Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 29.3s

Alternatives: 30

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = -2.3316804033443505e-276
  2. Ec = 1.1005898108852336e-147
  3. Vef = -4.472517965921282e+88
  4. EDonor = -2.86345238775068e+261
  5. mu = -1.1662474744028984e+156
  6. KbT = 2.713412252663881e+166
  7. NaChar = -5.317617222208774e-277
  8. Ev = -4.813642983999874e-154
  9. EAccept = -1.6085636772066583e+178

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({t_0}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)}\right)}^{3}} \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (cbrt (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
    (/
     NaChar
     (+
      1.0
      (pow
       (* (pow (pow t_0 2.0) 0.3333333333333333) (cbrt (expm1 (log1p t_0))))
       3.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = cbrt(exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	return (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + pow((pow(pow(t_0, 2.0), 0.3333333333333333) * cbrt(expm1(log1p(t_0)))), 3.0)));
}

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.cbrt(Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	return (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.pow((Math.pow(Math.pow(t_0, 2.0), 0.3333333333333333) * Math.cbrt(Math.expm1(Math.log1p(t_0)))), 3.0)));
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = cbrt(exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT)))
	return Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + (Float64(((t_0 ^ 2.0) ^ 0.3333333333333333) * cbrt(expm1(log1p(t_0)))) ^ 3.0))))
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Power[N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Power[N[(N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[Power[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $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%

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

   1. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

   3. div-inv 99.9%

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

   4. associate-*r/ 99.9%

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

   5. *-commutative 99.9%

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

   6. *-un-lft-identity 99.9%

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

   7. +-commutative 99.9%

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

   8. associate-+l- 99.9%

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

5. Applied egg-rr 99.9%

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

   1. pow1/3 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\color{blue}{\left({\left(e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}\right)}^{0.3333333333333333}\right)}}^{3}} \]

   2. add-cube-cbrt 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\color{blue}{\left(\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} \cdot \sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right) \cdot \sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}}^{0.3333333333333333}\right)}^{3}} \]

   3. unpow-prod-down 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\color{blue}{\left({\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} \cdot \sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333}\right)}}^{3}} \]

   4. pow2 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\color{blue}{\left({\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{2}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333}\right)}^{3}} \]

   5. rem-cbrt-cube 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{3}}\right)}}^{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333}\right)}^{3}} \]

   6. rem-cube-cbrt 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\left(\sqrt[3]{\color{blue}{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333}\right)}^{3}} \]

   7. associate--r- 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \color{blue}{\left(\left(Ev - mu\right) + EAccept\right)}}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333}\right)}^{3}} \]

   8. pow1/3 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}}}\right)}^{3}} \]

7. Applied egg-rr 99.9%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\color{blue}{\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}}^{3}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)\right)}}\right)}^{3}} \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)\right)}}\right)}^{3}} \]

10. Final simplification 100.0%

    \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)\right)}\right)}^{3}} \]

Alternative 2: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}\\ \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(e^{0.2222222222222222 \cdot t_0} \cdot \sqrt[3]{\sqrt[3]{e^{t_0}}}\right)}^{3}} \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
    (/
     NaChar
     (+
      1.0
      (pow
       (* (exp (* 0.2222222222222222 t_0)) (cbrt (cbrt (exp t_0))))
       3.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Vef + ((Ev - mu) + EAccept)) / KbT;
	return (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + pow((exp((0.2222222222222222 * t_0)) * cbrt(cbrt(exp(t_0)))), 3.0)));
}

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 = (Vef + ((Ev - mu) + EAccept)) / KbT;
	return (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.pow((Math.exp((0.2222222222222222 * t_0)) * Math.cbrt(Math.cbrt(Math.exp(t_0)))), 3.0)));
}

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT)
	return Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + (Float64(exp(Float64(0.2222222222222222 * t_0)) * cbrt(cbrt(exp(t_0)))) ^ 3.0))))
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]}, N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Power[N[(N[Exp[N[(0.2222222222222222 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[N[Exp[t$95$0], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $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%

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

   1. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

   3. div-inv 99.9%

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

   4. associate-*r/ 99.9%

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

   5. *-commutative 99.9%

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

   6. *-un-lft-identity 99.9%

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

   7. +-commutative 99.9%

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

   8. associate-+l- 99.9%

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

5. Applied egg-rr 99.9%

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

   1. pow1/3 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\color{blue}{\left({\left(e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}\right)}^{0.3333333333333333}\right)}}^{3}} \]

   2. add-cube-cbrt 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\color{blue}{\left(\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} \cdot \sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right) \cdot \sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}}^{0.3333333333333333}\right)}^{3}} \]

   3. unpow-prod-down 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\color{blue}{\left({\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}} \cdot \sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333}\right)}}^{3}} \]

   4. pow2 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\color{blue}{\left({\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{2}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333}\right)}^{3}} \]

   5. rem-cbrt-cube 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{3}}\right)}}^{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333}\right)}^{3}} \]

   6. rem-cube-cbrt 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\left(\sqrt[3]{\color{blue}{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333}\right)}^{3}} \]

   7. associate--r- 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \color{blue}{\left(\left(Ev - mu\right) + EAccept\right)}}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}\right)}^{0.3333333333333333}\right)}^{3}} \]

   8. pow1/3 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(Ev - \left(mu - EAccept\right)\right)}{KbT}}}}}\right)}^{3}} \]

7. Applied egg-rr 99.9%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\color{blue}{\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}}^{3}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

   2. expm1-udef 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\right)}^{2}\right)}^{0.3333333333333333}\right)} - 1\right)} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

9. Applied egg-rr 99.9%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(\color{blue}{\left(e^{\mathsf{log1p}\left(e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot \frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}\right)}\right)} - 1\right)} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]
10. Step-by-step derivation

    1. expm1-def 99.9%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot \frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}\right)}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

    2. expm1-log1p 99.9%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(\color{blue}{e^{0.6666666666666666 \cdot \left(0.3333333333333333 \cdot \frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}\right)}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

    3. associate-*r* 99.9%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(e^{\color{blue}{\left(0.6666666666666666 \cdot 0.3333333333333333\right) \cdot \frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

    4. metadata-eval 99.9%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(e^{\color{blue}{0.2222222222222222} \cdot \frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

    5. +-commutative 99.9%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(e^{0.2222222222222222 \cdot \frac{Vef + \color{blue}{\left(EAccept + \left(Ev - mu\right)\right)}}{KbT}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

    6. associate--l+ 99.9%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(e^{0.2222222222222222 \cdot \frac{Vef + \color{blue}{\left(\left(EAccept + Ev\right) - mu\right)}}{KbT}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

    7. +-commutative 99.9%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(e^{0.2222222222222222 \cdot \frac{\color{blue}{\left(\left(EAccept + Ev\right) - mu\right) + Vef}}{KbT}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

    8. associate--l+ 99.9%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(e^{0.2222222222222222 \cdot \frac{\color{blue}{\left(EAccept + \left(Ev - mu\right)\right)} + Vef}{KbT}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

11. Simplified 99.9%

    \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(\color{blue}{e^{0.2222222222222222 \cdot \frac{\left(EAccept + \left(Ev - mu\right)\right) + Vef}{KbT}}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

12. Final simplification 99.9%

    \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(e^{0.2222222222222222 \cdot \frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}}\right)}^{3}} \]

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{NdChar}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}\right)\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+
  (/ NdChar (+ 1.0 (log1p (expm1 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))))
  (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) 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 + log1p(expm1(exp(((mu + (EDonor + (Vef - Ec))) / KbT)))))) + (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT))));
}

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.log1p(Math.expm1(Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))))) + (NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT))));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / (1.0 + math.log1p(math.expm1(math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))))) + (NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT))))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar / Float64(1.0 + log1p(expm1(exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT)))))
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / N[(1.0 + N[Log[1 + N[(Exp[N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $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%

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

   1. log1p-expm1-u 99.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}\right)\right)}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

5. Applied egg-rr 99.9%

   \[\leadsto \frac{NdChar}{1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}\right)\right)}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

6. Final simplification 99.9%

   \[\leadsto \frac{NdChar}{1 + \mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}\right)\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} \]

Alternative 4: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{if}\;mu \leq -6.4 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;mu \leq 3.1 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;mu \leq 1.26 \cdot 10^{+19}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.3 \cdot 10^{+200}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NaChar (+ 1.0 (exp (/ (- mu) KbT))))
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))))
        (t_1
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT))))
          (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))))
   (if (<= mu -6.4e+74)
     t_0
     (if (<= mu 3.1e-63)
       t_1
       (if (<= mu 1.26e+19)
         (+
          (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
          (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT)))))
         (if (<= mu 1.3e+200) t_1 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 / (1.0 + exp((-mu / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	double t_1 = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	double tmp;
	if (mu <= -6.4e+74) {
		tmp = t_0;
	} else if (mu <= 3.1e-63) {
		tmp = t_1;
	} else if (mu <= 1.26e+19) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT))));
	} else if (mu <= 1.3e+200) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (nachar / (1.0d0 + exp((-mu / kbt)))) + (ndchar / (1.0d0 + exp((mu / kbt))))
    t_1 = (nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))) + (ndchar / (1.0d0 + exp((edonor / kbt))))
    if (mu <= (-6.4d+74)) then
        tmp = t_0
    else if (mu <= 3.1d-63) then
        tmp = t_1
    else if (mu <= 1.26d+19) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp((((mu + vef) - ec) / kbt))))
    else if (mu <= 1.3d+200) then
        tmp = t_1
    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 / (1.0 + Math.exp((-mu / KbT)))) + (NdChar / (1.0 + Math.exp((mu / KbT))));
	double t_1 = (NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	double tmp;
	if (mu <= -6.4e+74) {
		tmp = t_0;
	} else if (mu <= 3.1e-63) {
		tmp = t_1;
	} else if (mu <= 1.26e+19) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp((((mu + Vef) - Ec) / KbT))));
	} else if (mu <= 1.3e+200) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (1.0 + math.exp((-mu / KbT)))) + (NdChar / (1.0 + math.exp((mu / KbT))))
	t_1 = (NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	tmp = 0
	if mu <= -6.4e+74:
		tmp = t_0
	elif mu <= 3.1e-63:
		tmp = t_1
	elif mu <= 1.26e+19:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp((((mu + Vef) - Ec) / KbT))))
	elif mu <= 1.3e+200:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))))
	tmp = 0.0
	if (mu <= -6.4e+74)
		tmp = t_0;
	elseif (mu <= 3.1e-63)
		tmp = t_1;
	elseif (mu <= 1.26e+19)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))));
	elseif (mu <= 1.3e+200)
		tmp = t_1;
	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 / (1.0 + exp((-mu / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	t_1 = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	tmp = 0.0;
	if (mu <= -6.4e+74)
		tmp = t_0;
	elseif (mu <= 3.1e-63)
		tmp = t_1;
	elseif (mu <= 1.26e+19)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT))));
	elseif (mu <= 1.3e+200)
		tmp = t_1;
	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[(N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -6.4e+74], t$95$0, If[LessEqual[mu, 3.1e-63], t$95$1, If[LessEqual[mu, 1.26e+19], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.3e+200], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -6.39999999999999989e74 or 1.3000000000000001e200 < 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%

