You are planning checks of the statements of a document. Each target below is a statement whose
truth is to be assessed, with the check types that may be run on it. Propose the checks that can be
run on each target from the document itself, at most one of each type per target, and none where
no check fits. Do not judge whether any statement is true: the checks decide.

Check types:
- recompute-numbers: the target reports a number that follows from other numbers the document
  reports. Give "nodes", the ids of the chunks whose numerals you use (the target's own included),
  "expression", an arithmetic expression of those numerals with + - * / ** and parentheses, and
  "expected", the reported result. Write every number exactly as a plain decimal numeral that occurs
  in the text of one of "nodes" (a percentage such as 12.5% may be written 0.125); write no other
  number, not even 100 or 2.
- rederive-maths: the target rests on a step of mathematics from one equation to another. Give
  "nodes", the ids of the equation chunks the step goes between, "lhs" and "rhs", two expressions the
  step claims are equal for all values of the symbols, "symbols", each symbol with its sympy
  assumptions (real, positive, negative, nonnegative, nonpositive, nonzero, integer) and a finite
  domain [low, high] to sample it from, and "assumption", the step's assumptions in words. Use only
  the declared symbols, numerals, + - * / ** and the functions sqrt, exp, log, sin, cos, tan, asin,
  acos, atan, sinh, cosh, tanh, Abs and the constants pi, E and I.
- citation-support: the target states what a cited work found. Give "reference", the id of the
  bibliography entry it cites.

Targets, as id (type): text, then the check types that may be run on it:
s-2 (result): In total we analyse 90000 segments.
  checks: recompute-numbers
s-3 (result): Of the 400 realisations, 12.5\% are rejected, which leaves 350.
  checks: recompute-numbers
e-1 (derivation): \begin{equation} b = \frac{c}{N} \end{equation}
  checks: rederive-maths
e-2 (derivation): \begin{equation} b N = c \end{equation}
  checks: rederive-maths
e-3 (derivation): \begin{equation} \arctan x + \arctan(1/x) = \pi/2 \end{equation}
  checks: rederive-maths
s-7 (literature_claim): Smith reports a bias below 1\% in comparable settings~\cite{smith2020}.
  checks: citation-support
s-8 (literature_claim): Jones finds that the bias grows with the segment length~\cite{jones2021}.
  checks: citation-support

Chunks of the document, in document order, as id (type): text
s-1 (method): We simulate 400 noise realisations of 250 segments each.
s-2 (result): In total we analyse 90000 segments.
s-3 (result): Of the 400 realisations, 12.5\% are rejected, which leaves 350.
s-4 (context): The bias of the estimator is
e-1 (derivation) [label eq:bias]: \begin{equation} b = \frac{c}{N} \end{equation}
s-5 (context): and multiplying by the number of segments gives
e-2 (derivation) [label eq:product]: \begin{equation} b N = c \end{equation}
s-6 (context): For the phase we use
e-3 (derivation) [label eq:phase]: \begin{equation} \arctan x + \arctan(1/x) = \pi/2 \end{equation}
s-7 (literature_claim): Smith reports a bias below 1\% in comparable settings~\cite{smith2020}.
s-8 (literature_claim): Jones finds that the bias grows with the segment length~\cite{jones2021}.
s-9 (claim): The estimator is therefore consistent.
r-1 (reference) [key smith2020]: A. Smith, Estimators of noise, J. Stat. 12 (2020) 1.
r-2 (reference) [key jones2021]: B. Jones, Long segments, arXiv:2101.01234.

Proposals from your previous answer that were refused, each with the error it got and what to change
("fix"). If this list is not empty, return corrected proposals for these targets only, and nothing
else:
[]
