Bulmash initializePoisson

Average Error: 0.0 → 0.0

Time: 8.0s

Precision: binary64

Math

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

↓

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

Error

Bits error versus NdChar

Bits error versus Ec

Bits error versus Vef

Bits error versus EDonor

Bits error versus mu

Bits error versus KbT

Bits error versus NaChar

Bits error versus Ev

Bits error versus EAccept

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 *-un-lft-identity 0.0

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

4. Applied *-un-lft-identity 0.0

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

5. Applied times-frac 0.0

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

6. Applied exp-prod 0.0

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

7. Simplified 0.0

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

9. Applied add-sqr-sqrt 0.0

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

10. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020173 
(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
  :name "Bulmash initializePoisson"
  :precision binary64
  (+ (/ NdChar (+ 1.0 (exp (/ (neg (- (- (- Ec Vef) EDonor) mu)) KbT)))) (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (neg mu)) KbT))))))