library ABDKMathQuad {
    /*
     * 0.
     */
    bytes16 private constant POSITIVE_ZERO = 0x00000000000000000000000000000000;

    /*
     * -0.
     */
    bytes16 private constant NEGATIVE_ZERO = 0x80000000000000000000000000000000;

    /*
     * +Infinity.
     */
    bytes16 private constant POSITIVE_INFINITY = 0x7FFF0000000000000000000000000000;

    /*
     * -Infinity.
     */
    bytes16 private constant NEGATIVE_INFINITY = 0xFFFF0000000000000000000000000000;

    /*
     * Canonical NaN value.
     */
    bytes16 private constant NaN = 0x7FFF8000000000000000000000000000;

    /**
     * Convert signed 256-bit integer number into quadruple precision number.
     *
     * @param x signed 256-bit integer number
     * @return quadruple precision number
     */
    function fromInt (int256 x) internal pure returns (bytes16) {
        if (x == 0) return bytes16 (0);
        else {
            // We rely on overflow behavior here
            uint256 result = uint256 (x > 0 ? x : -x);

            uint256 msb = msb (result);
            if (msb < 112) result <<= 112 - msb;
            else if (msb > 112) result >>= msb - 112;

            result = result & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF | 16383 + msb << 112;
            if (x < 0) result |= 0x80000000000000000000000000000000;

            return bytes16 (uint128 (result));
        }
    }

    /**
     * Convert quadruple precision number into signed 256-bit integer number
     * rounding towards zero.  Revert on overflow.
     *
     * @param x quadruple precision number
     * @return signed 256-bit integer number
     */
    function toInt (bytes16 x) internal pure returns (int256) {
        uint256 exponent = uint128 (x) >> 112 & 0x7FFF;

        require (exponent <= 16638); // Overflow
        if (exponent < 16383) return 0; // Underflow

        uint256 result = uint256 (uint128 (x)) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF |
        0x10000000000000000000000000000;

        if (exponent < 16495) result >>= 16495 - exponent;
        else if (exponent > 16495) result <<= exponent - 16495;

        if (uint128 (x) >= 0x80000000000000000000000000000000) { // Negative
            require (result <= 0x8000000000000000000000000000000000000000000000000000000000000000);
            return -int256 (result); // We rely on overflow behavior here
        } else {
            require (result <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
            return int256 (result);
        }
    }

    /**
     * Convert unsigned 256-bit integer number into quadruple precision number.
     *
     * @param x unsigned 256-bit integer number
     * @return quadruple precision number
     */
    function fromUInt (uint256 x) internal pure returns (bytes16) {
        if (x == 0) return bytes16 (0);
        else {
            uint256 result = x;

            uint256 msb = msb (result);
            if (msb < 112) result <<= 112 - msb;
            else if (msb > 112) result >>= msb - 112;

            result = result & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF | 16383 + msb << 112;

            return bytes16 (uint128 (result));
        }
    }

    /**
     * Convert quadruple precision number into unsigned 256-bit integer number
     * rounding towards zero.  Revert on underflow.  Note, that negative floating
     * point numbers in range (-1.0 .. 0.0) may be converted to unsigned integer
     * without error, because they are rounded to zero.
     *
     * @param x quadruple precision number
     * @return unsigned 256-bit integer number
     */
    function toUInt (bytes16 x) internal pure returns (uint256) {
        uint256 exponent = uint128 (x) >> 112 & 0x7FFF;

        if (exponent < 16383) return 0; // Underflow

        require (uint128 (x) < 0x80000000000000000000000000000000); // Negative

        require (exponent <= 16638); // Overflow
        uint256 result = uint256 (uint128 (x)) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF |
        0x10000000000000000000000000000;

        if (exponent < 16495) result >>= 16495 - exponent;
        else if (exponent > 16495) result <<= exponent - 16495;

        return result;
    }

    /**
     * Convert signed 128.128 bit fixed point number into quadruple precision
     * number.
     *
     * @param x signed 128.128 bit fixed point number
     * @return quadruple precision number
     */
    function from128x128 (int256 x) internal pure returns (bytes16) {
        if (x == 0) return bytes16 (0);
        else {
            // We rely on overflow behavior here
            uint256 result = uint256 (x > 0 ? x : -x);

            uint256 msb = msb (result);
            if (msb < 112) result <<= 112 - msb;
            else if (msb > 112) result >>= msb - 112;

            result = result & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF | 16255 + msb << 112;
            if (x < 0) result |= 0x80000000000000000000000000000000;

            return bytes16 (uint128 (result));
        }
    }

    /**
     * Convert quadruple precision number into signed 128.128 bit fixed point
     * number.  Revert on overflow.
     *
     * @param x quadruple precision number
     * @return signed 128.128 bit fixed point number
     */
    function to128x128 (bytes16 x) internal pure returns (int256) {
        uint256 exponent = uint128 (x) >> 112 & 0x7FFF;

        require (exponent <= 16510); // Overflow
        if (exponent < 16255) return 0; // Underflow

        uint256 result = uint256 (uint128 (x)) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF |
        0x10000000000000000000000000000;

        if (exponent < 16367) result >>= 16367 - exponent;
        else if (exponent > 16367) result <<= exponent - 16367;

        if (uint128 (x) >= 0x80000000000000000000000000000000) { // Negative
            require (result <= 0x8000000000000000000000000000000000000000000000000000000000000000);
            return -int256 (result); // We rely on overflow behavior here
        } else {
            require (result <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
            return int256 (result);
        }
    }

