Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 20.9s

Alternatives: 20

Speedup: 1.0×

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

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

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (-
  (/ NdChar (+ (/ 1.0 (exp (/ (- (- Ec (+ mu EDonor)) Vef) KbT))) 1.0))
  (/ NaChar (- -1.0 (exp (/ (+ Vef (+ EAccept (- Ev 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 - (mu + EDonor)) - Vef) / KbT))) + 1.0)) - (NaChar / (-1.0 - exp(((Vef + (EAccept + (Ev - 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 - (mu + edonor)) - vef) / kbt))) + 1.0d0)) - (nachar / ((-1.0d0) - exp(((vef + (eaccept + (ev - 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 - (mu + EDonor)) - Vef) / KbT))) + 1.0)) - (NaChar / (-1.0 - Math.exp(((Vef + (EAccept + (Ev - mu))) / KbT))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. frac-2neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   2. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   3. exp-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   5. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   7. neg-sub0 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   10. associate-+r- N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

7. Final simplification 99.9%

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

Alternative 2: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Ec - EDonor\right) - \left(Vef + mu\right)\\ t_1 := \frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -2.1 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq 2.45 \cdot 10^{+36}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{\frac{-1}{-1 + \frac{\frac{-0.5 \cdot \left(t\_0 \cdot t\_0\right)}{KbT} + \left(\left(Vef + mu\right) + \left(EDonor - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (- Ec EDonor) (+ Vef mu)))
        (t_1
         (+
          (/ NdChar (+ (exp (/ (+ Vef (+ mu (- EDonor Ec))) KbT)) 1.0))
          (/ NaChar (+ (exp (/ Vef KbT)) 1.0)))))
   (if (<= Vef -2.1e+133)
     t_1
     (if (<= Vef 2.45e+36)
       (+
        (/ NaChar (+ (exp (/ (+ Vef (+ EAccept (- Ev mu))) KbT)) 1.0))
        (/
         NdChar
         (+
          (/
           -1.0
           (+
            -1.0
            (/
             (+ (/ (* -0.5 (* t_0 t_0)) KbT) (+ (+ Vef mu) (- EDonor Ec)))
             KbT)))
          1.0)))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Ec - EDonor) - (Vef + mu);
	double t_1 = (NdChar / (exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((Vef / KbT)) + 1.0));
	double tmp;
	if (Vef <= -2.1e+133) {
		tmp = t_1;
	} else if (Vef <= 2.45e+36) {
		tmp = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / ((-1.0 / (-1.0 + ((((-0.5 * (t_0 * t_0)) / KbT) + ((Vef + mu) + (EDonor - Ec))) / KbT))) + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (ec - edonor) - (vef + mu)
    t_1 = (ndchar / (exp(((vef + (mu + (edonor - ec))) / kbt)) + 1.0d0)) + (nachar / (exp((vef / kbt)) + 1.0d0))
    if (vef <= (-2.1d+133)) then
        tmp = t_1
    else if (vef <= 2.45d+36) then
        tmp = (nachar / (exp(((vef + (eaccept + (ev - mu))) / kbt)) + 1.0d0)) + (ndchar / (((-1.0d0) / ((-1.0d0) + (((((-0.5d0) * (t_0 * t_0)) / kbt) + ((vef + mu) + (edonor - ec))) / kbt))) + 1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
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) {
	double t_0 = (Ec - EDonor) - (Vef + mu);
	double t_1 = (NdChar / (Math.exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0)) + (NaChar / (Math.exp((Vef / KbT)) + 1.0));
	double tmp;
	if (Vef <= -2.1e+133) {
		tmp = t_1;
	} else if (Vef <= 2.45e+36) {
		tmp = (NaChar / (Math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / ((-1.0 / (-1.0 + ((((-0.5 * (t_0 * t_0)) / KbT) + ((Vef + mu) + (EDonor - Ec))) / KbT))) + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (Ec - EDonor) - (Vef + mu)
	t_1 = (NdChar / (math.exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0)) + (NaChar / (math.exp((Vef / KbT)) + 1.0))
	tmp = 0
	if Vef <= -2.1e+133:
		tmp = t_1
	elif Vef <= 2.45e+36:
		tmp = (NaChar / (math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / ((-1.0 / (-1.0 + ((((-0.5 * (t_0 * t_0)) / KbT) + ((Vef + mu) + (EDonor - Ec))) / KbT))) + 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Ec - EDonor) - Float64(Vef + mu))
	t_1 = Float64(Float64(NdChar / Float64(exp(Float64(Float64(Vef + Float64(mu + Float64(EDonor - Ec))) / KbT)) + 1.0)) + Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0)))
	tmp = 0.0
	if (Vef <= -2.1e+133)
		tmp = t_1;
	elseif (Vef <= 2.45e+36)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Vef + Float64(EAccept + Float64(Ev - mu))) / KbT)) + 1.0)) + Float64(NdChar / Float64(Float64(-1.0 / Float64(-1.0 + Float64(Float64(Float64(Float64(-0.5 * Float64(t_0 * t_0)) / KbT) + Float64(Float64(Vef + mu) + Float64(EDonor - Ec))) / KbT))) + 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (Ec - EDonor) - (Vef + mu);
	t_1 = (NdChar / (exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0)) + (NaChar / (exp((Vef / KbT)) + 1.0));
	tmp = 0.0;
	if (Vef <= -2.1e+133)
		tmp = t_1;
	elseif (Vef <= 2.45e+36)
		tmp = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / ((-1.0 / (-1.0 + ((((-0.5 * (t_0 * t_0)) / KbT) + ((Vef + mu) + (EDonor - Ec))) / KbT))) + 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Ec - EDonor), $MachinePrecision] - N[(Vef + mu), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(N[Exp[N[(N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -2.1e+133], t$95$1, If[LessEqual[Vef, 2.45e+36], N[(N[(NaChar / N[(N[Exp[N[(N[(Vef + N[(EAccept + N[(Ev - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[(-1.0 / N[(-1.0 + N[(N[(N[(N[(-0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(Vef + mu), $MachinePrecision] + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -2.1e133 or 2.4499999999999999e36 < Vef

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 84.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right), KbT\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right)\right) \]

   7. Simplified 84.5%

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

   if -2.1e133 < Vef < 2.4499999999999999e36

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 81.7%

      \[\leadsto \frac{NdChar}{1 + \frac{1}{\color{blue}{1 + \left(-\frac{\frac{-0.5 \cdot \left(\left(\left(Ec - EDonor\right) - \left(mu + Vef\right)\right) \cdot \left(\left(Ec - EDonor\right) - \left(mu + Vef\right)\right)\right)}{KbT} + \left(-\left(\left(Ec - EDonor\right) - \left(mu + Vef\right)\right)\right)}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -2.1 \cdot 10^{+133}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 2.45 \cdot 10^{+36}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{\frac{-1}{-1 + \frac{\frac{-0.5 \cdot \left(\left(\left(Ec - EDonor\right) - \left(Vef + mu\right)\right) \cdot \left(\left(Ec - EDonor\right) - \left(Vef + mu\right)\right)\right)}{KbT} + \left(\left(Vef + mu\right) + \left(EDonor - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{KbT}} + 1} + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+
  (/ NaChar (+ (exp (/ (+ Vef (+ EAccept (- Ev mu))) KbT)) 1.0))
  (/ NdChar (+ (exp (/ (+ Vef (+ mu (- EDonor Ec))) 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 (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (exp(((Vef + (mu + (EDonor - Ec))) / 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 = (nachar / (exp(((vef + (eaccept + (ev - mu))) / kbt)) + 1.0d0)) + (ndchar / (exp(((vef + (mu + (edonor - ec))) / 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 (NaChar / (Math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / (Math.exp(((Vef + (mu + (EDonor - Ec))) / KbT)) + 1.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 69.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(Ec - EDonor\right) - \left(Vef + mu\right)\\ t_1 := \frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -1.1 \cdot 10^{+184}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq 6 \cdot 10^{+233}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{\frac{-1}{-1 + \frac{\frac{-0.5 \cdot \left(t\_0 \cdot t\_0\right)}{KbT} + \left(\left(Vef + mu\right) + \left(EDonor - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (- Ec EDonor) (+ Vef mu)))
        (t_1 (/ NaChar (+ (exp (/ (- (+ Vef (+ EAccept Ev)) mu) KbT)) 1.0))))
   (if (<= Vef -1.1e+184)
     t_1
     (if (<= Vef 6e+233)
       (+
        (/ NaChar (+ (exp (/ (+ Vef (+ EAccept (- Ev mu))) KbT)) 1.0))
        (/
         NdChar
         (+
          (/
           -1.0
           (+
            -1.0
            (/
             (+ (/ (* -0.5 (* t_0 t_0)) KbT) (+ (+ Vef mu) (- EDonor Ec)))
             KbT)))
          1.0)))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Ec - EDonor) - (Vef + mu);
	double t_1 = NaChar / (exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	double tmp;
	if (Vef <= -1.1e+184) {
		tmp = t_1;
	} else if (Vef <= 6e+233) {
		tmp = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / ((-1.0 / (-1.0 + ((((-0.5 * (t_0 * t_0)) / KbT) + ((Vef + mu) + (EDonor - Ec))) / KbT))) + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (ec - edonor) - (vef + mu)
    t_1 = nachar / (exp((((vef + (eaccept + ev)) - mu) / kbt)) + 1.0d0)
    if (vef <= (-1.1d+184)) then
        tmp = t_1
    else if (vef <= 6d+233) then
        tmp = (nachar / (exp(((vef + (eaccept + (ev - mu))) / kbt)) + 1.0d0)) + (ndchar / (((-1.0d0) / ((-1.0d0) + (((((-0.5d0) * (t_0 * t_0)) / kbt) + ((vef + mu) + (edonor - ec))) / kbt))) + 1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
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) {
	double t_0 = (Ec - EDonor) - (Vef + mu);
	double t_1 = NaChar / (Math.exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	double tmp;
	if (Vef <= -1.1e+184) {
		tmp = t_1;
	} else if (Vef <= 6e+233) {
		tmp = (NaChar / (Math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / ((-1.0 / (-1.0 + ((((-0.5 * (t_0 * t_0)) / KbT) + ((Vef + mu) + (EDonor - Ec))) / KbT))) + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (Ec - EDonor) - (Vef + mu)
	t_1 = NaChar / (math.exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0)
	tmp = 0
	if Vef <= -1.1e+184:
		tmp = t_1
	elif Vef <= 6e+233:
		tmp = (NaChar / (math.exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / ((-1.0 / (-1.0 + ((((-0.5 * (t_0 * t_0)) / KbT) + ((Vef + mu) + (EDonor - Ec))) / KbT))) + 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Ec - EDonor) - Float64(Vef + mu))
	t_1 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Float64(EAccept + Ev)) - mu) / KbT)) + 1.0))
	tmp = 0.0
	if (Vef <= -1.1e+184)
		tmp = t_1;
	elseif (Vef <= 6e+233)
		tmp = Float64(Float64(NaChar / Float64(exp(Float64(Float64(Vef + Float64(EAccept + Float64(Ev - mu))) / KbT)) + 1.0)) + Float64(NdChar / Float64(Float64(-1.0 / Float64(-1.0 + Float64(Float64(Float64(Float64(-0.5 * Float64(t_0 * t_0)) / KbT) + Float64(Float64(Vef + mu) + Float64(EDonor - Ec))) / KbT))) + 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (Ec - EDonor) - (Vef + mu);
	t_1 = NaChar / (exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	tmp = 0.0;
	if (Vef <= -1.1e+184)
		tmp = t_1;
	elseif (Vef <= 6e+233)
		tmp = (NaChar / (exp(((Vef + (EAccept + (Ev - mu))) / KbT)) + 1.0)) + (NdChar / ((-1.0 / (-1.0 + ((((-0.5 * (t_0 * t_0)) / KbT) + ((Vef + mu) + (EDonor - Ec))) / KbT))) + 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Ec - EDonor), $MachinePrecision] - N[(Vef + mu), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + N[(EAccept + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -1.1e+184], t$95$1, If[LessEqual[Vef, 6e+233], N[(N[(NaChar / N[(N[Exp[N[(N[(Vef + N[(EAccept + N[(Ev - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[(-1.0 / N[(-1.0 + N[(N[(N[(N[(-0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(Vef + mu), $MachinePrecision] + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -1.1e184 or 6.00000000000000028e233 < Vef

