMBERT_POS2_SAMPLE;
        uint256 a = LAMBERT_POS2_SAMPLE * i;
        uint256 b = LAMBERT_POS2_SAMPLE * (i + 1);
        uint256 c = lambertArray[i];
        uint256 d = lambertArray[i + 1];
        return (c * (b - x) + d * (x - a)) / LAMBERT_POS2_SAMPLE;
    }

    /**
      * @dev computes W(x / FIXED_1) / (x / FIXED_1) * FIXED_1
      * input range: LAMBERT_POS2_MAXVAL + 1 <= x <= LAMBERT_POS3_MAXVAL
    */
    function lambertPos3(uint256 _x) internal pure returns (uint256) {
        uint256 l1 = _x < OPT_LOG_MAX_VAL ? optimalLog(_x) : generalLog(_x);
        uint256 l2 = l1 < OPT_LOG_MAX_VAL ? optimalLog(l1) : generalLog(l1);
        return (l1 - l2 + l2 * FIXED_1 / l1) * FIXED_1 / _x;
    }

    /**
      * @dev computes W(-x / FIXED_1) / (-x / FIXED_1) * FIXED_1
      * input range: 1 <= x <= 1 / e * FIXED_1
      * auto-generated via 'PrintFunctionLambertNeg1.py'
    */
    function lambertNeg1(uint256 _x) internal pure returns (uint256) {
        uint256 xi = _x;
        uint256 res = 0;

        xi = (xi * _x) / FIXED_1; res += xi * 0x00000000014d29a73a6e7b02c3668c7b0880000000; // add x^(03-1) * (34! * 03^(03-1) / 03!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000000002504a0cd9a7f7215b60f9be4800000000; // add x^(04-1) * (34! * 04^(04-1) / 04!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000000000484d0a1191c0ead267967c7a4a0000000; // add x^(05-1) * (34! * 05^(05-1) / 05!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x00000000095ec580d7e8427a4baf26a90a00000000; // add x^(06-1) * (34! * 06^(06-1) / 06!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000000001440b0be1615a47dba6e5b3b1f10000000; // add x^(07-1) * (34! * 07^(07-1) / 07!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000000002d207601f46a99b4112418400000000000; // add x^(08-1) * (34! * 08^(08-1) / 08!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000000066ebaac4c37c622dd8288a7eb1b2000000; // add x^(09-1) * (34! * 09^(09-1) / 09!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x00000000ef17240135f7dbd43a1ba10cf200000000; // add x^(10-1) * (34! * 10^(10-1) / 10!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000000233c33c676a5eb2416094a87b3657000000; // add x^(11-1) * (34! * 11^(11-1) / 11!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000000541cde48bc0254bed49a9f8700000000000; // add x^(12-1) * (34! * 12^(12-1) / 12!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000000cae1fad2cdd4d4cb8d73abca0d19a400000; // add x^(13-1) * (34! * 13^(13-1) / 13!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000001edb2aa2f760d15c41ceedba956400000000; // add x^(14-1) * (34! * 14^(14-1) / 14!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000004ba8d20d2dabd386c9529659841a2e200000; // add x^(15-1) * (34! * 15^(15-1) / 15!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000000bac08546b867cdaa20000000000000000000; // add x^(16-1) * (34! * 16^(16-1) / 16!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000001cfa8e70c03625b9db76c8ebf5bbf24820000; // add x^(17-1) * (34! * 17^(17-1) / 17!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000004851d99f82060df265f3309b26f8200000000; // add x^(18-1) * (34! * 18^(18-1) / 18!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x00000b550d19b129d270c44f6f55f027723cbb0000; // add x^(19-1) * (34! * 19^(19-1) / 19!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x00001c877dadc761dc272deb65d4b0000000000000; // add x^(20-1) * (34! * 20^(20-1) / 20!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000048178ece97479f33a77f2ad22a81b64406c000; // add x^(21-1) * (34! * 21^(21-1) / 21!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0000b6ca8268b9d810fedf6695ef2f8a6c00000000; // add x^(22-1) * (34! * 22^(22-1) / 22!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0001d0e76631a5b05d007b8cb72a7c7f11ec36e000; // add x^(23-1) * (34! * 23^(23-1) / 23!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0004a1c37bd9f85fd9c6c780000000000000000000; // add x^(24-1) * (34! * 24^(24-1) / 24!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x000bd8369f1b702bf491e2ebfcee08250313b65400; // add x^(25-1) * (34! * 25^(25-1) / 25!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x001e5c7c32a9f6c70ab2cb59d9225764d400000000; // add x^(26-1) * (34! * 26^(26-1) / 26!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x004dff5820e165e910f95120a708e742496221e600; // add x^(27-1) * (34! * 27^(27-1) / 27!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x00c8c8f66db1fced378ee50e536000000000000000; // add x^(28-1) * (34! * 28^(28-1) / 28!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0205db8dffff45bfa2938f128f599dbf16eb11d880; // add x^(29-1) * (34! * 29^(29-1) / 29!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x053a044ebd984351493e1786af38d39a0800000000; // add x^(30-1) * (34! * 30^(30-1) / 30!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x0d86dae2a4cc0f47633a544479735869b487b59c40; // add x^(31-1) * (34! * 31^(31-1) / 31!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x231000000000000000000000000000000000000000; // add x^(32-1) * (34! * 32^(32-1) / 32!)
        xi = (xi * _x) / FIXED_1; res += xi * 0x5b0485a76f6646c2039db1507cdd51b08649680822; // add x^(33-1) * (34! * 33^(33-1) / 33!)
        xi = (xi * _x) / FIXED_1; res += xi * 0xec983c46c49545bc17efa6b5b0055e242200000000; // add x^(34-1) * (34! * 34^(34-1) / 34!)

