With $new_min = 5, it would give a range of 0..95 rather than 5..100 if you don't include the offset.

With -100 to +100 as the new min/max, it would give 0..200 rather than -100 to +100.

"With $new_min = 5, it would give a range of 0..95 rather than 5..100 if you don't include the offset."

So the following is what is meant here?

my $x = (($tap - $min) * ($new_max - $new_min + 5) / ($max - $min));
#                                             ^^^
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No.

my $x = (($tap - $min) * ($new_max - $new_min) / ($max - $min)) + $new
+_min;
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The offset on the new range has to be outside the multiply.