This calculation is made on the assumption that there is a compounding involved, that the improvement is consistent every day without gaps, and no regression happens during that time. With $\Delta p$ as the improvement in percent and $P(t)$ as the cumulative improvement over time, we arrive at the following:
$$ \begin{aligned} \left( 1 + \Delta p \right)^t &= P(t) \ \left( 1 + 0.01 \right)^{365} &= 37.7834 \end{aligned} $$
Here is where we get the 37% cumulative improvement in the 1% Rule.
Now, what if we have regression on some days? As such a much more general equation where regression can be expected is as follows:
$$P(t) = \left( 1 + \Delta p + \Delta r \right)^{t}$$
where
This equation, however, only applies to improvement and regression increments that are constant. Now what if I do not consistently improve at 1% but have days where I improve 2%, 3%, or 0.25%? Or days where things did not go well and I regress by 0.5%, 1%, or 5%?
To convert it to discrete variable improvement and regression percentages per day we have the following (which we can further take its limits to arrive at a product integral):
$$P(t) = \prod_{i=1}^{t} \left( 1 + \Delta p_i + \Delta r_i \right)$$
Given some data (maybe a CSV file with date, $\Delta p$, and $\Delta r$ fields for each date in a given time period), we can code up a quick calculation for $P(t)$. For example in python
:

