Exp64x64.sol

// SPDX-License-Identifier: GPL-3.0-or-later
pragma solidity >=0.8.15; /*
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      | |  | |  __/ | (_| | ██╔══╝   ██╔██╗ ██╔═══╝ ██╔═══██╗╚════██║ ██╔██╗ ██╔═══██╗╚════██║
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       yieldprotocol.com    ╚══════╝╚═╝  ╚═╝╚═╝      ╚═════╝      ╚═╝╚═╝  ╚═╝ ╚═════╝      ╚═╝
                            Gas optimized math library custom-built by ABDK -- Copyright © 2019 */

import "./Math64x64.sol";

library Exp64x64 {
    using Math64x64 for int128;

    /// @dev Raises a 64.64 number to the power of another 64.64 number
    /// x^y = 2^(y*log_2(x))
    /// https://ethereum.stackexchange.com/questions/79903/exponential-function-with-fractional-numbers
    function pow(int128 x, int128 y) internal pure returns (int128) {
        return y.mul(x.log_2()).exp_2();
    }


    /* Mikhail Vladimirov, [Jul 6, 2022 at 12:26:12 PM (Jul 6, 2022 at 12:28:29 PM)]:
        In simple words, when have an n-bits wide number x and raise it to a power α, then the result would be α*n bits wide.  This, if α<1, the result will loose precision, and if α>1, the result could exceed range.

        So, the pow function multiplies the result by 2^(n * (1 - α)).  We have:

        x ∈ [0; 2^n)
        x^α ∈ [0; 2^(α*n))
        x^α * 2^(n * (1 - α)) ∈ [0; 2^(α*n) * 2^(n * (1 - α))) = [0; 2^(α*n + n * (1 - α))) = [0; 2^(n * (α +  (1 - α)))) =  [0; 2^n)

        So the normalization returns the result back into the proper range.

        Now note, that:

        pow (pow (x, α), 1/α) =
        pow (x^α * 2^(n * (1 -α)) , 1/α) =
        (x^α * 2^(n * (1 -α)))^(1/α) * 2^(n * (1 -1/α)) =
        x^(α * (1/α)) * 2^(n * (1 -α) * (1/α)) * 2^(n * (1 -1/α)) =
        x * 2^(n * (1/α -1)) * 2^(n * (1 -1/α)) =
        x * 2^(n * (1/α -1) + n * (1 -1/α)) =
        x

        So, for formulas that look like:

        (a x^α + b y^α + ...)^(1/α)

        The pow function could be used instead of normal power. */
    /// @dev Raise given number x into power specified as a simple fraction y/z and then
    /// multiply the result by the normalization factor 2^(128 /// (1 - y/z)).
    /// Revert if z is zero, or if both x and y are zeros.
    /// @param x number to raise into given power y/z -- integer
    /// @param y numerator of the power to raise x into  -- 64.64
    /// @param z denominator of the power to raise x into  -- 64.64
    /// @return x raised into power y/z and then multiplied by 2^(128 * (1 - y/z)) -- integer
    function pow(
        uint128 x,
        uint128 y,
        uint128 z
    ) internal pure returns (uint128) {
        unchecked {
            require(z != 0);

            if (x == 0) {
                require(y != 0);
                return 0;
            } else {
                uint256 l = (uint256(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF - log_2(x)) * y) / z;
                if (l > 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF) return 0;
                else return pow_2(uint128(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF - l));
            }
        }
    }

    /// @dev Calculate base 2 logarithm of an unsigned 128-bit integer number.  Revert
    /// in case x is zero.
    /// @param x number to calculate base 2 logarithm of
    /// @return base 2 logarithm of x, multiplied by 2^121
    function log_2(uint128 x) internal pure returns (uint128) {
        unchecked {
            require(x != 0);

            uint256 b = x;

            uint256 l = 0xFE000000000000000000000000000000;

            if (b < 0x10000000000000000) {
                l -= 0x80000000000000000000000000000000;
                b <<= 64;
            }
            if (b < 0x1000000000000000000000000) {
                l -= 0x40000000000000000000000000000000;
                b <<= 32;
            }
            if (b < 0x10000000000000000000000000000) {
                l -= 0x20000000000000000000000000000000;
                b <<= 16;
            }
            if (b < 0x1000000000000000000000000000000) {
                l -= 0x10000000000000000000000000000000;
                b <<= 8;
            }
            if (b < 0x10000000000000000000000000000000) {
                l -= 0x8000000000000000000000000000000;
                b <<= 4;
            }
            if (b < 0x40000000000000000000000000000000) {
                l -= 0x4000000000000000000000000000000;
                b <<= 2;
            }
            if (b < 0x80000000000000000000000000000000) {
                l -= 0x2000000000000000000000000000000;
                b <<= 1;
            }

