// 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 >= 0x8) { 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);
      }
    }
  }
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts v4.4.1 (utils/Context.sol)

pragma solidity ^0.8.0;

/**
 * @dev Provides information about the current execution context, including the
 * sender of the transaction and its data. While these are generally available
 * via msg.sender and msg.data, they should not be accessed in such a direct
 * manner, since when dealing with meta-transactions the account sending and
 * paying for execution may not be the actual sender (as far as an application
 * is concerned).
 *
 * This contract is only required for intermediate, library-like contracts.
 */
abstract contract Context {
    function _msgSender() internal view virtual returns (address) {
        return msg.sender;
    }

    function _msgData() internal view virtual returns (bytes calldata) {
        return msg.data;
    }
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.7.0) (token/ERC20/ERC20.sol)

pragma solidity ^0.8.0;

import "./IERC20.sol";
import "./extensions/IERC20Metadata.sol";
import "../../utils/Context.sol";

/**
 * @dev Implementation of the {IERC20} interface.
 *
 * This implementation is agnostic to the way tokens are created. This means
 * that a supply mechanism has to be added in a derived contract using {_mint}.
 * For a generic mechanism see {ERC20PresetMinterPauser}.
 *
 * TIP: For a detailed writeup see our guide
 * https://forum.zeppelin.solutions/t/how-to-implement-erc20-supply-mechanisms/226[How
 * to implement supply mechanisms].
 *
 * We have followed general OpenZeppelin Contracts guidelines: functions revert
 * instead returning `false` on failure. This behavior is nonetheless
 * conventional and does not conflict with the expectations of ERC20
 * applications.
 *
 * Additionally, an {Approval} event is emitted on calls to {transferFrom}.
 * This allows applications to reconstruct the allowance for all accounts just
 * by listening to said events. Other implementations of the EIP may not emit
 * these events, as it isn't required by the specification.
 *
 * Finally, the non-standard {decreaseAllowance} and {increaseAllowance}
 * functions have been added to mitigate the well-known issues around setting
 * allowances. See {IERC20-approve}.
 */
contract ERC20 is Context, IERC20, IERC20Metadata {
    mapping(address => uint256) private _balances;

    mapping(address => mapping(address => uint256)) private _allowances;

    uint256 private _totalSupply;

    string private _name;
    string private _symbol;

    /**
     * @dev Sets the values for {name} and {symbol}.
     *
     * The default value of {decimals} is 18. To select a different value for
     * {decimals} you should overload it.
     *
     * All two of these values are immutable: they can only be set once during
     * construction.
     */
    constructor(string memory name_, string memory symbol_) {
        _name = name_;
        _symbol = symbol_;
    }

    /**
     * @dev Returns the name of the token.
     */
    function name() public view virtual override returns (string memory) {
        return _name;
    }

    /**
     * @dev Returns the symbol of the token, usually a shorter version of the
     * name.
     */
    function symbol() public view virtual override returns (string memory) {
        return _symbol;
    }

    /**
     * @dev Returns the number of decimals used to get its user representation.
     * For example, if `decimals` equals `2`, a balance of `505` tokens should
     * be displayed to a user as `5.05` (`505 / 10 ** 2`).
     *
     * Tokens usually opt for a value of 18, imitating the relationship between
     * Ether and Wei. This is the value {ERC20} uses, unless this function is
     * overridden;
     *
     * NOTE: This information is only used for _display_ purposes: it in
     * no way affects any of the arithmetic of the contract, including
     * {IERC20-balanceOf} and {IERC20-transfer}.
     */
    function decimals() public view virtual override returns (uint8) {
        return 18;
    }

    /**
     * @dev See {IERC20-totalSupply}.
     */
    function totalSupply() public view virtual override returns (uint256) {
        return _totalSupply;
    }

    /**
     * @dev See {IERC20-balanceOf}.
     */
    function balanceOf(address account) public view virtual override returns (uint256) {
        return _balances[account];
    }

