library XNum {
    uint256 public constant BONE = 10**18;
    uint256 public constant MIN_BPOW_BASE = 1 wei;
    uint256 public constant MAX_BPOW_BASE = (2 * BONE) - 1 wei;
    uint256 public constant BPOW_PRECISION = BONE / 10**10;

    function btoi(uint256 a) internal pure returns (uint256) {
        return a / BONE;
    }

    function bfloor(uint256 a) internal pure returns (uint256) {
        return btoi(a) * BONE;
    }

    function badd(uint256 a, uint256 b) internal pure returns (uint256) {
        uint256 c = a + b;
        require(c >= a, "ERR_ADD_OVERFLOW");
        return c;
    }

    function bsub(uint256 a, uint256 b) internal pure returns (uint256) {
        (uint256 c, bool flag) = bsubSign(a, b);
        require(!flag, "ERR_SUB_UNDERFLOW");
        return c;
    }

    function bsubSign(uint256 a, uint256 b)
        internal
        pure
        returns (uint256, bool)
    {
        if (a >= b) {
            return (a - b, false);
        } else {
            return (b - a, true);
        }
    }

    function bmul(uint256 a, uint256 b) internal pure returns (uint256) {
        uint256 c0 = a * b;
        require(a == 0 || c0 / a == b, "ERR_MUL_OVERFLOW");
        uint256 c1 = c0 + (BONE / 2);
        require(c1 >= c0, "ERR_MUL_OVERFLOW");
        uint256 c2 = c1 / BONE;
        return c2;
    }

    function bdiv(uint256 a, uint256 b) internal pure returns (uint256) {
        require(b != 0, "ERR_DIV_ZERO");
        uint256 c0 = a * BONE;
        require(a == 0 || c0 / a == BONE, "ERR_DIV_INTERNAL"); // bmul overflow
        uint256 c1 = c0 + (b / 2);
        require(c1 >= c0, "ERR_DIV_INTERNAL"); //  badd require
        uint256 c2 = c1 / b;
        return c2;
    }

    // DSMath.wpow
    function bpowi(uint256 a, uint256 n) internal pure returns (uint256) {
        uint256 z = n % 2 != 0 ? a : BONE;

        for (n /= 2; n != 0; n /= 2) {
            a = bmul(a, a);

            if (n % 2 != 0) {
                z = bmul(z, a);
            }
        }
        return z;
    }

    // Compute b^(e.w) by splitting it into (b^e)*(b^0.w).
    // Use `bpowi` for `b^e` and `bpowK` for k iterations
    // of approximation of b^0.w
    function bpow(uint256 base, uint256 exp) internal pure returns (uint256) {
        require(base >= MIN_BPOW_BASE, "ERR_BPOW_BASE_TOO_LOW");
        require(base <= MAX_BPOW_BASE, "ERR_BPOW_BASE_TOO_HIGH");

        uint256 whole = bfloor(exp);
        uint256 remain = bsub(exp, whole);

        uint256 wholePow = bpowi(base, btoi(whole));

        if (remain == 0) {
            return wholePow;
        }

        uint256 partialResult = bpowApprox(base, remain, BPOW_PRECISION);
        return bmul(wholePow, partialResult);
    }

    function bpowApprox(
        uint256 base,
        uint256 exp,
        uint256 precision
    ) internal pure returns (uint256) {
        // term 0:
        uint256 a = exp;
        (uint256 x, bool xneg) = bsubSign(base, BONE);
        uint256 term = BONE;
        uint256 sum = term;
        bool negative = false;

        // term(k) = numer / denom
        //         = (product(a - i + 1, i=1-->k) * x^k) / (k!)
        // each iteration, multiply previous term by (a-(k-1)) * x / k
        // continue until term is less than precision
        for (uint256 i = 1; term >= precision; i++) {
            uint256 bigK = i * BONE;
            (uint256 c, bool cneg) = bsubSign(a, bsub(bigK, BONE));
            term = bmul(term, bmul(c, x));
            term = bdiv(term, bigK);
            if (term == 0) break;

            if (xneg) negative = !negative;
            if (cneg) negative = !negative;
            if (negative) {
                sum = bsub(sum, term);
            } else {
                sum = badd(sum, term);
            }
        }

        return sum;
    }
}
function bpowi(uint256 a, uint256 n) internal pure returns (uint256) {
    uint256 z = n % 2 != 0 ? a : BONE;

    for (n /= 2; n != 0; n /= 2) {
        a = bmul(a, a);

        if (n % 2 != 0) {
            z = bmul(z, a);
        }
    }
    return z;
}

// DSMath.wpow