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This paper studies the quantitative refinements of Abramsky's applicativesimilarity and bisimilarity in the context of a generalisation of Fuzz, acall-by-value $\lambda$-calculus with a linear type system that can expressprograms sensitivity, enriched with algebraic operations \emph{\`a la} Plotkinand Power. To do so a general, abstract framework for studying behaviouralrelations taking values over quantales is defined according to Lawvere'sanalysis of generalised metric spaces. Barr's notion of relator (or laxextension) is then extended to quantale-valued relations adapting and extendingresults from the field of monoidal topology. Abstract notions ofquantale-valued effectful applicative similarity and bisimilarity are thendefined and proved to be a compatible generalised metric (in the sense ofLawvere) and pseudometric, respectively, under mild conditions.

Quantitative Behavioural Reasoning for Higher-order Effectful Programs: Applicative Distances (Extended Version)