As classical exponential scaling faltered, Zhang (2022, arXiv:2205.15011) proposed a quantitative successor to Moore's and Koomey's laws: transistor density, processor performance, and energy efficiency grow sub-exponentially, with the logarithm of the growing quantity following a logarithmic-integral-type curve y=a x′/ln⁡x′+by = a\,x'/\ln x' + b (with x′=x−1943x' = x - 1943), i.e., growth on the order of 2t/log⁡t2^{t/\log t} rather than 2t/T2^{t/T}. The proposal is purely phenomenological: a two-parameter curve fitted to x86 AMD/Intel CPU data, sparse historical energy-efficiency series, and Top500 supercomputer aggregates (a Top10 subset for energy, since records were incomplete), and it conjectures the modified growth can last about another 20 years before approaching the thermodynamic Landauer limit. The open problem is whether this law is real: (a) does the logarithmic-integral form survive statistically defensible fitting on broader, higher-quality, cross-architecture data — ARM, RISC-V, domain-specific accelerators, chiplet packaging, complete supercomputer energy records — against exponential, logistic, and learning-curve alternatives; and (b) can the x/ln⁡xx/\ln x form be derived from manufacturing physics, the economics of production and R&D, or software/parallelism constraints, or is it refuted by them? The proposal is essentially untested: the only later empirical study of the same observables reports continued exponential energy-efficiency growth with a lengthened doubling time but performs no functional-form comparison, and no mechanistic derivation of a t/ln⁡tt/\ln t-type scaling law exists in the technology-forecasting literature.

Background

Moore's law — Gordon Moore's 1965 observation that the number of transistors on an integrated circuit doubles roughly every one to two years — served for decades as the semiconductor industry's central planning heuristic and as the default assumption behind forecasts of computer performance. Two corollaries converted transistor scaling into performance. Denard scaling (the circa-1974 observation that shrinking transistors also allowed supply voltage and clock frequency to rise while power density stayed roughly constant) linked density to speed until it broke down around 2005, after which single-thread gains slowed markedly (Waldrop 2016). Energy efficiency was captured by Koomey's law: the empirical finding of Koomey et al. (2011), building on Nordhaus's (2007) two-century productivity series, that computations per kilowatt-hour doubled about every 1.5 years, with later reports of the doubling time lengthening (to roughly 2.6 years after 2000, as recounted in Zhang 2022). As exponential extrapolations overshot reality, the field split between qualitative accounts of what replaces them (Leiserson et al. 2020) and quantitative alternatives.

The standard quantitative alternative to time-exponential laws is the experience curve, or Wright's law: cost (or performance) improves as a power law of cumulative production rather than of time. Nagy et al. (2013) showed statistically that Wright's-law and Moore-type time-exponential forecasts are roughly equally accurate and nearly coincide when production grows exponentially; Farmer & Lafond (2016) and Lafond et al. (2018) developed these into probabilistic forecasting methods. This literature works with exponential and power-law forms; it does not consider growth laws intermediate between exponential and linear.

Zhang (2022) proposed exactly such an intermediate form. Observing that the annual improvement ratio of computing metrics declines over time — the logarithm of the improvement ratio fits 1/ln⁡t1/\ln t — he fitted a logarithmic-integral-type curve, y=a x′/ln⁡x′+by = a\,x'/\ln x' + b with x′=x−1943x' = x - 1943 (the offset coincides with the start of the ENIAC project). The logarithmic integral function li(x)\mathrm{li}(x) satisfies li(x)≈x/ln⁡x\mathrm{li}(x) \approx x/\ln x for large xx, so the fitted quantity grows sub-exponentially, on the order of 2t/log⁡t2^{t/\log t}: for small-to-moderate horizons x/ln⁡xx/\ln x is nearly linear on a semilog plot, which retroactively explains why Moore's law held for decades, while at large xx it bends away from the exponential straight line. Zhang fitted this form (self-styled 'Nick's Law') to transistor density; to x86 AMD and Intel CPU performance, deliberately restricted to one architecture family to avoid mixing architectural effects; to Koomey-type energy-efficiency data; to the aggregate performance of the Top500 list (the biannual ranking of the world's fastest supercomputers); and to supercomputer power efficiency, substituting the Top10 for the energy-efficiency fits because Top500 power records were incomplete and the samples small. For other series he instead reported slower log⁡t\log t-type fits: the logarithm of SPECint results (an industry-standard single-thread benchmark suite), x86 clock-frequency improvement over the past 20 years, and the total energy consumption of supercomputers. He noted that the modified fits postpone the projected arrival at the Landauer limit — the thermodynamic minimum energy kTln⁡2kT\ln 2 per logically irreversible bit operation (Landauer 1961) — beyond the roughly 2050 date implied by the original Koomey extrapolation, and conjectured the modified law can last another 20 years. He also stressed, invoking Amdahl's law (the bound a computation's serial fraction places on parallel speedup) and measured algorithmic-improvement rates (Sherry & Thompson 2021), that realized performance depends on parallel software in ways a hardware density law does not capture.

The proposal thus stands as a purely phenomenological two-parameter curve fitted to sparse, partly incomplete series. Whether it constitutes a genuine law of computing progress — robust across data, architectures, and packaging styles, and grounded in a mechanism rather than curve flexibility — is what remains unresolved.

