erc20-demurrage-token

ABDKMath64x64.sol (25551B)

---

 // SPDX-License-Identifier: BSD-4-Clause
 /*
  * ABDK Math 64.64 Smart Contract Library.  Copyright © 2019 by ABDK Consulting.
  * Author: Mikhail Vladimirov <mikhail.vladimirov@gmail.com>
  */
 pragma solidity ^0.8.0;
 
 /**
  * Smart contract library of mathematical functions operating with signed
  * 64.64-bit fixed point numbers.  Signed 64.64-bit fixed point number is
  * basically a simple fraction whose numerator is signed 128-bit integer and
  * denominator is 2^64.  As long as denominator is always the same, there is no
  * need to store it, thus in Solidity signed 64.64-bit fixed point numbers are
  * represented by int128 type holding only the numerator.
  */
 library ABDKMath64x64 {
   /*
    * Minimum value signed 64.64-bit fixed point number may have. 
    */
   int128 private constant MIN_64x64 = -0x80000000000000000000000000000000;
 
   /*
    * Maximum value signed 64.64-bit fixed point number may have. 
    */
   int128 private constant MAX_64x64 = 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
 
   /**
    * Convert signed 256-bit integer number into signed 64.64-bit fixed point
    * number.  Revert on overflow.
    *
    * @param x signed 256-bit integer number
    * @return signed 64.64-bit fixed point number
    */
   function fromInt (int256 x) internal pure returns (int128) {
     unchecked {
       require (x >= -0x8000000000000000 && x <= 0x7FFFFFFFFFFFFFFF);
       return int128 (x << 64);
     }
   }
 
   /**
    * Convert signed 64.64 fixed point number into signed 64-bit integer number
    * rounding down.
    *
    * @param x signed 64.64-bit fixed point number
    * @return signed 64-bit integer number
    */
   function toInt (int128 x) internal pure returns (int64) {
     unchecked {
       return int64 (x >> 64);
     }
   }
 
   /**
    * Convert unsigned 256-bit integer number into signed 64.64-bit fixed point
    * number.  Revert on overflow.
    *
    * @param x unsigned 256-bit integer number
    * @return signed 64.64-bit fixed point number
    */
   function fromUInt (uint256 x) internal pure returns (int128) {
     unchecked {
       require (x <= 0x7FFFFFFFFFFFFFFF);
       return int128 (int256 (x << 64));
     }
   }
 
   /**
    * Convert signed 64.64 fixed point number into unsigned 64-bit integer
    * number rounding down.  Revert on underflow.
    *
    * @param x signed 64.64-bit fixed point number
    * @return unsigned 64-bit integer number
    */
   function toUInt (int128 x) internal pure returns (uint64) {
     unchecked {
       require (x >= 0);
       return uint64 (uint128 (x >> 64));
     }
   }
 
   /**
    * Convert signed 128.128 fixed point number into signed 64.64-bit fixed point
    * number rounding down.  Revert on overflow.
    *
    * @param x signed 128.128-bin fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function from128x128 (int256 x) internal pure returns (int128) {
     unchecked {
       int256 result = x >> 64;
       require (result >= MIN_64x64 && result <= MAX_64x64);
       return int128 (result);
     }
   }
 
   /**
    * Convert signed 64.64 fixed point number into signed 128.128 fixed point
    * number.
    *
    * @param x signed 64.64-bit fixed point number
    * @return signed 128.128 fixed point number
    */
   function to128x128 (int128 x) internal pure returns (int256) {
     unchecked {
       return int256 (x) << 64;
     }
   }
 
   /**
    * Calculate x + y.  Revert on overflow.
    *
    * @param x signed 64.64-bit fixed point number
    * @param y signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function add (int128 x, int128 y) internal pure returns (int128) {
     unchecked {
       int256 result = int256(x) + y;
       require (result >= MIN_64x64 && result <= MAX_64x64);
       return int128 (result);
     }
   }
 