            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x1000000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x800000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x400000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x200000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x100000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x80000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x40000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x20000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x10000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x8000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x4000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x2000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x1000000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x800000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x400000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x200000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x100000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x80000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x40000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x20000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x10000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x8000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x4000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x2000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x1000000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x800000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x400000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x200000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x100000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x80000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x40000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x20000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x10000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x8000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x4000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x2000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x1000000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x800000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x400000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x200000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x100000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x80000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x40000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x20000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x10000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x8000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x4000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x2000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x1000000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x800000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x400000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x200000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x100000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x80000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x40000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x20000000000000000;
            }
            b = (b * b) >> 127;
            if (b >= 0x100000000000000000000000000000000) {
                b >>= 1;
                l |= 0x10000000000000000;
            } /*
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x8000000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x4000000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x2000000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x1000000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x800000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x400000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x200000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x100000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x80000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x40000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x20000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x10000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x8000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x4000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x2000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x1000000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x800000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x400000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x200000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x100000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x80000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x40000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x20000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x10000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x8000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x4000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x2000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x1000000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x800000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x400000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x200000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x100000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x80000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x40000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x20000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x10000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x8000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x4000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x2000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x1000000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x800000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x400000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x200000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x100000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x80000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x40000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x20000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x10000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x8000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x4000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x2000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x1000;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x800;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x400;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x200;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x100;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x80;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x40;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x20;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x10;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x8;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x4;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) {b >>= 1; l |= 0x2;}
      b = b * b >> 127; if(b >= 0x100000000000000000000000000000000) l |= 0x1; */

