aave-protocol-v2/contracts/libraries/math/MathUtils.sol

// SPDX-License-Identifier: agpl-3.0
pragma solidity ^0.6.8;

import {SafeMath} from '@openzeppelin/contracts/math/SafeMath.sol';
import {WadRayMath} from './WadRayMath.sol';

library MathUtils {
  using SafeMath for uint256;
  using WadRayMath for uint256;

  uint256 internal constant SECONDS_PER_YEAR = 365 days;

  /**
   * @dev function to calculate the interest using a linear interest rate formula
   * @param _rate the interest rate, in ray
   * @param _lastUpdateTimestamp the timestamp of the last update of the interest
   * @return the interest rate linearly accumulated during the timeDelta, in ray
   **/

  function calculateLinearInterest(uint256 _rate, uint40 _lastUpdateTimestamp)
    internal
    view
    returns (uint256)
  {
    //solium-disable-next-line
    uint256 timeDifference = block.timestamp.sub(uint256(_lastUpdateTimestamp));

    uint256 timeDelta = timeDifference.wadToRay().rayDiv(SECONDS_PER_YEAR.wadToRay());

    return _rate.rayMul(timeDelta).add(WadRayMath.ray());
  }

  /**
   * @dev function to calculate the interest using a compounded interest rate formula.
   * To avoid expensive exponentiation, the calculation is performed using a binomial approximation:
   *
   *  (1+x)^n = 1+n*x+[n/2*(n-1)]*x^2+[n/6*(n-1)*(n-2)*x^3...
   *
   * The approximation slightly underpays liquidity providers, with the advantage of great gas cost reductions.
   * The whitepaper contains reference to the approximation and a table showing the margin of error per different time periods.
   *
   * @param _rate the interest rate, in ray
   * @param _lastUpdateTimestamp the timestamp of the last update of the interest
   * @return the interest rate compounded during the timeDelta, in ray
   **/
  function calculateCompoundedInterest(uint256 _rate, uint40 _lastUpdateTimestamp)
    internal
    view
    returns (uint256)
  {
    //solium-disable-next-line
    uint256 exp = block.timestamp.sub(uint256(_lastUpdateTimestamp));

    if (exp == 0) {
      return WadRayMath.ray();
    }

    uint256 expMinusOne = exp.sub(1);

    uint256 expMinusTwo = exp > 2 ? exp.sub(2) : 0;

    uint256 ratePerSecond = _rate.div(31536000);

    uint256 basePowerTwo = ratePerSecond.rayMul(ratePerSecond);
    uint256 basePowerThree = basePowerTwo.rayMul(ratePerSecond);

    uint256 secondTerm = exp.mul(expMinusOne).mul(basePowerTwo).div(2);
    uint256 thirdTerm = exp.mul(expMinusOne).mul(expMinusTwo).mul(basePowerThree).div(6);

    return WadRayMath.ray().add(ratePerSecond.mul(exp)).add(secondTerm).add(thirdTerm);
  }
}