    /**
     * Convert signed 64.64 bit fixed point number into quadruple precision
     * number.
     *
     * @param x signed 64.64 bit fixed point number
     * @return quadruple precision number
     */
    function from64x64 (int128 x) internal pure returns (bytes16) {
        if (x == 0) return bytes16 (0);
        else {
            // We rely on overflow behavior here
            uint256 result = uint128 (x > 0 ? x : -x);

            uint256 msb = msb (result);
            if (msb < 112) result <<= 112 - msb;
            else if (msb > 112) result >>= msb - 112;

            result = result & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF | 16319 + msb << 112;
            if (x < 0) result |= 0x80000000000000000000000000000000;

            return bytes16 (uint128 (result));
        }
    }

    /**
     * Convert quadruple precision number into signed 64.64 bit fixed point
     * number.  Revert on overflow.
     *
     * @param x quadruple precision number
     * @return signed 64.64 bit fixed point number
     */
    function to64x64 (bytes16 x) internal pure returns (int128) {
        uint256 exponent = uint128 (x) >> 112 & 0x7FFF;

        require (exponent <= 16446); // Overflow
        if (exponent < 16319) return 0; // Underflow

        uint256 result = uint256 (uint128 (x)) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF |
        0x10000000000000000000000000000;

        if (exponent < 16431) result >>= 16431 - exponent;
        else if (exponent > 16431) result <<= exponent - 16431;

        if (uint128 (x) >= 0x80000000000000000000000000000000) { // Negative
            require (result <= 0x80000000000000000000000000000000);
            return -int128 (result); // We rely on overflow behavior here
        } else {
            require (result <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
            return int128 (result);
        }
    }

    /**
     * Convert octuple precision number into quadruple precision number.
     *
     * @param x octuple precision number
     * @return quadruple precision number
     */
    function fromOctuple (bytes32 x) internal pure returns (bytes16) {
        bool negative = x & 0x8000000000000000000000000000000000000000000000000000000000000000 > 0;

        uint256 exponent = uint256 (x) >> 236 & 0x7FFFF;
        uint256 significand = uint256 (x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

        if (exponent == 0x7FFFF) {
            if (significand > 0) return NaN;
            else return negative ? NEGATIVE_INFINITY : POSITIVE_INFINITY;
        }

        if (exponent > 278526)
            return negative ? NEGATIVE_INFINITY : POSITIVE_INFINITY;
        else if (exponent < 245649)
            return negative ? NEGATIVE_ZERO : POSITIVE_ZERO;
        else if (exponent < 245761) {
            significand = (significand | 0x100000000000000000000000000000000000000000000000000000000000) >> 245885 - exponent;
            exponent = 0;
        } else {
            significand >>= 124;
            exponent -= 245760;
        }

        uint128 result = uint128 (significand | exponent << 112);
        if (negative) result |= 0x80000000000000000000000000000000;

        return bytes16 (result);
    }

    /**
     * Convert quadruple precision number into octuple precision number.
     *
     * @param x quadruple precision number
     * @return octuple precision number
     */
    function toOctuple (bytes16 x) internal pure returns (bytes32) {
        uint256 exponent = uint128 (x) >> 112 & 0x7FFF;

        uint256 result = uint128 (x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

        if (exponent == 0x7FFF) exponent = 0x7FFFF; // Infinity or NaN
        else if (exponent == 0) {
            if (result > 0) {
                uint256 msb = msb (result);
                result = result << 236 - msb & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                exponent = 245649 + msb;
            }
        } else {
            result <<= 124;
            exponent += 245760;
        }

        result |= exponent << 236;
        if (uint128 (x) >= 0x80000000000000000000000000000000)
            result |= 0x8000000000000000000000000000000000000000000000000000000000000000;

        return bytes32 (result);
    }

    /**
     * Convert double precision number into quadruple precision number.
     *
     * @param x double precision number
     * @return quadruple precision number
     */
    function fromDouble (bytes8 x) internal pure returns (bytes16) {
        uint256 exponent = uint64 (x) >> 52 & 0x7FF;

        uint256 result = uint64 (x) & 0xFFFFFFFFFFFFF;

        if (exponent == 0x7FF) exponent = 0x7FFF; // Infinity or NaN
        else if (exponent == 0) {
            if (result > 0) {
                uint256 msb = msb (result);
                result = result << 112 - msb & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                exponent = 15309 + msb;
            }
        } else {
            result <<= 60;
            exponent += 15360;
        }

        result |= exponent << 112;
        if (x & 0x8000000000000000 > 0)
            result |= 0x80000000000000000000000000000000;

        return bytes16 (uint128 (result));
    }

    /**
     * Convert quadruple precision number into double precision number.
     *
     * @param x quadruple precision number
     * @return double precision number
     */
    function toDouble (bytes16 x) internal pure returns (bytes8) {
        bool negative = uint128 (x) >= 0x80000000000000000000000000000000;

        uint256 exponent = uint128 (x) >> 112 & 0x7FFF;
        uint256 significand = uint128 (x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

        if (exponent == 0x7FFF) {
            if (significand > 0) return 0x7FF8000000000000; // NaN
            else return negative ?
            bytes8 (0xFFF0000000000000) : // -Infinity
            bytes8 (0x7FF0000000000000); // Infinity
        }

        if (exponent > 17406)
            return negative ?
            bytes8 (0xFFF0000000000000) : // -Infinity
            bytes8 (0x7FF0000000000000); // Infinity
        else if (exponent < 15309)
            return negative ?
            bytes8 (0x8000000000000000) : // -0
            bytes8 (0x0000000000000000); // 0
        else if (exponent < 15361) {
            significand = (significand | 0x10000000000000000000000000000) >> 15421 - exponent;
            exponent = 0;
        } else {
            significand >>= 60;
            exponent -= 15360;
        }

        uint64 result = uint64 (significand | exponent << 52);
        if (negative) result |= 0x8000000000000000;

        return bytes8 (result);
    }

    /**
     * Test whether given quadruple precision number is NaN.
     *
     * @param x quadruple precision number
     * @return true if x is NaN, false otherwise
     */
    function isNaN (bytes16 x) internal pure returns (bool) {
        return uint128 (x) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF >
        0x7FFF0000000000000000000000000000;
    }