        return res / 0xde1bc4d19efcac82445da75b00000000 + _x + FIXED_1; // divide by 34! and then add x^(2-1) * (34! * 2^(2-1) / 2!) + x^(1-1) * (34! * 1^(1-1) / 1!)
    }

    /**
      * @dev computes the weights based on "W(log(hi / lo) * tq / rp) * tq / rp", where "W" is a variation of the Lambert W function.
    */
    function balancedWeightsByStake(uint256 _hi, uint256 _lo, uint256 _tq, uint256 _rp, bool _lowerStake) internal view returns (uint32, uint32) {
        (_tq, _rp) = safeFactors(_tq, _rp);
        uint256 f = _hi.mul(FIXED_1) / _lo;
        uint256 g = f < OPT_LOG_MAX_VAL ? optimalLog(f) : generalLog(f);
        uint256 x = g.mul(_tq) / _rp;
        uint256 y = _lowerStake ? lowerStake(x) : higherStake(x);
        return normalizedWeights(y.mul(_tq), _rp.mul(FIXED_1));
    }

    /**
      * @dev reduces "a" and "b" while maintaining their ratio.
    */
    function safeFactors(uint256 _a, uint256 _b) internal pure returns (uint256, uint256) {
        if (_a <= FIXED_2 && _b <= FIXED_2)
            return (_a, _b);
        if (_a < FIXED_2)
            return (_a * FIXED_2 / _b, FIXED_2);
        if (_b < FIXED_2)
            return (FIXED_2, _b * FIXED_2 / _a);
        uint256 c = _a > _b ? _a : _b;
        uint256 n = floorLog2(c / FIXED_1);
        return (_a >> n, _b >> n);
    }

    /**
      * @dev computes "MAX_WEIGHT * a / (a + b)" and "MAX_WEIGHT * b / (a + b)".
    */
    function normalizedWeights(uint256 _a, uint256 _b) internal pure returns (uint32, uint32) {
        if (_a <= _b)
            return accurateWeights(_a, _b);
        (uint32 y, uint32 x) = accurateWeights(_b, _a);
        return (x, y);
    }

    /**
      * @dev computes "MAX_WEIGHT * a / (a + b)" and "MAX_WEIGHT * b / (a + b)", assuming that "a <= b".
    */
    function accurateWeights(uint256 _a, uint256 _b) internal pure returns (uint32, uint32) {
        if (_a > MAX_UNF_WEIGHT) {
            uint256 c = _a / (MAX_UNF_WEIGHT + 1) + 1;
            _a /= c;
            _b /= c;
        }
        uint256 x = roundDiv(_a * MAX_WEIGHT, _a.add(_b));
        uint256 y = MAX_WEIGHT - x;
        return (uint32(x), uint32(y));
    }

