How to check if a string represent an elisp number (including float), without trailing garbage

[Garreau, Alexandre]

`number-to-string' ignores garbage so that falsify the output for any
string beginning with a number but not being one, `cl-parse-integer'
raise an error instead of returning nil, and doesn’t support float,
`read' ignores trailing-garbage.

I’d like something that would work depending on elisp reading of
numbers, not regexps or some redundant specification of it, and
preferably cleaner than “(let ((num (string-to-number str))) (and num
(equal (number-to-string num))))” or anything converting back-and-forth.