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EAccept + Ev\right) + Vef\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), Vef\right), mu\right), KbT\right)\right)\right)\right) \]

      8. +-lowering-+.f64 87.7%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right)\right)\right) \]

   7. Simplified 87.7%

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

   if -1.1e184 < Vef < 6.00000000000000028e233

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 76.9%

      \[\leadsto \frac{NdChar}{1 + \frac{1}{\color{blue}{1 + \left(-\frac{\frac{-0.5 \cdot \left(\left(\left(Ec - EDonor\right) - \left(mu + Vef\right)\right) \cdot \left(\left(Ec - EDonor\right) - \left(mu + Vef\right)\right)\right)}{KbT} + \left(-\left(\left(Ec - EDonor\right) - \left(mu + Vef\right)\right)\right)}{KbT}\right)}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.1 \cdot 10^{+184}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 6 \cdot 10^{+233}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}} + 1} + \frac{NdChar}{\frac{-1}{-1 + \frac{\frac{-0.5 \cdot \left(\left(\left(Ec - EDonor\right) - \left(Vef + mu\right)\right) \cdot \left(\left(Ec - EDonor\right) - \left(Vef + mu\right)\right)\right)}{KbT} + \left(\left(Vef + mu\right) + \left(EDonor - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := NdChar + \frac{NaChar}{e^{\frac{EAccept + \left(Ev - \left(mu - Vef\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;NdChar \leq -9 \cdot 10^{-162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 1.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 620000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\frac{1}{e^{\frac{\left(\left(Ec - EDonor\right) - mu\right) - Vef}{KbT}}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          NdChar
          (/ NaChar (+ (exp (/ (+ EAccept (- Ev (- mu Vef))) KbT)) 1.0)))))
   (if (<= NdChar -9e-162)
     t_0
     (if (<= NdChar 1.2e-126)
       (/ NaChar (+ (exp (/ (- (+ Vef (+ EAccept Ev)) mu) KbT)) 1.0))
       (if (<= NdChar 620000000000.0)
         t_0
         (/
          NdChar
          (+ (/ 1.0 (exp (/ (- (- (- Ec EDonor) mu) Vef) KbT))) 1.0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar + (NaChar / (exp(((EAccept + (Ev - (mu - Vef))) / KbT)) + 1.0));
	double tmp;
	if (NdChar <= -9e-162) {
		tmp = t_0;
	} else if (NdChar <= 1.2e-126) {
		tmp = NaChar / (exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	} else if (NdChar <= 620000000000.0) {
		tmp = t_0;
	} else {
		tmp = NdChar / ((1.0 / exp(((((Ec - EDonor) - mu) - Vef) / KbT))) + 1.0);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar + (nachar / (exp(((eaccept + (ev - (mu - vef))) / kbt)) + 1.0d0))
    if (ndchar <= (-9d-162)) then
        tmp = t_0
    else if (ndchar <= 1.2d-126) then
        tmp = nachar / (exp((((vef + (eaccept + ev)) - mu) / kbt)) + 1.0d0)
    else if (ndchar <= 620000000000.0d0) then
        tmp = t_0
    else
        tmp = ndchar / ((1.0d0 / exp(((((ec - edonor) - mu) - vef) / kbt))) + 1.0d0)
    end if
    code = tmp
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) {
	double t_0 = NdChar + (NaChar / (Math.exp(((EAccept + (Ev - (mu - Vef))) / KbT)) + 1.0));
	double tmp;
	if (NdChar <= -9e-162) {
		tmp = t_0;
	} else if (NdChar <= 1.2e-126) {
		tmp = NaChar / (Math.exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	} else if (NdChar <= 620000000000.0) {
		tmp = t_0;
	} else {
		tmp = NdChar / ((1.0 / Math.exp(((((Ec - EDonor) - mu) - Vef) / KbT))) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar + (NaChar / (math.exp(((EAccept + (Ev - (mu - Vef))) / KbT)) + 1.0))
	tmp = 0
	if NdChar <= -9e-162:
		tmp = t_0
	elif NdChar <= 1.2e-126:
		tmp = NaChar / (math.exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0)
	elif NdChar <= 620000000000.0:
		tmp = t_0
	else:
		tmp = NdChar / ((1.0 / math.exp(((((Ec - EDonor) - mu) - Vef) / KbT))) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev - Float64(mu - Vef))) / KbT)) + 1.0)))
	tmp = 0.0
	if (NdChar <= -9e-162)
		tmp = t_0;
	elseif (NdChar <= 1.2e-126)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Float64(EAccept + Ev)) - mu) / KbT)) + 1.0));
	elseif (NdChar <= 620000000000.0)
		tmp = t_0;
	else
		tmp = Float64(NdChar / Float64(Float64(1.0 / exp(Float64(Float64(Float64(Float64(Ec - EDonor) - mu) - Vef) / KbT))) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar + (NaChar / (exp(((EAccept + (Ev - (mu - Vef))) / KbT)) + 1.0));
	tmp = 0.0;
	if (NdChar <= -9e-162)
		tmp = t_0;
	elseif (NdChar <= 1.2e-126)
		tmp = NaChar / (exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	elseif (NdChar <= 620000000000.0)
		tmp = t_0;
	else
		tmp = NdChar / ((1.0 / exp(((((Ec - EDonor) - mu) - Vef) / KbT))) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar + N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev - N[(mu - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -9e-162], t$95$0, If[LessEqual[NdChar, 1.2e-126], N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + N[(EAccept + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 620000000000.0], t$95$0, N[(NdChar / N[(N[(1.0 / N[Exp[N[(N[(N[(N[(Ec - EDonor), $MachinePrecision] - mu), $MachinePrecision] - Vef), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -9.00000000000000045e-162 or 1.20000000000000003e-126 < NdChar < 6.2e11

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 78.3%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 72.6%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 72.6%

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

   if -9.00000000000000045e-162 < NdChar < 1.20000000000000003e-126

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EAccept + Ev\right) + Vef\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), Vef\right), mu\right), KbT\right)\right)\right)\right) \]

      8. +-lowering-+.f64 82.0%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right)\right)\right) \]

   7. Simplified 82.0%

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

   if 6.2e11 < NdChar

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 75.6%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 75.6%

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

      1. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\left(EDonor + Vef\right) + \left(mu - Ec\right)}{KbT}}\right)\right)\right) \]

      2. associate-+r- N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\left(\left(EDonor + Vef\right) + mu\right) - Ec}{KbT}}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\left(\left(Vef + EDonor\right) + mu\right) - Ec}{KbT}}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\left(Vef + \left(mu + EDonor\right)\right) - Ec}{KbT}}\right)\right)\right) \]

      6. associate-+r- N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{Vef + \left(\left(mu + EDonor\right) - Ec\right)}{KbT}}\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right) \]

      8. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right) \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}\right)}\right)\right)\right) \]

      10. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}\right)\right)\right)\right)\right) \]

      13. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}\right)\right)\right)\right)\right) \]

