Does homomorphic encryption hide the algorithm itself?
FHE does not offer circuit privacy by default, but can it can be upgraded to do so:


https://eprint.iacr.org/2016/164.pdf
https://eprint.iacr.org/2016/381.pdf


In brief, the extra cost for circuit privacy is actually very small (just a few noisy-dummy bootstraps at the end of your circuit)... in the honest-but-curious model ! For stronger variants of circuit privacy, you need to certify that the FHE public key is correctly generated; and FHE public key are already quite large, so I let you imagine how proving ZK statements about them would be.