Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 16.7s

Alternatives: 19

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = -5.551265807646144e+216
  2. Ec = 3.204819365746333e-8
  3. Vef = 7.6291589799853e+126
  4. EDonor = -3.468643535829322e+172
  5. mu = -4.6770310384545794e-66
  6. KbT = 9.803009680548712e-303
  7. NaChar = 1.0314102903263173e+216
  8. Ev = -8.554612020924232e+130
  9. EAccept = -7.48163247100146e+52

Specification

Math

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

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+
  (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
  (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+
  (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT))))
  (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (ndchar / (1.0d0 + exp((-(((ec - vef) - edonor) - mu) / kbt)))) + (nachar / (1.0d0 + exp(((((ev + vef) + eaccept) + -mu) / kbt))))
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + Math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + Math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / (1.0 + math.exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + math.exp(((((Ev + Vef) + EAccept) + -mu) / KbT))))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Float64(Float64(Float64(Ec - Vef) - EDonor) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Float64(Ev + Vef) + EAccept) + Float64(-mu)) / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / (1.0 + exp((-(((Ec - Vef) - EDonor) - mu) / KbT)))) + (NaChar / (1.0 + exp(((((Ev + Vef) + EAccept) + -mu) / KbT))));
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / N[(1.0 + N[Exp[N[((-N[(N[(N[(Ec - Vef), $MachinePrecision] - EDonor), $MachinePrecision] - mu), $MachinePrecision]) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(N[(Ev + Vef), $MachinePrecision] + EAccept), $MachinePrecision] + (-mu)), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+
  (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
  (/ NaChar (+ (/ 1.0 (exp (/ (- (- mu (+ Vef Ev)) EAccept) KbT))) 1.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / ((1.0 / exp((((mu - (Vef + Ev)) - EAccept) / KbT))) + 1.0));
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (ndchar / (exp(((mu + (edonor + (vef - ec))) / kbt)) + 1.0d0)) + (nachar / ((1.0d0 / exp((((mu - (vef + ev)) - eaccept) / kbt))) + 1.0d0))
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / ((1.0 / Math.exp((((mu - (Vef + Ev)) - EAccept) / KbT))) + 1.0));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / (math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / ((1.0 / math.exp((((mu - (Vef + Ev)) - EAccept) / KbT))) + 1.0))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(Float64(1.0 / exp(Float64(Float64(Float64(mu - Float64(Vef + Ev)) - EAccept) / KbT))) + 1.0)))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / ((1.0 / exp((((mu - (Vef + Ev)) - EAccept) / KbT))) + 1.0));
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(1.0 / N[Exp[N[(N[(N[(mu - N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - EAccept), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. associate--l+ N/A

