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Equation 4 · Part 1 · Astra's 63%-Cheaper Claim Has to Beat a Price Disadvantage Nobody Mentions

Symbol s

(1−s)(10)+s(1.00)(1−s)(10)+s(0.25)=10−9s10−9.75s\frac{(1-s)(10) + s(1.00)}{(1-s)(10) + s(0.25)} = \frac{10 - 9s}{10 - 9.75s}
ss

What this part means

s is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Its job in the formula

s is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

The passage around this formula

Call the cache-hit share s — the fraction of input tokens that are cache reads rather than fresh reads. The blended input-token cost per model, in dollars per million tokens, is (1-s) ×\times 10 + s ×\times r , where r is each model’s cache-read rate: $1.00 for Astra, $0.25 for Fable 5.1. The ratio of Astra’s blended input cost to Fable 5.1’s, at a given cache-hit share, is: (1−s)(10)+s(1.00)(1−s)(10)+s(0.25)=10−9s10−9.75s\frac{(1-s)(10) + s(1.00)}{(1-s)(10) + s(0.25)} = \frac{10 - 9s}{10 - 9.75s}. Run that ratio across a stated bracket rather than picking one number and presenting it as fact:

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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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