portfolio management - Delta hedge value formula

When we delta hedge with implied volatility, and dynamically adjust every day, I believe the PnL theoretically is

$$PnL = 0.5 \Gamma S^2 (\sigma_r^2 - \sigma_i^2)dt$$ where $\sigma_r$ is realized volatility.

My question is, how accurate is this? I am trying to do a delta hedge experiment, and I find that my daily PnLs range wildly, yet the values given by above formula remain somewhat small (< 1)?

I compute my daily PnL as

$$'\text{change in call price'} + \Delta \cdot '\text{change in spot price}'$$ since the first term gives us what we lost/gained through the call, and the second gives us what we earned shorting the stock. But these two formulas don't match .... however the aggregate results do?