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

   4. Taylor expanded in mu around inf 85.5%

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

   5. Taylor expanded in mu around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. mul-1-neg 81.5%

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

   7. Simplified 81.5%

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

   if -6.39999999999999989e74 < mu < 3.09999999999999984e-63 or 1.26e19 < mu < 1.3000000000000001e200

   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%

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

   4. Taylor expanded in EDonor around inf 80.9%

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

   if 3.09999999999999984e-63 < mu < 1.26e19

   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%

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

   4. Taylor expanded in Ev around inf 77.7%

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

   5. Taylor expanded in EDonor around 0 77.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -6.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 3.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.26 \cdot 10^{+19}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.3 \cdot 10^{+200}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \]

Alternative 5: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ t_1 := t_0 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -6.6 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;mu \leq -1.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 3.6 \cdot 10^{-63}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))))
        (t_1 (+ t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))))
   (if (<= mu -6.6e+38)
     t_1
     (if (<= mu -1.9e-17)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
        (/ NaChar (+ 1.0 (+ 1.0 (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)))))
       (if (<= mu 3.6e-63)
         (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
         (if (<= mu 3.4e+15)
           (+
            (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
            (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT)))))
           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 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	double tmp;
	if (mu <= -6.6e+38) {
		tmp = t_1;
	} else if (mu <= -1.9e-17) {
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	} else if (mu <= 3.6e-63) {
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (mu <= 3.4e+15) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT))));
	} 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 = nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + exp((mu / kbt))))
    if (mu <= (-6.6d+38)) then
        tmp = t_1
    else if (mu <= (-1.9d-17)) then
        tmp = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + (1.0d0 + (((ev + eaccept) + (vef - mu)) / kbt))))
    else if (mu <= 3.6d-63) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (mu <= 3.4d+15) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp((((mu + vef) - ec) / kbt))))
    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 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	double tmp;
	if (mu <= -6.6e+38) {
		tmp = t_1;
	} else if (mu <= -1.9e-17) {
		tmp = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	} else if (mu <= 3.6e-63) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (mu <= 3.4e+15) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp((((mu + Vef) - Ec) / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	t_1 = t_0 + (NdChar / (1.0 + math.exp((mu / KbT))))
	tmp = 0
	if mu <= -6.6e+38:
		tmp = t_1
	elif mu <= -1.9e-17:
		tmp = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))))
	elif mu <= 3.6e-63:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif mu <= 3.4e+15:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp((((mu + Vef) - Ec) / KbT))))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))))
	tmp = 0.0
	if (mu <= -6.6e+38)
		tmp = t_1;
	elseif (mu <= -1.9e-17)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)))));
	elseif (mu <= 3.6e-63)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (mu <= 3.4e+15)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	t_1 = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	tmp = 0.0;
	if (mu <= -6.6e+38)
		tmp = t_1;
	elseif (mu <= -1.9e-17)
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	elseif (mu <= 3.6e-63)
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (mu <= 3.4e+15)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT))));
	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[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -6.6e+38], t$95$1, If[LessEqual[mu, -1.9e-17], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(1.0 + N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 3.6e-63], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 3.4e+15], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -6.5999999999999998e38 or 3.4e15 < mu

   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%

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

   4. Taylor expanded in mu around inf 85.5%

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

   if -6.5999999999999998e38 < mu < -1.9000000000000001e-17

   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%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. div-inv 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+l- 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in KbT around inf 80.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(1 + \left(0.3333333333333333 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} + 0.6666666666666666 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)}} \]
   7. Step-by-step derivation

      1. distribute-rgt-out 80.1%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot \left(0.3333333333333333 + 0.6666666666666666\right)}\right)} \]

      2. metadata-eval 80.1%

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

      3. *-rgt-identity 80.1%

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

      4. associate--l+ 80.1%

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

      5. sub-neg 80.1%

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

      6. associate-+r+ 80.1%

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

      7. mul-1-neg 80.1%

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

      8. associate-+r+ 80.1%

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

      9. mul-1-neg 80.1%

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

      10. sub-neg 80.1%

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

   8. Simplified 80.1%

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

   if -1.9000000000000001e-17 < mu < 3.60000000000000008e-63

   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%

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

   4. Taylor expanded in EDonor around inf 87.7%

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

   if 3.60000000000000008e-63 < mu < 3.4e15

   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%

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

   4. Taylor expanded in Ev around inf 75.4%

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

   5. Taylor expanded in EDonor around 0 75.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -6.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;mu \leq -1.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 3.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \]

Alternative 6: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ t_1 := t_0 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;mu \leq -1.35:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.1 \cdot 10^{-63}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.06 \cdot 10^{+15}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))))
        (t_1 (+ t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))))
   (if (<= mu -4.6e+150)
     t_1
     (if (<= mu -1.35)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
        (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
       (if (<= mu 2.1e-63)
         (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
         (if (<= mu 1.06e+15)
           (+
            (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
            (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT)))))
           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 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	double tmp;
	if (mu <= -4.6e+150) {
		tmp = t_1;
	} else if (mu <= -1.35) {
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (mu <= 2.1e-63) {
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (mu <= 1.06e+15) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT))));
	} 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 = nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + exp((mu / kbt))))
    if (mu <= (-4.6d+150)) then
        tmp = t_1
    else if (mu <= (-1.35d0)) then
        tmp = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else if (mu <= 2.1d-63) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (mu <= 1.06d+15) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp((((mu + vef) - ec) / kbt))))
    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 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	double tmp;
	if (mu <= -4.6e+150) {
		tmp = t_1;
	} else if (mu <= -1.35) {
		tmp = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else if (mu <= 2.1e-63) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (mu <= 1.06e+15) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp((((mu + Vef) - Ec) / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	t_1 = t_0 + (NdChar / (1.0 + math.exp((mu / KbT))))
	tmp = 0
	if mu <= -4.6e+150:
		tmp = t_1
	elif mu <= -1.35:
		tmp = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	elif mu <= 2.1e-63:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif mu <= 1.06e+15:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp((((mu + Vef) - Ec) / KbT))))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))))
	tmp = 0.0
	if (mu <= -4.6e+150)
		tmp = t_1;
	elseif (mu <= -1.35)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (mu <= 2.1e-63)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (mu <= 1.06e+15)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	t_1 = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	tmp = 0.0;
	if (mu <= -4.6e+150)
		tmp = t_1;
	elseif (mu <= -1.35)
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	elseif (mu <= 2.1e-63)
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (mu <= 1.06e+15)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT))));
	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[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -4.6e+150], t$95$1, If[LessEqual[mu, -1.35], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2.1e-63], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.06e+15], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -4.60000000000000002e150 or 1.06e15 < mu

   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%

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

   4. Taylor expanded in mu around inf 86.7%

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

   if -4.60000000000000002e150 < mu < -1.3500000000000001

   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%

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

   4. Taylor expanded in EAccept around inf 79.4%

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

   if -1.3500000000000001 < mu < 2.1e-63

   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%

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

   4. Taylor expanded in EDonor around inf 85.8%

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

   if 2.1e-63 < mu < 1.06e15

   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%

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

   4. Taylor expanded in Ev around inf 75.4%

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

   5. Taylor expanded in EDonor around 0 75.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;mu \leq -1.35:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.06 \cdot 10^{+15}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \]

Alternative 7: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ t_2 := t_1 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -3.8 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;mu \leq -7.2:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.3 \cdot 10^{-161}:\\ \;\;\;\;t_1 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT)))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))))
        (t_2 (+ t_1 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))))
   (if (<= mu -3.8e+151)
     t_2
     (if (<= mu -7.2)
       (+ t_0 (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
       (if (<= mu 2.3e-161)
         (+ t_1 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
         (if (<= mu 4.2e+15)
           (+ t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
           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 = NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)));
	double t_1 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_2 = t_1 + (NdChar / (1.0 + exp((mu / KbT))));
	double tmp;
	if (mu <= -3.8e+151) {
		tmp = t_2;
	} else if (mu <= -7.2) {
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (mu <= 2.3e-161) {
		tmp = t_1 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (mu <= 4.2e+15) {
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	} 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 = ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))
    t_1 = nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    t_2 = t_1 + (ndchar / (1.0d0 + exp((mu / kbt))))
    if (mu <= (-3.8d+151)) then
        tmp = t_2
    else if (mu <= (-7.2d0)) then
        tmp = t_0 + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else if (mu <= 2.3d-161) then
        tmp = t_1 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (mu <= 4.2d+15) then
        tmp = t_0 + (nachar / (1.0d0 + exp((ev / kbt))))
    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 = NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_2 = t_1 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	double tmp;
	if (mu <= -3.8e+151) {
		tmp = t_2;
	} else if (mu <= -7.2) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else if (mu <= 2.3e-161) {
		tmp = t_1 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (mu <= 4.2e+15) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))
	t_1 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	t_2 = t_1 + (NdChar / (1.0 + math.exp((mu / KbT))))
	tmp = 0
	if mu <= -3.8e+151:
		tmp = t_2
	elif mu <= -7.2:
		tmp = t_0 + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	elif mu <= 2.3e-161:
		tmp = t_1 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif mu <= 4.2e+15:
		tmp = t_0 + (NaChar / (1.0 + math.exp((Ev / KbT))))
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	t_2 = Float64(t_1 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))))
	tmp = 0.0
	if (mu <= -3.8e+151)
		tmp = t_2;
	elseif (mu <= -7.2)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (mu <= 2.3e-161)
		tmp = Float64(t_1 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (mu <= 4.2e+15)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)));
	t_1 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	t_2 = t_1 + (NdChar / (1.0 + exp((mu / KbT))));
	tmp = 0.0;
	if (mu <= -3.8e+151)
		tmp = t_2;
	elseif (mu <= -7.2)
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / KbT))));
	elseif (mu <= 2.3e-161)
		tmp = t_1 + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (mu <= 4.2e+15)
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	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[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -3.8e+151], t$95$2, If[LessEqual[mu, -7.2], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2.3e-161], N[(t$95$1 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 4.2e+15], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -3.8e151 or 4.2e15 < mu

   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%

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

   4. Taylor expanded in mu around inf 86.7%

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

   if -3.8e151 < mu < -7.20000000000000018

   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%

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

   4. Taylor expanded in EAccept around inf 79.4%

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

   if -7.20000000000000018 < mu < 2.3e-161

   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%

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

   4. Taylor expanded in EDonor around inf 87.4%

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

   if 2.3e-161 < mu < 4.2e15

   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%

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

   4. Taylor expanded in Ev around inf 73.2%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -3.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;mu \leq -7.2:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \]