      14. sub0-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}\right)\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{Vef + \left(\left(mu + EDonor\right) - Ec\right)}{\mathsf{neg}\left(KbT\right)}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right), \left(\mathsf{neg}\left(KbT\right)\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 75.6%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{Vef + \left(mu + \left(EDonor - Ec\right)\right)}{0 - KbT}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -9 \cdot 10^{-162}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{EAccept + \left(Ev - \left(mu - Vef\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 1.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 620000000000:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{EAccept + \left(Ev - \left(mu - Vef\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{\frac{1}{e^{\frac{\left(\left(Ec - EDonor\right) - mu\right) - Vef}{KbT}}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := NdChar + \frac{NaChar}{e^{\frac{EAccept + \left(Ev - \left(mu - Vef\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;NdChar \leq -2.45 \cdot 10^{-169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 3.9 \cdot 10^{-122}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 1.3:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          NdChar
          (/ NaChar (+ (exp (/ (+ EAccept (- Ev (- mu Vef))) KbT)) 1.0)))))
   (if (<= NdChar -2.45e-169)
     t_0
     (if (<= NdChar 3.9e-122)
       (/ NaChar (+ (exp (/ (- (+ Vef (+ EAccept Ev)) mu) KbT)) 1.0))
       (if (<= NdChar 1.3)
         t_0
         (/ NdChar (+ (exp (/ (+ EDonor (+ Vef (- mu Ec))) KbT)) 1.0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar + (NaChar / (exp(((EAccept + (Ev - (mu - Vef))) / KbT)) + 1.0));
	double tmp;
	if (NdChar <= -2.45e-169) {
		tmp = t_0;
	} else if (NdChar <= 3.9e-122) {
		tmp = NaChar / (exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	} else if (NdChar <= 1.3) {
		tmp = t_0;
	} else {
		tmp = NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar + (nachar / (exp(((eaccept + (ev - (mu - vef))) / kbt)) + 1.0d0))
    if (ndchar <= (-2.45d-169)) then
        tmp = t_0
    else if (ndchar <= 3.9d-122) then
        tmp = nachar / (exp((((vef + (eaccept + ev)) - mu) / kbt)) + 1.0d0)
    else if (ndchar <= 1.3d0) then
        tmp = t_0
    else
        tmp = ndchar / (exp(((edonor + (vef + (mu - ec))) / kbt)) + 1.0d0)
    end if
    code = tmp
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) {
	double t_0 = NdChar + (NaChar / (Math.exp(((EAccept + (Ev - (mu - Vef))) / KbT)) + 1.0));
	double tmp;
	if (NdChar <= -2.45e-169) {
		tmp = t_0;
	} else if (NdChar <= 3.9e-122) {
		tmp = NaChar / (Math.exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	} else if (NdChar <= 1.3) {
		tmp = t_0;
	} else {
		tmp = NdChar / (Math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar + (NaChar / (math.exp(((EAccept + (Ev - (mu - Vef))) / KbT)) + 1.0))
	tmp = 0
	if NdChar <= -2.45e-169:
		tmp = t_0
	elif NdChar <= 3.9e-122:
		tmp = NaChar / (math.exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0)
	elif NdChar <= 1.3:
		tmp = t_0
	else:
		tmp = NdChar / (math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar + Float64(NaChar / Float64(exp(Float64(Float64(EAccept + Float64(Ev - Float64(mu - Vef))) / KbT)) + 1.0)))
	tmp = 0.0
	if (NdChar <= -2.45e-169)
		tmp = t_0;
	elseif (NdChar <= 3.9e-122)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Float64(EAccept + Ev)) - mu) / KbT)) + 1.0));
	elseif (NdChar <= 1.3)
		tmp = t_0;
	else
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar + (NaChar / (exp(((EAccept + (Ev - (mu - Vef))) / KbT)) + 1.0));
	tmp = 0.0;
	if (NdChar <= -2.45e-169)
		tmp = t_0;
	elseif (NdChar <= 3.9e-122)
		tmp = NaChar / (exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	elseif (NdChar <= 1.3)
		tmp = t_0;
	else
		tmp = NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar + N[(NaChar / N[(N[Exp[N[(N[(EAccept + N[(Ev - N[(mu - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -2.45e-169], t$95$0, If[LessEqual[NdChar, 3.9e-122], N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + N[(EAccept + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.3], t$95$0, N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -2.45e-169 or 3.8999999999999999e-122 < NdChar < 1.30000000000000004

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 78.3%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 72.6%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 72.6%

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

   if -2.45e-169 < NdChar < 3.8999999999999999e-122

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EAccept + Ev\right) + Vef\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), Vef\right), mu\right), KbT\right)\right)\right)\right) \]

      8. +-lowering-+.f64 82.0%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right)\right)\right) \]

   7. Simplified 82.0%

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

   if 1.30000000000000004 < NdChar

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 75.6%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 75.6%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.45 \cdot 10^{-169}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{EAccept + \left(Ev - \left(mu - Vef\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 3.9 \cdot 10^{-122}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 1.3:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{EAccept + \left(Ev - \left(mu - Vef\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ t_1 := NdChar + \frac{NaChar}{t\_0}\\ \mathbf{if}\;Vef \leq -0.0005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq -2.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{0 - KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 4.5 \cdot 10^{+76}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0)) (t_1 (+ NdChar (/ NaChar t_0))))
   (if (<= Vef -0.0005)
     t_1
     (if (<= Vef -2.5e-91)
       (/ NdChar (+ (exp (/ Ec (- 0.0 KbT))) 1.0))
       (if (<= Vef 4.5e+76)
         (+ NdChar (/ NaChar (+ (exp (/ Ev KbT)) 1.0)))
         (if (<= Vef 5.2e+141) (/ NdChar t_0) t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((Vef / KbT)) + 1.0;
	double t_1 = NdChar + (NaChar / t_0);
	double tmp;
	if (Vef <= -0.0005) {
		tmp = t_1;
	} else if (Vef <= -2.5e-91) {
		tmp = NdChar / (exp((Ec / (0.0 - KbT))) + 1.0);
	} else if (Vef <= 4.5e+76) {
		tmp = NdChar + (NaChar / (exp((Ev / KbT)) + 1.0));
	} else if (Vef <= 5.2e+141) {
		tmp = NdChar / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp((vef / kbt)) + 1.0d0
    t_1 = ndchar + (nachar / t_0)
    if (vef <= (-0.0005d0)) then
        tmp = t_1
    else if (vef <= (-2.5d-91)) then
        tmp = ndchar / (exp((ec / (0.0d0 - kbt))) + 1.0d0)
    else if (vef <= 4.5d+76) then
        tmp = ndchar + (nachar / (exp((ev / kbt)) + 1.0d0))
    else if (vef <= 5.2d+141) then
        tmp = ndchar / t_0
    else
        tmp = t_1
    end if
    code = tmp
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) {
	double t_0 = Math.exp((Vef / KbT)) + 1.0;
	double t_1 = NdChar + (NaChar / t_0);
	double tmp;
	if (Vef <= -0.0005) {
		tmp = t_1;
	} else if (Vef <= -2.5e-91) {
		tmp = NdChar / (Math.exp((Ec / (0.0 - KbT))) + 1.0);
	} else if (Vef <= 4.5e+76) {
		tmp = NdChar + (NaChar / (Math.exp((Ev / KbT)) + 1.0));
	} else if (Vef <= 5.2e+141) {
		tmp = NdChar / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp((Vef / KbT)) + 1.0
	t_1 = NdChar + (NaChar / t_0)
	tmp = 0
	if Vef <= -0.0005:
		tmp = t_1
	elif Vef <= -2.5e-91:
		tmp = NdChar / (math.exp((Ec / (0.0 - KbT))) + 1.0)
	elif Vef <= 4.5e+76:
		tmp = NdChar + (NaChar / (math.exp((Ev / KbT)) + 1.0))
	elif Vef <= 5.2e+141:
		tmp = NdChar / t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	t_1 = Float64(NdChar + Float64(NaChar / t_0))
	tmp = 0.0
	if (Vef <= -0.0005)
		tmp = t_1;
	elseif (Vef <= -2.5e-91)
		tmp = Float64(NdChar / Float64(exp(Float64(Ec / Float64(0.0 - KbT))) + 1.0));
	elseif (Vef <= 4.5e+76)
		tmp = Float64(NdChar + Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0)));
	elseif (Vef <= 5.2e+141)
		tmp = Float64(NdChar / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp((Vef / KbT)) + 1.0;
	t_1 = NdChar + (NaChar / t_0);
	tmp = 0.0;
	if (Vef <= -0.0005)
		tmp = t_1;
	elseif (Vef <= -2.5e-91)
		tmp = NdChar / (exp((Ec / (0.0 - KbT))) + 1.0);
	elseif (Vef <= 4.5e+76)
		tmp = NdChar + (NaChar / (exp((Ev / KbT)) + 1.0));
	elseif (Vef <= 5.2e+141)
		tmp = NdChar / t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar + N[(NaChar / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -0.0005], t$95$1, If[LessEqual[Vef, -2.5e-91], N[(NdChar / N[(N[Exp[N[(Ec / N[(0.0 - KbT), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 4.5e+76], N[(NdChar + N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 5.2e+141], N[(NdChar / t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -5.0000000000000001e-4 or 5.1999999999999999e141 < Vef

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 64.5%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 63.7%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 63.7%

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

   13. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. /-lowering-/.f64 58.1%

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right)\right) \]

   15. Simplified 58.1%

       \[\leadsto NdChar + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if -5.0000000000000001e-4 < Vef < -2.49999999999999999e-91

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 71.6%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 71.6%

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

   8. Taylor expanded in Ec around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \frac{Ec}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{Ec}{KbT}\right)\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\left(\frac{Ec}{KbT}\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 66.8%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(Ec, KbT\right)\right)\right)\right)\right) \]

   10. Simplified 66.8%

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

   if -2.49999999999999999e-91 < Vef < 4.4999999999999997e76

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 79.2%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 70.1%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 70.1%

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

   13. Taylor expanded in Ev around inf

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. /-lowering-/.f64 49.4%

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right)\right)\right)\right)\right) \]

   15. Simplified 49.4%

       \[\leadsto NdChar + \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   if 4.4999999999999997e76 < Vef < 5.1999999999999999e141

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 77.6%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 77.6%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 69.9%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 69.9%

       \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -0.0005:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq -2.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Ec}{0 - KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 4.5 \cdot 10^{+76}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{if}\;NaChar \leq -1.9 \cdot 10^{-96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ (- (+ Vef (+ EAccept Ev)) mu) KbT)) 1.0))))
   (if (<= NaChar -1.9e-96)
     t_0
     (if (<= NaChar 5e+29)
       (/ NdChar (+ (exp (/ (+ EDonor (+ Vef (- mu Ec))) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.9e-96) {
		tmp = t_0;
	} else if (NaChar <= 5e+29) {
		tmp = NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp((((vef + (eaccept + ev)) - mu) / kbt)) + 1.0d0)
    if (nachar <= (-1.9d-96)) then
        tmp = t_0
    else if (nachar <= 5d+29) then
        tmp = ndchar / (exp(((edonor + (vef + (mu - ec))) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
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) {
	double t_0 = NaChar / (Math.exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	double tmp;
	if (NaChar <= -1.9e-96) {
		tmp = t_0;
	} else if (NaChar <= 5e+29) {
		tmp = NdChar / (Math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0)
	tmp = 0
	if NaChar <= -1.9e-96:
		tmp = t_0
	elif NaChar <= 5e+29:
		tmp = NdChar / (math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Float64(EAccept + Ev)) - mu) / KbT)) + 1.0))
	tmp = 0.0
	if (NaChar <= -1.9e-96)
		tmp = t_0;
	elseif (NaChar <= 5e+29)
		tmp = Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	tmp = 0.0;
	if (NaChar <= -1.9e-96)
		tmp = t_0;
	elseif (NaChar <= 5e+29)
		tmp = NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + N[(EAccept + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.9e-96], t$95$0, If[LessEqual[NaChar, 5e+29], N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.9e-96 or 5.0000000000000001e29 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EAccept + Ev\right) + Vef\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), Vef\right), mu\right), KbT\right)\right)\right)\right) \]