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

   15. lower-+.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 61.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := mu + \left(Vef + EDonor\right)\\ t_1 := t\_0 - Ec\\ t_2 := \frac{NaChar}{\frac{1}{\frac{\left(mu - \left(Vef + Ev\right)\right) - EAccept}{KbT} + 1} + 1}\\ t_3 := t\_2 + NdChar \cdot 0.5\\ t_4 := \frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{-27}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{t\_1 \cdot t\_1}{KbT}, Ec - t\_0\right)}{KbT}}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+53}:\\ \;\;\;\;t\_2 + \frac{NdChar}{2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \left(\frac{mu}{KbT} - \frac{Ec}{KbT}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ mu (+ Vef EDonor)))
        (t_1 (- t_0 Ec))
        (t_2
         (/
          NaChar
          (+ (/ 1.0 (+ (/ (- (- mu (+ Vef Ev)) EAccept) KbT) 1.0)) 1.0)))
        (t_3 (+ t_2 (* NdChar 0.5)))
        (t_4
         (+
          (/ NdChar (+ (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT)) 1.0)))))
   (if (<= t_4 -2e-27)
     t_3
     (if (<= t_4 -5e-305)
       (/ NdChar (+ (exp (/ Ec (- KbT))) 1.0))
       (if (<= t_4 0.0)
         (/ NdChar (- 2.0 (/ (fma -0.5 (/ (* t_1 t_1) KbT) (- Ec t_0)) KbT)))
         (if (<= t_4 5e+53)
           (+
            t_2
            (/
             NdChar
             (+
              2.0
              (+ (/ EDonor KbT) (+ (/ Vef KbT) (- (/ mu KbT) (/ Ec KbT)))))))
           t_3))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = mu + (Vef + EDonor);
	double t_1 = t_0 - Ec;
	double t_2 = NaChar / ((1.0 / ((((mu - (Vef + Ev)) - EAccept) / KbT) + 1.0)) + 1.0);
	double t_3 = t_2 + (NdChar * 0.5);
	double t_4 = (NdChar / (exp(((mu + (EDonor + (Vef - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((((EAccept + (Vef + Ev)) - mu) / KbT)) + 1.0));
	double tmp;
	if (t_4 <= -2e-27) {
		tmp = t_3;
	} else if (t_4 <= -5e-305) {
		tmp = NdChar / (exp((Ec / -KbT)) + 1.0);
	} else if (t_4 <= 0.0) {
		tmp = NdChar / (2.0 - (fma(-0.5, ((t_1 * t_1) / KbT), (Ec - t_0)) / KbT));
	} else if (t_4 <= 5e+53) {
		tmp = t_2 + (NdChar / (2.0 + ((EDonor / KbT) + ((Vef / KbT) + ((mu / KbT) - (Ec / KbT))))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(mu + Float64(Vef + EDonor))
	t_1 = Float64(t_0 - Ec)
	t_2 = Float64(NaChar / Float64(Float64(1.0 / Float64(Float64(Float64(Float64(mu - Float64(Vef + Ev)) - EAccept) / KbT) + 1.0)) + 1.0))
	t_3 = Float64(t_2 + Float64(NdChar * 0.5))
	t_4 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT)) + 1.0)))
	tmp = 0.0
	if (t_4 <= -2e-27)
		tmp = t_3;
	elseif (t_4 <= -5e-305)
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(-KbT))) + 1.0));
	elseif (t_4 <= 0.0)
		tmp = Float64(NdChar / Float64(2.0 - Float64(fma(-0.5, Float64(Float64(t_1 * t_1) / KbT), Float64(Ec - t_0)) / KbT)));
	elseif (t_4 <= 5e+53)
		tmp = Float64(t_2 + Float64(NdChar / Float64(2.0 + Float64(Float64(EDonor / KbT) + Float64(Float64(Vef / KbT) + Float64(Float64(mu / KbT) - Float64(Ec / KbT)))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(mu + N[(Vef + EDonor), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - Ec), $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(N[(1.0 / N[(N[(N[(N[(mu - N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - EAccept), $MachinePrecision] / KbT), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(NdChar / N[(N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e-27], t$95$3, If[LessEqual[t$95$4, -5e-305], N[(NdChar / N[(N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(NdChar / N[(2.0 - N[(N[(-0.5 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / KbT), $MachinePrecision] + N[(Ec - t$95$0), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+53], N[(t$95$2 + N[(NdChar / N[(2.0 + N[(N[(EDonor / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(N[(mu / KbT), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -2.0000000000000001e-27 or 5.0000000000000004e53 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT)))))

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate--l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + \frac{1}{\color{blue}{1 + -1 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 94.2

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NdChar} + \frac{NaChar}{1 + \frac{1}{1 - \frac{\left(\left(Ev + Vef\right) - mu\right) + EAccept}{KbT}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 64.4

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + \frac{1}{1 - \frac{\left(\left(Ev + Vef\right) - mu\right) + EAccept}{KbT}}} \]

   10. Applied rewrites 64.4%

       \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + \frac{1}{1 - \frac{\left(\left(Ev + Vef\right) - mu\right) + EAccept}{KbT}}} \]

   if -2.0000000000000001e-27 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < -4.99999999999999985e-305

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate--l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(Vef + EDonor\right) + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)}{KbT}}} \]