Alternative 8: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ t_2 := t_1 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -2.6 \cdot 10^{+156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;mu \leq -2.6 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;mu \leq 2.25 \cdot 10^{-67}:\\ \;\;\;\;t_1 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ Vef KbT))))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))))
        (t_2 (+ t_1 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))))
   (if (<= mu -2.6e+156)
     t_2
     (if (<= mu -2.6e-19)
       t_0
       (if (<= mu 2.25e-67)
         (+ t_1 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
         (if (<= mu 2.45e-11) t_0 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 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp((Vef / KbT))));
	double t_1 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_2 = t_1 + (NdChar / (1.0 + exp((mu / KbT))));
	double tmp;
	if (mu <= -2.6e+156) {
		tmp = t_2;
	} else if (mu <= -2.6e-19) {
		tmp = t_0;
	} else if (mu <= 2.25e-67) {
		tmp = t_1 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (mu <= 2.45e-11) {
		tmp = t_0;
	} 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 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + exp((vef / kbt))))
    t_1 = nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    t_2 = t_1 + (ndchar / (1.0d0 + exp((mu / kbt))))
    if (mu <= (-2.6d+156)) then
        tmp = t_2
    else if (mu <= (-2.6d-19)) then
        tmp = t_0
    else if (mu <= 2.25d-67) then
        tmp = t_1 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (mu <= 2.45d-11) then
        tmp = t_0
    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 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.exp((Vef / KbT))));
	double t_1 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_2 = t_1 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	double tmp;
	if (mu <= -2.6e+156) {
		tmp = t_2;
	} else if (mu <= -2.6e-19) {
		tmp = t_0;
	} else if (mu <= 2.25e-67) {
		tmp = t_1 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (mu <= 2.45e-11) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + math.exp((Vef / KbT))))
	t_1 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	t_2 = t_1 + (NdChar / (1.0 + math.exp((mu / KbT))))
	tmp = 0
	if mu <= -2.6e+156:
		tmp = t_2
	elif mu <= -2.6e-19:
		tmp = t_0
	elif mu <= 2.25e-67:
		tmp = t_1 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif mu <= 2.45e-11:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	t_2 = Float64(t_1 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))))
	tmp = 0.0
	if (mu <= -2.6e+156)
		tmp = t_2;
	elseif (mu <= -2.6e-19)
		tmp = t_0;
	elseif (mu <= 2.25e-67)
		tmp = Float64(t_1 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (mu <= 2.45e-11)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp((Vef / KbT))));
	t_1 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	t_2 = t_1 + (NdChar / (1.0 + exp((mu / KbT))));
	tmp = 0.0;
	if (mu <= -2.6e+156)
		tmp = t_2;
	elseif (mu <= -2.6e-19)
		tmp = t_0;
	elseif (mu <= 2.25e-67)
		tmp = t_1 + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (mu <= 2.45e-11)
		tmp = t_0;
	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[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -2.6e+156], t$95$2, If[LessEqual[mu, -2.6e-19], t$95$0, If[LessEqual[mu, 2.25e-67], N[(t$95$1 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2.45e-11], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -2.60000000000000019e156 or 2.4499999999999999e-11 < mu

   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%

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

   4. Taylor expanded in mu around inf 86.5%

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

   if -2.60000000000000019e156 < mu < -2.60000000000000013e-19 or 2.25000000000000008e-67 < mu < 2.4499999999999999e-11

   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%

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

   4. Taylor expanded in Vef around inf 89.7%

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

   if -2.60000000000000013e-19 < mu < 2.25000000000000008e-67

   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%

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

   4. Taylor expanded in EDonor around inf 87.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -2.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;mu \leq -2.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.25 \cdot 10^{-67}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \]

Alternative 9: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -6.2 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;mu \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{elif}\;mu \leq -1.7 \cdot 10^{-135}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;mu \leq 8 \cdot 10^{+197}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NaChar (+ 1.0 (exp (/ (- mu) KbT))))
          (/ NdChar (+ 1.0 (exp (/ mu KbT)))))))
   (if (<= mu -6.2e+52)
     t_0
     (if (<= mu -1.9e-36)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
        (/ NaChar (+ 1.0 (+ 1.0 (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)))))
       (if (<= mu -1.7e-135)
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT))))
          (/ NdChar 2.0))
         (if (<= mu 8e+197)
           (+
            (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
            (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT)))))
           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 / (1.0 + exp((-mu / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	double tmp;
	if (mu <= -6.2e+52) {
		tmp = t_0;
	} else if (mu <= -1.9e-36) {
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	} else if (mu <= -1.7e-135) {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0);
	} else if (mu <= 8e+197) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT))));
	} 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 / (1.0d0 + exp((-mu / kbt)))) + (ndchar / (1.0d0 + exp((mu / kbt))))
    if (mu <= (-6.2d+52)) then
        tmp = t_0
    else if (mu <= (-1.9d-36)) then
        tmp = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + (1.0d0 + (((ev + eaccept) + (vef - mu)) / kbt))))
    else if (mu <= (-1.7d-135)) then
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))) + (ndchar / 2.0d0)
    else if (mu <= 8d+197) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp((((mu + vef) - ec) / kbt))))
    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 / (1.0 + Math.exp((-mu / KbT)))) + (NdChar / (1.0 + Math.exp((mu / KbT))));
	double tmp;
	if (mu <= -6.2e+52) {
		tmp = t_0;
	} else if (mu <= -1.9e-36) {
		tmp = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	} else if (mu <= -1.7e-135) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0);
	} else if (mu <= 8e+197) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp((((mu + Vef) - Ec) / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (1.0 + math.exp((-mu / KbT)))) + (NdChar / (1.0 + math.exp((mu / KbT))))
	tmp = 0
	if mu <= -6.2e+52:
		tmp = t_0
	elif mu <= -1.9e-36:
		tmp = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))))
	elif mu <= -1.7e-135:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0)
	elif mu <= 8e+197:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp((((mu + Vef) - Ec) / KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))))
	tmp = 0.0
	if (mu <= -6.2e+52)
		tmp = t_0;
	elseif (mu <= -1.9e-36)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)))));
	elseif (mu <= -1.7e-135)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT)))) + Float64(NdChar / 2.0));
	elseif (mu <= 8e+197)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))));
	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 / (1.0 + exp((-mu / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	tmp = 0.0;
	if (mu <= -6.2e+52)
		tmp = t_0;
	elseif (mu <= -1.9e-36)
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	elseif (mu <= -1.7e-135)
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0);
	elseif (mu <= 8e+197)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT))));
	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[(N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -6.2e+52], t$95$0, If[LessEqual[mu, -1.9e-36], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(1.0 + N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, -1.7e-135], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 8e+197], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -6.2e52 or 7.9999999999999996e197 < 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%

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

   4. Taylor expanded in mu around inf 85.1%

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

   5. Taylor expanded in mu around inf 80.2%

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

      1. associate-*r/ 80.2%

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

      2. mul-1-neg 80.2%

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

   7. Simplified 80.2%

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

   if -6.2e52 < mu < -1.89999999999999985e-36

   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%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. div-inv 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+l- 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in KbT around inf 79.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(1 + \left(0.3333333333333333 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} + 0.6666666666666666 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)}} \]
   7. Step-by-step derivation

      1. distribute-rgt-out 79.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot \left(0.3333333333333333 + 0.6666666666666666\right)}\right)} \]

      2. metadata-eval 79.2%

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

      3. *-rgt-identity 79.2%

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

      4. associate--l+ 79.2%

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

      5. sub-neg 79.2%

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

      6. associate-+r+ 79.2%

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

      7. mul-1-neg 79.2%

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

      8. associate-+r+ 79.2%

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

      9. mul-1-neg 79.2%

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

      10. sub-neg 79.2%

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

   8. Simplified 79.2%

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

   if -1.89999999999999985e-36 < mu < -1.69999999999999995e-135

   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%

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

   4. Taylor expanded in KbT around inf 85.3%

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

      1. associate-+r+ 85.3%

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

   6. Simplified 85.3%

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

   7. Taylor expanded in KbT around inf 87.1%

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

   if -1.69999999999999995e-135 < mu < 7.9999999999999996e197

   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%

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

   4. Taylor expanded in Ev around inf 76.7%

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

   5. Taylor expanded in EDonor around 0 68.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -6.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;mu \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{elif}\;mu \leq -1.7 \cdot 10^{-135}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;mu \leq 8 \cdot 10^{+197}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \]

Alternative 10: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{if}\;mu \leq -8.4 \cdot 10^{-19}:\\ \;\;\;\;t_1 + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.1 \cdot 10^{-65}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 3.3 \cdot 10^{-11}:\\ \;\;\;\;t_1 + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))))
   (if (<= mu -8.4e-19)
     (+ t_1 (/ NaChar (+ 1.0 (exp (/ (- mu) KbT)))))
     (if (<= mu 2.1e-65)
       (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
       (if (<= mu 3.3e-11)
         (+ t_1 (/ NaChar (+ 1.0 (exp (/ Vef KbT)))))
         (+ t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT))))))))))

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 / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)));
	double tmp;
	if (mu <= -8.4e-19) {
		tmp = t_1 + (NaChar / (1.0 + exp((-mu / KbT))));
	} else if (mu <= 2.1e-65) {
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (mu <= 3.3e-11) {
		tmp = t_1 + (NaChar / (1.0 + exp((Vef / KbT))));
	} else {
		tmp = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	}
	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 = nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    t_1 = ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))
    if (mu <= (-8.4d-19)) then
        tmp = t_1 + (nachar / (1.0d0 + exp((-mu / kbt))))
    else if (mu <= 2.1d-65) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (mu <= 3.3d-11) then
        tmp = t_1 + (nachar / (1.0d0 + exp((vef / kbt))))
    else
        tmp = t_0 + (ndchar / (1.0d0 + exp((mu / kbt))))
    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 / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)));
	double tmp;
	if (mu <= -8.4e-19) {
		tmp = t_1 + (NaChar / (1.0 + Math.exp((-mu / KbT))));
	} else if (mu <= 2.1e-65) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (mu <= 3.3e-11) {
		tmp = t_1 + (NaChar / (1.0 + Math.exp((Vef / KbT))));
	} else {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	t_1 = NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))
	tmp = 0
	if mu <= -8.4e-19:
		tmp = t_1 + (NaChar / (1.0 + math.exp((-mu / KbT))))
	elif mu <= 2.1e-65:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif mu <= 3.3e-11:
		tmp = t_1 + (NaChar / (1.0 + math.exp((Vef / KbT))))
	else:
		tmp = t_0 + (NdChar / (1.0 + math.exp((mu / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT))))
	tmp = 0.0
	if (mu <= -8.4e-19)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))));
	elseif (mu <= 2.1e-65)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (mu <= 3.3e-11)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	t_1 = NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)));
	tmp = 0.0;
	if (mu <= -8.4e-19)
		tmp = t_1 + (NaChar / (1.0 + exp((-mu / KbT))));
	elseif (mu <= 2.1e-65)
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (mu <= 3.3e-11)
		tmp = t_1 + (NaChar / (1.0 + exp((Vef / KbT))));
	else
		tmp = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -8.4e-19], N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2.1e-65], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 3.3e-11], N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -8.3999999999999996e-19