   /**
    * Calculate x - y.  Revert on overflow.
    *
    * @param x signed 64.64-bit fixed point number
    * @param y signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function sub (int128 x, int128 y) internal pure returns (int128) {
     unchecked {
       int256 result = int256(x) - y;
       require (result >= MIN_64x64 && result <= MAX_64x64);
       return int128 (result);
     }
   }
 
   /**
    * Calculate x * y rounding down.  Revert on overflow.
    *
    * @param x signed 64.64-bit fixed point number
    * @param y signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function mul (int128 x, int128 y) internal pure returns (int128) {
     unchecked {
       int256 result = int256(x) * y >> 64;
       require (result >= MIN_64x64 && result <= MAX_64x64);
       return int128 (result);
     }
   }
 
   /**
    * Calculate x * y rounding towards zero, where x is signed 64.64 fixed point
    * number and y is signed 256-bit integer number.  Revert on overflow.
    *
    * @param x signed 64.64 fixed point number
    * @param y signed 256-bit integer number
    * @return signed 256-bit integer number
    */
   function muli (int128 x, int256 y) internal pure returns (int256) {
     unchecked {
       if (x == MIN_64x64) {
         require (y >= -0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF &&
           y <= 0x1000000000000000000000000000000000000000000000000);
         return -y << 63;
       } else {
         bool negativeResult = false;
         if (x < 0) {
           x = -x;
           negativeResult = true;
         }
         if (y < 0) {
           y = -y; // We rely on overflow behavior here
           negativeResult = !negativeResult;
         }
         uint256 absoluteResult = mulu (x, uint256 (y));
         if (negativeResult) {
           require (absoluteResult <=
             0x8000000000000000000000000000000000000000000000000000000000000000);
           return -int256 (absoluteResult); // We rely on overflow behavior here
         } else {
           require (absoluteResult <=
             0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
           return int256 (absoluteResult);
         }
       }
     }
   }
 
   /**
    * Calculate x * y rounding down, where x is signed 64.64 fixed point number
    * and y is unsigned 256-bit integer number.  Revert on overflow.
    *
    * @param x signed 64.64 fixed point number
    * @param y unsigned 256-bit integer number
    * @return unsigned 256-bit integer number
    */
   function mulu (int128 x, uint256 y) internal pure returns (uint256) {
     unchecked {
       if (y == 0) return 0;
 
       require (x >= 0);
 
       uint256 lo = (uint256 (int256 (x)) * (y & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)) >> 64;
       uint256 hi = uint256 (int256 (x)) * (y >> 128);
 
       require (hi <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
       hi <<= 64;
 
       require (hi <=
         0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF - lo);
       return hi + lo;
     }
   }
 
   /**
    * Calculate x / y rounding towards zero.  Revert on overflow or when y is
    * zero.
    *
    * @param x signed 64.64-bit fixed point number
    * @param y signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function div (int128 x, int128 y) internal pure returns (int128) {
     unchecked {
       require (y != 0);
       int256 result = (int256 (x) << 64) / y;
       require (result >= MIN_64x64 && result <= MAX_64x64);
       return int128 (result);
     }
   }
 
   /**
    * Calculate x / y rounding towards zero, where x and y are signed 256-bit
    * integer numbers.  Revert on overflow or when y is zero.
    *
    * @param x signed 256-bit integer number
    * @param y signed 256-bit integer number
    * @return signed 64.64-bit fixed point number
    */
   function divi (int256 x, int256 y) internal pure returns (int128) {
     unchecked {
       require (y != 0);
 
       bool negativeResult = false;
       if (x < 0) {
         x = -x; // We rely on overflow behavior here
         negativeResult = true;
       }
       if (y < 0) {
         y = -y; // We rely on overflow behavior here
         negativeResult = !negativeResult;
       }
       uint128 absoluteResult = divuu (uint256 (x), uint256 (y));
       if (negativeResult) {
         require (absoluteResult <= 0x80000000000000000000000000000000);
         return -int128 (absoluteResult); // We rely on overflow behavior here
       } else {
         require (absoluteResult <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
         return int128 (absoluteResult); // We rely on overflow behavior here
       }
     }
   }
 
