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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `NdChar`

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Derivation

Initial program 0.0
\[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]
Using strategy rm
Applied log1p-expm1-u0.1
\[\leadsto \frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\log_* (1 + (e^{e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} - 1)^*)}}\]
Final simplification0.1
\[\leadsto \frac{NdChar}{e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}} + 1} + \frac{NaChar}{1 + \log_* (1 + (e^{e^{\frac{\left(-mu\right) + \left(\left(Ev + Vef\right) + EAccept\right)}{KbT}}} - 1)^*)}\]

Reproduce

herbie shell --seed 2019030 +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))))))