            return uint128(l);
        }
    }

    /// @dev Calculate 2 raised into given power.
    /// @param x power to raise 2 into, multiplied by 2^121
    /// @return 2 raised into given power
    function pow_2(uint128 x) internal pure returns (uint128) {
        unchecked {
            uint256 r = 0x80000000000000000000000000000000;
            if (x & 0x1000000000000000000000000000000 > 0) r = (r * 0xb504f333f9de6484597d89b3754abe9f) >> 127;
            if (x & 0x800000000000000000000000000000 > 0) r = (r * 0x9837f0518db8a96f46ad23182e42f6f6) >> 127;
            if (x & 0x400000000000000000000000000000 > 0) r = (r * 0x8b95c1e3ea8bd6e6fbe4628758a53c90) >> 127;
            if (x & 0x200000000000000000000000000000 > 0) r = (r * 0x85aac367cc487b14c5c95b8c2154c1b2) >> 127;
            if (x & 0x100000000000000000000000000000 > 0) r = (r * 0x82cd8698ac2ba1d73e2a475b46520bff) >> 127;
            if (x & 0x80000000000000000000000000000 > 0) r = (r * 0x8164d1f3bc0307737be56527bd14def4) >> 127;
            if (x & 0x40000000000000000000000000000 > 0) r = (r * 0x80b1ed4fd999ab6c25335719b6e6fd20) >> 127;
            if (x & 0x20000000000000000000000000000 > 0) r = (r * 0x8058d7d2d5e5f6b094d589f608ee4aa2) >> 127;
            if (x & 0x10000000000000000000000000000 > 0) r = (r * 0x802c6436d0e04f50ff8ce94a6797b3ce) >> 127;
            if (x & 0x8000000000000000000000000000 > 0) r = (r * 0x8016302f174676283690dfe44d11d008) >> 127;
            if (x & 0x4000000000000000000000000000 > 0) r = (r * 0x800b179c82028fd0945e54e2ae18f2f0) >> 127;
            if (x & 0x2000000000000000000000000000 > 0) r = (r * 0x80058baf7fee3b5d1c718b38e549cb93) >> 127;
            if (x & 0x1000000000000000000000000000 > 0) r = (r * 0x8002c5d00fdcfcb6b6566a58c048be1f) >> 127;
            if (x & 0x800000000000000000000000000 > 0) r = (r * 0x800162e61bed4a48e84c2e1a463473d9) >> 127;
            if (x & 0x400000000000000000000000000 > 0) r = (r * 0x8000b17292f702a3aa22beacca949013) >> 127;
            if (x & 0x200000000000000000000000000 > 0) r = (r * 0x800058b92abbae02030c5fa5256f41fe) >> 127;
            if (x & 0x100000000000000000000000000 > 0) r = (r * 0x80002c5c8dade4d71776c0f4dbea67d6) >> 127;
            if (x & 0x80000000000000000000000000 > 0) r = (r * 0x8000162e44eaf636526be456600bdbe4) >> 127;
            if (x & 0x40000000000000000000000000 > 0) r = (r * 0x80000b1721fa7c188307016c1cd4e8b6) >> 127;
            if (x & 0x20000000000000000000000000 > 0) r = (r * 0x8000058b90de7e4cecfc487503488bb1) >> 127;
            if (x & 0x10000000000000000000000000 > 0) r = (r * 0x800002c5c8678f36cbfce50a6de60b14) >> 127;
            if (x & 0x8000000000000000000000000 > 0) r = (r * 0x80000162e431db9f80b2347b5d62e516) >> 127;
            if (x & 0x4000000000000000000000000 > 0) r = (r * 0x800000b1721872d0c7b08cf1e0114152) >> 127;
            if (x & 0x2000000000000000000000000 > 0) r = (r * 0x80000058b90c1aa8a5c3736cb77e8dff) >> 127;
            if (x & 0x1000000000000000000000000 > 0) r = (r * 0x8000002c5c8605a4635f2efc2362d978) >> 127;
            if (x & 0x800000000000000000000000 > 0) r = (r * 0x800000162e4300e635cf4a109e3939bd) >> 127;
            if (x & 0x400000000000000000000000 > 0) r = (r * 0x8000000b17217ff81bef9c551590cf83) >> 127;
            if (x & 0x200000000000000000000000 > 0) r = (r * 0x800000058b90bfdd4e39cd52c0cfa27c) >> 127;
            if (x & 0x100000000000000000000000 > 0) r = (r * 0x80000002c5c85fe6f72d669e0e76e411) >> 127;
            if (x & 0x80000000000000000000000 > 0) r = (r * 0x8000000162e42ff18f9ad35186d0df28) >> 127;
            if (x & 0x40000000000000000000000 > 0) r = (r * 0x80000000b17217f84cce71aa0dcfffe7) >> 127;
            if (x & 0x20000000000000000000000 > 0) r = (r * 0x8000000058b90bfc07a77ad56ed22aaa) >> 127;
            if (x & 0x10000000000000000000000 > 0) r = (r * 0x800000002c5c85fdfc23cdead40da8d6) >> 127;