   /**
    * Calculate x / y rounding towards zero, where x and y are unsigned 256-bit
    * integer numbers.  Revert on overflow or when y is zero.
    *
    * @param x unsigned 256-bit integer number
    * @param y unsigned 256-bit integer number
    * @return signed 64.64-bit fixed point number
    */
   function divu (uint256 x, uint256 y) internal pure returns (int128) {
     unchecked {
       require (y != 0);
       uint128 result = divuu (x, y);
       require (result <= uint128 (MAX_64x64));
       return int128 (result);
     }
   }
 
   /**
    * Calculate -x.  Revert on overflow.
    *
    * @param x signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function neg (int128 x) internal pure returns (int128) {
     unchecked {
       require (x != MIN_64x64);
       return -x;
     }
   }
 
   /**
    * Calculate |x|.  Revert on overflow.
    *
    * @param x signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function abs (int128 x) internal pure returns (int128) {
     unchecked {
       require (x != MIN_64x64);
       return x < 0 ? -x : x;
     }
   }
 
   /**
    * Calculate 1 / x rounding towards zero.  Revert on overflow or when x is
    * zero.
    *
    * @param x signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function inv (int128 x) internal pure returns (int128) {
     unchecked {
       require (x != 0);
       int256 result = int256 (0x100000000000000000000000000000000) / x;
       require (result >= MIN_64x64 && result <= MAX_64x64);
       return int128 (result);
     }
   }
 
   /**
    * Calculate arithmetics average of x and y, i.e. (x + y) / 2 rounding down.
    *
    * @param x signed 64.64-bit fixed point number
    * @param y signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function avg (int128 x, int128 y) internal pure returns (int128) {
     unchecked {
       return int128 ((int256 (x) + int256 (y)) >> 1);
     }
   }
 
   /**
    * Calculate geometric average of x and y, i.e. sqrt (x * y) rounding down.
    * Revert on overflow or in case x * y is negative.
    *
    * @param x signed 64.64-bit fixed point number
    * @param y signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function gavg (int128 x, int128 y) internal pure returns (int128) {
     unchecked {
       int256 m = int256 (x) * int256 (y);
       require (m >= 0);
       require (m <
           0x4000000000000000000000000000000000000000000000000000000000000000);
       return int128 (sqrtu (uint256 (m)));
     }
   }
 
   /**
    * Calculate x^y assuming 0^0 is 1, where x is signed 64.64 fixed point number
    * and y is unsigned 256-bit integer number.  Revert on overflow.
    *
    * @param x signed 64.64-bit fixed point number
    * @param y uint256 value
    * @return signed 64.64-bit fixed point number
    */
   function pow (int128 x, uint256 y) internal pure returns (int128) {
     unchecked {
       bool negative = x < 0 && y & 1 == 1;
 
       uint256 absX = uint128 (x < 0 ? -x : x);
       uint256 absResult;
       absResult = 0x100000000000000000000000000000000;
 
       if (absX <= 0x10000000000000000) {
         absX <<= 63;
         while (y != 0) {
           if (y & 0x1 != 0) {
             absResult = absResult * absX >> 127;
           }
           absX = absX * absX >> 127;
 
           if (y & 0x2 != 0) {
             absResult = absResult * absX >> 127;
           }
           absX = absX * absX >> 127;
 
           if (y & 0x4 != 0) {
             absResult = absResult * absX >> 127;
           }
           absX = absX * absX >> 127;
 
           if (y & 0x8 != 0) {
             absResult = absResult * absX >> 127;
           }
           absX = absX * absX >> 127;
 