    /**
     * Test whether given quadruple precision number is positive or negative
     * infinity.
     *
     * @param x quadruple precision number
     * @return true if x is positive or negative infinity, false otherwise
     */
    function isInfinity (bytes16 x) internal pure returns (bool) {
        return uint128 (x) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF ==
        0x7FFF0000000000000000000000000000;
    }

    /**
     * Calculate sign of x, i.e. -1 if x is negative, 0 if x if zero, and 1 if x
     * is positive.  Note that sign (-0) is zero.  Revert if x is NaN.
     *
     * @param x quadruple precision number
     * @return sign of x
     */
    function sign (bytes16 x) internal pure returns (int8) {
        uint128 absoluteX = uint128 (x) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

        require (absoluteX <= 0x7FFF0000000000000000000000000000); // Not NaN

        if (absoluteX == 0) return 0;
        else if (uint128 (x) >= 0x80000000000000000000000000000000) return -1;
        else return 1;
    }

    /**
     * Calculate sign (x - y).  Revert if either argument is NaN, or both
     * arguments are infinities of the same sign.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return sign (x - y)
     */
    function cmp (bytes16 x, bytes16 y) internal pure returns (int8) {
        uint128 absoluteX = uint128 (x) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

        require (absoluteX <= 0x7FFF0000000000000000000000000000); // Not NaN

        uint128 absoluteY = uint128 (y) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

        require (absoluteY <= 0x7FFF0000000000000000000000000000); // Not NaN

        // Not infinities of the same sign
        require (x != y || absoluteX < 0x7FFF0000000000000000000000000000);

        if (x == y) return 0;
        else {
            bool negativeX = uint128 (x) >= 0x80000000000000000000000000000000;
            bool negativeY = uint128 (y) >= 0x80000000000000000000000000000000;

            if (negativeX) {
                if (negativeY) return absoluteX > absoluteY ? -1 : int8 (1);
                else return -1;
            } else {
                if (negativeY) return 1;
                else return absoluteX > absoluteY ? int8 (1) : -1;
            }
        }
    }

    /**
     * Test whether x equals y.  NaN, infinity, and -infinity are not equal to
     * anything.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return true if x equals to y, false otherwise
     */
    function eq (bytes16 x, bytes16 y) internal pure returns (bool) {
        if (x == y) {
            return uint128 (x) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF <
            0x7FFF0000000000000000000000000000;
        } else return false;
    }

    /**
     * Calculate x + y.  Special values behave in the following way:
     *
     * NaN + x = NaN for any x.
     * Infinity + x = Infinity for any finite x.
     * -Infinity + x = -Infinity for any finite x.
     * Infinity + Infinity = Infinity.
     * -Infinity + -Infinity = -Infinity.
     * Infinity + -Infinity = -Infinity + Infinity = NaN.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return quadruple precision number
     */
    function add (bytes16 x, bytes16 y) internal pure returns (bytes16) {
        uint256 xExponent = uint128 (x) >> 112 & 0x7FFF;
        uint256 yExponent = uint128 (y) >> 112 & 0x7FFF;

        if (xExponent == 0x7FFF) {
            if (yExponent == 0x7FFF) {
                if (x == y) return x;
                else return NaN;
            } else return x;
        } else if (yExponent == 0x7FFF) return y;
        else {
            bool xSign = uint128 (x) >= 0x80000000000000000000000000000000;
            uint256 xSignifier = uint128 (x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
            if (xExponent == 0) xExponent = 1;
            else xSignifier |= 0x10000000000000000000000000000;

            bool ySign = uint128 (y) >= 0x80000000000000000000000000000000;
            uint256 ySignifier = uint128 (y) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
            if (yExponent == 0) yExponent = 1;
            else ySignifier |= 0x10000000000000000000000000000;

            if (xSignifier == 0) return y == NEGATIVE_ZERO ? POSITIVE_ZERO : y;
            else if (ySignifier == 0) return x == NEGATIVE_ZERO ? POSITIVE_ZERO : x;
            else {
                int256 delta = int256 (xExponent) - int256 (yExponent);

                if (xSign == ySign) {
                    if (delta > 112) return x;
                    else if (delta > 0) ySignifier >>= uint256 (delta);
                    else if (delta < -112) return y;
                    else if (delta < 0) {
                        xSignifier >>= uint256 (-delta);
                        xExponent = yExponent;
                    }

                    xSignifier += ySignifier;

                    if (xSignifier >= 0x20000000000000000000000000000) {
                        xSignifier >>= 1;
                        xExponent += 1;
                    }

                    if (xExponent == 0x7FFF)
                        return xSign ? NEGATIVE_INFINITY : POSITIVE_INFINITY;
                    else {
                        if (xSignifier < 0x10000000000000000000000000000) xExponent = 0;
                        else xSignifier &= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

                        return bytes16 (uint128 (
                                (xSign ? 0x80000000000000000000000000000000 : 0) |
                                (xExponent << 112) |
                                xSignifier));
                    }
                } else {
                    if (delta > 0) {
                        xSignifier <<= 1;
                        xExponent -= 1;
                    } else if (delta < 0) {
                        ySignifier <<= 1;
                        xExponent = yExponent - 1;
                    }

                    if (delta > 112) ySignifier = 1;
                    else if (delta > 1) ySignifier = (ySignifier - 1 >> uint256 (delta - 1)) + 1;
                    else if (delta < -112) xSignifier = 1;
                    else if (delta < -1) xSignifier = (xSignifier - 1 >> uint256 (-delta - 1)) + 1;

                    if (xSignifier >= ySignifier) xSignifier -= ySignifier;
                    else {
                        xSignifier = ySignifier - xSignifier;
                        xSign = ySign;
                    }

                    if (xSignifier == 0)
                        return POSITIVE_ZERO;

                    uint256 msb = msb (xSignifier);

                    if (msb == 113) {
                        xSignifier = xSignifier >> 1 & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                        xExponent += 1;
                    } else if (msb < 112) {
                        uint256 shift = 112 - msb;
                        if (xExponent > shift) {
                            xSignifier = xSignifier << shift & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                            xExponent -= shift;
                        } else {
                            xSignifier <<= xExponent - 1;
                            xExponent = 0;
                        }
                    } else xSignifier &= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

                    if (xExponent == 0x7FFF)
                        return xSign ? NEGATIVE_INFINITY : POSITIVE_INFINITY;
                    else return bytes16 (uint128 (
                            (xSign ? 0x80000000000000000000000000000000 : 0) |
                            (xExponent << 112) |
                            xSignifier));
                }
            }
        }
    }

