// SPDX-License-Identifier: MIT
pragma solidity ^0.8.6;

// taken from https://medium.com/coinmonks/math-in-solidity-part-3-percents-and-proportions-4db014e080b1
// license is CC-BY-4.0
library FullMath {
    function fullMul(uint256 x, uint256 y) internal pure returns (uint256 l, uint256 h) {
        uint256 mm = mulmod(x, y, uint256(int256(-1)));
        l = x * y;
        h = mm - l;
        if (mm < l) h -= 1;
    }

    function fullDiv(
        uint256 l,
        uint256 h,
        uint256 d
    ) private pure returns (uint256) {
        uint256 pow2 = uint256(int256(d) & -int256(d));
        d /= pow2;
        l /= pow2;
        l += h * ((uint256(-int256(pow2))) / pow2 + 1);
        uint256 r = 1;
        r *= 2 - d * r;
        r *= 2 - d * r;
        r *= 2 - d * r;
        r *= 2 - d * r;
        r *= 2 - d * r;
        r *= 2 - d * r;
        r *= 2 - d * r;
        r *= 2 - d * r;
        return l * r;
    }

    function mulDiv(
        uint256 x,
        uint256 y,
        uint256 d
    ) internal pure returns (uint256) {
        (uint256 l, uint256 h) = fullMul(x, y);

        uint256 mm = mulmod(x, y, d);
        if (mm > l) h -= 1;
        l -= mm;

        if (h == 0) return l / d;

        require(h < d, 'FullMath: FULLDIV_OVERFLOW');
        return fullDiv(l, h, d);
    }
}