           y >>= 4;
         }
 
         absResult >>= 64;
       } else {
         uint256 absXShift = 63;
         if (absX < 0x1000000000000000000000000) { absX <<= 32; absXShift -= 32; }
         if (absX < 0x10000000000000000000000000000) { absX <<= 16; absXShift -= 16; }
         if (absX < 0x1000000000000000000000000000000) { absX <<= 8; absXShift -= 8; }
         if (absX < 0x10000000000000000000000000000000) { absX <<= 4; absXShift -= 4; }
         if (absX < 0x40000000000000000000000000000000) { absX <<= 2; absXShift -= 2; }
         if (absX < 0x80000000000000000000000000000000) { absX <<= 1; absXShift -= 1; }
 
         uint256 resultShift = 0;
         while (y != 0) {
           require (absXShift < 64);
 
           if (y & 0x1 != 0) {
             absResult = absResult * absX >> 127;
             resultShift += absXShift;
             if (absResult > 0x100000000000000000000000000000000) {
               absResult >>= 1;
               resultShift += 1;
             }
           }
           absX = absX * absX >> 127;
           absXShift <<= 1;
           if (absX >= 0x100000000000000000000000000000000) {
               absX >>= 1;
               absXShift += 1;
           }
 
           y >>= 1;
         }
 
         require (resultShift < 64);
         absResult >>= 64 - resultShift;
       }
       int256 result = negative ? -int256 (absResult) : int256 (absResult);
       require (result >= MIN_64x64 && result <= MAX_64x64);
       return int128 (result);
     }
   }
 
   /**
    * Calculate sqrt (x) rounding down.  Revert if x < 0.
    *
    * @param x signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function sqrt (int128 x) internal pure returns (int128) {
     unchecked {
       require (x >= 0);
       return int128 (sqrtu (uint256 (int256 (x)) << 64));
     }
   }
 
   /**
    * Calculate binary logarithm of x.  Revert if x <= 0.
    *
    * @param x signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function log_2 (int128 x) internal pure returns (int128) {
     unchecked {
       require (x > 0);
 
       int256 msb = 0;
       int256 xc = x;
       if (xc >= 0x10000000000000000) { xc >>= 64; msb += 64; }
       if (xc >= 0x100000000) { xc >>= 32; msb += 32; }
       if (xc >= 0x10000) { xc >>= 16; msb += 16; }
       if (xc >= 0x100) { xc >>= 8; msb += 8; }
       if (xc >= 0x10) { xc >>= 4; msb += 4; }
       if (xc >= 0x4) { xc >>= 2; msb += 2; }
       if (xc >= 0x2) msb += 1;  // No need to shift xc anymore
 
       int256 result = msb - 64 << 64;
       uint256 ux = uint256 (int256 (x)) << uint256 (127 - msb);
       for (int256 bit = 0x8000000000000000; bit > 0; bit >>= 1) {
         ux *= ux;
         uint256 b = ux >> 255;
         ux >>= 127 + b;
         result += bit * int256 (b);
       }
 
       return int128 (result);
     }
   }
 
   /**
    * Calculate natural logarithm of x.  Revert if x <= 0.
    *
    * @param x signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function ln (int128 x) internal pure returns (int128) {
     unchecked {
       require (x > 0);
 
       return int128 (int256 (
           uint256 (int256 (log_2 (x))) * 0xB17217F7D1CF79ABC9E3B39803F2F6AF >> 128));
     }
   }
 
   /**
    * Calculate binary exponent of x.  Revert on overflow.
    *
    * @param x signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function exp_2 (int128 x) internal pure returns (int128) {
     unchecked {
       require (x < 0x400000000000000000); // Overflow
 
       if (x < -0x400000000000000000) return 0; // Underflow
 
       uint256 result = 0x80000000000000000000000000000000;
 