    /**
     * Calculate x - y.  Special values behave in the following way:
     *
     * NaN - x = NaN for any x.
     * Infinity - x = Infinity for any finite x.
     * -Infinity - x = -Infinity for any finite x.
     * Infinity - -Infinity = Infinity.
     * -Infinity - Infinity = -Infinity.
     * Infinity - Infinity = -Infinity - -Infinity = NaN.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return quadruple precision number
     */
    function sub (bytes16 x, bytes16 y) internal pure returns (bytes16) {
        return add (x, y ^ 0x80000000000000000000000000000000);
    }

    /**
     * Calculate x * y.  Special values behave in the following way:
     *
     * NaN * x = NaN for any x.
     * Infinity * x = Infinity for any finite positive x.
     * Infinity * x = -Infinity for any finite negative x.
     * -Infinity * x = -Infinity for any finite positive x.
     * -Infinity * x = Infinity for any finite negative x.
     * Infinity * 0 = NaN.
     * -Infinity * 0 = NaN.
     * Infinity * Infinity = Infinity.
     * Infinity * -Infinity = -Infinity.
     * -Infinity * Infinity = -Infinity.
     * -Infinity * -Infinity = Infinity.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return quadruple precision number
     */
    function mul (bytes16 x, bytes16 y) internal pure returns (bytes16) {
        uint256 xExponent = uint128 (x) >> 112 & 0x7FFF;
        uint256 yExponent = uint128 (y) >> 112 & 0x7FFF;

        if (xExponent == 0x7FFF) {
            if (yExponent == 0x7FFF) {
                if (x == y) return x ^ y & 0x80000000000000000000000000000000;
                else if (x ^ y == 0x80000000000000000000000000000000) return x | y;
                else return NaN;
            } else {
                if (y & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF == 0) return NaN;
                else return x ^ y & 0x80000000000000000000000000000000;
            }
        } else if (yExponent == 0x7FFF) {
            if (x & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF == 0) return NaN;
            else return y ^ x & 0x80000000000000000000000000000000;
        } else {
            uint256 xSignifier = uint128 (x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
            if (xExponent == 0) xExponent = 1;
            else xSignifier |= 0x10000000000000000000000000000;

            uint256 ySignifier = uint128 (y) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
            if (yExponent == 0) yExponent = 1;
            else ySignifier |= 0x10000000000000000000000000000;

            xSignifier *= ySignifier;
            if (xSignifier == 0)
                return (x ^ y) & 0x80000000000000000000000000000000 > 0 ?
                NEGATIVE_ZERO : POSITIVE_ZERO;

            xExponent += yExponent;

            uint256 msb =
            xSignifier >= 0x200000000000000000000000000000000000000000000000000000000 ? 225 :
            xSignifier >= 0x100000000000000000000000000000000000000000000000000000000 ? 224 :
            msb (xSignifier);

            if (xExponent + msb < 16496) { // Underflow
                xExponent = 0;
                xSignifier = 0;
            } else if (xExponent + msb < 16608) { // Subnormal
                if (xExponent < 16496)
                    xSignifier >>= 16496 - xExponent;
                else if (xExponent > 16496)
                    xSignifier <<= xExponent - 16496;
                xExponent = 0;
            } else if (xExponent + msb > 49373) {
                xExponent = 0x7FFF;
                xSignifier = 0;
            } else {
                if (msb > 112)
                    xSignifier >>= msb - 112;
                else if (msb < 112)
                    xSignifier <<= 112 - msb;

                xSignifier &= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

                xExponent = xExponent + msb - 16607;
            }

            return bytes16 (uint128 (uint128 ((x ^ y) & 0x80000000000000000000000000000000) |
            xExponent << 112 | xSignifier));
        }
    }

    /**
     * Calculate x / y.  Special values behave in the following way:
     *
     * NaN / x = NaN for any x.
     * x / NaN = NaN for any x.
     * Infinity / x = Infinity for any finite non-negative x.
     * Infinity / x = -Infinity for any finite negative x including -0.
     * -Infinity / x = -Infinity for any finite non-negative x.
     * -Infinity / x = Infinity for any finite negative x including -0.
     * x / Infinity = 0 for any finite non-negative x.
     * x / -Infinity = -0 for any finite non-negative x.
     * x / Infinity = -0 for any finite non-negative x including -0.
     * x / -Infinity = 0 for any finite non-negative x including -0.
     *
     * Infinity / Infinity = NaN.
     * Infinity / -Infinity = -NaN.
     * -Infinity / Infinity = -NaN.
     * -Infinity / -Infinity = NaN.
     *
     * Division by zero behaves in the following way:
     *
     * x / 0 = Infinity for any finite positive x.
     * x / -0 = -Infinity for any finite positive x.
     * x / 0 = -Infinity for any finite negative x.
     * x / -0 = Infinity for any finite negative x.
     * 0 / 0 = NaN.
     * 0 / -0 = NaN.
     * -0 / 0 = NaN.
     * -0 / -0 = NaN.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return quadruple precision number
     */
    function div (bytes16 x, bytes16 y) internal pure returns (bytes16) {
        uint256 xExponent = uint128 (x) >> 112 & 0x7FFF;
        uint256 yExponent = uint128 (y) >> 112 & 0x7FFF;