            if (x & 0x8000000000000000000000 > 0) r = (r * 0x80000000162e42fefc25eb1571853a66) >> 127;
            if (x & 0x4000000000000000000000 > 0) r = (r * 0x800000000b17217f7d97f692baacded5) >> 127;
            if (x & 0x2000000000000000000000 > 0) r = (r * 0x80000000058b90bfbead3b8b5dd254d7) >> 127;
            if (x & 0x1000000000000000000000 > 0) r = (r * 0x8000000002c5c85fdf4eedd62f084e67) >> 127;
            if (x & 0x800000000000000000000 > 0) r = (r * 0x800000000162e42fefa58aef378bf586) >> 127;
            if (x & 0x400000000000000000000 > 0) r = (r * 0x8000000000b17217f7d24a78a3c7ef02) >> 127;
            if (x & 0x200000000000000000000 > 0) r = (r * 0x800000000058b90bfbe9067c93e474a6) >> 127;
            if (x & 0x100000000000000000000 > 0) r = (r * 0x80000000002c5c85fdf47b8e5a72599f) >> 127;
            if (x & 0x80000000000000000000 > 0) r = (r * 0x8000000000162e42fefa3bdb315934a2) >> 127;
            if (x & 0x40000000000000000000 > 0) r = (r * 0x80000000000b17217f7d1d7299b49c46) >> 127;
            if (x & 0x20000000000000000000 > 0) r = (r * 0x8000000000058b90bfbe8e9a8d1c4ea0) >> 127;
            if (x & 0x10000000000000000000 > 0) r = (r * 0x800000000002c5c85fdf4745969ea76f) >> 127;
            if (x & 0x8000000000000000000 > 0) r = (r * 0x80000000000162e42fefa3a0df5373bf) >> 127;
            if (x & 0x4000000000000000000 > 0) r = (r * 0x800000000000b17217f7d1cff4aac1e1) >> 127;
            if (x & 0x2000000000000000000 > 0) r = (r * 0x80000000000058b90bfbe8e7db95a2f1) >> 127;
            if (x & 0x1000000000000000000 > 0) r = (r * 0x8000000000002c5c85fdf473e61ae1f8) >> 127;
            if (x & 0x800000000000000000 > 0) r = (r * 0x800000000000162e42fefa39f121751c) >> 127;
            if (x & 0x400000000000000000 > 0) r = (r * 0x8000000000000b17217f7d1cf815bb96) >> 127;
            if (x & 0x200000000000000000 > 0) r = (r * 0x800000000000058b90bfbe8e7bec1e0d) >> 127;
            if (x & 0x100000000000000000 > 0) r = (r * 0x80000000000002c5c85fdf473dee5f17) >> 127;
            if (x & 0x80000000000000000 > 0) r = (r * 0x8000000000000162e42fefa39ef5438f) >> 127;
            if (x & 0x40000000000000000 > 0) r = (r * 0x80000000000000b17217f7d1cf7a26c8) >> 127;
            if (x & 0x20000000000000000 > 0) r = (r * 0x8000000000000058b90bfbe8e7bcf4a4) >> 127;
            if (x & 0x10000000000000000 > 0) r = (r * 0x800000000000002c5c85fdf473de72a2) >> 127; /*
      if(x & 0x8000000000000000 > 0) r = r * 0x80000000000000162e42fefa39ef3765 >> 127;
      if(x & 0x4000000000000000 > 0) r = r * 0x800000000000000b17217f7d1cf79b37 >> 127;
      if(x & 0x2000000000000000 > 0) r = r * 0x80000000000000058b90bfbe8e7bcd7d >> 127;
      if(x & 0x1000000000000000 > 0) r = r * 0x8000000000000002c5c85fdf473de6b6 >> 127;
      if(x & 0x800000000000000 > 0) r = r * 0x800000000000000162e42fefa39ef359 >> 127;
      if(x & 0x400000000000000 > 0) r = r * 0x8000000000000000b17217f7d1cf79ac >> 127;
      if(x & 0x200000000000000 > 0) r = r * 0x800000000000000058b90bfbe8e7bcd6 >> 127;
      if(x & 0x100000000000000 > 0) r = r * 0x80000000000000002c5c85fdf473de6a >> 127;
      if(x & 0x80000000000000 > 0) r = r * 0x8000000000000000162e42fefa39ef35 >> 127;
      if(x & 0x40000000000000 > 0) r = r * 0x80000000000000000b17217f7d1cf79a >> 127;
      if(x & 0x20000000000000 > 0) r = r * 0x8000000000000000058b90bfbe8e7bcd >> 127;
      if(x & 0x10000000000000 > 0) r = r * 0x800000000000000002c5c85fdf473de6 >> 127;
      if(x & 0x8000000000000 > 0) r = r * 0x80000000000000000162e42fefa39ef3 >> 127;
      if(x & 0x4000000000000 > 0) r = r * 0x800000000000000000b17217f7d1cf79 >> 127;
      if(x & 0x2000000000000 > 0) r = r * 0x80000000000000000058b90bfbe8e7bc >> 127;
      if(x & 0x1000000000000 > 0) r = r * 0x8000000000000000002c5c85fdf473de >> 127;