    /**
     * @dev See {IERC20-transfer}.
     *
     * Requirements:
     *
     * - `to` cannot be the zero address.
     * - the caller must have a balance of at least `amount`.
     */
    function transfer(address to, uint256 amount) public virtual override returns (bool) {
        address owner = _msgSender();
        _transfer(owner, to, amount);
        return true;
    }

    /**
     * @dev See {IERC20-allowance}.
     */
    function allowance(address owner, address spender) public view virtual override returns (uint256) {
        return _allowances[owner][spender];
    }

    /**
     * @dev See {IERC20-approve}.
     *
     * NOTE: If `amount` is the maximum `uint256`, the allowance is not updated on
     * `transferFrom`. This is semantically equivalent to an infinite approval.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     */
    function approve(address spender, uint256 amount) public virtual override returns (bool) {
        address owner = _msgSender();
        _approve(owner, spender, amount);
        return true;
    }

    /**
     * @dev See {IERC20-transferFrom}.
     *
     * Emits an {Approval} event indicating the updated allowance. This is not
     * required by the EIP. See the note at the beginning of {ERC20}.
     *
     * NOTE: Does not update the allowance if the current allowance
     * is the maximum `uint256`.
     *
     * Requirements:
     *
     * - `from` and `to` cannot be the zero address.
     * - `from` must have a balance of at least `amount`.
     * - the caller must have allowance for ``from``'s tokens of at least
     * `amount`.
     */
    function transferFrom(
        address from,
        address to,
        uint256 amount
    ) public virtual override returns (bool) {
        address spender = _msgSender();
        _spendAllowance(from, spender, amount);
        _transfer(from, to, amount);
        return true;
    }

    /**
     * @dev Atomically increases the allowance granted to `spender` by the caller.
     *
     * This is an alternative to {approve} that can be used as a mitigation for
     * problems described in {IERC20-approve}.
     *
     * Emits an {Approval} event indicating the updated allowance.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     */
    function increaseAllowance(address spender, uint256 addedValue) public virtual returns (bool) {
        address owner = _msgSender();
        _approve(owner, spender, allowance(owner, spender) + addedValue);
        return true;
    }

    /**
     * @dev Atomically decreases the allowance granted to `spender` by the caller.
     *
     * This is an alternative to {approve} that can be used as a mitigation for
     * problems described in {IERC20-approve}.
     *
     * Emits an {Approval} event indicating the updated allowance.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     * - `spender` must have allowance for the caller of at least
     * `subtractedValue`.
     */
    function decreaseAllowance(address spender, uint256 subtractedValue) public virtual returns (bool) {
        address owner = _msgSender();
        uint256 currentAllowance = allowance(owner, spender);
        require(currentAllowance >= subtractedValue, "ERC20: decreased allowance below zero");
        unchecked {
            _approve(owner, spender, currentAllowance - subtractedValue);
        }

        return true;
    }

    /**
     * @dev Moves `amount` of tokens from `from` to `to`.
     *
     * This internal function is equivalent to {transfer}, and can be used to
     * e.g. implement automatic token fees, slashing mechanisms, etc.
     *
     * Emits a {Transfer} event.
     *
     * Requirements:
     *
     * - `from` cannot be the zero address.
     * - `to` cannot be the zero address.
     * - `from` must have a balance of at least `amount`.
     */
    function _transfer(
        address from,
        address to,
        uint256 amount
    ) internal virtual {
        require(from != address(0), "ERC20: transfer from the zero address");
        require(to != address(0), "ERC20: transfer to the zero address");

        _beforeTokenTransfer(from, to, amount);

        uint256 fromBalance = _balances[from];
        require(fromBalance >= amount, "ERC20: transfer amount exceeds balance");
        unchecked {
            _balances[from] = fromBalance - amount;
        }
        _balances[to] += amount;

        emit Transfer(from, to, amount);

        _afterTokenTransfer(from, to, amount);
    }

    /** @dev Creates `amount` tokens and assigns them to `account`, increasing
     * the total supply.
     *
     * Emits a {Transfer} event with `from` set to the zero address.
     *
     * Requirements:
     *
     * - `account` cannot be the zero address.
     */
    function _mint(address account, uint256 amount) internal virtual {
        require(account != address(0), "ERC20: mint to the zero address");

