AroundReplace

AroundReplace[expr,{s₁Around[x₁,δ₁],s₂Around[x₂,δ₂],…}]

propagates uncertainty in expr by replacing all occurrences of s_i by Around[x_i,δ_i].

AroundReplace[expr,rules,n]

propagates uncertainty in expr using a series expansion to order n.

Details

In AroundReplace[expr,{s₁->Around[x_i,…],…}], each occurrence of a given s_i in expr is assumed to represent the same value, but different replacements Around[x_i,…] are assumed to be uncorrelated.
In AroundReplace[expr,rules], rules can contain scalar rules of the form sAround[…] for an uncorrelated variable s, or vector rules of the form {s,r,…}VectorAround[…] for correlated variables s,r,….
In AroundReplace[f[s,r],rules], uncertainty propagation is based on the following formula:

In AroundReplace[f[s,r],rules,n], uncertainty propagation involves derivatives up to order n.
For n=1, the result of AroundReplace[f[s],sAround[x,δ],n] is centered at f[x], but for higher n orders the central value may depend on δ as well.

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Perform an operation with Around objects:

Converting x into two uncorrelated variables makes the result have smaller uncertainty:

Specify an intermediate level of correlation with VectorAround:

By default, uncertainty propagation uses first-order expansions:

Use a higher-order expansion to obtain better results:

AroundReplace[expr,rules] is not equivalent to ReplaceAll[expr,rules] in general:

They coincide if each replaced variable appears once in the original expression:

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