Prometheus calculates rate(count[d]) at timestamp t in the following way:

1. It obtains raw samples per each time series with count name on the time range (t-d ... t]. Note that t-d timestamp isn't included in the range, while t timestamp is included in the range. For example, when calculating rate(count[1m]) at a timestamp t=2m the following raw samples are selected: 4423 @ 2m, 4402 @ 1m45s, 4402 @ 1m30s, 4381 @ 1m15s. Note that the 4381 @ 1m sample isn't included in calculations.
2. Then it calculates the difference between the last and the first sample on the selected time range per each time series with the name count. Prometheus can detect and remove time series resets to zero on the selected time range, but let's skip this for now for the sake of clarity. In the case above it calculates 4423 @ 2m - 4381 @ 1m15s = 42.
3. Then it divides results from step 2 by the duration d in seconds per each time series with name count. In the case above it calculates 42 / 1m = 42 / 60s = 0.7.

The actual result for rate(count[1m]) @ 2m - 0.700023 - differs from the calculated result - 0.7 - because of extrapolation, which can be applied to results calculated at step 2 if timestamps for the first and/or the last raw sample are located too far from the selected time range bounds. See more details about the extrapolation in this issue.

Note also that Prometheus misses possible counter increase on the time range [1m ... 1m15s] when calculating both rate() and increase(). See more details about this issue here and here.