      8. +-lowering-+.f64 72.1%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right)\right)\right) \]

   7. Simplified 72.1%

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

   if -1.9e-96 < NaChar < 5.0000000000000001e29

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 70.0%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 70.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.9 \cdot 10^{-96}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := NdChar + \frac{NaChar}{e^{\frac{mu}{0 - KbT}} + 1}\\ \mathbf{if}\;mu \leq -1.2 \cdot 10^{+177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq 2.15 \cdot 10^{+102}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ NdChar (/ NaChar (+ (exp (/ mu (- 0.0 KbT))) 1.0)))))
   (if (<= mu -1.2e+177)
     t_0
     (if (<= mu 2.15e+102)
       (/ NaChar (+ (exp (/ (- (+ Vef (+ EAccept Ev)) mu) KbT)) 1.0))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar + (NaChar / (exp((mu / (0.0 - KbT))) + 1.0));
	double tmp;
	if (mu <= -1.2e+177) {
		tmp = t_0;
	} else if (mu <= 2.15e+102) {
		tmp = NaChar / (exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar + (nachar / (exp((mu / (0.0d0 - kbt))) + 1.0d0))
    if (mu <= (-1.2d+177)) then
        tmp = t_0
    else if (mu <= 2.15d+102) then
        tmp = nachar / (exp((((vef + (eaccept + ev)) - mu) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
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) {
	double t_0 = NdChar + (NaChar / (Math.exp((mu / (0.0 - KbT))) + 1.0));
	double tmp;
	if (mu <= -1.2e+177) {
		tmp = t_0;
	} else if (mu <= 2.15e+102) {
		tmp = NaChar / (Math.exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar + (NaChar / (math.exp((mu / (0.0 - KbT))) + 1.0))
	tmp = 0
	if mu <= -1.2e+177:
		tmp = t_0
	elif mu <= 2.15e+102:
		tmp = NaChar / (math.exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar + Float64(NaChar / Float64(exp(Float64(mu / Float64(0.0 - KbT))) + 1.0)))
	tmp = 0.0
	if (mu <= -1.2e+177)
		tmp = t_0;
	elseif (mu <= 2.15e+102)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(Vef + Float64(EAccept + Ev)) - mu) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar + (NaChar / (exp((mu / (0.0 - KbT))) + 1.0));
	tmp = 0.0;
	if (mu <= -1.2e+177)
		tmp = t_0;
	elseif (mu <= 2.15e+102)
		tmp = NaChar / (exp((((Vef + (EAccept + Ev)) - mu) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar + N[(NaChar / N[(N[Exp[N[(mu / N[(0.0 - KbT), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -1.2e+177], t$95$0, If[LessEqual[mu, 2.15e+102], N[(NaChar / N[(N[Exp[N[(N[(N[(Vef + N[(EAccept + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if mu < -1.2e177 or 2.15e102 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 76.0%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 74.9%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 74.9%

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

   13. Taylor expanded in mu around inf

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \frac{mu}{KbT}\right)}\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{-1 \cdot mu}{KbT}\right)\right)\right)\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(mu\right)}{KbT}\right)\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - mu\right), KbT\right)\right)\right)\right)\right) \]

       5. --lowering--.f64 71.9%

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, mu\right), KbT\right)\right)\right)\right)\right) \]

   15. Simplified 71.9%

       \[\leadsto NdChar + \frac{NaChar}{1 + e^{\color{blue}{\frac{0 - mu}{KbT}}}} \]

   if -1.2e177 < mu < 2.15e102

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EAccept + Ev\right) + Vef\right), mu\right), KbT\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), Vef\right), mu\right), KbT\right)\right)\right)\right) \]

      8. +-lowering-+.f64 68.0%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right)\right)\right) \]

   7. Simplified 68.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.2 \cdot 10^{+177}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{mu}{0 - KbT}} + 1}\\ \mathbf{elif}\;mu \leq 2.15 \cdot 10^{+102}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{mu}{0 - KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(EDonor + \left(Vef + mu\right)\right) - Ec\\ t_1 := \frac{NdChar}{\frac{-1}{\left(\left(\left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right) + \frac{mu}{KbT}\right) + \frac{-0.5 \cdot \left(t\_0 \cdot t\_0\right)}{KbT \cdot KbT}\right) + \left(-1 - \frac{Ec}{KbT}\right)} + 1}\\ t_2 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(-0.25 \cdot \frac{NaChar}{\frac{KbT}{Vef}} + t\_2\right)\\ \mathbf{elif}\;KbT \leq -2.9 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq 1.6 \cdot 10^{-205}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 9.6 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (+ EDonor (+ Vef mu)) Ec))
        (t_1
         (/
          NdChar
          (+
           (/
            -1.0
            (+
             (+
              (+ (+ (/ Vef KbT) (/ EDonor KbT)) (/ mu KbT))
              (/ (* -0.5 (* t_0 t_0)) (* KbT KbT)))
             (- -1.0 (/ Ec KbT))))
           1.0)))
        (t_2 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.35e+154)
     (+
      (/ (* NdChar -0.25) (/ KbT (+ Vef (+ mu (- EDonor Ec)))))
      (+ (* -0.25 (/ NaChar (/ KbT Vef))) t_2))
     (if (<= KbT -2.9e-71)
       t_1
       (if (<= KbT 1.6e-205)
         (/ NdChar (+ (exp (/ Vef KbT)) 1.0))
         (if (<= KbT 9.6e+150) t_1 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (EDonor + (Vef + mu)) - Ec;
	double t_1 = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0);
	double t_2 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.35e+154) {
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_2);
	} else if (KbT <= -2.9e-71) {
		tmp = t_1;
	} else if (KbT <= 1.6e-205) {
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	} else if (KbT <= 9.6e+150) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (edonor + (vef + mu)) - ec
    t_1 = ndchar / (((-1.0d0) / (((((vef / kbt) + (edonor / kbt)) + (mu / kbt)) + (((-0.5d0) * (t_0 * t_0)) / (kbt * kbt))) + ((-1.0d0) - (ec / kbt)))) + 1.0d0)
    t_2 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.35d+154)) then
        tmp = ((ndchar * (-0.25d0)) / (kbt / (vef + (mu + (edonor - ec))))) + (((-0.25d0) * (nachar / (kbt / vef))) + t_2)
    else if (kbt <= (-2.9d-71)) then
        tmp = t_1
    else if (kbt <= 1.6d-205) then
        tmp = ndchar / (exp((vef / kbt)) + 1.0d0)
    else if (kbt <= 9.6d+150) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
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) {
	double t_0 = (EDonor + (Vef + mu)) - Ec;
	double t_1 = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0);
	double t_2 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.35e+154) {
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_2);
	} else if (KbT <= -2.9e-71) {
		tmp = t_1;
	} else if (KbT <= 1.6e-205) {
		tmp = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	} else if (KbT <= 9.6e+150) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (EDonor + (Vef + mu)) - Ec
	t_1 = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0)
	t_2 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -1.35e+154:
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_2)
	elif KbT <= -2.9e-71:
		tmp = t_1
	elif KbT <= 1.6e-205:
		tmp = NdChar / (math.exp((Vef / KbT)) + 1.0)
	elif KbT <= 9.6e+150:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(EDonor + Float64(Vef + mu)) - Ec)
	t_1 = Float64(NdChar / Float64(Float64(-1.0 / Float64(Float64(Float64(Float64(Float64(Vef / KbT) + Float64(EDonor / KbT)) + Float64(mu / KbT)) + Float64(Float64(-0.5 * Float64(t_0 * t_0)) / Float64(KbT * KbT))) + Float64(-1.0 - Float64(Ec / KbT)))) + 1.0))
	t_2 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.35e+154)
		tmp = Float64(Float64(Float64(NdChar * -0.25) / Float64(KbT / Float64(Vef + Float64(mu + Float64(EDonor - Ec))))) + Float64(Float64(-0.25 * Float64(NaChar / Float64(KbT / Vef))) + t_2));
	elseif (KbT <= -2.9e-71)
		tmp = t_1;
	elseif (KbT <= 1.6e-205)
		tmp = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	elseif (KbT <= 9.6e+150)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (EDonor + (Vef + mu)) - Ec;
	t_1 = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0);
	t_2 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.35e+154)
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_2);
	elseif (KbT <= -2.9e-71)
		tmp = t_1;
	elseif (KbT <= 1.6e-205)
		tmp = NdChar / (exp((Vef / KbT)) + 1.0);
	elseif (KbT <= 9.6e+150)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(N[(-1.0 / N[(N[(N[(N[(N[(Vef / KbT), $MachinePrecision] + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.35e+154], N[(N[(N[(NdChar * -0.25), $MachinePrecision] / N[(KbT / N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(NaChar / N[(KbT / Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -2.9e-71], t$95$1, If[LessEqual[KbT, 1.6e-205], N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 9.6e+150], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} \cdot \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(NaChar \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EAccept + Ev\right) + Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) + \color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)}\right)\right) \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\left(NaChar \cdot \frac{\left(\left(EAccept + Ev\right) + Vef\right) - mu}{KbT}\right) \cdot -0.25 + \left(0.5 \cdot \left(NaChar + NdChar\right) + \frac{\left(-0.25 \cdot NdChar\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}\right)} \]

   9. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(0.5 \cdot \left(NdChar + NaChar\right) + \frac{NaChar}{\frac{KbT}{Vef + \left(\left(EAccept + Ev\right) - mu\right)}} \cdot -0.25\right)} \]

