Diodes have a logarithmic relation between current through the diode and the voltage across the diode. A ten:1 increase in current causes 0.058 volts increase across the diode. (the 0.058 V depends on several parameters, but you can see that number in lots of on-chip-silicon bandgap voltage-references].
What if the current changes 1,000:1, either increasing or decreasing? You should expect to see (at least) 3 * 0.058 volts change in Vdiode.
What if the current changes 10,000:1? Expect at least 4 * 0.058 volts.
At high currents (1 mA or higher), the bulk resistance of the silicon starts to affect the logarithmic behavior, and you get more of a straight line relation between Idiode and Vdiode.
The standard equation for this behavior involves "e", 2.718, thusly
$$Idiode = Is * [e^-(q*Vdiode/K*T*n) - 1]$$
and at room temperature and ideal doping profiles (n=1)
$$Idiode= Is *[e^-Vdiode/0.026 -1]$$
By the way, this same behavior exists for bipolar transistor emitter-base diodes. Assuming 0.60000000 volts at 1 mA, at 1 µA, expect 3 * 0.058 V = 0.174 V less. At 1 nanoampere, expect 6 * 0.058 V = 0.348 V less. At 1 picoampere, expect 9 * 0.058 volts = 0.522 volts less (ending up with only 78 millivolts across the diode); perhaps this pure-log behavior ceases to be an accurate tool, near zero volts Vdiode.
Here is Vbe plot over 3 decades of Ic; we expect at least 3*0.058 volts or 0.174 volts; reality for this bipolar transistor is 0.23 volts.