    /**
      * @dev computes the nearest integer to a given quotient without overflowing or underflowing.
    */
    function roundDiv(uint256 _n, uint256 _d) internal pure returns (uint256) {
        return _n / _d + _n % _d / (_d - _d / 2);
    }

    /**
      * @dev deprecated, backward compatibility
    */
    function calculatePurchaseReturn(uint256 _supply,
                                     uint256 _reserveBalance,
                                     uint32 _reserveWeight,
                                     uint256 _amount)
                                     public view returns (uint256)
    {
        return purchaseTargetAmount(_supply, _reserveBalance, _reserveWeight, _amount);
    }

    /**
      * @dev deprecated, backward compatibility
    */
    function calculateSaleReturn(uint256 _supply,
                                 uint256 _reserveBalance,
                                 uint32 _reserveWeight,
                                 uint256 _amount)
                                 public view returns (uint256)
    {
        return saleTargetAmount(_supply, _reserveBalance, _reserveWeight, _amount);
    }

    /**
      * @dev deprecated, backward compatibility
    */
    function calculateCrossReserveReturn(uint256 _sourceReserveBalance,
                                         uint32 _sourceReserveWeight,
                                         uint256 _targetReserveBalance,
                                         uint32 _targetReserveWeight,
                                         uint256 _amount)
                                         public view returns (uint256)
    {
        return crossReserveTargetAmount(_sourceReserveBalance, _sourceReserveWeight, _targetReserveBalance, _targetReserveWeight, _amount);
    }

    /**
      * @dev deprecated, backward compatibility
    */
    function calculateCrossConnectorReturn(uint256 _sourceReserveBalance,
                                           uint32 _sourceReserveWeight,
                                           uint256 _targetReserveBalance,
                                           uint32 _targetReserveWeight,
                                           uint256 _amount)
                                           public view returns (uint256)
    {
        return crossReserveTargetAmount(_sourceReserveBalance, _sourceReserveWeight, _targetReserveBalance, _targetReserveWeight, _amount);
    }

    /**
      * @dev deprecated, backward compatibility
    */
    function calculateFundCost(uint256 _supply,
                               uint256 _reserveBalance,
                               uint32 _reserveRatio,
                               uint256 _amount)
                               public view returns (uint256)
    {
        return fundCost(_supply, _reserveBalance, _reserveRatio, _amount);
    }

    /**
      * @dev deprecated, backward compatibility
    */
    function calculateLiquidateReturn(uint256 _supply,
                                      uint256 _reserveBalance,
                                      uint32 _reserveRatio,
                                      uint256 _amount)
                                      public view returns (uint256)
    {
        return liquidateReserveAmount(_supply, _reserveBalance, _reserveRatio, _amount);
    }

    /**
      * @dev deprecated, backward compatibility
    */
    function purchaseRate(uint256 _supply,
                          uint256 _reserveBalance,
                          uint32 _reserveWeight,
                          uint256 _amount)
                          public view returns (uint256)
    {
        return purchaseTargetAmount(_supply, _reserveBalance, _reserveWeight, _amount);
    }

    /**
      * @dev deprecated, backward compatibility
    */
    function saleRate(uint256 _supply,
                      uint256 _reserveBalance,
                      uint32 _reserveWeight,
                      uint256 _amount)
                      public view returns (uint256)
    {
        return saleTargetAmount(_supply, _reserveBalance, _reserveWeight, _amount);
    }

    /**
      * @dev deprecated, backward compatibility
    */
    function crossReserveRate(uint256 _sourceReserveBalance,
                              uint32 _sourceReserveWeight,
                              uint256 _targetReserveBalance,
                              uint32 _targetReserveWeight,
                              uint256 _amount)
                              public view returns (uint256)
    {
        return crossReserveTargetAmount(_sourceReserveBalance, _sourceReserveWeight, _targetReserveBalance, _targetReserveWeight, _amount);
    }

    /**
      * @dev deprecated, backward compatibility
    */
    function liquidateRate(uint256 _supply,
                           uint256 _reserveBalance,
                           uint32 _reserveRatio,
                           uint256 _amount)
                           public view returns (uint256)
    {
        return liquidateReserveAmount(_supply, _reserveBalance, _reserveRatio, _amount);
    }
}