       if (x & 0x8000000000000000 > 0)
         result = result * 0x16A09E667F3BCC908B2FB1366EA957D3E >> 128;
       if (x & 0x4000000000000000 > 0)
         result = result * 0x1306FE0A31B7152DE8D5A46305C85EDEC >> 128;
       if (x & 0x2000000000000000 > 0)
         result = result * 0x1172B83C7D517ADCDF7C8C50EB14A791F >> 128;
       if (x & 0x1000000000000000 > 0)
         result = result * 0x10B5586CF9890F6298B92B71842A98363 >> 128;
       if (x & 0x800000000000000 > 0)
         result = result * 0x1059B0D31585743AE7C548EB68CA417FD >> 128;
       if (x & 0x400000000000000 > 0)
         result = result * 0x102C9A3E778060EE6F7CACA4F7A29BDE8 >> 128;
       if (x & 0x200000000000000 > 0)
         result = result * 0x10163DA9FB33356D84A66AE336DCDFA3F >> 128;
       if (x & 0x100000000000000 > 0)
         result = result * 0x100B1AFA5ABCBED6129AB13EC11DC9543 >> 128;
       if (x & 0x80000000000000 > 0)
         result = result * 0x10058C86DA1C09EA1FF19D294CF2F679B >> 128;
       if (x & 0x40000000000000 > 0)
         result = result * 0x1002C605E2E8CEC506D21BFC89A23A00F >> 128;
       if (x & 0x20000000000000 > 0)
         result = result * 0x100162F3904051FA128BCA9C55C31E5DF >> 128;
       if (x & 0x10000000000000 > 0)
         result = result * 0x1000B175EFFDC76BA38E31671CA939725 >> 128;
       if (x & 0x8000000000000 > 0)
         result = result * 0x100058BA01FB9F96D6CACD4B180917C3D >> 128;
       if (x & 0x4000000000000 > 0)
         result = result * 0x10002C5CC37DA9491D0985C348C68E7B3 >> 128;
       if (x & 0x2000000000000 > 0)
         result = result * 0x1000162E525EE054754457D5995292026 >> 128;
       if (x & 0x1000000000000 > 0)
         result = result * 0x10000B17255775C040618BF4A4ADE83FC >> 128;
       if (x & 0x800000000000 > 0)
         result = result * 0x1000058B91B5BC9AE2EED81E9B7D4CFAB >> 128;
       if (x & 0x400000000000 > 0)
         result = result * 0x100002C5C89D5EC6CA4D7C8ACC017B7C9 >> 128;
       if (x & 0x200000000000 > 0)
         result = result * 0x10000162E43F4F831060E02D839A9D16D >> 128;
       if (x & 0x100000000000 > 0)
         result = result * 0x100000B1721BCFC99D9F890EA06911763 >> 128;
       if (x & 0x80000000000 > 0)
         result = result * 0x10000058B90CF1E6D97F9CA14DBCC1628 >> 128;
       if (x & 0x40000000000 > 0)
         result = result * 0x1000002C5C863B73F016468F6BAC5CA2B >> 128;
       if (x & 0x20000000000 > 0)
         result = result * 0x100000162E430E5A18F6119E3C02282A5 >> 128;
       if (x & 0x10000000000 > 0)
         result = result * 0x1000000B1721835514B86E6D96EFD1BFE >> 128;
       if (x & 0x8000000000 > 0)
         result = result * 0x100000058B90C0B48C6BE5DF846C5B2EF >> 128;
       if (x & 0x4000000000 > 0)
         result = result * 0x10000002C5C8601CC6B9E94213C72737A >> 128;
       if (x & 0x2000000000 > 0)
         result = result * 0x1000000162E42FFF037DF38AA2B219F06 >> 128;
       if (x & 0x1000000000 > 0)
         result = result * 0x10000000B17217FBA9C739AA5819F44F9 >> 128;
       if (x & 0x800000000 > 0)
         result = result * 0x1000000058B90BFCDEE5ACD3C1CEDC823 >> 128;
       if (x & 0x400000000 > 0)
         result = result * 0x100000002C5C85FE31F35A6A30DA1BE50 >> 128;
       if (x & 0x200000000 > 0)
         result = result * 0x10000000162E42FF0999CE3541B9FFFCF >> 128;
