Once again: if you to try to design a circuit to drive an LED with some specific voltage you are doomed to failure.
The story is actually far more complicated than that.
Some data:
For 1n4001 (Vishay, diode), If = 20ma @ Vfwd = 0.65v; If = 50ma @ Vfwd = 0.7v (delta Vfwd = 50mv); dynamic Rd = 50mv / 30ma = 1.6ohm.
For CLA1B (Cree, LED), If = 20ma @ Vfwd = 3.05v; If = 22ma @ Vfwd = 3.10v (delta Vfwd = 50mv); dynamic Rd = 50mv / 2ma = 25ohm.
For the same change in Vfwd, the led's If increased much less than a diode's.
Contrary to conventional "wisdom", high power LEDs behave far more like a resistor than a diode.