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Equation 1 · Part 11 · How AI Memory Systems and the Bandwidth Wall Actually Work

Denominator: bytes moved

P  ≤  min⁡ ⁣(Pmax⁡,  I⋅B),I=FLOPsbytes movedP \;\le\; \min\!\left(P_{\max},\; I \cdot B\right), \qquad I = \frac{\text{FLOPs}}{\text{bytes moved}}
bytes moved\text{bytes moved}

What this part means

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

Its job in the formula

bytes moved occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

Given a memory system built this way, the question for any specific piece of computation is simple to state and consequential to answer: for this kernel, is the bottleneck the arithmetic units or the memory system that feeds them? The quantity that answers it is arithmetic intensity , defined as the number of floating-point operations a kernel performs per byte it moves across the memory boundary that matters. Williams, Waterman and Patterson formalized the relationship between intensity and attainable performance as the roofline model: P  ≤  min⁡ ⁣(Pmax⁡,  I⋅B),I=FLOPsbytes movedP \;\le\; \min\!\left(P_{\max},\; I \cdot B\right), \qquad I = \frac{\text{FLOPs}}{\text{bytes moved}}. with Pmax⁡P_{\max} the peak arithmetic rate of the device, B the achievable bandwidth of the memory tier supplying the operands, and P the…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.