The reverse-mode "cheap gradient" bound has two figures — "one to four times" (the book's prose) and "5×" (the Baur-Strassen formalization) — and they don't contradict

The reverse mode of automatic differentiation computes a full gradient — of a function of a million inputs — for a small constant multiple of the cost of evaluating the function once. This is the "cheap gradient principle," and it is why backpropagation is affordable at all. The literature quotes two different constants for it, which reads like a discrepancy and is not.

"Between one and four times" — the book's own prose. Griewank & Walther, Evaluating Derivatives (2008), Ch. 1: "an additional floating point operation count of between one and four times that required for a single evaluation of y." An example-motivated, informal statement introducing the principle.

"5 times" / 5·cost(f) — the number nearly every paper reaches for when it formalizes the same result through the Baur–Strassen theorem (1983), which the principle is built on. Bolte, Boustany, Pauwels & Pesquet-Popescu (ICLR 2023, arXiv:2206.01730, verified resolving): "According to the key complexity theory version of this result due to Baur and Strassen (1983), arithmetic complexity of the evaluation of the derivative of a rational function is at most 5 times the complexity of function evaluation." Independently, Kakade & Lee (NeurIPS 2018, arXiv:1809.08530, verified resolving): "for all rational functions, the striking Baur-Strassen theorem (Baur and Strassen, 1983; Griewank, 1989) shows that this increase in computational complexity is a (dimension free) factor of 5." (Attribution corrected 2026-07-07, audit correction 3: the source capture had cross-attributed these two papers' quotes and credited arXiv:1809.08530 to "Bolte & Pauwels.")

Same underlying result, two framings: the book states a tighter, example-specific range in its motivating prose; the "factor of 5" is the worst-case bound from the complexity-theory theorem the principle rests on. Neither is wrong; they answer slightly different questions (a typical operation count vs. a proven worst case). The recurring "5×" citation in ML writing is really a citation of Baur–Strassen, one rung below Griewank & Walther — the same pattern the cluster keeps finding, where the number travels detached from the paper that first stated it.

Two open verification threads remain queued: whether Baur & Strassen (1983) state the "factor of 3/5" bound in those words in their own paper, and whether Griewank's 1989 restatement is the direct source of the figure ML papers cite. Until those are read, the "5×" is grounded on the verified Bolte et al. (2023) and Kakade–Lee (2018) formalizations, not on the 1983 primary itself.