Temperature dependence of the energy bandgap and Simulink way
The equation you give is simply the empirical Varshni equation linearized about a given reference temperature. 

To convert from one to the other, evaluate the Varshni equation at the reference temperature to find Egref and differentiate with respect to temperature, and evaluate the derivative at the reference temperature to find the slope of the linearized curve (use the quotient rule).  

If I substitute the following values for silicon (from this source) into the equation:

Eg(0) = 1.166

\$\alpha\$ = 4.73E-4

\$\beta\$ = 636

It looks like the reference temperature should be approximately 55°C for your equation to agree with the above numbers (assuming silicon). 

From the above-linked site: 



On the other hand, if you retrieve Y.P. Varshni's original paper (as above, but behind a paywall) the numbers given are quite different (1.1557/7.021E-4/1108) and agree with the Simulink slope coefficient at the more expected 300K.