Dose arithmetic has three parts: concentration from vial content and diluent, volume from dose and concentration, and units from volume and syringe scale.
Worked example, because the general form is easier to trust once you have seen it once. Take a 10 mg vial and add 2 mL of diluent: the concentration is 10 ÷ 2 = 5 mg/mL. A 0.5 mg dose is 0.5 ÷ 5 = 0.1 mL. On a U-100 syringe, where 1 unit = 0.01 mL, that is 0.1 ÷ 0.01 = 10 units. Change the diluent to 1 mL and the same dose becomes 5 units — same dose, half the resolution.
Mechanically, room temperature before drawing is worth the ten minutes. Cold solution is more viscous, draws slower, and condensation on a cold barrel makes it harder to read the meniscus.
The limitation is that technique reduces risk, it does not remove it, and nothing you can do outside a controlled environment makes a non-sterile preparation sterile.
Do the arithmetic twice, ideally with someone else doing it independently.