        if (xExponent == 0x7FFF) {
            if (yExponent == 0x7FFF) return NaN;
            else return x ^ y & 0x80000000000000000000000000000000;
        } else if (yExponent == 0x7FFF) {
            if (y & 0x0000FFFFFFFFFFFFFFFFFFFFFFFFFFFF != 0) return NaN;
            else return POSITIVE_ZERO | (x ^ y) & 0x80000000000000000000000000000000;
        } else if (y & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF == 0) {
            if (x & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF == 0) return NaN;
            else return POSITIVE_INFINITY | (x ^ y) & 0x80000000000000000000000000000000;
        } else {
            uint256 ySignifier = uint128 (y) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
            if (yExponent == 0) yExponent = 1;
            else ySignifier |= 0x10000000000000000000000000000;

            uint256 xSignifier = uint128 (x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
            if (xExponent == 0) {
                if (xSignifier != 0) {
                    uint shift = 226 - msb (xSignifier);

                    xSignifier <<= shift;

                    xExponent = 1;
                    yExponent += shift - 114;
                }
            }
            else {
                xSignifier = (xSignifier | 0x10000000000000000000000000000) << 114;
            }

            xSignifier = xSignifier / ySignifier;
            if (xSignifier == 0)
                return (x ^ y) & 0x80000000000000000000000000000000 > 0 ?
                NEGATIVE_ZERO : POSITIVE_ZERO;

            assert (xSignifier >= 0x1000000000000000000000000000);

            uint256 msb =
            xSignifier >= 0x80000000000000000000000000000 ? msb (xSignifier) :
            xSignifier >= 0x40000000000000000000000000000 ? 114 :
            xSignifier >= 0x20000000000000000000000000000 ? 113 : 112;

            if (xExponent + msb > yExponent + 16497) { // Overflow
                xExponent = 0x7FFF;
                xSignifier = 0;
            } else if (xExponent + msb + 16380  < yExponent) { // Underflow
                xExponent = 0;
                xSignifier = 0;
            } else if (xExponent + msb + 16268  < yExponent) { // Subnormal
                if (xExponent + 16380 > yExponent)
                    xSignifier <<= xExponent + 16380 - yExponent;
                else if (xExponent + 16380 < yExponent)
                    xSignifier >>= yExponent - xExponent - 16380;

                xExponent = 0;
            } else { // Normal
                if (msb > 112)
                    xSignifier >>= msb - 112;

                xSignifier &= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

                xExponent = xExponent + msb + 16269 - yExponent;
            }

            return bytes16 (uint128 (uint128 ((x ^ y) & 0x80000000000000000000000000000000) |
            xExponent << 112 | xSignifier));
        }
    }

    /**
     * Calculate -x.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function neg (bytes16 x) internal pure returns (bytes16) {
        return x ^ 0x80000000000000000000000000000000;
    }

    /**
     * Calculate |x|.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function abs (bytes16 x) internal pure returns (bytes16) {
        return x & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
    }

    /**
     * Calculate square root of x.  Return NaN on negative x excluding -0.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function sqrt (bytes16 x) internal pure returns (bytes16) {
        if (uint128 (x) >  0x80000000000000000000000000000000) return NaN;
        else {
            uint256 xExponent = uint128 (x) >> 112 & 0x7FFF;
            if (xExponent == 0x7FFF) return x;
            else {
                uint256 xSignifier = uint128 (x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                if (xExponent == 0) xExponent = 1;
                else xSignifier |= 0x10000000000000000000000000000;

                if (xSignifier == 0) return POSITIVE_ZERO;

                bool oddExponent = xExponent & 0x1 == 0;
                xExponent = xExponent + 16383 >> 1;

                if (oddExponent) {
                    if (xSignifier >= 0x10000000000000000000000000000)
                        xSignifier <<= 113;
                    else {
                        uint256 msb = msb (xSignifier);
                        uint256 shift = (226 - msb) & 0xFE;
                        xSignifier <<= shift;
                        xExponent -= shift - 112 >> 1;
                    }
                } else {
                    if (xSignifier >= 0x10000000000000000000000000000)
                        xSignifier <<= 112;
                    else {
                        uint256 msb = msb (xSignifier);
                        uint256 shift = (225 - msb) & 0xFE;
                        xSignifier <<= shift;
                        xExponent -= shift - 112 >> 1;
                    }
                }

                uint256 r = 0x10000000000000000000000000000;
                r = (r + xSignifier / r) >> 1;
                r = (r + xSignifier / r) >> 1;
                r = (r + xSignifier / r) >> 1;
                r = (r + xSignifier / r) >> 1;
                r = (r + xSignifier / r) >> 1;
                r = (r + xSignifier / r) >> 1;
                r = (r + xSignifier / r) >> 1; // Seven iterations should be enough
                uint256 r1 = xSignifier / r;
                if (r1 < r) r = r1;

                return bytes16 (uint128 (xExponent << 112 | r & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF));
            }
        }
    }

    /**
     * Calculate binary logarithm of x.  Return NaN on negative x excluding -0.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function log_2 (bytes16 x) internal pure returns (bytes16) {
        if (uint128 (x) > 0x80000000000000000000000000000000) return NaN;
        else if (x == 0x3FFF0000000000000000000000000000) return POSITIVE_ZERO;
        else {
            uint256 xExponent = uint128 (x) >> 112 & 0x7FFF;
            if (xExponent == 0x7FFF) return x;
            else {
                uint256 xSignifier = uint128 (x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                if (xExponent == 0) xExponent = 1;
                else xSignifier |= 0x10000000000000000000000000000;

                if (xSignifier == 0) return NEGATIVE_INFINITY;

                bool resultNegative;
                uint256 resultExponent = 16495;
                uint256 resultSignifier;

                if (xExponent >= 0x3FFF) {
                    resultNegative = false;
                    resultSignifier = xExponent - 0x3FFF;
                    xSignifier <<= 15;
                } else {
                    resultNegative = true;
                    if (xSignifier >= 0x10000000000000000000000000000) {
                        resultSignifier = 0x3FFE - xExponent;
                        xSignifier <<= 15;
                    } else {
                        uint256 msb = msb (xSignifier);
                        resultSignifier = 16493 - msb;
                        xSignifier <<= 127 - msb;
                    }
                }

                if (xSignifier == 0x80000000000000000000000000000000) {
                    if (resultNegative) resultSignifier += 1;
                    uint256 shift = 112 - msb (resultSignifier);
                    resultSignifier <<= shift;
                    resultExponent -= shift;
                } else {
                    uint256 bb = resultNegative ? 1 : 0;
                    while (resultSignifier < 0x10000000000000000000000000000) {
                        resultSignifier <<= 1;
                        resultExponent -= 1;