   10. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(NdChar, \frac{-1}{4}\right), \mathsf{/.f64}\left(KbT, \mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NdChar, NaChar\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(NaChar, \color{blue}{\left(\frac{KbT}{Vef}\right)}\right), \frac{-1}{4}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 65.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(NdChar, \frac{-1}{4}\right), \mathsf{/.f64}\left(KbT, \mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NdChar, NaChar\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(NaChar, \mathsf{/.f64}\left(KbT, Vef\right)\right), \frac{-1}{4}\right)\right)\right) \]

   12. Simplified 65.8%

       \[\leadsto \frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(0.5 \cdot \left(NdChar + NaChar\right) + \frac{NaChar}{\color{blue}{\frac{KbT}{Vef}}} \cdot -0.25\right) \]

   if -1.35000000000000003e154 < KbT < -2.8999999999999999e-71 or 1.60000000000000005e-205 < KbT < 9.60000000000000011e150

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 77.6%

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

   10. Taylor expanded in NdChar around inf

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + \frac{1}{\left(1 + \frac{Ec}{KbT}\right) - \left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\left(1 + \frac{Ec}{KbT}\right) - \left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(1 + \frac{Ec}{KbT}\right) - \left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)\right)}\right)\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(1 + \frac{Ec}{KbT}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{Ec}{KbT}\right)\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(Ec, KbT\right)\right), \left(\frac{-1}{2} \cdot \color{blue}{\frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(Ec, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}}\right), \color{blue}{\left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)}\right)\right)\right)\right)\right) \]

   12. Simplified 48.4%

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

   if -2.8999999999999999e-71 < KbT < 1.60000000000000005e-205

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 60.6%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 60.6%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 45.2%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 45.2%

       \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if 9.60000000000000011e150 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 56.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(-0.25 \cdot \frac{NaChar}{\frac{KbT}{Vef}} + 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \mathbf{elif}\;KbT \leq -2.9 \cdot 10^{-71}:\\ \;\;\;\;\frac{NdChar}{\frac{-1}{\left(\left(\left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right) + \frac{mu}{KbT}\right) + \frac{-0.5 \cdot \left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT \cdot KbT}\right) + \left(-1 - \frac{Ec}{KbT}\right)} + 1}\\ \mathbf{elif}\;KbT \leq 1.6 \cdot 10^{-205}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 9.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{NdChar}{\frac{-1}{\left(\left(\left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right) + \frac{mu}{KbT}\right) + \frac{-0.5 \cdot \left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT \cdot KbT}\right) + \left(-1 - \frac{Ec}{KbT}\right)} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(EDonor + \left(Vef + mu\right)\right) - Ec\\ t_1 := \frac{NdChar}{\frac{-1}{\left(\left(\left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right) + \frac{mu}{KbT}\right) + \frac{-0.5 \cdot \left(t\_0 \cdot t\_0\right)}{KbT \cdot KbT}\right) + \left(-1 - \frac{Ec}{KbT}\right)} + 1}\\ t_2 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.75 \cdot 10^{+157}:\\ \;\;\;\;\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(-0.25 \cdot \frac{NaChar}{\frac{KbT}{Vef}} + t\_2\right)\\ \mathbf{elif}\;KbT \leq -1.16 \cdot 10^{-132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq 3.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 9.6 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (+ EDonor (+ Vef mu)) Ec))
        (t_1
         (/
          NdChar
          (+
           (/
            -1.0
            (+
             (+
              (+ (+ (/ Vef KbT) (/ EDonor KbT)) (/ mu KbT))
              (/ (* -0.5 (* t_0 t_0)) (* KbT KbT)))
             (- -1.0 (/ Ec KbT))))
           1.0)))
        (t_2 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.75e+157)
     (+
      (/ (* NdChar -0.25) (/ KbT (+ Vef (+ mu (- EDonor Ec)))))
      (+ (* -0.25 (/ NaChar (/ KbT Vef))) t_2))
     (if (<= KbT -1.16e-132)
       t_1
       (if (<= KbT 3.2e-205)
         (/ NdChar (+ (exp (/ EDonor KbT)) 1.0))
         (if (<= KbT 9.6e+150) t_1 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (EDonor + (Vef + mu)) - Ec;
	double t_1 = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0);
	double t_2 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.75e+157) {
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_2);
	} else if (KbT <= -1.16e-132) {
		tmp = t_1;
	} else if (KbT <= 3.2e-205) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else if (KbT <= 9.6e+150) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (edonor + (vef + mu)) - ec
    t_1 = ndchar / (((-1.0d0) / (((((vef / kbt) + (edonor / kbt)) + (mu / kbt)) + (((-0.5d0) * (t_0 * t_0)) / (kbt * kbt))) + ((-1.0d0) - (ec / kbt)))) + 1.0d0)
    t_2 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.75d+157)) then
        tmp = ((ndchar * (-0.25d0)) / (kbt / (vef + (mu + (edonor - ec))))) + (((-0.25d0) * (nachar / (kbt / vef))) + t_2)
    else if (kbt <= (-1.16d-132)) then
        tmp = t_1
    else if (kbt <= 3.2d-205) then
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    else if (kbt <= 9.6d+150) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
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) {
	double t_0 = (EDonor + (Vef + mu)) - Ec;
	double t_1 = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0);
	double t_2 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.75e+157) {
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_2);
	} else if (KbT <= -1.16e-132) {
		tmp = t_1;
	} else if (KbT <= 3.2e-205) {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	} else if (KbT <= 9.6e+150) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (EDonor + (Vef + mu)) - Ec
	t_1 = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0)
	t_2 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -1.75e+157:
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_2)
	elif KbT <= -1.16e-132:
		tmp = t_1
	elif KbT <= 3.2e-205:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	elif KbT <= 9.6e+150:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(EDonor + Float64(Vef + mu)) - Ec)
	t_1 = Float64(NdChar / Float64(Float64(-1.0 / Float64(Float64(Float64(Float64(Float64(Vef / KbT) + Float64(EDonor / KbT)) + Float64(mu / KbT)) + Float64(Float64(-0.5 * Float64(t_0 * t_0)) / Float64(KbT * KbT))) + Float64(-1.0 - Float64(Ec / KbT)))) + 1.0))
	t_2 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.75e+157)
		tmp = Float64(Float64(Float64(NdChar * -0.25) / Float64(KbT / Float64(Vef + Float64(mu + Float64(EDonor - Ec))))) + Float64(Float64(-0.25 * Float64(NaChar / Float64(KbT / Vef))) + t_2));
	elseif (KbT <= -1.16e-132)
		tmp = t_1;
	elseif (KbT <= 3.2e-205)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	elseif (KbT <= 9.6e+150)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (EDonor + (Vef + mu)) - Ec;
	t_1 = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0);
	t_2 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.75e+157)
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_2);
	elseif (KbT <= -1.16e-132)
		tmp = t_1;
	elseif (KbT <= 3.2e-205)
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	elseif (KbT <= 9.6e+150)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(N[(-1.0 / N[(N[(N[(N[(N[(Vef / KbT), $MachinePrecision] + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.75e+157], N[(N[(N[(NdChar * -0.25), $MachinePrecision] / N[(KbT / N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(NaChar / N[(KbT / Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -1.16e-132], t$95$1, If[LessEqual[KbT, 3.2e-205], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 9.6e+150], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -1.75000000000000001e157

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} \cdot \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(NaChar \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EAccept + Ev\right) + Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) + \color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)}\right)\right) \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\left(NaChar \cdot \frac{\left(\left(EAccept + Ev\right) + Vef\right) - mu}{KbT}\right) \cdot -0.25 + \left(0.5 \cdot \left(NaChar + NdChar\right) + \frac{\left(-0.25 \cdot NdChar\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}\right)} \]

   9. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(0.5 \cdot \left(NdChar + NaChar\right) + \frac{NaChar}{\frac{KbT}{Vef + \left(\left(EAccept + Ev\right) - mu\right)}} \cdot -0.25\right)} \]

   10. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(NdChar, \frac{-1}{4}\right), \mathsf{/.f64}\left(KbT, \mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NdChar, NaChar\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(NaChar, \color{blue}{\left(\frac{KbT}{Vef}\right)}\right), \frac{-1}{4}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 65.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(NdChar, \frac{-1}{4}\right), \mathsf{/.f64}\left(KbT, \mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NdChar, NaChar\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(NaChar, \mathsf{/.f64}\left(KbT, Vef\right)\right), \frac{-1}{4}\right)\right)\right) \]

   12. Simplified 65.8%

       \[\leadsto \frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(0.5 \cdot \left(NdChar + NaChar\right) + \frac{NaChar}{\color{blue}{\frac{KbT}{Vef}}} \cdot -0.25\right) \]

   if -1.75000000000000001e157 < KbT < -1.1599999999999999e-132 or 3.20000000000000009e-205 < KbT < 9.60000000000000011e150

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 76.1%

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

   10. Taylor expanded in NdChar around inf

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + \frac{1}{\left(1 + \frac{Ec}{KbT}\right) - \left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\left(1 + \frac{Ec}{KbT}\right) - \left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(1 + \frac{Ec}{KbT}\right) - \left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)\right)}\right)\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(1 + \frac{Ec}{KbT}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{Ec}{KbT}\right)\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(Ec, KbT\right)\right), \left(\frac{-1}{2} \cdot \color{blue}{\frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(Ec, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}}\right), \color{blue}{\left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)}\right)\right)\right)\right)\right) \]

   12. Simplified 45.2%

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

   if -1.1599999999999999e-132 < KbT < 3.20000000000000009e-205

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 65.5%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 65.5%

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

   8. Taylor expanded in EDonor around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EDonor}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 38.3%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(EDonor, KbT\right)\right)\right)\right) \]

