contract LibMath is
    SafeMath
{
    /// @dev Calculates partial value given a numerator and denominator rounded down.
    ///      Reverts if rounding error is >= 0.1%
    /// @param numerator Numerator.
    /// @param denominator Denominator.
    /// @param target Value to calculate partial of.
    /// @return Partial value of target rounded down.
    function safeGetPartialAmountFloor(
        uint256 numerator,
        uint256 denominator,
        uint256 target
    )
        internal
        pure
        returns (uint256 partialAmount)
    {
        require(
            denominator > 0,
            "DIVISION_BY_ZERO"
        );

        require(
            !isRoundingErrorFloor(
                numerator,
                denominator,
                target
            ),
            "ROUNDING_ERROR"
        );
        
        partialAmount = safeDiv(
            safeMul(numerator, target),
            denominator
        );
        return partialAmount;
    }

    /// @dev Calculates partial value given a numerator and denominator rounded down.
    ///      Reverts if rounding error is >= 0.1%
    /// @param numerator Numerator.
    /// @param denominator Denominator.
    /// @param target Value to calculate partial of.
    /// @return Partial value of target rounded up.
    function safeGetPartialAmountCeil(
        uint256 numerator,
        uint256 denominator,
        uint256 target
    )
        internal
        pure
        returns (uint256 partialAmount)
    {
        require(
            denominator > 0,
            "DIVISION_BY_ZERO"
        );

        require(
            !isRoundingErrorCeil(
                numerator,
                denominator,
                target
            ),
            "ROUNDING_ERROR"
        );
        
        // safeDiv computes `floor(a / b)`. We use the identity (a, b integer):
        //       ceil(a / b) = floor((a + b - 1) / b)
        // To implement `ceil(a / b)` using safeDiv.
        partialAmount = safeDiv(
            safeAdd(
                safeMul(numerator, target),
                safeSub(denominator, 1)
            ),
            denominator
        );
        return partialAmount;
    }

    /// @dev Calculates partial value given a numerator and denominator rounded down.
    /// @param numerator Numerator.
    /// @param denominator Denominator.
    /// @param target Value to calculate partial of.
    /// @return Partial value of target rounded down.
    function getPartialAmountFloor(
        uint256 numerator,
        uint256 denominator,
        uint256 target
    )
        internal
        pure
        returns (uint256 partialAmount)
    {
        require(
            denominator > 0,
            "DIVISION_BY_ZERO"
        );

        partialAmount = safeDiv(
            safeMul(numerator, target),
            denominator
        );
        return partialAmount;
    }
    
    /// @dev Calculates partial value given a numerator and denominator rounded down.
    /// @param numerator Numerator.
    /// @param denominator Denominator.
    /// @param target Value to calculate partial of.
    /// @return Partial value of target rounded up.
    function getPartialAmountCeil(
        uint256 numerator,
        uint256 denominator,
        uint256 target
    )
        internal
        pure
        returns (uint256 partialAmount)
    {
        require(
            denominator > 0,
            "DIVISION_BY_ZERO"
        );

        // safeDiv computes `floor(a / b)`. We use the identity (a, b integer):
        //       ceil(a / b) = floor((a + b - 1) / b)
        // To implement `ceil(a / b)` using safeDiv.
        partialAmount = safeDiv(
            safeAdd(
                safeMul(numerator, target),
                safeSub(denominator, 1)
            ),
            denominator
        );
        return partialAmount;
    }
    
    /// @dev Checks if rounding error >= 0.1% when rounding down.
    /// @param numerator Numerator.
    /// @param denominator Denominator.
    /// @param target Value to multiply with numerator/denominator.
    /// @return Rounding error is present.
    function isRoundingErrorFloor(
        uint256 numerator,
        uint256 denominator,
        uint256 target
    )
        internal
        pure
        returns (bool isError)
    {
        require(
            denominator > 0,
            "DIVISION_BY_ZERO"
        );
        