                        xSignifier *= xSignifier;
                        uint256 b = xSignifier >> 255;
                        resultSignifier += b ^ bb;
                        xSignifier >>= 127 + b;
                    }
                }

                return bytes16 (uint128 ((resultNegative ? 0x80000000000000000000000000000000 : 0) |
                resultExponent << 112 | resultSignifier & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF));
            }
        }
    }

    /**
     * Calculate natural logarithm of x.  Return NaN on negative x excluding -0.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function ln (bytes16 x) internal pure returns (bytes16) {
        return mul (log_2 (x), 0x3FFE62E42FEFA39EF35793C7673007E5);
    }

    /**
     * Calculate 2^x.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function pow_2 (bytes16 x) internal pure returns (bytes16) {
        bool xNegative = uint128 (x) > 0x80000000000000000000000000000000;
        uint256 xExponent = uint128 (x) >> 112 & 0x7FFF;
        uint256 xSignifier = uint128 (x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

        if (xExponent == 0x7FFF && xSignifier != 0) return NaN;
        else if (xExponent > 16397)
            return xNegative ? POSITIVE_ZERO : POSITIVE_INFINITY;
        else if (xExponent < 16255)
            return 0x3FFF0000000000000000000000000000;
        else {
            if (xExponent == 0) xExponent = 1;
            else xSignifier |= 0x10000000000000000000000000000;

            if (xExponent > 16367)
                xSignifier <<= xExponent - 16367;
            else if (xExponent < 16367)
                xSignifier >>= 16367 - xExponent;

            if (xNegative && xSignifier > 0x406E00000000000000000000000000000000)
                return POSITIVE_ZERO;

            if (!xNegative && xSignifier > 0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)
                return POSITIVE_INFINITY;

            uint256 resultExponent = xSignifier >> 128;
            xSignifier &= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
            if (xNegative && xSignifier != 0) {
                xSignifier = ~xSignifier;
                resultExponent += 1;
            }

