As a public school teacher let me answer this there are three types of regents diplomas currently.
Regents Diploma: 5 regents (English, Global History and Geography, U.S. History and Government, 1 Science and 1 Math (Preferably Integrated Algebra).
Advanced Regents Diploma (English,Global History and Geography, U.S. History and Government, and Foreign Language Exam (not a regents). Also you need either 2 science and 3 maths or three sciences and two maths.
Advanced Regents Diploma with Honors get above a 85 or 90 on all regents based on what the honors score for that regents (for math it usually is an 85 and other subjects a 90 but it changes from Mayra to year in some cases on some subject for different reasons.)
Now every high school in NY is required to give enough courses assuming every student will get an advanced regents diploma and can require it as an requirement to graduate from that school. They also have the option of forcing you take 3 or even 4 years of math and science if they like.