My modeling of
@leman's frequency vs. power data for the A17 Pro Performance Core (single thread):
The basic theoretical models for frequency vs. power follow a power law (real world is obviously more complicated), so I started by looking for that. The easiest way to check if your data follows a power law (something of the form f(x) = a x^b) is to plot it on a log-log plot, and see if your data follows a straight line (log-log plots linearize power laws). If it does, the slope will be equal to the value of the exponent.
When I did that, I didn't see a single straight line, but rather what appear to be three different scaling regimes. Fitting each of these to its own simple power law gave these results:
Low Frequency (1.09 GHz to 1.34 GHz, 344 data points, green): p(f) = 0.4 * f^1.2
Middle Frequency (2.73 GHz to 3.38 GHz, 295 data points, blue): p(f) = 0.2* f^2.4
High Frequency (3.45 GHz to 3.78 GHz, 55 data points, red): p(f) = 0.07 * f^3.2
Note1: I've rounded all the parameter values in this post for readability, but because of the sensitivity of these equations to those values (especially the equations shown later), you won't be able to recover these plots from these equations—you'll need a lot more digits. If anyone wants these, LMK, and I'll add them to this post.
Note 2: All equations were fitted using Mathematica's NonlinearModelFit function, with a Weighted Least Squares (WLS) minimization, and a (probably excessive) internal precision of 100 digits (to avoid rounding errors). For more details, see "Note 2, extended", at bottom.
If we expand the graph, we find the high-frequency curve extrapolates to 13 watts at 5 GHz:
Now you might argue, reasonably, that when you go from 4 GHz to 5 GHz, yet another scaling regime will come into effect, with an even higher slope, leading to a power consumption >13 watts at 5 GHz. And that's essentially what leman got when he fit the whole curve, which is effectively a prediction of how the scaling exponent will continue to increase as the frequency increases (yielding a predicted power consumption of 15 watts at 5 GHz).
So given that we have no knowledge of what the next scaling exponent will be, a polynomial fit, like what leman did, seems the best we can do at this point.
Having said that, just for fun, we can play with math to see if we can get a good overall fit with fewer paramaters than what leman used as a starting point. IIUC, he fit the data to a polynomial of the form:
p(f) = a + b f + c f^2 + d f^3, + e f^4. I.e., his model uses five parameters.
With a polynomial, the simplest equation I managed to find that gave a good fit had three parameters:
p(f) = 0.2 + 0.2 f^2 + 0.0006 f^6. Like leman's equation, this predicts 15 watts at 5 GHz. It's plotted immediately below.
Note that, unlike my first model, this one isn't directly physical (I don't think there is any f^6 power scaling going on); it's simply the math that gives the best polynomial fit with the fewest parameters (that I could find). I also don't think leman's quartic model is directly physical either—while it nicely tracks the values, I don't see any evidence of f^4 power scaling in this data. [Yes, given the trend, it may not be surprising to see that in the next scaling regime, since thus far we've gone from 1.2⟶2.4⟶3.2; but it's not present in this data.]
I.e., I think the best way to understand these two polynomial equations (my sixth-power and leman's quartic) are that they use a single higher-order polynomial to model what's actually going on, which is a successively-increasing set of lower-order power law behaviors. This also applies to the exponential I show at the end.
Note: The following are linear (i.e., not log-log) plots, which is why you can see the curve.
p(f) = 0.2 + 0.2 f^2 + 0.0006 f^6
If one is willing to accept a modest reduction in quality-of-fit, one can reduce the number of parameters even further, to two, using an exponential. The exponential is a bit stronger than the polynomial, giving a predicted power of 16 w at 5 GHz:
p(f) = 0.2 e^(0.9 f)
Note 2, extended: All equations were fitted using Mathematica's NonlinearModelFit function, with Weighted Least Squares (WLS) minimization. Specifically, instead of minimizing the sum of the squares of the residuals (OLS = ordinary least squares), I minimized Sum[(residual/value)^2]. I.e., I minimized the squares of the relative errors rather than the squares of the absolute errors. The latter is only appropriate when the error is expected to be independent of the size of the data (as is found in homoskedastic data). However, I've found that, more typically, the error increases in proportion to the size of the data. If I were being paid to do this I would have done a formal test of the distribution of the residuals. But since I'm not, I just did both and determined which gave the better-looking fit (or if they were comparable, I stuck with WLS for consistency). With this approach, I ended up using WLS for everything (not that it made much of a difference in these cases -- the visual differences between the two are subtle).
.