            uint256 resultSignifier = 0x80000000000000000000000000000000;
            if (xSignifier & 0x80000000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x16A09E667F3BCC908B2FB1366EA957D3E >> 128;
            if (xSignifier & 0x40000000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1306FE0A31B7152DE8D5A46305C85EDEC >> 128;
            if (xSignifier & 0x20000000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1172B83C7D517ADCDF7C8C50EB14A791F >> 128;
            if (xSignifier & 0x10000000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10B5586CF9890F6298B92B71842A98363 >> 128;
            if (xSignifier & 0x8000000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1059B0D31585743AE7C548EB68CA417FD >> 128;
            if (xSignifier & 0x4000000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x102C9A3E778060EE6F7CACA4F7A29BDE8 >> 128;
            if (xSignifier & 0x2000000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10163DA9FB33356D84A66AE336DCDFA3F >> 128;
            if (xSignifier & 0x1000000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100B1AFA5ABCBED6129AB13EC11DC9543 >> 128;
            if (xSignifier & 0x800000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10058C86DA1C09EA1FF19D294CF2F679B >> 128;
            if (xSignifier & 0x400000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1002C605E2E8CEC506D21BFC89A23A00F >> 128;
            if (xSignifier & 0x200000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100162F3904051FA128BCA9C55C31E5DF >> 128;
            if (xSignifier & 0x100000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000B175EFFDC76BA38E31671CA939725 >> 128;
            if (xSignifier & 0x80000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100058BA01FB9F96D6CACD4B180917C3D >> 128;
            if (xSignifier & 0x40000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10002C5CC37DA9491D0985C348C68E7B3 >> 128;
            if (xSignifier & 0x20000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000162E525EE054754457D5995292026 >> 128;
            if (xSignifier & 0x10000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000B17255775C040618BF4A4ADE83FC >> 128;
            if (xSignifier & 0x8000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000058B91B5BC9AE2EED81E9B7D4CFAB >> 128;
            if (xSignifier & 0x4000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100002C5C89D5EC6CA4D7C8ACC017B7C9 >> 128;
            if (xSignifier & 0x2000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000162E43F4F831060E02D839A9D16D >> 128;
            if (xSignifier & 0x1000000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000B1721BCFC99D9F890EA06911763 >> 128;
            if (xSignifier & 0x800000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000058B90CF1E6D97F9CA14DBCC1628 >> 128;
            if (xSignifier & 0x400000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000002C5C863B73F016468F6BAC5CA2B >> 128;
            if (xSignifier & 0x200000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000162E430E5A18F6119E3C02282A5 >> 128;
            if (xSignifier & 0x100000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000B1721835514B86E6D96EFD1BFE >> 128;
            if (xSignifier & 0x80000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000058B90C0B48C6BE5DF846C5B2EF >> 128;
            if (xSignifier & 0x40000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000002C5C8601CC6B9E94213C72737A >> 128;
            if (xSignifier & 0x20000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000162E42FFF037DF38AA2B219F06 >> 128;
            if (xSignifier & 0x10000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000B17217FBA9C739AA5819F44F9 >> 128;
            if (xSignifier & 0x8000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000058B90BFCDEE5ACD3C1CEDC823 >> 128;
            if (xSignifier & 0x4000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000002C5C85FE31F35A6A30DA1BE50 >> 128;
            if (xSignifier & 0x2000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000162E42FF0999CE3541B9FFFCF >> 128;
            if (xSignifier & 0x1000000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000B17217F80F4EF5AADDA45554 >> 128;
            if (xSignifier & 0x800000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000058B90BFBF8479BD5A81B51AD >> 128;
            if (xSignifier & 0x400000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000002C5C85FDF84BD62AE30A74CC >> 128;
            if (xSignifier & 0x200000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000162E42FEFB2FED257559BDAA >> 128;
            if (xSignifier & 0x100000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000B17217F7D5A7716BBA4A9AE >> 128;
            if (xSignifier & 0x80000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000058B90BFBE9DDBAC5E109CCE >> 128;
            if (xSignifier & 0x40000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000002C5C85FDF4B15DE6F17EB0D >> 128;
            if (xSignifier & 0x20000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000162E42FEFA494F1478FDE05 >> 128;
            if (xSignifier & 0x10000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000B17217F7D20CF927C8E94C >> 128;
            if (xSignifier & 0x8000000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000058B90BFBE8F71CB4E4B33D >> 128;
            if (xSignifier & 0x4000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000002C5C85FDF477B662B26945 >> 128;
            if (xSignifier & 0x2000000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000162E42FEFA3AE53369388C >> 128;
            if (xSignifier & 0x1000000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000B17217F7D1D351A389D40 >> 128;
            if (xSignifier & 0x800000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000058B90BFBE8E8B2D3D4EDE >> 128;
            if (xSignifier & 0x400000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000002C5C85FDF4741BEA6E77E >> 128;
            if (xSignifier & 0x200000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000162E42FEFA39FE95583C2 >> 128;
            if (xSignifier & 0x100000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000B17217F7D1CFB72B45E1 >> 128;
            if (xSignifier & 0x80000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000058B90BFBE8E7CC35C3F0 >> 128;
            if (xSignifier & 0x40000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000002C5C85FDF473E242EA38 >> 128;
            if (xSignifier & 0x20000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000162E42FEFA39F02B772C >> 128;
            if (xSignifier & 0x10000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000B17217F7D1CF7D83C1A >> 128;
            if (xSignifier & 0x8000000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000058B90BFBE8E7BDCBE2E >> 128;
            if (xSignifier & 0x4000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000002C5C85FDF473DEA871F >> 128;
            if (xSignifier & 0x2000000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000162E42FEFA39EF44D91 >> 128;
            if (xSignifier & 0x1000000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000B17217F7D1CF79E949 >> 128;
            if (xSignifier & 0x800000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000058B90BFBE8E7BCE544 >> 128;
            if (xSignifier & 0x400000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000002C5C85FDF473DE6ECA >> 128;
            if (xSignifier & 0x200000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000162E42FEFA39EF366F >> 128;
            if (xSignifier & 0x100000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000B17217F7D1CF79AFA >> 128;
            if (xSignifier & 0x80000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000058B90BFBE8E7BCD6D >> 128;
            if (xSignifier & 0x40000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000002C5C85FDF473DE6B2 >> 128;
            if (xSignifier & 0x20000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000162E42FEFA39EF358 >> 128;
            if (xSignifier & 0x10000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000B17217F7D1CF79AB >> 128;
            if (xSignifier & 0x8000000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000058B90BFBE8E7BCD5 >> 128;
            if (xSignifier & 0x4000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000002C5C85FDF473DE6A >> 128;
            if (xSignifier & 0x2000000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000162E42FEFA39EF34 >> 128;
            if (xSignifier & 0x1000000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000B17217F7D1CF799 >> 128;
            if (xSignifier & 0x800000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000058B90BFBE8E7BCC >> 128;
            if (xSignifier & 0x400000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000002C5C85FDF473DE5 >> 128;
            if (xSignifier & 0x200000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000162E42FEFA39EF2 >> 128;
            if (xSignifier & 0x100000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000B17217F7D1CF78 >> 128;
            if (xSignifier & 0x80000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000058B90BFBE8E7BB >> 128;
            if (xSignifier & 0x40000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000002C5C85FDF473DD >> 128;
            if (xSignifier & 0x20000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000162E42FEFA39EE >> 128;
            if (xSignifier & 0x10000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000B17217F7D1CF6 >> 128;
            if (xSignifier & 0x8000000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000058B90BFBE8E7A >> 128;
            if (xSignifier & 0x4000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000002C5C85FDF473C >> 128;
            if (xSignifier & 0x2000000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000162E42FEFA39D >> 128;
            if (xSignifier & 0x1000000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000B17217F7D1CE >> 128;
            if (xSignifier & 0x800000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000058B90BFBE8E6 >> 128;
            if (xSignifier & 0x400000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000002C5C85FDF472 >> 128;
            if (xSignifier & 0x200000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000162E42FEFA38 >> 128;
            if (xSignifier & 0x100000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000B17217F7D1B >> 128;
            if (xSignifier & 0x80000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000058B90BFBE8D >> 128;
            if (xSignifier & 0x40000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000002C5C85FDF46 >> 128;
            if (xSignifier & 0x20000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000162E42FEFA2 >> 128;
            if (xSignifier & 0x10000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000B17217F7D0 >> 128;
            if (xSignifier & 0x8000000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000058B90BFBE7 >> 128;
            if (xSignifier & 0x4000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000002C5C85FDF3 >> 128;
            if (xSignifier & 0x2000000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000162E42FEF9 >> 128;
            if (xSignifier & 0x1000000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000B17217F7C >> 128;
            if (xSignifier & 0x800000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000058B90BFBD >> 128;
            if (xSignifier & 0x400000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000002C5C85FDE >> 128;
            if (xSignifier & 0x200000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000162E42FEE >> 128;
            if (xSignifier & 0x100000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000000B17217F6 >> 128;
            if (xSignifier & 0x80000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000058B90BFA >> 128;
            if (xSignifier & 0x40000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000002C5C85FC >> 128;
            if (xSignifier & 0x20000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000000162E42FD >> 128;
            if (xSignifier & 0x10000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000000B17217E >> 128;
            if (xSignifier & 0x8000000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000000058B90BE >> 128;
            if (xSignifier & 0x4000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000002C5C85E >> 128;
            if (xSignifier & 0x2000000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000000162E42E >> 128;
            if (xSignifier & 0x1000000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000000B17216 >> 128;
            if (xSignifier & 0x800000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000000058B90A >> 128;
            if (xSignifier & 0x400000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000000002C5C84 >> 128;
            if (xSignifier & 0x200000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000000162E41 >> 128;
            if (xSignifier & 0x100000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000000000B1720 >> 128;
            if (xSignifier & 0x80000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000000058B8F >> 128;
            if (xSignifier & 0x40000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000000002C5C7 >> 128;
            if (xSignifier & 0x20000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000000000162E3 >> 128;
            if (xSignifier & 0x10000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000000000B171 >> 128;
            if (xSignifier & 0x8000 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000000000058B8 >> 128;
            if (xSignifier & 0x4000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000000002C5B >> 128;
            if (xSignifier & 0x2000 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000000000162D >> 128;
            if (xSignifier & 0x1000 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000000000B16 >> 128;
            if (xSignifier & 0x800 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000000000058A >> 128;
            if (xSignifier & 0x400 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000000000002C4 >> 128;
            if (xSignifier & 0x200 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000000000161 >> 128;
            if (xSignifier & 0x100 > 0) resultSignifier = resultSignifier * 0x1000000000000000000000000000000B0 >> 128;
            if (xSignifier & 0x80 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000000000057 >> 128;
            if (xSignifier & 0x40 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000000000002B >> 128;
            if (xSignifier & 0x20 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000000000015 >> 128;
            if (xSignifier & 0x10 > 0) resultSignifier = resultSignifier * 0x10000000000000000000000000000000A >> 128;
            if (xSignifier & 0x8 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000000000004 >> 128;
            if (xSignifier & 0x4 > 0) resultSignifier = resultSignifier * 0x100000000000000000000000000000001 >> 128;