      14. sub-neg N/A

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

      15. lower--.f64 54.7

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

   7. Applied rewrites 54.7%

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

   8. Taylor expanded in Ec around inf

      \[\leadsto \frac{NdChar}{1 + e^{-1 \cdot \frac{Ec}{KbT}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 34.7%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}} \]

   if -4.99999999999999985e-305 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 0.0

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate--l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in NdChar around inf

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)}}{KbT}}} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \color{blue}{\left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)}}{KbT}}} \]

      8. mul-1-neg N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \frac{NdChar}{1 + e^{\frac{\left(Vef + EDonor\right) + \left(mu + \color{blue}{\left(\mathsf{neg}\left(Ec\right)\right)}\right)}{KbT}}} \]

      14. sub-neg N/A

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

      15. lower--.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in KbT around -inf

      \[\leadsto \frac{NdChar}{2 + \color{blue}{-1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.3%

       \[\leadsto \frac{NdChar}{2 + \color{blue}{\left(-\frac{\mathsf{fma}\left(-0.5, \frac{\left(\left(\left(Vef + EDonor\right) + mu\right) - Ec\right) \cdot \left(\left(\left(Vef + EDonor\right) + mu\right) - Ec\right)}{KbT}, -\left(\left(\left(Vef + EDonor\right) + mu\right) - Ec\right)\right)}{KbT}\right)}} \]

   if 0.0 < (+.f64 (/.f64 NdChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (neg.f64 (-.f64 (-.f64 (-.f64 Ec Vef) EDonor) mu)) KbT)))) (/.f64 NaChar (+.f64 #s(literal 1 binary64) (exp.f64 (/.f64 (+.f64 (+.f64 (+.f64 Ev Vef) EAccept) (neg.f64 mu)) KbT))))) < 5.0000000000000004e53

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate--l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\mathsf{neg}\left(\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + \frac{1}{\color{blue}{1 + -1 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate--l+ N/A

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

      6. sub-neg N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 76.4

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

   7. Applied rewrites 76.4%

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

   8. Taylor expanded in KbT around inf

      \[\leadsto \frac{NdChar}{\color{blue}{\left(2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right) - \frac{Ec}{KbT}}} + \frac{NaChar}{1 + \frac{1}{1 - \frac{\left(\left(Ev + Vef\right) - mu\right) + EAccept}{KbT}}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 46.2

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

   10. Applied rewrites 46.2%

       \[\leadsto \frac{NdChar}{\color{blue}{2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \left(\frac{mu}{KbT} - \frac{Ec}{KbT}\right)\right)\right)}} + \frac{NaChar}{1 + \frac{1}{1 - \frac{\left(\left(Ev + Vef\right) - mu\right) + EAccept}{KbT}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -2 \cdot 10^{-27}:\\ \;\;\;\;\frac{NaChar}{\frac{1}{\frac{\left(mu - \left(Vef + Ev\right)\right) - EAccept}{KbT} + 1} + 1} + NdChar \cdot 0.5\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{-KbT}} + 1}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 0:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\mathsf{fma}\left(-0.5, \frac{\left(\left(mu + \left(Vef + EDonor\right)\right) - Ec\right) \cdot \left(\left(mu + \left(Vef + EDonor\right)\right) - Ec\right)}{KbT}, Ec - \left(mu + \left(Vef + EDonor\right)\right)\right)}{KbT}}\\ \mathbf{elif}\;\frac{NdChar}{e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}} + 1} \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{NaChar}{\frac{1}{\frac{\left(mu - \left(Vef + Ev\right)\right) - EAccept}{KbT} + 1} + 1} + \frac{NdChar}{2 + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \left(\frac{mu}{KbT} - \frac{Ec}{KbT}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\frac{1}{\frac{\left(mu - \left(Vef + Ev\right)\right) - EAccept}{KbT} + 1} + 1} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
  :name "Bulmash initializePoisson"
  :precision binary64
  (+ (/ NdChar (+ 1.0 (exp (/ (- (- (- (- Ec Vef) EDonor) mu)) KbT)))) (/ NaChar (+ 1.0 (exp (/ (+ (+ (+ Ev Vef) EAccept) (- mu)) KbT))))))