ad lib. (ad libidinem, ad libita) -> at one's whim, at will

[spiros]

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ad libitum (ad lib) -> toward pleasure

Loosely, "according to what pleases" or "as you wish"; libitum comes from the past participle of libere, "to please". It typically indicates in music and theatrical scripts that the performer has the liberty to change or omit something. Ad lib is specifically often used when someone improvises or ignores limitations.

Wikipedia—List of Latin phrases

[billberg23]

Ad lib. can be interpreted in two ways.  In classical ("golden age") Latin, it would be ad libidinem, "at your whim":
Lewis & Short s.v. libido:
Unlawful or inordinate desire, passion, caprice, wilfulness, wantonness: ingenium est omnium hominum ab labore proclive ad libidinem, Ter. And. 1, 1, 51: ad libidinem suam vexare aliquem, Cic. Rosc. Am. 49, 141: fortuna res cunctas ex lubidine magis, quam ex vero celebrat obscuratque, arbitrarily, according to pleasure or caprice, Sall. C. 8: quod positum est in alterius voluntate, ne dicam libidine, Cic. Fam. 9, 16, 3: ad libidinem aliorum judicare, id. Font. 12, 26; id. Fin. 1, 6, 19: instruitur acies ad libidinem militum, Liv. 25, 21.

In post-classical ("silver age") Latin, it would be ad libita, "at will":
Lewis & Short s.v. libet:
lĭbĭtus, a, um, P. a.; only plural as subst.: lĭbĭta, ōrum, n., lit., the things that please, one's pleasure, will, liking, humor (Tacitean): sua libita exercebant, Tac. A. 6, 1: ad libita Caesarum, id. ib. 12, 6: ad libita Pallantis, id. ib. 14, 2.