Bulmash initializePoisson

Average Error: 0.0 → 0.1

Time: 13.4s

Precision: binary64

Math

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

↓

\[\frac{NdChar}{1 + \frac{1}{\sqrt[3]{{\left(e^{\sqrt{2}}\right)}^{\left(\sqrt{2} \cdot \frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} \cdot \sqrt[3]{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

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 distribute-frac-neg 0.0

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

4. Applied exp-neg 0.0

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

6. Applied add-cube-cbrt 0.0

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

8. Applied cbrt-unprod 0.1

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

9. Simplified 0.1

   \[\leadsto \frac{NdChar}{1 + \frac{1}{\sqrt[3]{\color{blue}{{\left(e^{2}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} \cdot \sqrt[3]{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]
10. Using strategy rm

11. Applied add-sqr-sqrt 0.1

    \[\leadsto \frac{NdChar}{1 + \frac{1}{\sqrt[3]{{\left(e^{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}\right)}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} \cdot \sqrt[3]{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

12. Applied exp-prod 0.1

    \[\leadsto \frac{NdChar}{1 + \frac{1}{\sqrt[3]{{\color{blue}{\left({\left(e^{\sqrt{2}}\right)}^{\left(\sqrt{2}\right)}\right)}}^{\left(\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} \cdot \sqrt[3]{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

13. Applied pow-pow 0.1

    \[\leadsto \frac{NdChar}{1 + \frac{1}{\sqrt[3]{\color{blue}{{\left(e^{\sqrt{2}}\right)}^{\left(\sqrt{2} \cdot \frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}}} \cdot \sqrt[3]{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

14. Final simplification 0.1

    \[\leadsto \frac{NdChar}{1 + \frac{1}{\sqrt[3]{{\left(e^{\sqrt{2}}\right)}^{\left(\sqrt{2} \cdot \frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}\right)}} \cdot \sqrt[3]{e^{\frac{\left(\left(Ec - Vef\right) - EDonor\right) - mu}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}}\]

Reproduce

herbie shell --seed 2020169 
(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))))))