            if (!xNegative) {
                resultSignifier = resultSignifier >> 15 & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                resultExponent += 0x3FFF;
            } else if (resultExponent <= 0x3FFE) {
                resultSignifier = resultSignifier >> 15 & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                resultExponent = 0x3FFF - resultExponent;
            } else {
                resultSignifier = resultSignifier >> resultExponent - 16367;
                resultExponent = 0;
            }

            return bytes16 (uint128 (resultExponent << 112 | resultSignifier));
        }
    }

    /**
     * Calculate e^x.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function exp (bytes16 x) internal pure returns (bytes16) {
        return pow_2 (mul (x, 0x3FFF71547652B82FE1777D0FFDA0D23A));
    }

    /**
     * Get index of the most significant non-zero bit in binary representation of
     * x.  Reverts if x is zero.
     *
     * @return index of the most significant non-zero bit in binary representation
     *         of x
     */
    function msb (uint256 x) private pure returns (uint256) {
        require (x > 0);

        uint256 result = 0;

        if (x >= 0x100000000000000000000000000000000) { x >>= 128; result += 128; }
        if (x >= 0x10000000000000000) { x >>= 64; result += 64; }
        if (x >= 0x100000000) { x >>= 32; result += 32; }
        if (x >= 0x10000) { x >>= 16; result += 16; }
        if (x >= 0x100) { x >>= 8; result += 8; }
        if (x >= 0x10) { x >>= 4; result += 4; }
        if (x >= 0x4) { x >>= 2; result += 2; }
        if (x >= 0x2) result += 1; // No need to shift x anymore

        return result;
    }
}
/**
 * Smart contract library of mathematical functions operating with IEEE 754
 * quadruple-precision binary floating-point numbers (quadruple precision
 * numbers).  As long as quadruple precision numbers are 16-bytes long, they are
 * represented by bytes16 type.
 */
function log_2 (bytes16 x) internal pure returns (bytes16) {
    if (uint128 (x) > 0x80000000000000000000000000000000) return NaN;
    else if (x == 0x3FFF0000000000000000000000000000) return POSITIVE_ZERO;
    else {
        uint256 xExponent = uint128 (x) >> 112 & 0x7FFF;
        if (xExponent == 0x7FFF) return x;
        else {
            uint256 xSignifier = uint128 (x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
            if (xExponent == 0) xExponent = 1;
            else xSignifier |= 0x10000000000000000000000000000;

            if (xSignifier == 0) return NEGATIVE_INFINITY;

            bool resultNegative;
            uint256 resultExponent = 16495;
            uint256 resultSignifier;

            if (xExponent >= 0x3FFF) {
                resultNegative = false;
                resultSignifier = xExponent - 0x3FFF;
                xSignifier <<= 15;
            } else {
                resultNegative = true;
                if (xSignifier >= 0x10000000000000000000000000000) {
                    resultSignifier = 0x3FFE - xExponent;
                    xSignifier <<= 15;
                } else {
                    uint256 msb = msb (xSignifier);
                    resultSignifier = 16493 - msb;
                    xSignifier <<= 127 - msb;
                }
            }

            if (xSignifier == 0x80000000000000000000000000000000) {
                if (resultNegative) resultSignifier += 1;
                uint256 shift = 112 - msb (resultSignifier);
                resultSignifier <<= shift;
                resultExponent -= shift;
            } else {
                uint256 bb = resultNegative ? 1 : 0;
                while (resultSignifier < 0x10000000000000000000000000000) {
                    resultSignifier <<= 1;
                    resultExponent -= 1;

                    xSignifier *= xSignifier;
                    uint256 b = xSignifier >> 255;
                    resultSignifier += b ^ bb;
                    xSignifier >>= 127 + b;
                }
            }

            return bytes16 (uint128 ((resultNegative ? 0x80000000000000000000000000000000 : 0) |
            resultExponent << 112 | resultSignifier & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF));
        }
    }
}

/**
 * Calculate binary logarithm of x.  Return NaN on negative x excluding -0.
 *
 * @param x quadruple precision number
 * @return quadruple precision number
 */