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Derivation

1. Initial program 0.0

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]
2. Using strategy rm

3. Applied add-log-exp0.0

   \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\log \left(e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}\right)}}}\]

4. Applied rem-exp-log0.0

   \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}}\]

5. Simplified0.0

   \[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{\left(Vef + Ev\right) - \left(mu - EAccept\right)}{KbT}}}}\]

6. Final simplification0.0

   \[\leadsto \frac{NdChar}{e^{-\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}} + 1} + \frac{NaChar}{1 + e^{\frac{\left(Ev + Vef\right) - \left(mu - EAccept\right)}{KbT}}}\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
  :name "Bulmash initializePoisson"
  (+ (/ NdChar (+ 1 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))) (/ NaChar (+ 1 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT))))))

Details

Time bar (total: 44.0s)Debug log