cDAI_interest.sol

/*
 * @dev Calculates the utilization and borrow rates for use by getBorrowRate function
 */
function getUtilizationAndAnnualBorrowRate(uint cash, uint borrows) view internal returns (IRError, Exp memory, Exp memory) {
    (IRError err0, Exp memory utilizationRate) = getUtilizationRate(cash, borrows);
    if (err0 != IRError.NO_ERROR) {
        return (err0, Exp({mantissa: 0}), Exp({mantissa: 0}));
    }

    // Borrow Rate is 5% + UtilizationRate * 45% (baseRate + UtilizationRate * multiplier);
    // 45% of utilizationRate, is `rate * 45 / 100`
    (MathError err1, Exp memory utilizationRateMuled) = mulScalar(utilizationRate, multiplier);
    // `mulScalar` only overflows when the product is >= 2^256.
    // utilizationRate is a real number on the interval [0,1], which means that
    // utilizationRate.mantissa is in the interval [0e18,1e18], which means that 45 times
    // that is in the interval [0e18,45e18]. That interval has no intersection with 2^256, and therefore
    // this can never overflow for the standard rates.
    if (err1 != MathError.NO_ERROR) {
        return (IRError.FAILED_TO_MUL_UTILIZATION_RATE, Exp({mantissa: 0}), Exp({mantissa: 0}));
    }

    (MathError err2, Exp memory utilizationRateScaled) = divScalar(utilizationRateMuled, mantissaOne);
    // 100 is a constant, and therefore cannot be zero, which is the only error case of divScalar.
    assert(err2 == MathError.NO_ERROR);

    // Add the 5% for (5% + 45% * Ua)
    (MathError err3, Exp memory annualBorrowRate) = addExp(utilizationRateScaled, Exp({mantissa: baseRate}));
    // `addExp` only fails when the addition of mantissas overflow.
    // As per above, utilizationRateMuled is capped at 45e18,
    // and utilizationRateScaled is capped at 4.5e17. mantissaFivePercent = 0.5e17, and thus the addition
    // is capped at 5e17, which is less than 2^256. This only applies to the standard rates
    if (err3 != MathError.NO_ERROR) {
        return (IRError.FAILED_TO_ADD_BASE_RATE, Exp({mantissa: 0}), Exp({mantissa: 0}));
    }

    return (IRError.NO_ERROR, utilizationRate, annualBorrowRate);
}