   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%

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

   4. Taylor expanded in mu around inf 87.6%

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

      1. associate-*r/ 61.9%

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

      2. mul-1-neg 61.9%

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

   6. Simplified 87.6%

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

   if -8.3999999999999996e-19 < mu < 2.10000000000000003e-65

   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%

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

   4. Taylor expanded in EDonor around inf 87.6%

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

   if 2.10000000000000003e-65 < mu < 3.3000000000000002e-11

   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%

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

   4. Taylor expanded in Vef around inf 100.0%

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

   if 3.3000000000000002e-11 < mu

   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%

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

   4. Taylor expanded in mu around inf 89.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -8.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.1 \cdot 10^{-65}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 3.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \]

Alternative 11: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Final simplification 99.9%

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

Alternative 12: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -7 \cdot 10^{-74}:\\ \;\;\;\;t_1 + \frac{NdChar}{1 + \left(\left(\left(1 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;NaChar \leq 1.9 \cdot 10^{+108}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.3 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (+ 1.0 (/ (+ (+ Ev EAccept) (- Vef mu)) KbT))))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT))))))
   (if (<= NaChar -7e-74)
     (+
      t_1
      (/
       NdChar
       (+
        1.0
        (- (+ (+ 1.0 (/ EDonor KbT)) (+ (/ Vef KbT) (/ mu KbT))) (/ Ec KbT)))))
     (if (<= NaChar 1.5e-77)
       t_0
       (if (<= NaChar 1.9e+108)
         (+
          (/ NaChar (+ 1.0 (exp (/ (- mu) KbT))))
          (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
         (if (<= NaChar 1.3e+131) t_0 (+ t_1 (/ NdChar 2.0))))))))

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 / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	double t_1 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double tmp;
	if (NaChar <= -7e-74) {
		tmp = t_1 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT))));
	} else if (NaChar <= 1.5e-77) {
		tmp = t_0;
	} else if (NaChar <= 1.9e+108) {
		tmp = (NaChar / (1.0 + exp((-mu / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	} else if (NaChar <= 1.3e+131) {
		tmp = t_0;
	} else {
		tmp = t_1 + (NdChar / 2.0);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + (1.0d0 + (((ev + eaccept) + (vef - mu)) / kbt))))
    t_1 = nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    if (nachar <= (-7d-74)) then
        tmp = t_1 + (ndchar / (1.0d0 + (((1.0d0 + (edonor / kbt)) + ((vef / kbt) + (mu / kbt))) - (ec / kbt))))
    else if (nachar <= 1.5d-77) then
        tmp = t_0
    else if (nachar <= 1.9d+108) then
        tmp = (nachar / (1.0d0 + exp((-mu / kbt)))) + (ndchar / (1.0d0 + exp((mu / kbt))))
    else if (nachar <= 1.3d+131) then
        tmp = t_0
    else
        tmp = t_1 + (ndchar / 2.0d0)
    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 / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	double t_1 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double tmp;
	if (NaChar <= -7e-74) {
		tmp = t_1 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT))));
	} else if (NaChar <= 1.5e-77) {
		tmp = t_0;
	} else if (NaChar <= 1.9e+108) {
		tmp = (NaChar / (1.0 + Math.exp((-mu / KbT)))) + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else if (NaChar <= 1.3e+131) {
		tmp = t_0;
	} else {
		tmp = t_1 + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))))
	t_1 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	tmp = 0
	if NaChar <= -7e-74:
		tmp = t_1 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT))))
	elif NaChar <= 1.5e-77:
		tmp = t_0
	elif NaChar <= 1.9e+108:
		tmp = (NaChar / (1.0 + math.exp((-mu / KbT)))) + (NdChar / (1.0 + math.exp((mu / KbT))))
	elif NaChar <= 1.3e+131:
		tmp = t_0
	else:
		tmp = t_1 + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	tmp = 0.0
	if (NaChar <= -7e-74)
		tmp = Float64(t_1 + Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(EDonor / KbT)) + Float64(Float64(Vef / KbT) + Float64(mu / KbT))) - Float64(Ec / KbT)))));
	elseif (NaChar <= 1.5e-77)
		tmp = t_0;
	elseif (NaChar <= 1.9e+108)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	elseif (NaChar <= 1.3e+131)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	t_1 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	tmp = 0.0;
	if (NaChar <= -7e-74)
		tmp = t_1 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT))));
	elseif (NaChar <= 1.5e-77)
		tmp = t_0;
	elseif (NaChar <= 1.9e+108)
		tmp = (NaChar / (1.0 + exp((-mu / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	elseif (NaChar <= 1.3e+131)
		tmp = t_0;
	else
		tmp = t_1 + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(1.0 + N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -7e-74], N[(t$95$1 + N[(NdChar / N[(1.0 + N[(N[(N[(1.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.5e-77], t$95$0, If[LessEqual[NaChar, 1.9e+108], N[(N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.3e+131], t$95$0, N[(t$95$1 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -7.00000000000000029e-74

   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%

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

   4. Taylor expanded in KbT around inf 71.4%

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

      1. associate-+r+ 71.4%

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

   6. Simplified 71.4%

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

   if -7.00000000000000029e-74 < NaChar < 1.50000000000000008e-77 or 1.90000000000000004e108 < NaChar < 1.3e131

   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%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. div-inv 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+l- 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in KbT around inf 70.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(1 + \left(0.3333333333333333 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} + 0.6666666666666666 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)}} \]
   7. Step-by-step derivation

      1. distribute-rgt-out 70.7%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot \left(0.3333333333333333 + 0.6666666666666666\right)}\right)} \]

      2. metadata-eval 70.7%

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

      3. *-rgt-identity 70.7%

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

      4. associate--l+ 70.7%

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

      5. sub-neg 70.7%

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

      6. associate-+r+ 70.7%

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

      7. mul-1-neg 70.7%

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

      8. associate-+r+ 70.7%

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

      9. mul-1-neg 70.7%

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

      10. sub-neg 70.7%

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

   8. Simplified 70.7%

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

   if 1.50000000000000008e-77 < NaChar < 1.90000000000000004e108

   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%

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

   4. Taylor expanded in mu around inf 80.1%

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

   5. Taylor expanded in mu around inf 66.5%

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

      1. associate-*r/ 66.5%

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

      2. mul-1-neg 66.5%

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

   7. Simplified 66.5%

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

   if 1.3e131 < NaChar

   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%

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

   4. Taylor expanded in KbT around inf 77.2%

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

      1. associate-+r+ 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in KbT around inf 83.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -7 \cdot 10^{-74}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + \left(\left(\left(1 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 1.9 \cdot 10^{+108}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.3 \cdot 10^{+131}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 13: 65.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ t_1 := t_0 + \frac{KbT}{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{NdChar}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{if}\;NdChar \leq -1.45 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;NdChar \leq 3.7 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq 2.65 \cdot 10^{-107}:\\ \;\;\;\;t_0 + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq 1.25 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))))
        (t_1 (+ t_0 (/ KbT (/ (+ EDonor (+ Vef (- mu Ec))) NdChar))))
        (t_2
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (+ 1.0 (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)))))))
   (if (<= NdChar -1.45e-77)
     t_2
     (if (<= NdChar 3.7e-228)
       t_1
       (if (<= NdChar 2.65e-107)
         (+ t_0 (/ NdChar 2.0))
         (if (<= NdChar 1.25e-28) 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 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = t_0 + (KbT / ((EDonor + (Vef + (mu - Ec))) / NdChar));
	double t_2 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	double tmp;
	if (NdChar <= -1.45e-77) {
		tmp = t_2;
	} else if (NdChar <= 3.7e-228) {
		tmp = t_1;
	} else if (NdChar <= 2.65e-107) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (NdChar <= 1.25e-28) {
		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 = nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    t_1 = t_0 + (kbt / ((edonor + (vef + (mu - ec))) / ndchar))
    t_2 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + (1.0d0 + (((ev + eaccept) + (vef - mu)) / kbt))))
    if (ndchar <= (-1.45d-77)) then
        tmp = t_2
    else if (ndchar <= 3.7d-228) then
        tmp = t_1
    else if (ndchar <= 2.65d-107) then
        tmp = t_0 + (ndchar / 2.0d0)
    else if (ndchar <= 1.25d-28) 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 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = t_0 + (KbT / ((EDonor + (Vef + (mu - Ec))) / NdChar));
	double t_2 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	double tmp;
	if (NdChar <= -1.45e-77) {
		tmp = t_2;
	} else if (NdChar <= 3.7e-228) {
		tmp = t_1;
	} else if (NdChar <= 2.65e-107) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (NdChar <= 1.25e-28) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	t_1 = t_0 + (KbT / ((EDonor + (Vef + (mu - Ec))) / NdChar))
	t_2 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))))
	tmp = 0
	if NdChar <= -1.45e-77:
		tmp = t_2
	elif NdChar <= 3.7e-228:
		tmp = t_1
	elif NdChar <= 2.65e-107:
		tmp = t_0 + (NdChar / 2.0)
	elif NdChar <= 1.25e-28:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	t_1 = Float64(t_0 + Float64(KbT / Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / NdChar)))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -1.45e-77)
		tmp = t_2;
	elseif (NdChar <= 3.7e-228)
		tmp = t_1;
	elseif (NdChar <= 2.65e-107)
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	elseif (NdChar <= 1.25e-28)
		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 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	t_1 = t_0 + (KbT / ((EDonor + (Vef + (mu - Ec))) / NdChar));
	t_2 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -1.45e-77)
		tmp = t_2;
	elseif (NdChar <= 3.7e-228)
		tmp = t_1;
	elseif (NdChar <= 2.65e-107)
		tmp = t_0 + (NdChar / 2.0);
	elseif (NdChar <= 1.25e-28)
		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[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(KbT / N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(1.0 + N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.45e-77], t$95$2, If[LessEqual[NdChar, 3.7e-228], t$95$1, If[LessEqual[NdChar, 2.65e-107], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.25e-28], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.4499999999999999e-77 or 1.25e-28 < NdChar

   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%

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

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. div-inv 99.9%

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

      4. associate-*r/ 99.9%

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

      5. *-commutative 99.9%

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

      6. *-un-lft-identity 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l- 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in KbT around inf 65.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(1 + \left(0.3333333333333333 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} + 0.6666666666666666 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)}} \]
   7. Step-by-step derivation

      1. distribute-rgt-out 65.1%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot \left(0.3333333333333333 + 0.6666666666666666\right)}\right)} \]

      2. metadata-eval 65.1%

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

      3. *-rgt-identity 65.1%

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

      4. associate--l+ 65.1%

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

      5. sub-neg 65.1%

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

      6. associate-+r+ 65.1%

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

      7. mul-1-neg 65.1%

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

      8. associate-+r+ 65.1%

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

      9. mul-1-neg 65.1%

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

      10. sub-neg 65.1%

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

   8. Simplified 65.1%

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

   if -1.4499999999999999e-77 < NdChar < 3.7e-228 or 2.65e-107 < NdChar < 1.25e-28

   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%

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

   4. Taylor expanded in KbT around inf 73.6%

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

      1. associate-+r+ 73.6%

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

   6. Simplified 73.6%

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

   7. Taylor expanded in KbT around 0 76.5%

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

      1. associate-/l* 79.2%

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

      2. associate--l+ 79.2%

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

      3. associate--l+ 79.2%

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

   9. Simplified 79.2%

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

   if 3.7e-228 < NdChar < 2.65e-107

   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%

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

   4. Taylor expanded in KbT around inf 76.5%

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

      1. associate-+r+ 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in KbT around inf 87.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.45 \cdot 10^{-77}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 3.7 \cdot 10^{-228}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{KbT}{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{NdChar}}\\ \mathbf{elif}\;NdChar \leq 2.65 \cdot 10^{-107}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq 1.25 \cdot 10^{-28}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{KbT}{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \end{array} \]