       if (x & 0x100000000 > 0)
         result = result * 0x100000000B17217F80F4EF5AADDA45554 >> 128;
       if (x & 0x80000000 > 0)
         result = result * 0x10000000058B90BFBF8479BD5A81B51AD >> 128;
       if (x & 0x40000000 > 0)
         result = result * 0x1000000002C5C85FDF84BD62AE30A74CC >> 128;
       if (x & 0x20000000 > 0)
         result = result * 0x100000000162E42FEFB2FED257559BDAA >> 128;
       if (x & 0x10000000 > 0)
         result = result * 0x1000000000B17217F7D5A7716BBA4A9AE >> 128;
       if (x & 0x8000000 > 0)
         result = result * 0x100000000058B90BFBE9DDBAC5E109CCE >> 128;
       if (x & 0x4000000 > 0)
         result = result * 0x10000000002C5C85FDF4B15DE6F17EB0D >> 128;
       if (x & 0x2000000 > 0)
         result = result * 0x1000000000162E42FEFA494F1478FDE05 >> 128;
       if (x & 0x1000000 > 0)
         result = result * 0x10000000000B17217F7D20CF927C8E94C >> 128;
       if (x & 0x800000 > 0)
         result = result * 0x1000000000058B90BFBE8F71CB4E4B33D >> 128;
       if (x & 0x400000 > 0)
         result = result * 0x100000000002C5C85FDF477B662B26945 >> 128;
       if (x & 0x200000 > 0)
         result = result * 0x10000000000162E42FEFA3AE53369388C >> 128;
       if (x & 0x100000 > 0)
         result = result * 0x100000000000B17217F7D1D351A389D40 >> 128;
       if (x & 0x80000 > 0)
         result = result * 0x10000000000058B90BFBE8E8B2D3D4EDE >> 128;
       if (x & 0x40000 > 0)
         result = result * 0x1000000000002C5C85FDF4741BEA6E77E >> 128;
       if (x & 0x20000 > 0)
         result = result * 0x100000000000162E42FEFA39FE95583C2 >> 128;
       if (x & 0x10000 > 0)
         result = result * 0x1000000000000B17217F7D1CFB72B45E1 >> 128;
       if (x & 0x8000 > 0)
         result = result * 0x100000000000058B90BFBE8E7CC35C3F0 >> 128;
       if (x & 0x4000 > 0)
         result = result * 0x10000000000002C5C85FDF473E242EA38 >> 128;
       if (x & 0x2000 > 0)
         result = result * 0x1000000000000162E42FEFA39F02B772C >> 128;
       if (x & 0x1000 > 0)
         result = result * 0x10000000000000B17217F7D1CF7D83C1A >> 128;
       if (x & 0x800 > 0)
         result = result * 0x1000000000000058B90BFBE8E7BDCBE2E >> 128;
       if (x & 0x400 > 0)
         result = result * 0x100000000000002C5C85FDF473DEA871F >> 128;
       if (x & 0x200 > 0)
         result = result * 0x10000000000000162E42FEFA39EF44D91 >> 128;
       if (x & 0x100 > 0)
         result = result * 0x100000000000000B17217F7D1CF79E949 >> 128;
       if (x & 0x80 > 0)
         result = result * 0x10000000000000058B90BFBE8E7BCE544 >> 128;
       if (x & 0x40 > 0)
         result = result * 0x1000000000000002C5C85FDF473DE6ECA >> 128;
       if (x & 0x20 > 0)
         result = result * 0x100000000000000162E42FEFA39EF366F >> 128;
       if (x & 0x10 > 0)
         result = result * 0x1000000000000000B17217F7D1CF79AFA >> 128;
       if (x & 0x8 > 0)
         result = result * 0x100000000000000058B90BFBE8E7BCD6D >> 128;
       if (x & 0x4 > 0)
         result = result * 0x10000000000000002C5C85FDF473DE6B2 >> 128;
       if (x & 0x2 > 0)
         result = result * 0x1000000000000000162E42FEFA39EF358 >> 128;
       if (x & 0x1 > 0)
         result = result * 0x10000000000000000B17217F7D1CF79AB >> 128;
 
