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Equation 1 · The Deployment Envelope: Small Models Where the Power Is Not

What does this equation mean?

Rattainable=min⁡(Rpeak,B⋅I)R_{\mathrm{attainable}} = \min(R_{\mathrm{peak}}, B \cdot I)

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationsmin(R_peak, B × I)
Result or conditionR_attainable
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value 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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RattainableR_{\mathrm{attainable}}

Symbol R_attainable

RaR_attainable is the quantity selected or evaluated by the optimization written on the right.

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RpeakR_{\mathrm{peak}}

Symbol R_peak

the peak arithmetic rate.

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BB

Symbol B

the achievable memory bandwidth and I the operational intensity of the kernel.

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II

Symbol I

the operational intensity of the kernel.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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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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How to interpret it

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What the article says around this equation

The general form of the argument is old and well posed. Williams, Waterman and Patterson built the roofline model on the premise that “for the recent past and foreseeable future, off-chip memory bandwidth will often be the constraining resource”, and defined operational intensity as operations per byte of DRAM traffic [ 1 ] . Their bound is one line, and it is the single most useful line in this whole subject: Rattainable=min⁡(Rpeak,B⋅I)R_{\mathrm{attainable}} = \min(R_{\mathrm{peak}}, B \cdot I). with RpeakR_{\mathrm{peak}} the peak arithmetic rate, B the achievable memory bandwidth and I the operational intensity of the kernel. If I is small, the second term wins and the accelerator’s headline throughput is irrelevant.

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Sources cited in the surrounding passage

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