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P  ≤  min⁡(Pmax⁡,  I⋅B)P \;\le\; \min\left(P_{\max},\; I \cdot B\right)

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The quantity that decides which of these limits binds is arithmetic intensity : the number of arithmetic operations a kernel performs per byte it moves across a given level. Williams, Waterman and Patterson formalised the consequence as the roofline model, which relates attainable floating-point performance to operational intensity and memory bandwidth in a single visual bound [ 5 ] . Written as an inequality, attainable performance P satisfies P  ≤  min⁡(Pmax⁡,  I⋅B)P \;\le\; \min\left(P_{\max},\; I \cdot B\right). with Pmax⁡P_{\max} the peak arithmetic rate, I the arithmetic intensity in operations per byte, and B the achievable bandwidth at the level that dominates. The two terms cross at a ridge point I∗I^{*} = Pmax⁡P_{\max}/B . Below it a kernel is…

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PP

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P  ≤  min⁡(Pmax⁡,  I⋅B),P \;\le\; \min\left(P_{\max},\; I \cdot B\right),

Equation 2 · AI Hardware & Semiconductors

Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

The quantity that decides which of these limits binds is arithmetic intensity : the number of arithmetic operations a kernel performs per byte it moves across a given level. Williams, Waterman and Patterson formalised the consequence as the roofline model, which relates attainable floating-point performance to operational intensity and memory bandwidth in a single visual bound [ 5 ] . Written as an inequality, attainable performance P satisfies P  ≤  min⁡(Pmax⁡,  I⋅B)P \;\le\; \min\left(P_{\max},\; I \cdot B\right). with Pmax⁡P_{\max} the peak arithmetic rate, I the arithmetic intensity in operations per byte, and B the achievable bandwidth at the level that dominates. The two terms cross at a ridge point I∗I^{*} = Pmax⁡P_{\max}/B . Below it a kernel is…

Meanings in this article

  • Pmax⁡P_{\max}: the peak arithmetic rate.
  • II: the arithmetic intensity in operations per byte.
  • BB: the achievable bandwidth at the level that dominates.
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