       result >>= uint256 (int256 (63 - (x >> 64)));
       require (result <= uint256 (int256 (MAX_64x64)));
 
       return int128 (int256 (result));
     }
   }
 
   /**
    * Calculate natural exponent of x.  Revert on overflow.
    *
    * @param x signed 64.64-bit fixed point number
    * @return signed 64.64-bit fixed point number
    */
   function exp (int128 x) internal pure returns (int128) {
     unchecked {
       require (x < 0x400000000000000000); // Overflow
 
       if (x < -0x400000000000000000) return 0; // Underflow
 
       return exp_2 (
           int128 (int256 (x) * 0x171547652B82FE1777D0FFDA0D23A7D12 >> 128));
     }
   }
 
   /**
    * Calculate x / y rounding towards zero, where x and y are unsigned 256-bit
    * integer numbers.  Revert on overflow or when y is zero.
    *
    * @param x unsigned 256-bit integer number
    * @param y unsigned 256-bit integer number
    * @return unsigned 64.64-bit fixed point number
    */
   function divuu (uint256 x, uint256 y) private pure returns (uint128) {
     unchecked {
       require (y != 0);
 
       uint256 result;
 
       if (x <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)
         result = (x << 64) / y;
       else {
         uint256 msb = 192;
         uint256 xc = x >> 192;
         if (xc >= 0x100000000) { xc >>= 32; msb += 32; }
         if (xc >= 0x10000) { xc >>= 16; msb += 16; }
         if (xc >= 0x100) { xc >>= 8; msb += 8; }
         if (xc >= 0x10) { xc >>= 4; msb += 4; }
         if (xc >= 0x4) { xc >>= 2; msb += 2; }
         if (xc >= 0x2) msb += 1;  // No need to shift xc anymore
 
         result = (x << 255 - msb) / ((y - 1 >> msb - 191) + 1);
         require (result <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
 
         uint256 hi = result * (y >> 128);
         uint256 lo = result * (y & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
 
         uint256 xh = x >> 192;
         uint256 xl = x << 64;
 
         if (xl < lo) xh -= 1;
         xl -= lo; // We rely on overflow behavior here
         lo = hi << 128;
         if (xl < lo) xh -= 1;
         xl -= lo; // We rely on overflow behavior here
 
         assert (xh == hi >> 128);
 
         result += xl / y;
       }
 
       require (result <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
       return uint128 (result);
     }
   }
 
   /**
    * Calculate sqrt (x) rounding down, where x is unsigned 256-bit integer
    * number.
    *
    * @param x unsigned 256-bit integer number
    * @return unsigned 128-bit integer number
    */
   function sqrtu (uint256 x) private pure returns (uint128) {
     unchecked {
       if (x == 0) return 0;
       else {
         uint256 xx = x;
         uint256 r = 1;
         if (xx >= 0x100000000000000000000000000000000) { xx >>= 128; r <<= 64; }
         if (xx >= 0x10000000000000000) { xx >>= 64; r <<= 32; }
         if (xx >= 0x100000000) { xx >>= 32; r <<= 16; }
         if (xx >= 0x10000) { xx >>= 16; r <<= 8; }
         if (xx >= 0x100) { xx >>= 8; r <<= 4; }
         if (xx >= 0x10) { xx >>= 4; r <<= 2; }
         if (xx >= 0x4) { r <<= 1; }
         r = (r + x / r) >> 1;
         r = (r + x / r) >> 1;
         r = (r + x / r) >> 1;
         r = (r + x / r) >> 1;
         r = (r + x / r) >> 1;
         r = (r + x / r) >> 1;
         r = (r + x / r) >> 1; // Seven iterations should be enough
         uint256 r1 = x / r;
         return uint128 (r < r1 ? r : r1);
       }
     }
   }
 }