      if(x & 0x800000000000 > 0) r = r * 0x800000000000000000162e42fefa39ef >> 127;
      if(x & 0x400000000000 > 0) r = r * 0x8000000000000000000b17217f7d1cf7 >> 127;
      if(x & 0x200000000000 > 0) r = r * 0x800000000000000000058b90bfbe8e7b >> 127;
      if(x & 0x100000000000 > 0) r = r * 0x80000000000000000002c5c85fdf473d >> 127;
      if(x & 0x80000000000 > 0) r = r * 0x8000000000000000000162e42fefa39e >> 127;
      if(x & 0x40000000000 > 0) r = r * 0x80000000000000000000b17217f7d1cf >> 127;
      if(x & 0x20000000000 > 0) r = r * 0x8000000000000000000058b90bfbe8e7 >> 127;
      if(x & 0x10000000000 > 0) r = r * 0x800000000000000000002c5c85fdf473 >> 127;
      if(x & 0x8000000000 > 0) r = r * 0x80000000000000000000162e42fefa39 >> 127;
      if(x & 0x4000000000 > 0) r = r * 0x800000000000000000000b17217f7d1c >> 127;
      if(x & 0x2000000000 > 0) r = r * 0x80000000000000000000058b90bfbe8e >> 127;
      if(x & 0x1000000000 > 0) r = r * 0x8000000000000000000002c5c85fdf47 >> 127;
      if(x & 0x800000000 > 0) r = r * 0x800000000000000000000162e42fefa3 >> 127;
      if(x & 0x400000000 > 0) r = r * 0x8000000000000000000000b17217f7d1 >> 127;
      if(x & 0x200000000 > 0) r = r * 0x800000000000000000000058b90bfbe8 >> 127;
      if(x & 0x100000000 > 0) r = r * 0x80000000000000000000002c5c85fdf4 >> 127;
      if(x & 0x80000000 > 0) r = r * 0x8000000000000000000000162e42fefa >> 127;
      if(x & 0x40000000 > 0) r = r * 0x80000000000000000000000b17217f7d >> 127;
      if(x & 0x20000000 > 0) r = r * 0x8000000000000000000000058b90bfbe >> 127;
      if(x & 0x10000000 > 0) r = r * 0x800000000000000000000002c5c85fdf >> 127;
      if(x & 0x8000000 > 0) r = r * 0x80000000000000000000000162e42fef >> 127;
      if(x & 0x4000000 > 0) r = r * 0x800000000000000000000000b17217f7 >> 127;
      if(x & 0x2000000 > 0) r = r * 0x80000000000000000000000058b90bfb >> 127;
      if(x & 0x1000000 > 0) r = r * 0x8000000000000000000000002c5c85fd >> 127;
      if(x & 0x800000 > 0) r = r * 0x800000000000000000000000162e42fe >> 127;
      if(x & 0x400000 > 0) r = r * 0x8000000000000000000000000b17217f >> 127;
      if(x & 0x200000 > 0) r = r * 0x800000000000000000000000058b90bf >> 127;
      if(x & 0x100000 > 0) r = r * 0x80000000000000000000000002c5c85f >> 127;
      if(x & 0x80000 > 0) r = r * 0x8000000000000000000000000162e42f >> 127;
      if(x & 0x40000 > 0) r = r * 0x80000000000000000000000000b17217 >> 127;
      if(x & 0x20000 > 0) r = r * 0x8000000000000000000000000058b90b >> 127;
      if(x & 0x10000 > 0) r = r * 0x800000000000000000000000002c5c85 >> 127;
      if(x & 0x8000 > 0) r = r * 0x80000000000000000000000000162e42 >> 127;
      if(x & 0x4000 > 0) r = r * 0x800000000000000000000000000b1721 >> 127;
      if(x & 0x2000 > 0) r = r * 0x80000000000000000000000000058b90 >> 127;
      if(x & 0x1000 > 0) r = r * 0x8000000000000000000000000002c5c8 >> 127;
      if(x & 0x800 > 0) r = r * 0x800000000000000000000000000162e4 >> 127;
      if(x & 0x400 > 0) r = r * 0x8000000000000000000000000000b172 >> 127;
      if(x & 0x200 > 0) r = r * 0x800000000000000000000000000058b9 >> 127;
      if(x & 0x100 > 0) r = r * 0x80000000000000000000000000002c5c >> 127;
      if(x & 0x80 > 0) r = r * 0x8000000000000000000000000000162e >> 127;
      if(x & 0x40 > 0) r = r * 0x80000000000000000000000000000b17 >> 127;
      if(x & 0x20 > 0) r = r * 0x8000000000000000000000000000058b >> 127;
      if(x & 0x10 > 0) r = r * 0x800000000000000000000000000002c5 >> 127;
      if(x & 0x8 > 0) r = r * 0x80000000000000000000000000000162 >> 127;
      if(x & 0x4 > 0) r = r * 0x800000000000000000000000000000b1 >> 127;
      if(x & 0x2 > 0) r = r * 0x80000000000000000000000000000058 >> 127;
      if(x & 0x1 > 0) r = r * 0x8000000000000000000000000000002c >> 127; */

            r >>= 127 - (x >> 121);

            return uint128(r);
        }
    }
}