Problem Statement

Does the sub-exponential, logarithmic-integral-type scaling law proposed as a quantitative successor to Moore's and Koomey's laws — the logarithm yy of a computing metric growing as y∼li(t)≈t/ln⁡ty \sim \mathrm{li}(t) \approx t/\ln t, in the author's parameterization fitted as y=a x′/ln⁡x′+by = a\,x'/\ln x' + b with x′=x−1943x' = x - 1943, i.e., the metric itself growing on the order of 2t/log⁡t2^{t/\log t} — hold as a general law of computing progress? The central open question is empirical:

(a) Empirical validity and generality: when fitted on broader and higher-quality data than the original analyses — architectures beyond the x86 AMD/Intel CPUs used there (ARM, RISC-V, domain-specific accelerators), packaging variants such as chiplets and multi-die assemblies, and complete supercomputer energy-consumption records in place of the incomplete Top500 data that forced the original Top10 substitution — does the logarithmic-integral form continue to describe transistor density, performance, and energy efficiency better than the exponential Moore/Koomey forms? Because the source fits its form series by series and reports log⁡t\log t-type fits for some series (the logarithm of SPECint results, total supercomputer energy consumption), per-observable outcomes — including mixed ones — bear directly on the generality of the proposed law.

A complementary open facet, not itself part of the empirical question, concerns its mechanistic basis:

(b) Mechanistic basis: can the x/ln⁡xx/\ln x form be derived from manufacturing physics, the economics of production and R&D, or algorithm/software constraints — or is it demonstrably incompatible with them, marking the existing fits as phenomenological accident?

A settled answer would also fix, on principled ground, the quantities the source can only conjecture: how long the modified growth can plausibly last, the projected date at which energy-efficiency trends encounter the Landauer limit, and how much of the hardware-implied growth is realizable under parallelism and algorithmic limits (Amdahl-type constraints). A minimally complete answer must adjudicate (a) across the named observables and address (b); the conjectured quantities just listed are consequences of a settled answer rather than separate deliverables.

References

  1. Zhang, Nick (2022). 'Moore's Law is dead, long live Moore's Law!'. arXiv:2205.15011. DOI 10.48550/arXiv.2205.15011. https://arxiv.org/abs/2205.15011
  2. Moore, G. E. (1965). 'Cramming more components onto integrated circuits'. Electronics 38(8); reprinted in Proceedings of the IEEE 86(1), 82-85 (1998). DOI 10.1109/JPROC.1998.658762. https://doi.org/10.1109/JPROC.1998.658762
  3. Koomey, J.; Berard, S.; Sanchez, M.; Wong, H. (2011). 'Implications of Historical Trends in the Electrical Efficiency of Computing'. IEEE Annals of the History of Computing 33(3), 46-54. DOI 10.1109/MAHC.2010.28. https://doi.org/10.1109/MAHC.2010.28
  4. Nordhaus, W. (2007). 'Two Centuries of Productivity Growth in Computing'. The Journal of Economic History 67(1), 128-159. DOI 10.1017/S0022050707000058. https://doi.org/10.1017/S0022050707000058
  5. Landauer, R. (1961). 'Irreversibility and Heat Generation in the Computing Process'. IBM Journal of Research and Development 5(3), 183-191. DOI 10.1147/rd.53.0183. https://doi.org/10.1147/rd.53.0183
  6. Waldrop, M. M. (2016). 'The chips are down for Moore's law'. Nature 530(7589), 144-147. DOI 10.1038/530144a. https://doi.org/10.1038/530144a
  7. Nagy, B.; Farmer, J. D.; Bui, Q. M.; Trancik, J. E. (2013). 'Statistical Basis for Predicting Technological Progress'. PLoS ONE 8(2), e52669. DOI 10.1371/journal.pone.0052669. https://doi.org/10.1371/journal.pone.0052669
  8. Farmer, J. D.; Lafond, F. (2016). 'How predictable is technological progress?'. Research Policy 45(3), 647-665. DOI 10.1016/j.respol.2015.11.001. https://doi.org/10.1016/j.respol.2015.11.001
  9. Lafond, F.; Bailey, A. G.; Bakker, J. D.; Rebois, D.; Zadourian, R.; McSharry, P.; Farmer, J. D. (2018). 'How well do experience curves predict technological progress? A method for making distributional forecasts'. Technological Forecasting and Social Change 128, 104-117. DOI 10.1016/j.techfore.2017.11.001. https://doi.org/10.1016/j.techfore.2017.11.001
  10. Leiserson, C. E.; Thompson, N. C.; Emer, J. S.; Kuszmaul, B. C.; Lampson, B. W.; Sanchez, D.; Schardl, T. B. (2020). 'There's plenty of room at the Top: What will drive computer performance after Moore's law?'. Science 368(6495), eaam9744. DOI 10.1126/science.aam9744. https://doi.org/10.1126/science.aam9744
  11. Sherry, Y.; Thompson, N. (2021). 'How Fast Do Algorithms Improve? [Point of View]'. Proceedings of the IEEE 109(11), 1768-1777. DOI 10.1109/JPROC.2021.3107219. https://doi.org/10.1109/JPROC.2021.3107219
  12. Prieto, A.; Prieto, B.; Escobar, J. J.; Lampert, T. (2025). 'Evolution of computing energy efficiency: Koomey's law revisited'. Cluster Computing 28(1), published online 24 October 2024. DOI 10.1007/s10586-024-04767-y. https://doi.org/10.1007/s10586-024-04767-y
  13. Schwierz, F. (2025). 'Moore's Law at 60 - Still in Good Shape or Already Ailing?'. 2025 IEEE International Conference on Electron Devices and Solid-State Circuits (EDSSC), 249-252. DOI 10.1109/EDSSC64492.2025.11183007. https://doi.org/10.1109/EDSSC64492.2025.11183007
  14. Zhang, Nick (2022). 'Growth Rates of Knowledge'. arXiv:2208.11936. DOI 10.48550/arXiv.2208.11936. https://arxiv.org/abs/2208.11936