I do not know how your computer is doing it. But I know, it would take artificial intelligence to do it properly.
Unfortunately TeknoJunky confuses a few things and the wikipedia articles cited are also not the best.
There are a few fundamental things to keep in mind:
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Digital video is not encoded in RGB but in a different colour space in which a much more efficient data reduction is possible.
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In that color space the brightness information and the colour information are largely decoupled. Hence, one talks about luminance and chromance.
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The brightness reception of the human eye is not linear, but close to logarithmic.
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The relation between the encoded brightness value and the brightness output on the screen is non-linear. This is the “gamma,” or “gamma correction.” More precisely, there is a mathematical function mapping these two which historically contained a parameter that was named gamma.
As far as I understand the problem with the HDR to HD conversion is more with the luminance than the chromance. But I may be corrected.
In SDR video, 8 bits and the gamma correction allow to reproduce a brightness range where the brightest spot is 2^7 times brighter than the darkest one. I.e. one can double the brightness about 7 times before one runs out of the dynamic range the format allows us to encode. However, the human eye is able to resolve 10 such steps.
If one would use 8 bit to describe a larger dynamics range, the eye would see “bands,” i.e. a luminance gradient would not appear as a continuous change, but in discrete steps. (The eye is really good at recognizing these.)
In HDR, using 10 or 12 bit, a higher dynamic range can be encoded without running into banding. With it comes that a different gamma correction function must be used.
The offspring from that is that you cannot just chop off the least significant bits from the 10 or 12 bit information in order to obtain a good 8 bit representation of the image.
Of course you can define a stupid math function doing that semi-properly, but implicitly you have to make an artistic decision. Where do you want to leave detail away: in the darker or the brighter parts of the image? In the professional world a human would make that decision scene by scene.
Added later:
I should stress the following point: With motion pictures, you can not obtain good results by working frame after frame, as the camera might follow the action and the light conditions or the focus of the interest change. Hence, an algorithm has to analyse a complete scene or at least a larger time window.
Oh, this became long, but I hope it helps.