   10. Simplified 38.3%

       \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} \]

   if 9.60000000000000011e150 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 56.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.75 \cdot 10^{+157}:\\ \;\;\;\;\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(-0.25 \cdot \frac{NaChar}{\frac{KbT}{Vef}} + 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \mathbf{elif}\;KbT \leq -1.16 \cdot 10^{-132}:\\ \;\;\;\;\frac{NdChar}{\frac{-1}{\left(\left(\left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right) + \frac{mu}{KbT}\right) + \frac{-0.5 \cdot \left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT \cdot KbT}\right) + \left(-1 - \frac{Ec}{KbT}\right)} + 1}\\ \mathbf{elif}\;KbT \leq 3.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 9.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{NdChar}{\frac{-1}{\left(\left(\left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right) + \frac{mu}{KbT}\right) + \frac{-0.5 \cdot \left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT \cdot KbT}\right) + \left(-1 - \frac{Ec}{KbT}\right)} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ \mathbf{if}\;Vef \leq -9.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{elif}\;Vef \leq 1.55 \cdot 10^{+160}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{mu}{0 - KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0)))
   (if (<= Vef -9.5e+166)
     (/ NdChar t_0)
     (if (<= Vef 1.55e+160)
       (+ NdChar (/ NaChar (+ (exp (/ mu (- 0.0 KbT))) 1.0)))
       (+ NdChar (/ NaChar t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((Vef / KbT)) + 1.0;
	double tmp;
	if (Vef <= -9.5e+166) {
		tmp = NdChar / t_0;
	} else if (Vef <= 1.55e+160) {
		tmp = NdChar + (NaChar / (exp((mu / (0.0 - KbT))) + 1.0));
	} else {
		tmp = NdChar + (NaChar / t_0);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((vef / kbt)) + 1.0d0
    if (vef <= (-9.5d+166)) then
        tmp = ndchar / t_0
    else if (vef <= 1.55d+160) then
        tmp = ndchar + (nachar / (exp((mu / (0.0d0 - kbt))) + 1.0d0))
    else
        tmp = ndchar + (nachar / t_0)
    end if
    code = tmp
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) {
	double t_0 = Math.exp((Vef / KbT)) + 1.0;
	double tmp;
	if (Vef <= -9.5e+166) {
		tmp = NdChar / t_0;
	} else if (Vef <= 1.55e+160) {
		tmp = NdChar + (NaChar / (Math.exp((mu / (0.0 - KbT))) + 1.0));
	} else {
		tmp = NdChar + (NaChar / t_0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp((Vef / KbT)) + 1.0
	tmp = 0
	if Vef <= -9.5e+166:
		tmp = NdChar / t_0
	elif Vef <= 1.55e+160:
		tmp = NdChar + (NaChar / (math.exp((mu / (0.0 - KbT))) + 1.0))
	else:
		tmp = NdChar + (NaChar / t_0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	tmp = 0.0
	if (Vef <= -9.5e+166)
		tmp = Float64(NdChar / t_0);
	elseif (Vef <= 1.55e+160)
		tmp = Float64(NdChar + Float64(NaChar / Float64(exp(Float64(mu / Float64(0.0 - KbT))) + 1.0)));
	else
		tmp = Float64(NdChar + Float64(NaChar / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp((Vef / KbT)) + 1.0;
	tmp = 0.0;
	if (Vef <= -9.5e+166)
		tmp = NdChar / t_0;
	elseif (Vef <= 1.55e+160)
		tmp = NdChar + (NaChar / (exp((mu / (0.0 - KbT))) + 1.0));
	else
		tmp = NdChar + (NaChar / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[Vef, -9.5e+166], N[(NdChar / t$95$0), $MachinePrecision], If[LessEqual[Vef, 1.55e+160], N[(NdChar + N[(NaChar / N[(N[Exp[N[(mu / N[(0.0 - KbT), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar + N[(NaChar / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -9.49999999999999984e166

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 58.2%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 58.2%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 55.7%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 55.7%

       \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if -9.49999999999999984e166 < Vef < 1.5499999999999999e160

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 77.5%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 67.7%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 67.7%

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

   13. Taylor expanded in mu around inf

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \frac{mu}{KbT}\right)}\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{-1 \cdot mu}{KbT}\right)\right)\right)\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(mu\right)}{KbT}\right)\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - mu\right), KbT\right)\right)\right)\right)\right) \]

       5. --lowering--.f64 56.5%

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, mu\right), KbT\right)\right)\right)\right)\right) \]

   15. Simplified 56.5%

       \[\leadsto NdChar + \frac{NaChar}{1 + e^{\color{blue}{\frac{0 - mu}{KbT}}}} \]

   if 1.5499999999999999e160 < Vef

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 62.6%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 62.6%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 62.6%

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

   13. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. /-lowering-/.f64 62.6%

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right)\right) \]

   15. Simplified 62.6%

       \[\leadsto NdChar + \frac{NaChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -9.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 1.55 \cdot 10^{+160}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{mu}{0 - KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -9.5 \cdot 10^{+166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 5 \cdot 10^{+76}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ Vef KbT)) 1.0))))
   (if (<= Vef -9.5e+166)
     t_0
     (if (<= Vef 5e+76) (+ NdChar (/ NaChar (+ (exp (/ Ev KbT)) 1.0))) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -9.5e+166) {
		tmp = t_0;
	} else if (Vef <= 5e+76) {
		tmp = NdChar + (NaChar / (exp((Ev / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp((vef / kbt)) + 1.0d0)
    if (vef <= (-9.5d+166)) then
        tmp = t_0
    else if (vef <= 5d+76) then
        tmp = ndchar + (nachar / (exp((ev / kbt)) + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
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) {
	double t_0 = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -9.5e+166) {
		tmp = t_0;
	} else if (Vef <= 5e+76) {
		tmp = NdChar + (NaChar / (Math.exp((Ev / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((Vef / KbT)) + 1.0)
	tmp = 0
	if Vef <= -9.5e+166:
		tmp = t_0
	elif Vef <= 5e+76:
		tmp = NdChar + (NaChar / (math.exp((Ev / KbT)) + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0))
	tmp = 0.0
	if (Vef <= -9.5e+166)
		tmp = t_0;
	elseif (Vef <= 5e+76)
		tmp = Float64(NdChar + Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((Vef / KbT)) + 1.0);
	tmp = 0.0;
	if (Vef <= -9.5e+166)
		tmp = t_0;
	elseif (Vef <= 5e+76)
		tmp = NdChar + (NaChar / (exp((Ev / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -9.5e+166], t$95$0, If[LessEqual[Vef, 5e+76], N[(NdChar + N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -9.49999999999999984e166 or 4.99999999999999991e76 < Vef

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 58.3%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 58.3%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 50.3%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 50.3%

       \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if -9.49999999999999984e166 < Vef < 4.99999999999999991e76

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 79.2%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 68.7%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 68.7%

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

   13. Taylor expanded in Ev around inf

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. /-lowering-/.f64 48.5%

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right)\right)\right)\right)\right) \]

   15. Simplified 48.5%

       \[\leadsto NdChar + \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -9.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 5 \cdot 10^{+76}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -3.65 \cdot 10^{+168}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 5.1 \cdot 10^{+76}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ Vef KbT)) 1.0))))
   (if (<= Vef -3.65e+168)
     t_0
     (if (<= Vef 5.1e+76)
       (+ NdChar (/ NaChar (+ (exp (/ EAccept KbT)) 1.0)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -3.65e+168) {
		tmp = t_0;
	} else if (Vef <= 5.1e+76) {
		tmp = NdChar + (NaChar / (exp((EAccept / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp((vef / kbt)) + 1.0d0)
    if (vef <= (-3.65d+168)) then
        tmp = t_0
    else if (vef <= 5.1d+76) then
        tmp = ndchar + (nachar / (exp((eaccept / kbt)) + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
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) {
	double t_0 = NdChar / (Math.exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -3.65e+168) {
		tmp = t_0;
	} else if (Vef <= 5.1e+76) {
		tmp = NdChar + (NaChar / (Math.exp((EAccept / KbT)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp((Vef / KbT)) + 1.0)
	tmp = 0
	if Vef <= -3.65e+168:
		tmp = t_0
	elif Vef <= 5.1e+76:
		tmp = NdChar + (NaChar / (math.exp((EAccept / KbT)) + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Vef / KbT)) + 1.0))
	tmp = 0.0
	if (Vef <= -3.65e+168)
		tmp = t_0;
	elseif (Vef <= 5.1e+76)
		tmp = Float64(NdChar + Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp((Vef / KbT)) + 1.0);
	tmp = 0.0;
	if (Vef <= -3.65e+168)
		tmp = t_0;
	elseif (Vef <= 5.1e+76)
		tmp = NdChar + (NaChar / (exp((EAccept / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -3.65e+168], t$95$0, If[LessEqual[Vef, 5.1e+76], N[(NdChar + N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -3.6499999999999998e168 or 5.1000000000000002e76 < Vef

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 58.3%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 58.3%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 50.3%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 50.3%

       \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{Vef}{KbT}}}} \]

   if -3.6499999999999998e168 < Vef < 5.1000000000000002e76

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 79.2%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 68.7%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 68.7%

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

   13. Taylor expanded in EAccept around inf

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. /-lowering-/.f64 54.6%

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right)\right)\right)\right)\right) \]