Alternative 14: 64.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -2.3 \cdot 10^{-72}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(\left(\left(1 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 4.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT))))))
   (if (<= NaChar -2.3e-72)
     (+
      t_0
      (/
       NdChar
       (+
        1.0
        (- (+ (+ 1.0 (/ EDonor KbT)) (+ (/ Vef KbT) (/ mu KbT))) (/ Ec KbT)))))
     (if (<= NaChar 4.2e+131)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
        (/ NaChar (+ 1.0 (+ 1.0 (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)))))
       (+ t_0 (/ NdChar 2.0))))))

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 / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double tmp;
	if (NaChar <= -2.3e-72) {
		tmp = t_0 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT))));
	} else if (NaChar <= 4.2e+131) {
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	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 / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    if (nachar <= (-2.3d-72)) then
        tmp = t_0 + (ndchar / (1.0d0 + (((1.0d0 + (edonor / kbt)) + ((vef / kbt) + (mu / kbt))) - (ec / kbt))))
    else if (nachar <= 4.2d+131) then
        tmp = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + (1.0d0 + (((ev + eaccept) + (vef - mu)) / kbt))))
    else
        tmp = t_0 + (ndchar / 2.0d0)
    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 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double tmp;
	if (NaChar <= -2.3e-72) {
		tmp = t_0 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT))));
	} else if (NaChar <= 4.2e+131) {
		tmp = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	tmp = 0
	if NaChar <= -2.3e-72:
		tmp = t_0 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT))))
	elif NaChar <= 4.2e+131:
		tmp = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))))
	else:
		tmp = t_0 + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	tmp = 0.0
	if (NaChar <= -2.3e-72)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(EDonor / KbT)) + Float64(Float64(Vef / KbT) + Float64(mu / KbT))) - Float64(Ec / KbT)))));
	elseif (NaChar <= 4.2e+131)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	tmp = 0.0;
	if (NaChar <= -2.3e-72)
		tmp = t_0 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT))));
	elseif (NaChar <= 4.2e+131)
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	else
		tmp = t_0 + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -2.3e-72], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(N[(N[(1.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 4.2e+131], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(1.0 + N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -2.29999999999999995e-72

   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%

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

   4. Taylor expanded in KbT around inf 71.4%

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

      1. associate-+r+ 71.4%

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

   6. Simplified 71.4%

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

   if -2.29999999999999995e-72 < NaChar < 4.19999999999999971e131

   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%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. div-inv 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+l- 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in KbT around inf 66.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(1 + \left(0.3333333333333333 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} + 0.6666666666666666 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)}} \]
   7. Step-by-step derivation

      1. distribute-rgt-out 66.1%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot \left(0.3333333333333333 + 0.6666666666666666\right)}\right)} \]

      2. metadata-eval 66.1%

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

      3. *-rgt-identity 66.1%

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

      4. associate--l+ 66.1%

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

      5. sub-neg 66.1%

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

      6. associate-+r+ 66.1%

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

      7. mul-1-neg 66.1%

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

      8. associate-+r+ 66.1%

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

      9. mul-1-neg 66.1%

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

      10. sub-neg 66.1%

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

   8. Simplified 66.1%

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

   if 4.19999999999999971e131 < NaChar

   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%

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

   4. Taylor expanded in KbT around inf 77.2%

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

      1. associate-+r+ 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in KbT around inf 83.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{1 + \left(\left(\left(1 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 4.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 15: 62.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ t_1 := t_0 + \frac{KbT}{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{NdChar}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;NdChar \leq 1.25 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq 4.3 \cdot 10^{-108}:\\ \;\;\;\;t_0 + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq 9.5 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))))
        (t_1 (+ t_0 (/ KbT (/ (+ EDonor (+ Vef (- mu Ec))) NdChar))))
        (t_2
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 2.0 (/ Ev KbT))))))
   (if (<= NdChar -1.6e-77)
     t_2
     (if (<= NdChar 1.25e-225)
       t_1
       (if (<= NdChar 4.3e-108)
         (+ t_0 (/ NdChar 2.0))
         (if (<= NdChar 9.5e-31) 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 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = t_0 + (KbT / ((EDonor + (Vef + (mu - Ec))) / NdChar));
	double t_2 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (2.0 + (Ev / KbT)));
	double tmp;
	if (NdChar <= -1.6e-77) {
		tmp = t_2;
	} else if (NdChar <= 1.25e-225) {
		tmp = t_1;
	} else if (NdChar <= 4.3e-108) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (NdChar <= 9.5e-31) {
		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 = nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    t_1 = t_0 + (kbt / ((edonor + (vef + (mu - ec))) / ndchar))
    t_2 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (2.0d0 + (ev / kbt)))
    if (ndchar <= (-1.6d-77)) then
        tmp = t_2
    else if (ndchar <= 1.25d-225) then
        tmp = t_1
    else if (ndchar <= 4.3d-108) then
        tmp = t_0 + (ndchar / 2.0d0)
    else if (ndchar <= 9.5d-31) 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 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = t_0 + (KbT / ((EDonor + (Vef + (mu - Ec))) / NdChar));
	double t_2 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (2.0 + (Ev / KbT)));
	double tmp;
	if (NdChar <= -1.6e-77) {
		tmp = t_2;
	} else if (NdChar <= 1.25e-225) {
		tmp = t_1;
	} else if (NdChar <= 4.3e-108) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (NdChar <= 9.5e-31) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	t_1 = t_0 + (KbT / ((EDonor + (Vef + (mu - Ec))) / NdChar))
	t_2 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (2.0 + (Ev / KbT)))
	tmp = 0
	if NdChar <= -1.6e-77:
		tmp = t_2
	elif NdChar <= 1.25e-225:
		tmp = t_1
	elif NdChar <= 4.3e-108:
		tmp = t_0 + (NdChar / 2.0)
	elif NdChar <= 9.5e-31:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	t_1 = Float64(t_0 + Float64(KbT / Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / NdChar)))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))))
	tmp = 0.0
	if (NdChar <= -1.6e-77)
		tmp = t_2;
	elseif (NdChar <= 1.25e-225)
		tmp = t_1;
	elseif (NdChar <= 4.3e-108)
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	elseif (NdChar <= 9.5e-31)
		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 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	t_1 = t_0 + (KbT / ((EDonor + (Vef + (mu - Ec))) / NdChar));
	t_2 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (2.0 + (Ev / KbT)));
	tmp = 0.0;
	if (NdChar <= -1.6e-77)
		tmp = t_2;
	elseif (NdChar <= 1.25e-225)
		tmp = t_1;
	elseif (NdChar <= 4.3e-108)
		tmp = t_0 + (NdChar / 2.0);
	elseif (NdChar <= 9.5e-31)
		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[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(KbT / N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.6e-77], t$95$2, If[LessEqual[NdChar, 1.25e-225], t$95$1, If[LessEqual[NdChar, 4.3e-108], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 9.5e-31], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.6e-77 or 9.5000000000000008e-31 < NdChar

   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%

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

   4. Taylor expanded in Ev around inf 76.0%

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

   5. Taylor expanded in Ev around 0 57.7%

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

   if -1.6e-77 < NdChar < 1.25e-225 or 4.3e-108 < NdChar < 9.5000000000000008e-31

   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%

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

   4. Taylor expanded in KbT around inf 73.6%

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

      1. associate-+r+ 73.6%

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

   6. Simplified 73.6%

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

   7. Taylor expanded in KbT around 0 76.5%

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

      1. associate-/l* 79.2%

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

      2. associate--l+ 79.2%

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

      3. associate--l+ 79.2%

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

   9. Simplified 79.2%

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

   if 1.25e-225 < NdChar < 4.3e-108

   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%

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

   4. Taylor expanded in KbT around inf 76.5%

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

      1. associate-+r+ 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in KbT around inf 87.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;NdChar \leq 1.25 \cdot 10^{-225}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{KbT}{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{NdChar}}\\ \mathbf{elif}\;NdChar \leq 4.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq 9.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{KbT}{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \end{array} \]

Alternative 16: 64.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -1.75 \cdot 10^{-74}:\\ \;\;\;\;t_0 + \frac{NdChar}{\left(2 + \left(\frac{EDonor}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{elif}\;NaChar \leq 2.4 \cdot 10^{+133}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT))))))
   (if (<= NaChar -1.75e-74)
     (+ t_0 (/ NdChar (- (+ 2.0 (+ (/ EDonor KbT) (/ Vef KbT))) (/ Ec KbT))))
     (if (<= NaChar 2.4e+133)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
        (/ NaChar (+ 1.0 (+ 1.0 (/ (+ (+ Ev EAccept) (- Vef mu)) KbT)))))
       (+ t_0 (/ NdChar 2.0))))))

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 / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double tmp;
	if (NaChar <= -1.75e-74) {
		tmp = t_0 + (NdChar / ((2.0 + ((EDonor / KbT) + (Vef / KbT))) - (Ec / KbT)));
	} else if (NaChar <= 2.4e+133) {
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	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 / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    if (nachar <= (-1.75d-74)) then
        tmp = t_0 + (ndchar / ((2.0d0 + ((edonor / kbt) + (vef / kbt))) - (ec / kbt)))
    else if (nachar <= 2.4d+133) then
        tmp = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + (1.0d0 + (((ev + eaccept) + (vef - mu)) / kbt))))
    else
        tmp = t_0 + (ndchar / 2.0d0)
    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 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double tmp;
	if (NaChar <= -1.75e-74) {
		tmp = t_0 + (NdChar / ((2.0 + ((EDonor / KbT) + (Vef / KbT))) - (Ec / KbT)));
	} else if (NaChar <= 2.4e+133) {
		tmp = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	tmp = 0
	if NaChar <= -1.75e-74:
		tmp = t_0 + (NdChar / ((2.0 + ((EDonor / KbT) + (Vef / KbT))) - (Ec / KbT)))
	elif NaChar <= 2.4e+133:
		tmp = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))))
	else:
		tmp = t_0 + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	tmp = 0.0
	if (NaChar <= -1.75e-74)
		tmp = Float64(t_0 + Float64(NdChar / Float64(Float64(2.0 + Float64(Float64(EDonor / KbT) + Float64(Vef / KbT))) - Float64(Ec / KbT))));
	elseif (NaChar <= 2.4e+133)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(Ev + EAccept) + Float64(Vef - mu)) / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	tmp = 0.0;
	if (NaChar <= -1.75e-74)
		tmp = t_0 + (NdChar / ((2.0 + ((EDonor / KbT) + (Vef / KbT))) - (Ec / KbT)));
	elseif (NaChar <= 2.4e+133)
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + (1.0 + (((Ev + EAccept) + (Vef - mu)) / KbT))));
	else
		tmp = t_0 + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.75e-74], N[(t$95$0 + N[(NdChar / N[(N[(2.0 + N[(N[(EDonor / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 2.4e+133], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(1.0 + N[(N[(N[(Ev + EAccept), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -1.75000000000000007e-74

