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Published equation contexts

I=FLOPs performedbytes moved,Pachievable=min⁡(Ppeak compute,  I×BWpeak memory)I = \frac{\text{FLOPs performed}}{\text{bytes moved}}, \qquad P_{\text{achievable}} = \min\left(P_{\text{peak compute}},\; I \times BW_{\text{peak memory}}\right)

Why this formula appears here

The roofline model, introduced by Williams, Waterman, and Patterson in 2009, gives the concept its formal shape. It plots two independent ceilings on the same chart: the chip’s peak arithmetic throughput (operations per second) on one axis, and its peak memory bandwidth (bytes per second) combined with a workload-specific ratio on the other. That ratio — arithmetic intensity — is defined as the number of floating-point operations performed per byte of data moved between memory and the arithmetic units [ 4 ] . A workload’s achievable throughput is bounded by whichever ceiling it hits first: I=FLOPs performedbytes moved,Pachievable=min⁡(Ppeak compute,  I×BWpeak memory)I = \frac{\text{FLOPs performed}}{\text{bytes moved}}, \qquad P_{\text{achievable}} = \min\left(P_{\text{peak compute}},\; I \times BW_{\text{peak memory}}\right). When a workload’s intensity I is high enough that I ×\times BWpeak memoryW_{\text{peak memory}}…

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PachievableP_{\text{achievable}}

Symbol P_achievable

PaP_achievable appears in the objective or constraint used by the optimization on the right.

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Ppeak computeP_{\text{peak compute}}

Symbol P_peak compute

PpP_peak compute appears in the objective or constraint used by the optimization on the right.

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Wpeak memoryW_{\text{peak memory}}

Symbol W_peak memory

WpW_peak memory appears in the objective or constraint used by the optimization on the right.

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FLOPs performed\text{FLOPs performed}

Numerator: FLOPs performed

The complete quantity above the fraction bar. FLOPs count floating-point arithmetic operations.

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bytes moved\text{bytes moved}

Denominator: bytes moved

The complete quantity below the fraction bar; it must be nonzero for this division. Bytes measure the data moved or stored.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

I=FLOPs performedbytes moved,Pachievable=min⁡(Ppeak compute,  I×BWpeak memory)I = \frac{\text{FLOPs performed}}{\text{bytes moved}}, \qquad P_{\text{achievable}} = \min\left(P_{\text{peak compute}},\; I \times BW_{\text{peak memory}}\right)

Equation 1 · Semiconductors

How AI Memory Systems and the Bandwidth Wall Actually Work

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The roofline model, introduced by Williams, Waterman, and Patterson in 2009, gives the concept its formal shape. It plots two independent ceilings on the same chart: the chip’s peak arithmetic throughput (operations per second) on one axis, and its peak memory bandwidth (bytes per second) combined with a workload-specific ratio on the other. That ratio — arithmetic intensity — is defined as the number of floating-point operations performed per byte of data moved between memory and the arithmetic units [ 4 ] . A workload’s achievable throughput is bounded by whichever ceiling it hits first: I=FLOPs performedbytes moved,Pachievable=min⁡(Ppeak compute,  I×BWpeak memory)I = \frac{\text{FLOPs performed}}{\text{bytes moved}}, \qquad P_{\text{achievable}} = \min\left(P_{\text{peak compute}},\; I \times BW_{\text{peak memory}}\right). When a workload’s intensity I is high enough that I ×\times BWpeak memoryW_{\text{peak memory}}…

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