   15. Simplified 54.6%

       \[\leadsto NdChar + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -3.65 \cdot 10^{+168}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 5.1 \cdot 10^{+76}:\\ \;\;\;\;NdChar + \frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(EDonor + \left(Vef + mu\right)\right) - Ec\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(-0.25 \cdot \frac{NaChar}{\frac{KbT}{Vef}} + t\_1\right)\\ \mathbf{elif}\;KbT \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;\frac{NdChar}{\frac{-1}{\left(\left(\left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right) + \frac{mu}{KbT}\right) + \frac{-0.5 \cdot \left(t\_0 \cdot t\_0\right)}{KbT \cdot KbT}\right) + \left(-1 - \frac{Ec}{KbT}\right)} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- (+ EDonor (+ Vef mu)) Ec)) (t_1 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.35e+154)
     (+
      (/ (* NdChar -0.25) (/ KbT (+ Vef (+ mu (- EDonor Ec)))))
      (+ (* -0.25 (/ NaChar (/ KbT Vef))) t_1))
     (if (<= KbT 1.7e+153)
       (/
        NdChar
        (+
         (/
          -1.0
          (+
           (+
            (+ (+ (/ Vef KbT) (/ EDonor KbT)) (/ mu KbT))
            (/ (* -0.5 (* t_0 t_0)) (* KbT KbT)))
           (- -1.0 (/ Ec KbT))))
         1.0))
       t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (EDonor + (Vef + mu)) - Ec;
	double t_1 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.35e+154) {
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_1);
	} else if (KbT <= 1.7e+153) {
		tmp = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (edonor + (vef + mu)) - ec
    t_1 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.35d+154)) then
        tmp = ((ndchar * (-0.25d0)) / (kbt / (vef + (mu + (edonor - ec))))) + (((-0.25d0) * (nachar / (kbt / vef))) + t_1)
    else if (kbt <= 1.7d+153) then
        tmp = ndchar / (((-1.0d0) / (((((vef / kbt) + (edonor / kbt)) + (mu / kbt)) + (((-0.5d0) * (t_0 * t_0)) / (kbt * kbt))) + ((-1.0d0) - (ec / kbt)))) + 1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
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) {
	double t_0 = (EDonor + (Vef + mu)) - Ec;
	double t_1 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.35e+154) {
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_1);
	} else if (KbT <= 1.7e+153) {
		tmp = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (EDonor + (Vef + mu)) - Ec
	t_1 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -1.35e+154:
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_1)
	elif KbT <= 1.7e+153:
		tmp = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(EDonor + Float64(Vef + mu)) - Ec)
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.35e+154)
		tmp = Float64(Float64(Float64(NdChar * -0.25) / Float64(KbT / Float64(Vef + Float64(mu + Float64(EDonor - Ec))))) + Float64(Float64(-0.25 * Float64(NaChar / Float64(KbT / Vef))) + t_1));
	elseif (KbT <= 1.7e+153)
		tmp = Float64(NdChar / Float64(Float64(-1.0 / Float64(Float64(Float64(Float64(Float64(Vef / KbT) + Float64(EDonor / KbT)) + Float64(mu / KbT)) + Float64(Float64(-0.5 * Float64(t_0 * t_0)) / Float64(KbT * KbT))) + Float64(-1.0 - Float64(Ec / KbT)))) + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (EDonor + (Vef + mu)) - Ec;
	t_1 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.35e+154)
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_1);
	elseif (KbT <= 1.7e+153)
		tmp = NdChar / ((-1.0 / (((((Vef / KbT) + (EDonor / KbT)) + (mu / KbT)) + ((-0.5 * (t_0 * t_0)) / (KbT * KbT))) + (-1.0 - (Ec / KbT)))) + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(EDonor + N[(Vef + mu), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.35e+154], N[(N[(N[(NdChar * -0.25), $MachinePrecision] / N[(KbT / N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(NaChar / N[(KbT / Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.7e+153], N[(NdChar / N[(N[(-1.0 / N[(N[(N[(N[(N[(Vef / KbT), $MachinePrecision] + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} \cdot \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(NaChar \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EAccept + Ev\right) + Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) + \color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)}\right)\right) \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\left(NaChar \cdot \frac{\left(\left(EAccept + Ev\right) + Vef\right) - mu}{KbT}\right) \cdot -0.25 + \left(0.5 \cdot \left(NaChar + NdChar\right) + \frac{\left(-0.25 \cdot NdChar\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}\right)} \]

   9. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(0.5 \cdot \left(NdChar + NaChar\right) + \frac{NaChar}{\frac{KbT}{Vef + \left(\left(EAccept + Ev\right) - mu\right)}} \cdot -0.25\right)} \]

   10. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(NdChar, \frac{-1}{4}\right), \mathsf{/.f64}\left(KbT, \mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NdChar, NaChar\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(NaChar, \color{blue}{\left(\frac{KbT}{Vef}\right)}\right), \frac{-1}{4}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 65.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(NdChar, \frac{-1}{4}\right), \mathsf{/.f64}\left(KbT, \mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NdChar, NaChar\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(NaChar, \mathsf{/.f64}\left(KbT, Vef\right)\right), \frac{-1}{4}\right)\right)\right) \]

   12. Simplified 65.8%

       \[\leadsto \frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(0.5 \cdot \left(NdChar + NaChar\right) + \frac{NaChar}{\color{blue}{\frac{KbT}{Vef}}} \cdot -0.25\right) \]

   if -1.35000000000000003e154 < KbT < 1.6999999999999999e153

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 73.0%

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

   10. Taylor expanded in NdChar around inf

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + \frac{1}{\left(1 + \frac{Ec}{KbT}\right) - \left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{\left(1 + \frac{Ec}{KbT}\right) - \left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\right)}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(1 + \frac{Ec}{KbT}\right) - \left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)\right)}\right)\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(1 + \frac{Ec}{KbT}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{Ec}{KbT}\right)\right), \left(\color{blue}{\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(Ec, KbT\right)\right), \left(\frac{-1}{2} \cdot \color{blue}{\frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}}} + \left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(Ec, KbT\right)\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{{KbT}^{2}}\right), \color{blue}{\left(\frac{EDonor}{KbT} + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)}\right)\right)\right)\right)\right) \]

   12. Simplified 39.2%

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

   if 1.6999999999999999e153 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 56.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(-0.25 \cdot \frac{NaChar}{\frac{KbT}{Vef}} + 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \mathbf{elif}\;KbT \leq 1.7 \cdot 10^{+153}:\\ \;\;\;\;\frac{NdChar}{\frac{-1}{\left(\left(\left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right) + \frac{mu}{KbT}\right) + \frac{-0.5 \cdot \left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT \cdot KbT}\right) + \left(-1 - \frac{Ec}{KbT}\right)} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.5 \cdot 10^{+180}:\\ \;\;\;\;\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(-0.25 \cdot \frac{NaChar}{\frac{KbT}{Vef}} + t\_0\right)\\ \mathbf{elif}\;KbT \leq 1.35 \cdot 10^{+69}:\\ \;\;\;\;NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.5e+180)
     (+
      (/ (* NdChar -0.25) (/ KbT (+ Vef (+ mu (- EDonor Ec)))))
      (+ (* -0.25 (/ NaChar (/ KbT Vef))) t_0))
     (if (<= KbT 1.35e+69) NdChar t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.5e+180) {
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_0);
	} else if (KbT <= 1.35e+69) {
		tmp = NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.5d+180)) then
        tmp = ((ndchar * (-0.25d0)) / (kbt / (vef + (mu + (edonor - ec))))) + (((-0.25d0) * (nachar / (kbt / vef))) + t_0)
    else if (kbt <= 1.35d+69) then
        tmp = ndchar
    else
        tmp = t_0
    end if
    code = tmp
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) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.5e+180) {
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_0);
	} else if (KbT <= 1.35e+69) {
		tmp = NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -1.5e+180:
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_0)
	elif KbT <= 1.35e+69:
		tmp = NdChar
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.5e+180)
		tmp = Float64(Float64(Float64(NdChar * -0.25) / Float64(KbT / Float64(Vef + Float64(mu + Float64(EDonor - Ec))))) + Float64(Float64(-0.25 * Float64(NaChar / Float64(KbT / Vef))) + t_0));
	elseif (KbT <= 1.35e+69)
		tmp = NdChar;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.5e+180)
		tmp = ((NdChar * -0.25) / (KbT / (Vef + (mu + (EDonor - Ec))))) + ((-0.25 * (NaChar / (KbT / Vef))) + t_0);
	elseif (KbT <= 1.35e+69)
		tmp = NdChar;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.5e+180], N[(N[(N[(NdChar * -0.25), $MachinePrecision] / N[(KbT / N[(Vef + N[(mu + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(NaChar / N[(KbT / Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.35e+69], NdChar, t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.50000000000000001e180

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} \cdot \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(NaChar \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EAccept + Ev\right) + Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) + \color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)}\right)\right) \]

   7. Simplified 64.1%

      \[\leadsto \color{blue}{\left(NaChar \cdot \frac{\left(\left(EAccept + Ev\right) + Vef\right) - mu}{KbT}\right) \cdot -0.25 + \left(0.5 \cdot \left(NaChar + NdChar\right) + \frac{\left(-0.25 \cdot NdChar\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}\right)} \]

   9. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(0.5 \cdot \left(NdChar + NaChar\right) + \frac{NaChar}{\frac{KbT}{Vef + \left(\left(EAccept + Ev\right) - mu\right)}} \cdot -0.25\right)} \]

   10. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(NdChar, \frac{-1}{4}\right), \mathsf{/.f64}\left(KbT, \mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NdChar, NaChar\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(NaChar, \color{blue}{\left(\frac{KbT}{Vef}\right)}\right), \frac{-1}{4}\right)\right)\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 70.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(NdChar, \frac{-1}{4}\right), \mathsf{/.f64}\left(KbT, \mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(mu, \mathsf{\_.f64}\left(EDonor, Ec\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NdChar, NaChar\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(NaChar, \mathsf{/.f64}\left(KbT, Vef\right)\right), \frac{-1}{4}\right)\right)\right) \]

   12. Simplified 70.7%

       \[\leadsto \frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(0.5 \cdot \left(NdChar + NaChar\right) + \frac{NaChar}{\color{blue}{\frac{KbT}{Vef}}} \cdot -0.25\right) \]

   if -1.50000000000000001e180 < KbT < 1.3499999999999999e69

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 72.2%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 67.5%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 67.5%

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

   13. Taylor expanded in NdChar around inf

       \[\leadsto \color{blue}{NdChar} \]
   14. Step-by-step derivation

   15. Simplified 38.5%

       \[\leadsto \color{blue}{NdChar} \]

