Substitutive names are formed by adding a suffix such as ‘al’, ‘ol’, ‘yl’, ‘carbaldehyde’, ‘carboxylic acid’, etc., to the name of a parent hydride, with elision of the final letter ‘e’ of the parent hydride, if any, before ‘a’, ‘i’, ‘o’, ‘u’, and ‘y’.
The corresponding rule in Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book)) reads as follows.
P-16.7.1 Vowels are systematically elided as follows:
(a) the terminal letter ‘e’ in names of parent hydrides or endings ‘ene’ and ‘yne’
when followed by a suffix or ‘en’ ending beginning with ‘a’, ‘e’, ‘i’, ‘o’, ‘u’,
or ‘y’;
Accordingly, the correct name for the compound that is given in the question is (3Z)-3-methylpent-3-en-1-yne-1,4-diol since ‘yne’ is not followed by ‘a’, ‘e’, ‘i’, ‘o’, ‘u’,
or ‘y’.
Similar examples that can be found in the Blue Book are 2-methylpent-1-en-4-yn-3-ol
and undeca-2,4,7-trien-9-yne-1,11-diol.