   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%

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

   4. Taylor expanded in KbT around inf 71.4%

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

      1. associate-+r+ 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in mu around 0 70.5%

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

   if -1.75000000000000007e-74 < NaChar < 2.3999999999999999e133

   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%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. div-inv 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+l- 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in KbT around inf 66.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(1 + \left(0.3333333333333333 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} + 0.6666666666666666 \cdot \frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)}} \]
   7. Step-by-step derivation

      1. distribute-rgt-out 66.1%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \color{blue}{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT} \cdot \left(0.3333333333333333 + 0.6666666666666666\right)}\right)} \]

      2. metadata-eval 66.1%

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

      3. *-rgt-identity 66.1%

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

      4. associate--l+ 66.1%

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

      5. sub-neg 66.1%

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

      6. associate-+r+ 66.1%

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

      7. mul-1-neg 66.1%

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

      8. associate-+r+ 66.1%

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

      9. mul-1-neg 66.1%

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

      10. sub-neg 66.1%

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

   8. Simplified 66.1%

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

   if 2.3999999999999999e133 < NaChar

   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%

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

   4. Taylor expanded in KbT around inf 77.2%

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

      1. associate-+r+ 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in KbT around inf 83.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.75 \cdot 10^{-74}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{\left(2 + \left(\frac{EDonor}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{elif}\;NaChar \leq 2.4 \cdot 10^{+133}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{\left(Ev + EAccept\right) + \left(Vef - mu\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 17: 58.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{if}\;NdChar \leq -1.05 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq -1.45 \cdot 10^{-141}:\\ \;\;\;\;t_0 + \frac{KbT}{\frac{Vef}{NdChar}}\\ \mathbf{elif}\;NdChar \leq -3 \cdot 10^{-166} \lor \neg \left(NdChar \leq 1.08 \cdot 10^{+24}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
          (/ NaChar (+ 2.0 (/ Ev KbT))))))
   (if (<= NdChar -1.05e-77)
     t_1
     (if (<= NdChar -1.45e-141)
       (+ t_0 (/ KbT (/ Vef NdChar)))
       (if (or (<= NdChar -3e-166) (not (<= NdChar 1.08e+24)))
         t_1
         (+ t_0 (/ NdChar 2.0)))))))

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 / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (2.0 + (Ev / KbT)));
	double tmp;
	if (NdChar <= -1.05e-77) {
		tmp = t_1;
	} else if (NdChar <= -1.45e-141) {
		tmp = t_0 + (KbT / (Vef / NdChar));
	} else if ((NdChar <= -3e-166) || !(NdChar <= 1.08e+24)) {
		tmp = t_1;
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / (2.0d0 + (ev / kbt)))
    if (ndchar <= (-1.05d-77)) then
        tmp = t_1
    else if (ndchar <= (-1.45d-141)) then
        tmp = t_0 + (kbt / (vef / ndchar))
    else if ((ndchar <= (-3d-166)) .or. (.not. (ndchar <= 1.08d+24))) then
        tmp = t_1
    else
        tmp = t_0 + (ndchar / 2.0d0)
    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 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (2.0 + (Ev / KbT)));
	double tmp;
	if (NdChar <= -1.05e-77) {
		tmp = t_1;
	} else if (NdChar <= -1.45e-141) {
		tmp = t_0 + (KbT / (Vef / NdChar));
	} else if ((NdChar <= -3e-166) || !(NdChar <= 1.08e+24)) {
		tmp = t_1;
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (2.0 + (Ev / KbT)))
	tmp = 0
	if NdChar <= -1.05e-77:
		tmp = t_1
	elif NdChar <= -1.45e-141:
		tmp = t_0 + (KbT / (Vef / NdChar))
	elif (NdChar <= -3e-166) or not (NdChar <= 1.08e+24):
		tmp = t_1
	else:
		tmp = t_0 + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))))
	tmp = 0.0
	if (NdChar <= -1.05e-77)
		tmp = t_1;
	elseif (NdChar <= -1.45e-141)
		tmp = Float64(t_0 + Float64(KbT / Float64(Vef / NdChar)));
	elseif ((NdChar <= -3e-166) || !(NdChar <= 1.08e+24))
		tmp = t_1;
	else
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	t_1 = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / (2.0 + (Ev / KbT)));
	tmp = 0.0;
	if (NdChar <= -1.05e-77)
		tmp = t_1;
	elseif (NdChar <= -1.45e-141)
		tmp = t_0 + (KbT / (Vef / NdChar));
	elseif ((NdChar <= -3e-166) || ~((NdChar <= 1.08e+24)))
		tmp = t_1;
	else
		tmp = t_0 + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.05e-77], t$95$1, If[LessEqual[NdChar, -1.45e-141], N[(t$95$0 + N[(KbT / N[(Vef / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[NdChar, -3e-166], N[Not[LessEqual[NdChar, 1.08e+24]], $MachinePrecision]], t$95$1, N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.05000000000000008e-77 or -1.45e-141 < NdChar < -3.0000000000000003e-166 or 1.0799999999999999e24 < NdChar

   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%

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

   4. Taylor expanded in Ev around inf 76.2%

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

   5. Taylor expanded in Ev around 0 59.5%

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

   if -1.05000000000000008e-77 < NdChar < -1.45e-141

   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%

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

   4. Taylor expanded in KbT around inf 76.3%

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

      1. associate-+r+ 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in Vef around inf 68.1%

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

      1. associate-/l* 75.9%

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

   9. Simplified 75.9%

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

   if -3.0000000000000003e-166 < NdChar < 1.0799999999999999e24

   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%

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

   4. Taylor expanded in KbT around inf 72.7%

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

      1. associate-+r+ 72.7%

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

   6. Simplified 72.7%

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

   7. Taylor expanded in KbT around inf 73.0%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.05 \cdot 10^{-77}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;NdChar \leq -1.45 \cdot 10^{-141}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{KbT}{\frac{Vef}{NdChar}}\\ \mathbf{elif}\;NdChar \leq -3 \cdot 10^{-166} \lor \neg \left(NdChar \leq 1.08 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 18: 50.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ t_2 := t_0 + \frac{KbT}{\frac{Vef}{NdChar}}\\ \mathbf{if}\;NdChar \leq -1.1 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq -5.8 \cdot 10^{-288}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;NdChar \leq 9.6 \cdot 10^{-206}:\\ \;\;\;\;t_0 + \frac{KbT}{\frac{mu}{NdChar}}\\ \mathbf{elif}\;NdChar \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT)))))
        (t_1
         (+ (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT)))) (/ NaChar 2.0)))
        (t_2 (+ t_0 (/ KbT (/ Vef NdChar)))))
   (if (<= NdChar -1.1e-77)
     t_1
     (if (<= NdChar -5.8e-288)
       t_2
       (if (<= NdChar 9.6e-206)
         (+ t_0 (/ KbT (/ mu NdChar)))
         (if (<= NdChar 7.2e-75) t_2 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 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	double t_2 = t_0 + (KbT / (Vef / NdChar));
	double tmp;
	if (NdChar <= -1.1e-77) {
		tmp = t_1;
	} else if (NdChar <= -5.8e-288) {
		tmp = t_2;
	} else if (NdChar <= 9.6e-206) {
		tmp = t_0 + (KbT / (mu / NdChar));
	} else if (NdChar <= 7.2e-75) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((((mu + vef) - ec) / kbt)))) + (nachar / 2.0d0)
    t_2 = t_0 + (kbt / (vef / ndchar))
    if (ndchar <= (-1.1d-77)) then
        tmp = t_1
    else if (ndchar <= (-5.8d-288)) then
        tmp = t_2
    else if (ndchar <= 9.6d-206) then
        tmp = t_0 + (kbt / (mu / ndchar))
    else if (ndchar <= 7.2d-75) then
        tmp = t_2
    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 = NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	double t_2 = t_0 + (KbT / (Vef / NdChar));
	double tmp;
	if (NdChar <= -1.1e-77) {
		tmp = t_1;
	} else if (NdChar <= -5.8e-288) {
		tmp = t_2;
	} else if (NdChar <= 9.6e-206) {
		tmp = t_0 + (KbT / (mu / NdChar));
	} else if (NdChar <= 7.2e-75) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0)
	t_2 = t_0 + (KbT / (Vef / NdChar))
	tmp = 0
	if NdChar <= -1.1e-77:
		tmp = t_1
	elif NdChar <= -5.8e-288:
		tmp = t_2
	elif NdChar <= 9.6e-206:
		tmp = t_0 + (KbT / (mu / NdChar))
	elif NdChar <= 7.2e-75:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))) + Float64(NaChar / 2.0))
	t_2 = Float64(t_0 + Float64(KbT / Float64(Vef / NdChar)))
	tmp = 0.0
	if (NdChar <= -1.1e-77)
		tmp = t_1;
	elseif (NdChar <= -5.8e-288)
		tmp = t_2;
	elseif (NdChar <= 9.6e-206)
		tmp = Float64(t_0 + Float64(KbT / Float64(mu / NdChar)));
	elseif (NdChar <= 7.2e-75)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)));
	t_1 = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	t_2 = t_0 + (KbT / (Vef / NdChar));
	tmp = 0.0;
	if (NdChar <= -1.1e-77)
		tmp = t_1;
	elseif (NdChar <= -5.8e-288)
		tmp = t_2;
	elseif (NdChar <= 9.6e-206)
		tmp = t_0 + (KbT / (mu / NdChar));
	elseif (NdChar <= 7.2e-75)
		tmp = t_2;
	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[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(KbT / N[(Vef / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.1e-77], t$95$1, If[LessEqual[NdChar, -5.8e-288], t$95$2, If[LessEqual[NdChar, 9.6e-206], N[(t$95$0 + N[(KbT / N[(mu / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 7.2e-75], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.10000000000000003e-77 or 7.2000000000000001e-75 < NdChar

   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%

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

   4. Taylor expanded in KbT around inf 56.1%

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

   5. Taylor expanded in EDonor around 0 51.4%

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

   if -1.10000000000000003e-77 < NdChar < -5.8000000000000003e-288 or 9.5999999999999998e-206 < NdChar < 7.2000000000000001e-75

   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%

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

   4. Taylor expanded in KbT around inf 70.5%

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

      1. associate-+r+ 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in Vef around inf 60.6%

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

      1. associate-/l* 62.1%

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

   9. Simplified 62.1%

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

   if -5.8000000000000003e-288 < NdChar < 9.5999999999999998e-206

   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%

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

   4. Taylor expanded in KbT around inf 79.9%

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

      1. associate-+r+ 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in mu around inf 79.7%

