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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-cbrt-cube0.0

   \[\leadsto \frac{NdChar}{1 + \color{blue}{\sqrt[3]{\left(e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}} \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}\right) \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]
4. Using strategy rm

5. Applied distribute-frac-neg0.0

   \[\leadsto \frac{NdChar}{1 + \sqrt[3]{\left(e^{\color{blue}{-\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}} \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}\right) \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

6. Applied exp-neg0.0

   \[\leadsto \frac{NdChar}{1 + \sqrt[3]{\left(\color{blue}{\frac{1}{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}} \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}\right) \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

7. Applied associate-*l/0.0

   \[\leadsto \frac{NdChar}{1 + \sqrt[3]{\color{blue}{\frac{1 \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}} \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

8. Applied associate-*l/0.0

   \[\leadsto \frac{NdChar}{1 + \sqrt[3]{\color{blue}{\frac{\left(1 \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}\right) \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

9. Applied cbrt-div0.0

   \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{\sqrt[3]{\left(1 \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}\right) \cdot e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}}}{\sqrt[3]{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

10. Simplified0.0

    \[\leadsto \frac{NdChar}{1 + \frac{\color{blue}{\sqrt[3]{e^{\frac{\left(mu + EDonor\right) + \left(Vef - Ec\right)}{KbT}} \cdot e^{\frac{\left(mu + EDonor\right) + \left(Vef - Ec\right)}{KbT}}}}}{\sqrt[3]{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]
11. Using strategy rm

12. Applied expm1-log1p-u0.0

    \[\leadsto \frac{NdChar}{1 + \frac{\color{blue}{(e^{\log_* (1 + \sqrt[3]{e^{\frac{\left(mu + EDonor\right) + \left(Vef - Ec\right)}{KbT}} \cdot e^{\frac{\left(mu + EDonor\right) + \left(Vef - Ec\right)}{KbT}}})} - 1)^*}}{\sqrt[3]{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

13. Final simplification0.0

    \[\leadsto \frac{NdChar}{1 + \frac{(e^{\log_* (1 + \sqrt[3]{e^{\frac{\left(Vef - Ec\right) + \left(EDonor + mu\right)}{KbT}} \cdot e^{\frac{\left(Vef - Ec\right) + \left(EDonor + mu\right)}{KbT}}})} - 1)^*}{\sqrt[3]{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}} + \frac{NaChar}{e^{\frac{\left(\left(Vef + Ev\right) + EAccept\right) + \left(-mu\right)}{KbT}} + 1}\]

Runtime

Time bar (total: 57.4s)Debug logProfile

herbie shell --seed 2018295 +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))))))