AnalyticMath.sol

// SPDX-License-Identifier: SEE LICENSE IN LICENSE
pragma solidity 0.8.26;

import "./IntegralMath.sol";

contract AnalyticMath {
    uint8 internal constant MIN_PRECISION = 32;
    uint8 internal constant MAX_PRECISION = 127;

    uint256 internal constant FIXED_1 = 1 << MAX_PRECISION;
    uint256 internal constant FIXED_2 = 2 << MAX_PRECISION;

    // Auto-generated via 'PrintLn2ScalingFactors.py'
    uint256 internal constant LN2_NUMERATOR   = 0x3f80fe03f80fe03f80fe03f80fe03f8;
    uint256 internal constant LN2_DENOMINATOR = 0x5b9de1d10bf4103d647b0955897ba80;

    // Auto-generated via 'PrintOptimalThresholds.py'
    uint256 internal constant OPT_LOG_MAX_VAL = 0x15bf0a8b1457695355fb8ac404e7a79e4;
    uint256 internal constant OPT_EXP_MAX_VAL = 0x800000000000000000000000000000000;

    uint256[MAX_PRECISION + 1] private maxExpArray;

    /**
      * @dev Should be executed either during construction or after construction (if too large for the constructor)
    */
    function init() public virtual {
        initMaxExpArray();
    }

    /**
      * @dev Compute (a / b) ^ (c / d)
    */
    function pow(uint256 a, uint256 b, uint256 c, uint256 d) internal view returns (uint256, uint256) { unchecked {
        if (a >= b)
            return mulDivExp(mulDivLog(FIXED_1, a, b), c, d);
        (uint256 q, uint256 p) = mulDivExp(mulDivLog(FIXED_1, b, a), c, d);
        return (p, q);
    }}

    /**
      * @dev Compute log(a / b)
    */
    function log(uint256 a, uint256 b) internal pure returns (uint256, uint256) { unchecked {
        require(a >= b, "log: a < b");
        return (mulDivLog(FIXED_1, a, b), FIXED_1);
    }}

    /**
      * @dev Compute e ^ (a / b)
    */
    function exp(uint256 a, uint256 b) internal view returns (uint256, uint256) { unchecked {
        return mulDivExp(FIXED_1, a, b);
    }}

    /**
      * @dev Compute log(x / FIXED_1) * FIXED_1
    */
    function fixedLog(uint256 x) internal pure returns (uint256) { unchecked {
        if (x < OPT_LOG_MAX_VAL) {
            return optimalLog(x);
        }
        else {
            return generalLog(x);
        }
    }}

    /**
      * @dev Compute e ^ (x / FIXED_1) * FIXED_1
    */
    function fixedExp(uint256 x) internal view returns (uint256, uint256) { unchecked {
        if (x < OPT_EXP_MAX_VAL) {
            return (optimalExp(x), 1 << MAX_PRECISION);
        }
        else {
            uint8 precision = findPosition(x);
            return (generalExp(x >> (MAX_PRECISION - precision), precision), 1 << precision);
        }
    }}

    /**
      * @dev Compute 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) { unchecked {
        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 = IntegralMath.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 += 1 << (i - 1);
                }
            }
        }

        return res * LN2_NUMERATOR / LN2_DENOMINATOR;
    }}

    /**
      * @dev Approximate e ^ x as (x ^ 0) / 0! + (x ^ 1) / 1! + ... + (x ^ n) / n!
      * Auto-generated via 'PrintFunctionGeneralExp.py'
      * Detailed description:
      * - This function 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) { unchecked {
        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 + (1 << precision); // divide by 33! and then add x^1 / 1! + x^0 / 0!
    }}

    /**
      * @dev Compute 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) { unchecked {
        uint256 res = 0;

        uint256 y;
        uint256 z;
        uint256 w;

        if (x >= 0xd3094c70f034de4b96ff7d5b6f99fcd9) {res += 0x40000000000000000000000000000000; x = x * FIXED_1 / 0xd3094c70f034de4b96ff7d5b6f99fcd9;} // add 1 / 2^1
        if (x >= 0xa45af1e1f40c333b3de1db4dd55f29a8) {res += 0x20000000000000000000000000000000; x = x * FIXED_1 / 0xa45af1e1f40c333b3de1db4dd55f29a8;} // add 1 / 2^2
        if (x >= 0x910b022db7ae67ce76b441c27035c6a2) {res += 0x10000000000000000000000000000000; x = x * FIXED_1 / 0x910b022db7ae67ce76b441c27035c6a2;} // add 1 / 2^3
        if (x >= 0x88415abbe9a76bead8d00cf112e4d4a9) {res += 0x08000000000000000000000000000000; x = x * FIXED_1 / 0x88415abbe9a76bead8d00cf112e4d4a9;} // add 1 / 2^4
        if (x >= 0x84102b00893f64c705e841d5d4064bd4) {res += 0x04000000000000000000000000000000; x = x * FIXED_1 / 0x84102b00893f64c705e841d5d4064bd4;} // add 1 / 2^5
        if (x >= 0x8204055aaef1c8bd5c3259f4822735a3) {res += 0x02000000000000000000000000000000; x = x * FIXED_1 / 0x8204055aaef1c8bd5c3259f4822735a3;} // add 1 / 2^6
        if (x >= 0x810100ab00222d861931c15e39b44e9a) {res += 0x01000000000000000000000000000000; x = x * FIXED_1 / 0x810100ab00222d861931c15e39b44e9a;} // add 1 / 2^7
        if (x >= 0x808040155aabbbe9451521693554f734) {res += 0x00800000000000000000000000000000; x = x * FIXED_1 / 0x808040155aabbbe9451521693554f734;} // 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 Compute 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) { unchecked {
        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 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"]
      * This function supports the rational approximation of "(a / b) ^ (c / d)" via "e ^ (log(a / b) * c / d)".
      * The value of "log(a / b)" is represented with an integer slightly smaller than "log(a / b) * 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 (a 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 the result with maximum accuracy and without exceeding 256 bits in any of the intermediate computations.
    */
    function findPosition(uint256 x) internal view returns (uint8) { unchecked {
        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;

        revert("findPosition: x > max");
    }}

    /**
      * @dev Initialize internal data structure
      * Auto-generated via 'PrintMaxExpArray.py'
    */
    function initMaxExpArray() internal {
    //  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;
    }

    // auxiliary function
    function mulDivLog(uint256 x, uint256 y, uint256 z) private pure returns (uint256) {
        return fixedLog(IntegralMath.mulDivF(x, y, z));
    }

    // auxiliary function
    function mulDivExp(uint256 x, uint256 y, uint256 z) private view returns (uint256, uint256) {
        return fixedExp(IntegralMath.mulDivF(x, y, z));
    }
}