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Equation 2 · Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read

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

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This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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PP

Symbol P

P is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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Pmax⁡P_{\max}

Symbol P_max

the peak arithmetic rate.

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II

Symbol I

the arithmetic intensity in operations per byte.

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BB

Symbol B

the achievable bandwidth at the level that dominates.

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multiplication

multiplication

Multiply the quantities on either side.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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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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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 memory-bound and its runtime is set by traffic; above it the kernel is compute-bound and traffic is hidden.

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