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

      1. associate-/l* 79.4%

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

   9. Simplified 79.4%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.1 \cdot 10^{-77}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;NdChar \leq -5.8 \cdot 10^{-288}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{KbT}{\frac{Vef}{NdChar}}\\ \mathbf{elif}\;NdChar \leq 9.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{KbT}{\frac{mu}{NdChar}}\\ \mathbf{elif}\;NdChar \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{KbT}{\frac{Vef}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \]

Alternative 19: 49.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -5.8 \cdot 10^{-81} \lor \neg \left(NdChar \leq 5.6 \cdot 10^{-155}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + KbT \cdot \frac{NdChar}{EDonor}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -5.8e-81) (not (<= NdChar 5.6e-155)))
   (+ (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT)))) (/ NaChar 2.0))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT))))
    (* KbT (/ NdChar EDonor)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -5.8e-81) || !(NdChar <= 5.6e-155)) {
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (KbT * (NdChar / EDonor));
	}
	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 ((ndchar <= (-5.8d-81)) .or. (.not. (ndchar <= 5.6d-155))) then
        tmp = (ndchar / (1.0d0 + exp((((mu + vef) - ec) / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))) + (kbt * (ndchar / edonor))
    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 ((NdChar <= -5.8e-81) || !(NdChar <= 5.6e-155)) {
		tmp = (NdChar / (1.0 + Math.exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (KbT * (NdChar / EDonor));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -5.8e-81) or not (NdChar <= 5.6e-155):
		tmp = (NdChar / (1.0 + math.exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (KbT * (NdChar / EDonor))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -5.8e-81) || !(NdChar <= 5.6e-155))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT)))) + Float64(KbT * Float64(NdChar / EDonor)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -5.8e-81) || ~((NdChar <= 5.6e-155)))
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (KbT * (NdChar / EDonor));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -5.8e-81], N[Not[LessEqual[NdChar, 5.6e-155]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(KbT * N[(NdChar / EDonor), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -5.79999999999999978e-81 or 5.5999999999999999e-155 < NdChar

   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%

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

   4. Taylor expanded in KbT around inf 54.7%

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

   5. Taylor expanded in EDonor around 0 50.5%

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

   if -5.79999999999999978e-81 < NdChar < 5.5999999999999999e-155

   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%

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

   4. Taylor expanded in KbT around inf 74.0%

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

      1. associate-+r+ 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in EDonor around inf 57.2%

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

      1. expm1-log1p-u 55.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{KbT \cdot NdChar}{EDonor}\right)\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

      2. expm1-udef 62.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{KbT \cdot NdChar}{EDonor}\right)} - 1\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

      3. associate-/l* 62.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{KbT}{\frac{EDonor}{NdChar}}}\right)} - 1\right) + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

   9. Applied egg-rr 62.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{KbT}{\frac{EDonor}{NdChar}}\right)} - 1\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]
   10. Step-by-step derivation

       1. expm1-def 54.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{KbT}{\frac{EDonor}{NdChar}}\right)\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(EAccept + \left(Ev - mu\right)\right)}{KbT}}} \]

       2. expm1-log1p 56.0%

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

       3. associate-/r/ 50.1%

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

       4. *-commutative 50.1%

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

       5. associate-*r/ 57.2%

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

       6. *-commutative 57.2%

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

       7. associate-*r/ 54.7%

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

   11. Simplified 54.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -5.8 \cdot 10^{-81} \lor \neg \left(NdChar \leq 5.6 \cdot 10^{-155}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + KbT \cdot \frac{NdChar}{EDonor}\\ \end{array} \]

Alternative 20: 50.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{-77} \lor \neg \left(NdChar \leq 6.6 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{KbT}{\frac{Vef}{NdChar}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -1.6e-77) (not (<= NdChar 6.6e-75)))
   (+ (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT)))) (/ NaChar 2.0))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT))))
    (/ KbT (/ Vef NdChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.6e-77) || !(NdChar <= 6.6e-75)) {
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (KbT / (Vef / NdChar));
	}
	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 ((ndchar <= (-1.6d-77)) .or. (.not. (ndchar <= 6.6d-75))) then
        tmp = (ndchar / (1.0d0 + exp((((mu + vef) - ec) / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))) + (kbt / (vef / ndchar))
    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 ((NdChar <= -1.6e-77) || !(NdChar <= 6.6e-75)) {
		tmp = (NdChar / (1.0 + Math.exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (KbT / (Vef / NdChar));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -1.6e-77) or not (NdChar <= 6.6e-75):
		tmp = (NdChar / (1.0 + math.exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (KbT / (Vef / NdChar))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -1.6e-77) || !(NdChar <= 6.6e-75))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT)))) + Float64(KbT / Float64(Vef / NdChar)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -1.6e-77) || ~((NdChar <= 6.6e-75)))
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (KbT / (Vef / NdChar));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -1.6e-77], N[Not[LessEqual[NdChar, 6.6e-75]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(KbT / N[(Vef / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.6e-77 or 6.5999999999999999e-75 < NdChar

   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%

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

   4. Taylor expanded in KbT around inf 56.1%

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

   5. Taylor expanded in EDonor around 0 51.4%

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

   if -1.6e-77 < NdChar < 6.5999999999999999e-75

   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%

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

   4. Taylor expanded in KbT around inf 73.3%

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

      1. associate-+r+ 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in Vef around inf 61.1%

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

      1. associate-/l* 61.9%

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

   9. Simplified 61.9%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.6 \cdot 10^{-77} \lor \neg \left(NdChar \leq 6.6 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{KbT}{\frac{Vef}{NdChar}}\\ \end{array} \]

Alternative 21: 54.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -4.3 \cdot 10^{-29} \lor \neg \left(NdChar \leq 2.55 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -4.3e-29) (not (<= NdChar 2.55e+29)))
   (+ (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT)))) (/ NaChar 2.0))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT))))
    (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -4.3e-29) || !(NdChar <= 2.55e+29)) {
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0);
	}
	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) :: tmp
    if ((ndchar <= (-4.3d-29)) .or. (.not. (ndchar <= 2.55d+29))) then
        tmp = (ndchar / (1.0d0 + exp((((mu + vef) - ec) / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))) + (ndchar / 2.0d0)
    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 tmp;
	if ((NdChar <= -4.3e-29) || !(NdChar <= 2.55e+29)) {
		tmp = (NdChar / (1.0 + Math.exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -4.3e-29) or not (NdChar <= 2.55e+29):
		tmp = (NdChar / (1.0 + math.exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -4.3e-29) || !(NdChar <= 2.55e+29))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -4.3e-29) || ~((NdChar <= 2.55e+29)))
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -4.3e-29], N[Not[LessEqual[NdChar, 2.55e+29]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -4.2999999999999998e-29 or 2.55e29 < NdChar

   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%

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

   4. Taylor expanded in KbT around inf 55.2%

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

   5. Taylor expanded in EDonor around 0 50.5%

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

   if -4.2999999999999998e-29 < NdChar < 2.55e29

   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%

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

   4. Taylor expanded in KbT around inf 72.3%

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

      1. associate-+r+ 72.3%

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

   6. Simplified 72.3%

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

   7. Taylor expanded in KbT around inf 70.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -4.3 \cdot 10^{-29} \lor \neg \left(NdChar \leq 2.55 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 22: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -5.4 \cdot 10^{-52} \lor \neg \left(NdChar \leq 2.4 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -5.4e-52) (not (<= NdChar 2.4e+24)))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ mu (+ EDonor (- Vef Ec))) KbT))))
    (/ NaChar 2.0))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ (- Ev mu) EAccept)) KbT))))
    (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -5.4e-52) || !(NdChar <= 2.4e+24)) {
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0);
	}
	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) :: tmp
    if ((ndchar <= (-5.4d-52)) .or. (.not. (ndchar <= 2.4d+24))) then
        tmp = (ndchar / (1.0d0 + exp(((mu + (edonor + (vef - ec))) / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev - mu) + eaccept)) / kbt)))) + (ndchar / 2.0d0)
    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 tmp;
	if ((NdChar <= -5.4e-52) || !(NdChar <= 2.4e+24)) {
		tmp = (NdChar / (1.0 + Math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -5.4e-52) or not (NdChar <= 2.4e+24):
		tmp = (NdChar / (1.0 + math.exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -5.4e-52) || !(NdChar <= 2.4e+24))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(EDonor + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev - mu) + EAccept)) / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -5.4e-52) || ~((NdChar <= 2.4e+24)))
		tmp = (NdChar / (1.0 + exp(((mu + (EDonor + (Vef - Ec))) / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev - mu) + EAccept)) / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -5.4e-52], N[Not[LessEqual[NdChar, 2.4e+24]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(EDonor + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev - mu), $MachinePrecision] + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -5.40000000000000019e-52 or 2.4000000000000001e24 < NdChar

   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%

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

   4. Taylor expanded in KbT around inf 55.8%

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

   if -5.40000000000000019e-52 < NdChar < 2.4000000000000001e24

   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%

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

   4. Taylor expanded in KbT around inf 72.6%

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

      1. associate-+r+ 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in KbT around inf 70.2%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -5.4 \cdot 10^{-52} \lor \neg \left(NdChar \leq 2.4 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(EDonor + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev - mu\right) + EAccept\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 23: 46.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -6 \cdot 10^{-178} \lor \neg \left(NdChar \leq 5.5 \cdot 10^{-135}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -6e-178) (not (<= NdChar 5.5e-135)))
   (+ (/ NdChar (+ 1.0 (exp (/ (- (+ mu Vef) Ec) KbT)))) (/ NaChar 2.0))
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -6e-178) || !(NdChar <= 5.5e-135)) {
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	}
	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) :: tmp
    if ((ndchar <= (-6d-178)) .or. (.not. (ndchar <= 5.5d-135))) then
        tmp = (ndchar / (1.0d0 + exp((((mu + vef) - ec) / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
    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 tmp;
	if ((NdChar <= -6e-178) || !(NdChar <= 5.5e-135)) {
		tmp = (NdChar / (1.0 + Math.exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -6e-178) or not (NdChar <= 5.5e-135):
		tmp = (NdChar / (1.0 + math.exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -6e-178) || !(NdChar <= 5.5e-135))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(mu + Vef) - Ec) / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -6e-178) || ~((NdChar <= 5.5e-135)))
		tmp = (NdChar / (1.0 + exp((((mu + Vef) - Ec) / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -6e-178], N[Not[LessEqual[NdChar, 5.5e-135]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(mu + Vef), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -5.9999999999999997e-178 or 5.4999999999999999e-135 < NdChar

   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%

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

   4. Taylor expanded in KbT around inf 52.3%

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

   5. Taylor expanded in EDonor around 0 48.4%

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

   if -5.9999999999999997e-178 < NdChar < 5.4999999999999999e-135

   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%

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

   4. Taylor expanded in Ev around inf 67.4%

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

   5. Taylor expanded in KbT around inf 59.7%

      \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -6 \cdot 10^{-178} \lor \neg \left(NdChar \leq 5.5 \cdot 10^{-135}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(mu + Vef\right) - Ec}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 24: 36.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq 4.8 \cdot 10^{-304} \lor \neg \left(KbT \leq 7.2 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar \cdot KbT}{EDonor}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= KbT 4.8e-304) (not (<= KbT 7.2e+24)))
   (+ (/ NaChar 2.0) (/ NdChar (+ 1.0 (exp (/ (- Ec) KbT)))))
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ (* NdChar KbT) EDonor))))