        _beforeTokenTransfer(address(0), account, amount);

        _totalSupply += amount;
        _balances[account] += amount;
        emit Transfer(address(0), account, amount);

        _afterTokenTransfer(address(0), account, amount);
    }

    /**
     * @dev Destroys `amount` tokens from `account`, reducing the
     * total supply.
     *
     * Emits a {Transfer} event with `to` set to the zero address.
     *
     * Requirements:
     *
     * - `account` cannot be the zero address.
     * - `account` must have at least `amount` tokens.
     */
    function _burn(address account, uint256 amount) internal virtual {
        require(account != address(0), "ERC20: burn from the zero address");

        _beforeTokenTransfer(account, address(0), amount);

        uint256 accountBalance = _balances[account];
        require(accountBalance >= amount, "ERC20: burn amount exceeds balance");
        unchecked {
            _balances[account] = accountBalance - amount;
        }
        _totalSupply -= amount;

        emit Transfer(account, address(0), amount);

        _afterTokenTransfer(account, address(0), amount);
    }

    /**
     * @dev Sets `amount` as the allowance of `spender` over the `owner` s tokens.
     *
     * This internal function is equivalent to `approve`, and can be used to
     * e.g. set automatic allowances for certain subsystems, etc.
     *
     * Emits an {Approval} event.
     *
     * Requirements:
     *
     * - `owner` cannot be the zero address.
     * - `spender` cannot be the zero address.
     */
    function _approve(
        address owner,
        address spender,
        uint256 amount
    ) internal virtual {
        require(owner != address(0), "ERC20: approve from the zero address");
        require(spender != address(0), "ERC20: approve to the zero address");

        _allowances[owner][spender] = amount;
        emit Approval(owner, spender, amount);
    }

    /**
     * @dev Updates `owner` s allowance for `spender` based on spent `amount`.
     *
     * Does not update the allowance amount in case of infinite allowance.
     * Revert if not enough allowance is available.
     *
     * Might emit an {Approval} event.
     */
    function _spendAllowance(
        address owner,
        address spender,
        uint256 amount
    ) internal virtual {
        uint256 currentAllowance = allowance(owner, spender);
        if (currentAllowance != type(uint256).max) {
            require(currentAllowance >= amount, "ERC20: insufficient allowance");
            unchecked {
                _approve(owner, spender, currentAllowance - amount);
            }
        }
    }

    /**
     * @dev Hook that is called before any transfer of tokens. This includes
     * minting and burning.
     *
     * Calling conditions:
     *
     * - when `from` and `to` are both non-zero, `amount` of ``from``'s tokens
     * will be transferred to `to`.
     * - when `from` is zero, `amount` tokens will be minted for `to`.
     * - when `to` is zero, `amount` of ``from``'s tokens will be burned.
     * - `from` and `to` are never both zero.
     *
     * To learn more about hooks, head to xref:ROOT:extending-contracts.adoc#using-hooks[Using Hooks].
     */
    function _beforeTokenTransfer(
        address from,
        address to,
        uint256 amount
    ) internal virtual {}

    /**
     * @dev Hook that is called after any transfer of tokens. This includes
     * minting and burning.
     *
     * Calling conditions:
     *
     * - when `from` and `to` are both non-zero, `amount` of ``from``'s tokens
     * has been transferred to `to`.
     * - when `from` is zero, `amount` tokens have been minted for `to`.
     * - when `to` is zero, `amount` of ``from``'s tokens have been burned.
     * - `from` and `to` are never both zero.
     *
     * To learn more about hooks, head to xref:ROOT:extending-contracts.adoc#using-hooks[Using Hooks].
     */
    function _afterTokenTransfer(
        address from,
        address to,
        uint256 amount
    ) internal virtual {}
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts v4.4.1 (utils/introspection/IERC165.sol)

pragma solidity ^0.8.0;