   if 1.3499999999999999e69 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 48.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 48.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.5 \cdot 10^{+180}:\\ \;\;\;\;\frac{NdChar \cdot -0.25}{\frac{KbT}{Vef + \left(mu + \left(EDonor - Ec\right)\right)}} + \left(-0.25 \cdot \frac{NaChar}{\frac{KbT}{Vef}} + 0.5 \cdot \left(NdChar + NaChar\right)\right)\\ \mathbf{elif}\;KbT \leq 1.35 \cdot 10^{+69}:\\ \;\;\;\;NdChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -2.1 \cdot 10^{+180}:\\ \;\;\;\;-0.25 \cdot \left(NaChar \cdot \frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}\right) + \left(t\_0 + -0.25 \cdot \left(NdChar \cdot \frac{mu}{KbT}\right)\right)\\ \mathbf{elif}\;KbT \leq 2.5 \cdot 10^{+69}:\\ \;\;\;\;NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -2.1e+180)
     (+
      (* -0.25 (* NaChar (/ (- (+ Vef (+ EAccept Ev)) mu) KbT)))
      (+ t_0 (* -0.25 (* NdChar (/ mu KbT)))))
     (if (<= KbT 2.5e+69) NdChar t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -2.1e+180) {
		tmp = (-0.25 * (NaChar * (((Vef + (EAccept + Ev)) - mu) / KbT))) + (t_0 + (-0.25 * (NdChar * (mu / KbT))));
	} else if (KbT <= 2.5e+69) {
		tmp = NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-2.1d+180)) then
        tmp = ((-0.25d0) * (nachar * (((vef + (eaccept + ev)) - mu) / kbt))) + (t_0 + ((-0.25d0) * (ndchar * (mu / kbt))))
    else if (kbt <= 2.5d+69) then
        tmp = ndchar
    else
        tmp = t_0
    end if
    code = tmp
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) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -2.1e+180) {
		tmp = (-0.25 * (NaChar * (((Vef + (EAccept + Ev)) - mu) / KbT))) + (t_0 + (-0.25 * (NdChar * (mu / KbT))));
	} else if (KbT <= 2.5e+69) {
		tmp = NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -2.1e+180:
		tmp = (-0.25 * (NaChar * (((Vef + (EAccept + Ev)) - mu) / KbT))) + (t_0 + (-0.25 * (NdChar * (mu / KbT))))
	elif KbT <= 2.5e+69:
		tmp = NdChar
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -2.1e+180)
		tmp = Float64(Float64(-0.25 * Float64(NaChar * Float64(Float64(Float64(Vef + Float64(EAccept + Ev)) - mu) / KbT))) + Float64(t_0 + Float64(-0.25 * Float64(NdChar * Float64(mu / KbT)))));
	elseif (KbT <= 2.5e+69)
		tmp = NdChar;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -2.1e+180)
		tmp = (-0.25 * (NaChar * (((Vef + (EAccept + Ev)) - mu) / KbT))) + (t_0 + (-0.25 * (NdChar * (mu / KbT))));
	elseif (KbT <= 2.5e+69)
		tmp = NdChar;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -2.1e+180], N[(N[(-0.25 * N[(NaChar * N[(N[(N[(Vef + N[(EAccept + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(-0.25 * N[(NdChar * N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.5e+69], NdChar, t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -2.1e180

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} + \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{4} \cdot \frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT} \cdot \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)}{KbT}\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(NaChar \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right), \frac{-1}{4}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(EAccept + \left(Ev + Vef\right)\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(EAccept + Ev\right) + Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \left(\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right) + \color{blue}{\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}{KbT}\right)}\right)\right) \]

   7. Simplified 64.1%

      \[\leadsto \color{blue}{\left(NaChar \cdot \frac{\left(\left(EAccept + Ev\right) + Vef\right) - mu}{KbT}\right) \cdot -0.25 + \left(0.5 \cdot \left(NaChar + NdChar\right) + \frac{\left(-0.25 \cdot NdChar\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)}{KbT}\right)} \]

   8. Taylor expanded in mu around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{NdChar \cdot mu}{KbT}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{NdChar \cdot mu}{KbT}\right)}\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \left(NdChar \cdot \color{blue}{\frac{mu}{KbT}}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(NdChar, \color{blue}{\left(\frac{mu}{KbT}\right)}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 69.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), Vef\right), mu\right), KbT\right)\right), \frac{-1}{4}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(NdChar, \mathsf{/.f64}\left(mu, \color{blue}{KbT}\right)\right)\right)\right)\right) \]

   10. Simplified 69.0%

       \[\leadsto \left(NaChar \cdot \frac{\left(\left(EAccept + Ev\right) + Vef\right) - mu}{KbT}\right) \cdot -0.25 + \left(0.5 \cdot \left(NaChar + NdChar\right) + \color{blue}{-0.25 \cdot \left(NdChar \cdot \frac{mu}{KbT}\right)}\right) \]

   if -2.1e180 < KbT < 2.50000000000000018e69

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 72.2%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 67.5%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 67.5%

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

   13. Taylor expanded in NdChar around inf

       \[\leadsto \color{blue}{NdChar} \]
   14. Step-by-step derivation

   15. Simplified 38.5%

       \[\leadsto \color{blue}{NdChar} \]

   if 2.50000000000000018e69 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 48.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 48.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.1 \cdot 10^{+180}:\\ \;\;\;\;-0.25 \cdot \left(NaChar \cdot \frac{\left(Vef + \left(EAccept + Ev\right)\right) - mu}{KbT}\right) + \left(0.5 \cdot \left(NdChar + NaChar\right) + -0.25 \cdot \left(NdChar \cdot \frac{mu}{KbT}\right)\right)\\ \mathbf{elif}\;KbT \leq 2.5 \cdot 10^{+69}:\\ \;\;\;\;NdChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -2.75 \cdot 10^{+180}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 6.2 \cdot 10^{+69}:\\ \;\;\;\;NdChar\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -2.75e+180) t_0 (if (<= KbT 6.2e+69) NdChar t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -2.75e+180) {
		tmp = t_0;
	} else if (KbT <= 6.2e+69) {
		tmp = NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-2.75d+180)) then
        tmp = t_0
    else if (kbt <= 6.2d+69) then
        tmp = ndchar
    else
        tmp = t_0
    end if
    code = tmp
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) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -2.75e+180) {
		tmp = t_0;
	} else if (KbT <= 6.2e+69) {
		tmp = NdChar;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -2.75e+180:
		tmp = t_0
	elif KbT <= 6.2e+69:
		tmp = NdChar
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -2.75e+180)
		tmp = t_0;
	elseif (KbT <= 6.2e+69)
		tmp = NdChar;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -2.75e+180)
		tmp = t_0;
	elseif (KbT <= 6.2e+69)
		tmp = NdChar;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -2.75e+180], t$95$0, If[LessEqual[KbT, 6.2e+69], NdChar, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -2.7500000000000001e180 or 6.1999999999999997e69 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 54.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 54.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   if -2.7500000000000001e180 < KbT < 6.1999999999999997e69

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 72.2%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 67.5%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 67.5%

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

   13. Taylor expanded in NdChar around inf

       \[\leadsto \color{blue}{NdChar} \]
   14. Step-by-step derivation

   15. Simplified 38.5%

       \[\leadsto \color{blue}{NdChar} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.75 \cdot 10^{+180}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 6.2 \cdot 10^{+69}:\\ \;\;\;\;NdChar\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -1.8 \cdot 10^{+191}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 6.2 \cdot 10^{+69}:\\ \;\;\;\;NdChar\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -1.8e+191)
   (* NaChar 0.5)
   (if (<= KbT 6.2e+69) NdChar (* NaChar 0.5))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.8e+191) {
		tmp = NaChar * 0.5;
	} else if (KbT <= 6.2e+69) {
		tmp = NdChar;
	} else {
		tmp = NaChar * 0.5;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (kbt <= (-1.8d+191)) then
        tmp = nachar * 0.5d0
    else if (kbt <= 6.2d+69) then
        tmp = ndchar
    else
        tmp = nachar * 0.5d0
    end if
    code = tmp
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) {
	double tmp;
	if (KbT <= -1.8e+191) {
		tmp = NaChar * 0.5;
	} else if (KbT <= 6.2e+69) {
		tmp = NdChar;
	} else {
		tmp = NaChar * 0.5;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -1.8e+191:
		tmp = NaChar * 0.5
	elif KbT <= 6.2e+69:
		tmp = NdChar
	else:
		tmp = NaChar * 0.5
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -1.8e+191)
		tmp = Float64(NaChar * 0.5);
	elseif (KbT <= 6.2e+69)
		tmp = NdChar;
	else
		tmp = Float64(NaChar * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -1.8e+191)
		tmp = NaChar * 0.5;
	elseif (KbT <= 6.2e+69)
		tmp = NdChar;
	else
		tmp = NaChar * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -1.8e+191], N[(NaChar * 0.5), $MachinePrecision], If[LessEqual[KbT, 6.2e+69], NdChar, N[(NaChar * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -1.8e191 or 6.1999999999999997e69 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 54.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 54.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   8. Taylor expanded in NaChar around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot NaChar} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 42.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{NaChar}\right) \]

   10. Simplified 42.0%

       \[\leadsto \color{blue}{0.5 \cdot NaChar} \]

   if -1.8e191 < KbT < 6.1999999999999997e69

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      10. associate-+r- N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   7. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. Simplified 72.7%

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

   10. Taylor expanded in Ec around inf

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

       6. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

       10. --lowering--.f64 67.6%

           \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. Simplified 67.6%

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

   13. Taylor expanded in NdChar around inf

       \[\leadsto \color{blue}{NdChar} \]
   14. Step-by-step derivation

   15. Simplified 38.5%

       \[\leadsto \color{blue}{NdChar} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.8 \cdot 10^{+191}:\\ \;\;\;\;NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 6.2 \cdot 10^{+69}:\\ \;\;\;\;NdChar\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 33.0% accurate, 229.0× speedup

Math

\[\begin{array}{l} \\ NdChar \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 NdChar)

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return NdChar;
}

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
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;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return NdChar

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return NdChar
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = NdChar;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := NdChar

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. frac-2neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{\mathsf{neg}\left(KbT\right)}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   2. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   3. exp-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}}\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   5. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   7. neg-sub0 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(Vef + \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(mu + \left(EDonor - Ec\right)\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   10. associate-+r- N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \left(\left(mu + EDonor\right) - Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\left(mu + EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(mu, EDonor\right), Ec\right)\right)\right), KbT\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \frac{NdChar}{1 + \color{blue}{\frac{1}{e^{\frac{0 - \left(Vef + \left(\left(mu + EDonor\right) - Ec\right)\right)}{KbT}}}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

7. Taylor expanded in KbT around -inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)}\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(-1 \cdot \frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right) + \frac{-1}{2} \cdot \frac{{\left(Ec - \left(EDonor + \left(Vef + mu\right)\right)\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

9. Simplified 72.2%

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

10. Taylor expanded in Ec around inf

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

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(NdChar, \color{blue}{\left(\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}\right)}\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right)\right) \]

    4. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right)\right) \]

    6. associate--l+ N/A

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right)\right) \]

    8. associate--l+ N/A

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \left(Ev + \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

    9. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \left(Vef - mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

    10. --lowering--.f64 65.1%

        \[\leadsto \mathsf{+.f64}\left(NdChar, \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EAccept, \mathsf{+.f64}\left(Ev, \mathsf{\_.f64}\left(Vef, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]

12. Simplified 65.1%

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

13. Taylor expanded in NdChar around inf

    \[\leadsto \color{blue}{NdChar} \]
14. Step-by-step derivation

15. Simplified 31.0%

    \[\leadsto \color{blue}{NdChar} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(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))))))