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Published equation contexts

Q(b)  ≈  2N3b+N2,subject to3b2≤MQ(b) \;\approx\; \frac{2N^3}{b} + N^2, \qquad \text{subject to} \quad 3b^2 \le M

Why this formula appears here

The repair is to restructure the loops so that the working set of the innermost computation is small enough to sit in a fast store and is used many times before being evicted. Partition the operands into b ×\times b blocks and compute block by block. Three such blocks — two inputs and an accumulator — must be resident simultaneously, so the tile size is bounded by the fast-store capacity M in words: Q(b)  ≈  2N3b+N2,subject to3b2≤MQ(b) \;\approx\; \frac{2N^3}{b} + N^2, \qquad \text{subject to} \quad 3b^2 \le M . Taking b as large as the constraint allows gives

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QQ

Symbol Q

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MM

Symbol M

M 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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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Its accuracy depends on the assumptions and range of use described in the article.

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Published contexts (1)

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Q(b)  ≈  2N3b+N2,subject to3b2≤M.Q(b) \;\approx\; \frac{2N^3}{b} + N^2, \qquad \text{subject to} \quad 3b^2 \le M .

Equation 17 · AI Hardware & Semiconductors

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

This equation gives an approximation: it relates the quantities while allowing an approximation.

The repair is to restructure the loops so that the working set of the innermost computation is small enough to sit in a fast store and is used many times before being evicted. Partition the operands into b ×\times b blocks and compute block by block. Three such blocks — two inputs and an accumulator — must be resident simultaneously, so the tile size is bounded by the fast-store capacity M in words: Q(b)  ≈  2N3b+N2,subject to3b2≤MQ(b) \;\approx\; \frac{2N^3}{b} + N^2, \qquad \text{subject to} \quad 3b^2 \le M . Taking b as large as the constraint allows gives

Meanings in this article

  • bb: the traffic falls linearly in.
  • b2b^2: the square of b; the traffic falls linearly in.
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