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Equation 17 · Part 7 · Locality Is the Whole Game: The Memory Hierarchy and What a Kernel Does Not Read

fraction

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 .
fraction

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

The passage around this formula

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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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 article section

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