/**
 * @dev Interface of the ERC165 standard, as defined in the
 * https://eips.ethereum.org/EIPS/eip-165[EIP].
 *
 * Implementers can declare support of contract interfaces, which can then be
 * queried by others ({ERC165Checker}).
 *
 * For an implementation, see {ERC165}.
 */
interface IERC165 {
    /**
     * @dev Returns true if this contract implements the interface defined by
     * `interfaceId`. See the corresponding
     * https://eips.ethereum.org/EIPS/eip-165#how-interfaces-are-identified[EIP section]
     * to learn more about how these ids are created.
     *
     * This function call must use less than 30 000 gas.
     */
    function supportsInterface(bytes4 interfaceId) external view returns (bool);
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.6.0) (token/ERC20/IERC20.sol)

pragma solidity ^0.8.0;

/**
 * @dev Interface of the ERC20 standard as defined in the EIP.
 */
interface IERC20 {
    /**
     * @dev Emitted when `value` tokens are moved from one account (`from`) to
     * another (`to`).
     *
     * Note that `value` may be zero.
     */
    event Transfer(address indexed from, address indexed to, uint256 value);

    /**
     * @dev Emitted when the allowance of a `spender` for an `owner` is set by
     * a call to {approve}. `value` is the new allowance.
     */
    event Approval(address indexed owner, address indexed spender, uint256 value);

    /**
     * @dev Returns the amount of tokens in existence.
     */
    function totalSupply() external view returns (uint256);

    /**
     * @dev Returns the amount of tokens owned by `account`.
     */
    function balanceOf(address account) external view returns (uint256);

    /**
     * @dev Moves `amount` tokens from the caller's account to `to`.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * Emits a {Transfer} event.
     */
    function transfer(address to, uint256 amount) external returns (bool);

    /**
     * @dev Returns the remaining number of tokens that `spender` will be
     * allowed to spend on behalf of `owner` through {transferFrom}. This is
     * zero by default.
     *
     * This value changes when {approve} or {transferFrom} are called.
     */
    function allowance(address owner, address spender) external view returns (uint256);

    /**
     * @dev Sets `amount` as the allowance of `spender` over the caller's tokens.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * IMPORTANT: Beware that changing an allowance with this method brings the risk
     * that someone may use both the old and the new allowance by unfortunate
     * transaction ordering. One possible solution to mitigate this race
     * condition is to first reduce the spender's allowance to 0 and set the
     * desired value afterwards:
     * https://github.com/ethereum/EIPs/issues/20#issuecomment-263524729
     *
     * Emits an {Approval} event.
     */
    function approve(address spender, uint256 amount) external returns (bool);

    /**
     * @dev Moves `amount` tokens from `from` to `to` using the
     * allowance mechanism. `amount` is then deducted from the caller's
     * allowance.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * Emits a {Transfer} event.
     */
    function transferFrom(
        address from,
        address to,
        uint256 amount
    ) external returns (bool);
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts v4.4.1 (token/ERC20/extensions/IERC20Metadata.sol)

pragma solidity ^0.8.0;

import "../IERC20.sol";

/**
 * @dev Interface for the optional metadata functions from the ERC20 standard.
 *
 * _Available since v4.1._
 */
interface IERC20Metadata is IERC20 {
    /**
     * @dev Returns the name of the token.
     */
    function name() external view returns (string memory);

    /**
     * @dev Returns the symbol of the token.
     */
    function symbol() external view returns (string memory);

    /**
     * @dev Returns the decimals places of the token.
     */
    function decimals() external view returns (uint8);
}

{
  "compilationTarget": {
    "/contracts/XENCrypto.sol": "XENCrypto"
  },
  "evmVersion": "london",
  "libraries": {
    "/contracts/Math.sol:Math": "0xeb585163debb1e637c6d617de3bef99347cd75c8"
  },
  "metadata": {
    "bytecodeHash": "ipfs"
  },
  "optimizer": {
    "enabled": true,
    "runs": 20
  },
  "remappings": []
}