A brief note on Moore’s law

One variation on Moore’s law says that computer performance doubles every 18 months. Another says performance improves by a factor of 100 every decade. The two are equivalent, though the latter sounds more impressive.

Exponential growth doesn’t mesh well with human intuition. Scott Berkun said “Technological progress is overestimated in the short term, underestimated in the long term.” This is probably not limited to technology but true of exponential growth in general.

(Why are the two statements of Moore’s law equivalent? The first says performance grows like exp( log(2) t / 1.5 ) and the second says it grows like exp( log(100) t / 10 ). And log(2) / 1.5 = 0.462 and log(100)/10 = 0.461.)

Related post: Moore’s law and software bloat

11 thoughts on “A brief note on Moore’s law

  1. If you represent the growing quantity, for example the number of transistors, by a decimal number, then you can say that we need an extra digit every 5 years. If you assume people are familiar with the positional number system, this could be slightly more intuitive than saying a factor 10 every 5 years.

    A similar trick is used in numerical analysis when representing the convergence of the error for certain iterative methods. Rather than the convergence factor r, you can use the convergence rate R = -log[10](r), which gives the average number of extra digits of accuracy acquired per iteration.

    The “divergence” factor for Moore’s law is about 1.585.
    The “divergence” rate is 0.2.

    I think the number of digits needed to write down a number is an interesting way to illustrate logs. Exponential growth for the number translates into linear growth for the number of digits. This idea also comes up in complexity theory, for example.

  2. Frédéric Grosshans

    If you think of your logs of being base 10^{1/10}, (or in dBs) the equivalence is easier to compute : Log 2 = 3 dB (actually 3.01) while log 100 = 20 dB. So both rates correspond to a 2 dB increase per year.

    I guess you computed natural logarithms, but decimal ones are more suited to ’round numbers’

  3. You can also use binary, to stay closer to the original doubling representation.
    1 bit per 1.5 years = 1 digit every 5 years
    0.67 bits per year = 0.2 digits per year = 2 decidigits per year
    The latter corresponds directly to dB.

  4. For me, exponential growth is difficult to intuitively understand because it is easier to envision linear rates of expansion rather than exponential rates. I wonder if this is due to the notion that I feel that fewer iterations of expansion are required to see trends which can be misleading. As a child I was first amazed by this through a story based on the power of two where a wise mouse negotiated with an oppressive dictator mouse to get grain for his people starting with one grain on the first day and exponentially growing thereafter (f(x)=2^x)in exchange for linear growth harvests ((f(x)=Nx). Chalk this up to myopia but I was always amazed (and partly still am!) by how much the grain grew due to the base 2 exponentiation.

    Therefore, when it comes to Moore’s law, it is hard to fathom the progress technology will make in both the near and long term. The former due to horrendous overestimation and the latter due to extreme underestimation. One way this impacts me is through the analysis of what phone to buy next. I have an Motorola Droid today and have been eligible to upgrade since June of this year but have been holding off. Part of the reason for not upgrading yet and committing to another two year contract is because I am under the continual impression that a phone will come out that will retain its edge in speed and power significantly better than others. I think this is an example of over estimation because I feel that if I get a phone today it will be completely outclassed in a month. While this appears to be a common consideration when I talk to friends I think that it’s a trap because it takes time for Moore’s law to kick in to the degree that it makes a notable difference in a phone’s user experience. For instance, most new phone on the market today would be a relatively huge upgrade for me because it would triple to quadruple my current processor speeds, RAM capacity, and storage space. Having waited since June I will now be able to, technically, get a better phone than I could have then but it will only be marginally better based on my analysis. The exchange in this case is that I have an older phone that is dying a bit more and is more frustrating to use every day. I think this may be a good example of not really taking into account exponential growth because no matter what phone I get it will be comparatively obsolete in two years time.

    So, I appreciate your post as it made me think about how I understand exponential growth and even what my objectives are in a little thing like buying a new phone. By the way, how did you arrive at your numbers of 0.461/2? When rounding to three decimal places I get the following:
    log(2) / 1.5 = 0.201
    log(100) / 10 = 0.2
    What am I missing?


  5. human mathematics

    Why do you think exponential growth is overestimated in the short-term? The linear approximation out of 1 is e per period, that’s underneath the arc to the right. (Using 1 because there is usually some small history whence people are projecting.)

  6. A professor I have at Cleveland State University says it is still true, performance doubles every 18 months. What he wondered (it seemed) is if it has not ended yet, when or will this come to an end? Just dropping the comment because he just said this last week, not arguing with anyone, I just try and get what I can from here- way past me but if I can pull 1% of new learning from “The Endeavour” it adds up to more! : )

  7. @human mathematics, I believe the short term overestimation is probable because the gains accelerate over time. When I hear, “processor speeds double every 18 months” I think of it as a linear progression from today to 18 months from now. Much of the exponential growth happens at the tail of log(2)/1.5 instead of being a direct progression. This leaves me with unmet expectations part way through the cycle as the gains forecast have not materialized to the degree I would expect. I believe this contrast between projected and expected growth rates is why exponential growth is overestimated in the near term, would you agree? What other ideas would do you have regarding this?

  8. human mathematics

    @T Fogarty So you are saying people draw a chord from A to B rather than extrapolating out of 0 → A. I guess that makes sense … but to me it’s intuitive that most of exponential speed gains come later. Like a large % of the people who have ever lived living today, that’s just the way geometric growth works.

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