        // The absolute rounding error is the difference between the rounded
        // value and the ideal value. The relative rounding error is the
        // absolute rounding error divided by the absolute value of the
        // ideal value. This is undefined when the ideal value is zero.
        //
        // The ideal value is `numerator * target / denominator`.
        // Let's call `numerator * target % denominator` the remainder.
        // The absolute error is `remainder / denominator`.
        //
        // When the ideal value is zero, we require the absolute error to
        // be zero. Fortunately, this is always the case. The ideal value is
        // zero iff `numerator == 0` and/or `target == 0`. In this case the
        // remainder and absolute error are also zero. 
        if (target == 0 || numerator == 0) {
            return false;
        }
        
        // Otherwise, we want the relative rounding error to be strictly
        // less than 0.1%.
        // The relative error is `remainder / (numerator * target)`.
        // We want the relative error less than 1 / 1000:
        //        remainder / (numerator * denominator)  <  1 / 1000
        // or equivalently:
        //        1000 * remainder  <  numerator * target
        // so we have a rounding error iff:
        //        1000 * remainder  >=  numerator * target
        uint256 remainder = mulmod(
            target,
            numerator,
            denominator
        );
        isError = safeMul(1000, remainder) >= safeMul(numerator, target);
        return isError;
    }
    
    /// @dev Checks if rounding error >= 0.1% when rounding up.
    /// @param numerator Numerator.
    /// @param denominator Denominator.
    /// @param target Value to multiply with numerator/denominator.
    /// @return Rounding error is present.
    function isRoundingErrorCeil(
        uint256 numerator,
        uint256 denominator,
        uint256 target
    )
        internal
        pure
        returns (bool isError)
    {
        require(
            denominator > 0,
            "DIVISION_BY_ZERO"
        );
        
        // See the comments in `isRoundingError`.
        if (target == 0 || numerator == 0) {
            // When either is zero, the ideal value and rounded value are zero
            // and there is no rounding error. (Although the relative error
            // is undefined.)
            return false;
        }
        // Compute remainder as before
        uint256 remainder = mulmod(
            target,
            numerator,
            denominator
        );
        remainder = safeSub(denominator, remainder) % denominator;
        isError = safeMul(1000, remainder) >= safeMul(numerator, target);
        return isError;
    }
}
function isRoundingErrorFloor(
        uint256 numerator,
        uint256 denominator,
        uint256 target
    )
        internal
        pure
        returns (bool isError)
    {
        require(
            denominator > 0,
            "DIVISION_BY_ZERO"
        );
        
        // The absolute rounding error is the difference between the rounded
        // value and the ideal value. The relative rounding error is the
        // absolute rounding error divided by the absolute value of the
        // ideal value. This is undefined when the ideal value is zero.
        //
        // The ideal value is `numerator * target / denominator`.
        // Let's call `numerator * target % denominator` the remainder.
        // The absolute error is `remainder / denominator`.
        //
        // When the ideal value is zero, we require the absolute error to
        // be zero. Fortunately, this is always the case. The ideal value is
        // zero iff `numerator == 0` and/or `target == 0`. In this case the
        // remainder and absolute error are also zero. 
        if (target == 0 || numerator == 0) {
            return false;
        }
        
        // Otherwise, we want the relative rounding error to be strictly
        // less than 0.1%.
        // The relative error is `remainder / (numerator * target)`.
        // We want the relative error less than 1 / 1000:
        //        remainder / (numerator * denominator)  <  1 / 1000
        // or equivalently:
        //        1000 * remainder  <  numerator * target
        // so we have a rounding error iff:
        //        1000 * remainder  >=  numerator * target
        uint256 remainder = mulmod(
            target,
            numerator,
            denominator
        );
        isError = safeMul(1000, remainder) >= safeMul(numerator, target);
        return isError;
    }
/// @dev Checks if rounding error >= 0.1% when rounding down.
/// @param numerator Numerator.
/// @param denominator Denominator.
/// @param target Value to multiply with numerator/denominator.
/// @return Rounding error is present.