A prototype implementation of excel's PMT function in Solidity, using the ABDKMath library for 64 bit fixed numbers. Verified to work with principal amounts with 18 decimals (e.g. Dai contract)

pmt-solidity.sol

pragma solidity ^0.5.7;

contract PMT {
    using ABDKMath64x64 for uint128;
    
    // rate is in basis points (0.5% = 50)
    function pmt(int128 rate, uint256 numPayments, int128 principal) 
        public 
        returns (int128) 
    {
        // Step 1: Calculate normalized rate by converting basis point to decimal
        int128 one64x64 = 0x10000000000000000;
        int128 rateDivisor = 10000;
        int128 rate64x64 = ABDKMath64x64.div(rate,rateDivisor);
        
        // Step 2: Calculate Numerator
        int128 numerator = ABDKMath64x64.mul(principal,rate64x64);
        
        // Step 3: Calculate discount series
        int128 discount = ABDKMath64x64.add(one64x64,rate64x64);
        int128 inverseDiscount = ABDKMath64x64.div(one64x64,discount);
        int128 series = ABDKMath64x64.pow(inverseDiscount,numPayments);
        
        // Step 4: Denominator
        int128 denominator = ABDKMath64x64.sub(one64x64,series);
        
        // Step 5: Calculate PMT
        int128 pmt = ABDKMath64x64.div(numerator,denominator);
        
        // Step 4: Denominator
        return pmt;
    }
}

/*
 * ABDK Math 64.64 Smart Contract Library.  Copyright © 2019 by ABDK Consulting.
 * Author: Mikhail Vladimirov <[email protected]>
 */

/**
 * 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) {
    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) {
    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) {
    require (x <= 0x7FFFFFFFFFFFFFFF);
    return int128 (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) {
    require (x >= 0);
    return uint64 (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) {
    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) {
    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) {
    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) {
    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) {
    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) {
    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) {
    if (y == 0) return 0;

    require (x >= 0);

    uint256 lo = (uint256 (x) * (y & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)) >> 64;
    uint256 hi = uint256 (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) {
    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) {
    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) {
    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) {
    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) {
    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) {
    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) {
    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) {
    int256 m = int256 (x) * int256 (y);
    require (m >= 0);
    require (m <
        0x4000000000000000000000000000000000000000000000000000000000000000);
    return int128 (sqrtu (uint256 (m), uint256 (x) + uint256 (y) >> 1));
  }

  /**
   * 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) {
    uint256 absoluteResult;
    bool negativeResult = false;
    if (x >= 0) {
      absoluteResult = powu (uint256 (x) << 63, y);
    } else {
      // We rely on overflow behavior here
      absoluteResult = powu (uint256 (uint128 (-x)) << 63, y);
      negativeResult = y & 1 > 0;
    }

    absoluteResult >>= 63;

    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 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) {
    require (x >= 0);
    return int128 (sqrtu (uint256 (x) << 64, 0x10000000000000000));
  }

  /**
   * 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) {
    require (x > 0);

    int128 a = 0;
    int128 b = 126;
    while (a < b) {
      int128 m = a + b >> 1;
      int128 t = x >> m;
      if (t == 0) b = m - 1;
      else if (t > 1) a = m + 1;
      else {
        a = m;
        break;
      }
    }

    int128 result = a - 64 << 64;
    uint256 ux = uint256 (x) << 127 - a;
    for (int128 bit = 0x8000000000000000; bit > 0; bit >>= 1) {
      ux *= ux;
      if (ux >=
        0x8000000000000000000000000000000000000000000000000000000000000000) {
        ux >>= 128;
        result += bit;
      } else ux >>= 127;
    }

    return 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) {
    require (x > 0);

    return int128 (
        uint256 (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) {
    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 >>= 63 - (x >> 64);
    require (result <= uint256 (MAX_64x64));

    return int128 (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) {
    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) {
    require (y != 0);

    uint256 result;

    if (x <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)
      result = (x << 64) / y;
    else {
      uint256 a = 192;
      uint256 b = 255;
      while (a < b) {
        uint256 m = a + b >> 1;
        uint256 t = x >> m;
        if (t == 0) b = m - 1;
        else if (t > 1) a = m + 1;
        else {
          a = m;
          break;
        }
      }

      result = (x << 255 - a) / ((y - 1 >> a - 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 x^y assuming 0^0 is 1, where x is unsigned 129.127 fixed point
   * number and y is unsigned 256-bit integer number.  Revert on overflow.
   *
   * @param x unsigned 129.127-bit fixed point number
   * @param y uint256 value
   * @return unsigned 129.127-bit fixed point number
   */
  function powu (uint256 x, uint256 y) private pure returns (uint256) {
    if (y == 0) return 0x80000000000000000000000000000000;
    else if (x == 0) return 0;
    else {
      uint256 a = 0;
      uint256 b = 255;
      while (a < b) {
        uint256 m = a + b >> 1;
        uint256 t = x >> m;
        if (t == 0) b = m - 1;
        else if (t > 1) a = m + 1;
        else {
          a = m;
          break;
        }
      }

      int256 xe = int256 (a) - 127;
      if (xe > 0) x >>= xe;
      else x <<= -xe;

      uint256 result = 0x80000000000000000000000000000000;
      int256 re = 0;

      while (y > 0) {
        if (y & 1 > 0) {
          result = result * x;
          y -= 1;
          re += xe;
          if (result >=
            0x8000000000000000000000000000000000000000000000000000000000000000) {
            result >>= 128;
            re += 1;
          } else result >>= 127;
          if (re < -127) return 0; // Underflow
          require (re < 128); // Overflow
        } else {
          x = x * x;
          y >>= 1;
          xe <<= 1;
          if (x >=
            0x8000000000000000000000000000000000000000000000000000000000000000) {
            x >>= 128;
            xe += 1;
          } else x >>= 127;
          if (xe < -127) return 0; // Underflow
          require (xe < 128); // Overflow
        }
      }

      if (re > 0) result <<= re;
      else if (re < 0) result >>= -re;

      return 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, uint256 r) private pure returns (uint128) {
    if (x == 0) return 0;
    else {
      require (r > 0);
      while (true) {
        uint256 rr = x / r;
        if (r == rr || r + 1 == rr) return uint128 (r);
        else if (r == rr + 1) return uint128 (rr);
        r = r + rr + 1 >> 1;
      }
    }
  }
}