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 <= 4.8e-304) || !(KbT <= 7.2e+24)) {
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + exp((-Ec / KbT))));
	} else {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + ((NdChar * KbT) / EDonor);
	}
	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 <= 4.8d-304) .or. (.not. (kbt <= 7.2d+24))) then
        tmp = (nachar / 2.0d0) + (ndchar / (1.0d0 + exp((-ec / kbt))))
    else
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + ((ndchar * kbt) / edonor)
    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 <= 4.8e-304) || !(KbT <= 7.2e+24)) {
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + Math.exp((-Ec / KbT))));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + ((NdChar * KbT) / EDonor);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (KbT <= 4.8e-304) or not (KbT <= 7.2e+24):
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + math.exp((-Ec / KbT))))
	else:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + ((NdChar * KbT) / EDonor)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((KbT <= 4.8e-304) || !(KbT <= 7.2e+24))
		tmp = Float64(Float64(NaChar / 2.0) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Ec) / KbT)))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(Float64(NdChar * KbT) / EDonor));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((KbT <= 4.8e-304) || ~((KbT <= 7.2e+24)))
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + exp((-Ec / KbT))));
	else
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + ((NdChar * KbT) / EDonor);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[KbT, 4.8e-304], N[Not[LessEqual[KbT, 7.2e+24]], $MachinePrecision]], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(NdChar * KbT), $MachinePrecision] / EDonor), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < 4.8000000000000002e-304 or 7.19999999999999966e24 < KbT

   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%

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

   4. Taylor expanded in KbT around inf 54.8%

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

   5. Taylor expanded in Ec around inf 44.7%

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

      1. associate-*r/ 44.7%

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

      2. mul-1-neg 44.7%

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

   7. Simplified 44.7%

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

   if 4.8000000000000002e-304 < KbT < 7.19999999999999966e24

   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%

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

   4. Taylor expanded in KbT around inf 48.8%

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

      1. associate-+r+ 48.8%

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

   6. Simplified 48.8%

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

   7. Taylor expanded in EDonor around inf 44.8%

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

   8. Taylor expanded in Ev around inf 33.2%

      \[\leadsto \frac{KbT \cdot NdChar}{EDonor} + \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq 4.8 \cdot 10^{-304} \lor \neg \left(KbT \leq 7.2 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar \cdot KbT}{EDonor}\\ \end{array} \]

Alternative 25: 39.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -9.4 \cdot 10^{-178} \lor \neg \left(NdChar \leq 1.5 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -9.4e-178) (not (<= NdChar 1.5e+96)))
   (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (/ NaChar 2.0))
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -9.4e-178) || !(NdChar <= 1.5e+96)) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	}
	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) :: tmp
    if ((ndchar <= (-9.4d-178)) .or. (.not. (ndchar <= 1.5d+96))) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
    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 tmp;
	if ((NdChar <= -9.4e-178) || !(NdChar <= 1.5e+96)) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -9.4e-178) or not (NdChar <= 1.5e+96):
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -9.4e-178) || !(NdChar <= 1.5e+96))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -9.4e-178) || ~((NdChar <= 1.5e+96)))
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -9.4e-178], N[Not[LessEqual[NdChar, 1.5e+96]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -9.39999999999999999e-178 or 1.5e96 < NdChar

   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%

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

   4. Taylor expanded in KbT around inf 54.4%

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

   5. Taylor expanded in EDonor around inf 39.1%

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

   if -9.39999999999999999e-178 < NdChar < 1.5e96

   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%

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

   4. Taylor expanded in Ev around inf 69.0%

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

   5. Taylor expanded in KbT around inf 51.9%

      \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -9.4 \cdot 10^{-178} \lor \neg \left(NdChar \leq 1.5 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 26: 36.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -1.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -1.05e+44)
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar 2.0))
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* 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 (Ev <= -1.05e+44) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (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 (ev <= (-1.05d+44)) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
    else
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (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 (Ev <= -1.05e+44) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -1.05e+44:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / 2.0)
	else:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -1.05e+44)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / 2.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + 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 (Ev <= -1.05e+44)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	else
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -1.05e+44], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Ev < -1.04999999999999993e44

   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%

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

   4. Taylor expanded in Ev around inf 84.8%

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

   5. Taylor expanded in KbT around inf 48.4%

      \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} \]

   if -1.04999999999999993e44 < Ev

   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%

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

   4. Taylor expanded in mu around inf 73.0%

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

   5. Taylor expanded in KbT around inf 36.5%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \color{blue}{0.5 \cdot NaChar} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -1.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 27: 35.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar 2.0)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
}

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 / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
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 / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / 2.0);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / 2.0)

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / 2.0))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $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%

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

4. Taylor expanded in Ev around inf 72.0%

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

5. Taylor expanded in KbT around inf 41.4%

   \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} \]

6. Final simplification 41.4%

   \[\leadsto \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2} \]

Alternative 28: 27.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq -5 \cdot 10^{+266}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(\left(1 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 + \left(\left(\left(1 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Vef -5e+266)
   (+
    (/
     NdChar
     (+
      1.0
      (- (+ (+ 1.0 (/ EDonor KbT)) (+ (/ Vef KbT) (/ mu KbT))) (/ Ec KbT))))
    (/
     NaChar
     (+
      1.0
      (- (+ (+ 1.0 (/ EAccept KbT)) (+ (/ Vef KbT) (/ Ev KbT))) (/ mu KbT)))))
   (* 0.5 (+ NdChar NaChar))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Vef <= -5e+266) {
		tmp = (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)))) + (NaChar / (1.0 + (((1.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT))));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	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 (vef <= (-5d+266)) then
        tmp = (ndchar / (1.0d0 + (((1.0d0 + (edonor / kbt)) + ((vef / kbt) + (mu / kbt))) - (ec / kbt)))) + (nachar / (1.0d0 + (((1.0d0 + (eaccept / kbt)) + ((vef / kbt) + (ev / kbt))) - (mu / kbt))))
    else
        tmp = 0.5d0 * (ndchar + nachar)
    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 (Vef <= -5e+266) {
		tmp = (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)))) + (NaChar / (1.0 + (((1.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT))));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Vef <= -5e+266:
		tmp = (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)))) + (NaChar / (1.0 + (((1.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT))))
	else:
		tmp = 0.5 * (NdChar + NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Vef <= -5e+266)
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(EDonor / KbT)) + Float64(Float64(Vef / KbT) + Float64(mu / KbT))) - Float64(Ec / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(EAccept / KbT)) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) - Float64(mu / KbT)))));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Vef <= -5e+266)
		tmp = (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)))) + (NaChar / (1.0 + (((1.0 + (EAccept / KbT)) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT))));
	else
		tmp = 0.5 * (NdChar + NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Vef, -5e+266], N[(N[(NdChar / N[(1.0 + N[(N[(N[(1.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(N[(1.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Vef < -4.9999999999999999e266

   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%

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

   4. Taylor expanded in KbT around inf 76.5%

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

      1. associate-+r+ 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in KbT around inf 64.2%

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

      1. associate-+r+ 64.2%

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

   9. Simplified 64.2%

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

   if -4.9999999999999999e266 < Vef

   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%

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

   4. Taylor expanded in KbT around inf 49.3%

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

   5. Taylor expanded in Ec around inf 38.8%

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

      1. associate-*r/ 38.8%

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

      2. mul-1-neg 38.8%

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

   7. Simplified 38.8%

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

   8. Taylor expanded in Ec around 0 29.6%

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

      1. mul-1-neg 29.6%

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

      2. unsub-neg 29.6%

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{Ec}{KbT}}} + \frac{NaChar}{1 + 1} \]

   10. Simplified 29.6%

       \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{Ec}{KbT}}} + \frac{NaChar}{1 + 1} \]

   11. Taylor expanded in Ec around 0 30.4%

       \[\leadsto \color{blue}{0.5 \cdot NaChar + 0.5 \cdot NdChar} \]
   12. Step-by-step derivation

       1. distribute-lft-out 30.4%

          \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

   13. Simplified 30.4%

       \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -5 \cdot 10^{+266}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(\left(1 + \frac{EDonor}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 + \left(\left(\left(1 + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]

Alternative 29: 27.7% accurate, 45.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(NdChar + NaChar\right) \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* 0.5 (+ NdChar NaChar)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return 0.5 * (NdChar + NaChar);
}

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 = 0.5d0 * (ndchar + nachar)
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 0.5 * (NdChar + NaChar);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return 0.5 * (NdChar + NaChar)

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(0.5 * Float64(NdChar + NaChar))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.5 * (NdChar + NaChar);
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(0.5 * N[(NdChar + NaChar), $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%

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

4. Taylor expanded in KbT around inf 48.3%

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

5. Taylor expanded in Ec around inf 38.2%

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

   1. associate-*r/ 38.2%

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

   2. mul-1-neg 38.2%

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

7. Simplified 38.2%

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

8. Taylor expanded in Ec around 0 28.8%

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

   1. mul-1-neg 28.8%

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

   2. unsub-neg 28.8%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{Ec}{KbT}}} + \frac{NaChar}{1 + 1} \]

10. Simplified 28.8%

    \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{Ec}{KbT}}} + \frac{NaChar}{1 + 1} \]

11. Taylor expanded in Ec around 0 29.7%

    \[\leadsto \color{blue}{0.5 \cdot NaChar + 0.5 \cdot NdChar} \]
12. Step-by-step derivation

    1. distribute-lft-out 29.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

13. Simplified 29.7%

    \[\leadsto \color{blue}{0.5 \cdot \left(NaChar + NdChar\right)} \]

14. Final simplification 29.7%

    \[\leadsto 0.5 \cdot \left(NdChar + NaChar\right) \]

Alternative 30: 18.6% accurate, 76.3× speedup

Math

\[\begin{array}{l} \\ NaChar \cdot 0.5 \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* NaChar 0.5))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return NaChar * 0.5;
}

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 * 0.5d0
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 * 0.5;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return NaChar * 0.5

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(NaChar * 0.5)
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = NaChar * 0.5;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(NaChar * 0.5), $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%

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

4. Taylor expanded in KbT around inf 48.3%

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

5. Taylor expanded in Ec around inf 38.2%

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

   1. associate-*r/ 38.2%

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

   2. mul-1-neg 38.2%

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

7. Simplified 38.2%

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

8. Taylor expanded in Ec around 0 28.8%

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

   1. mul-1-neg 28.8%

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

   2. unsub-neg 28.8%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{Ec}{KbT}}} + \frac{NaChar}{1 + 1} \]

10. Simplified 28.8%

    \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{Ec}{KbT}}} + \frac{NaChar}{1 + 1} \]

11. Taylor expanded in NdChar around 0 19.8%

    \[\leadsto \color{blue}{0.5 \cdot NaChar} \]

12. Final simplification 19.8%

    \[\leadsto NaChar \cdot 0.5 \]

Reproduce

herbie shell --seed 2023333 
(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))))))