Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 39.2s

Alternatives: 38

Speedup: 1.0×

• could not determine a ground truth

  1. branch3970 = -3.56304870039969e-299
  2. NdChar = 5.927750898229982e-300
  3. Ec = 1.9625832266950479e-277
  4. Vef = -1.3428363846574586e+231
  5. EDonor = -4.618135979952894e-124
  6. mu = 9.177935123796655e+72
  7. KbT = 1.173289838317765e+120
  8. NaChar = 1.4887312415341055e+178
  9. Ev = -1.2029713517274522e-169
  10. EAccept = 4.14957447587175e+169

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}\\ \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + {\left(e^{{t\_0}^{2}}\right)}^{t\_0}} \end{array} \end{array} \]

FPCore

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

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = cbrt(((Vef - ((mu - Ev) - EAccept)) / KbT));
	return (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + pow(exp(pow(t_0, 2.0)), t_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(((Vef - ((mu - Ev) - EAccept)) / KbT));
	return (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.pow(Math.exp(Math.pow(t_0, 2.0)), t_0)));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-cube-cbrt 100.0%

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

   2. exp-prod 100.0%

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

   3. pow2 100.0%

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

   4. +-commutative 100.0%

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

   5. associate-+l- 100.0%

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

   6. +-commutative 100.0%

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

   7. associate-+l- 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. log1p-define 100.0%

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

   3. log1p-expm1-u 100.0%

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

   4. log1p-define 100.0%

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

   5. expm1-log1p-u 100.0%

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

   6. +-commutative 100.0%

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

   7. associate-+l- 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 67.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -0.37:\\ \;\;\;\;t\_2 + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.42 \cdot 10^{-216}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + mu \cdot \left(\frac{1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 9.5 \cdot 10^{-150}:\\ \;\;\;\;t\_1 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;mu \leq 6.6 \cdot 10^{-117}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + \left(\left(1 + Ev \cdot \left(\frac{1}{KbT} + \left(\frac{EAccept}{KbT \cdot Ev} + \frac{Vef}{KbT \cdot Ev}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 31.5:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + t\_1\\ \mathbf{elif}\;mu \leq 2.35 \cdot 10^{+173}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
        (t_2 (/ NdChar (+ 1.0 (exp (/ mu KbT))))))
   (if (<= mu -0.37)
     (+ t_2 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef Ev) mu) KbT)))))
     (if (<= mu 1.42e-216)
       (+
        t_0
        (/
         NaChar
         (+
          1.0
          (*
           mu
           (+
            (/ (+ 1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT)))) mu)
            (/ -1.0 KbT))))))
       (if (<= mu 9.5e-150)
         (+ t_1 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
         (if (<= mu 6.6e-117)
           (+
            t_0
            (/
             NaChar
             (+
              1.0
              (-
               (+
                1.0
                (*
                 Ev
                 (+
                  (/ 1.0 KbT)
                  (+ (/ EAccept (* KbT Ev)) (/ Vef (* KbT Ev))))))
               (/ mu KbT)))))
           (if (<= mu 31.5)
             (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) t_1)
             (if (<= mu 2.35e+173)
               (+
                (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
                (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))
               (+ t_2 (/ NaChar (+ 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 = NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = NdChar / (1.0 + exp((EDonor / KbT)));
	double t_2 = NdChar / (1.0 + exp((mu / KbT)));
	double tmp;
	if (mu <= -0.37) {
		tmp = t_2 + (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT))));
	} else if (mu <= 1.42e-216) {
		tmp = t_0 + (NaChar / (1.0 + (mu * (((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu) + (-1.0 / KbT)))));
	} else if (mu <= 9.5e-150) {
		tmp = t_1 + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (mu <= 6.6e-117) {
		tmp = t_0 + (NaChar / (1.0 + ((1.0 + (Ev * ((1.0 / KbT) + ((EAccept / (KbT * Ev)) + (Vef / (KbT * Ev)))))) - (mu / KbT))));
	} else if (mu <= 31.5) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + t_1;
	} else if (mu <= 2.35e+173) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else {
		tmp = t_2 + (NaChar / (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) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))
    t_1 = ndchar / (1.0d0 + exp((edonor / kbt)))
    t_2 = ndchar / (1.0d0 + exp((mu / kbt)))
    if (mu <= (-0.37d0)) then
        tmp = t_2 + (nachar / (1.0d0 + exp((((vef + ev) - mu) / kbt))))
    else if (mu <= 1.42d-216) then
        tmp = t_0 + (nachar / (1.0d0 + (mu * (((1.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) / mu) + ((-1.0d0) / kbt)))))
    else if (mu <= 9.5d-150) then
        tmp = t_1 + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (mu <= 6.6d-117) then
        tmp = t_0 + (nachar / (1.0d0 + ((1.0d0 + (ev * ((1.0d0 / kbt) + ((eaccept / (kbt * ev)) + (vef / (kbt * ev)))))) - (mu / kbt))))
    else if (mu <= 31.5d0) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + t_1
    else if (mu <= 2.35d+173) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    else
        tmp = t_2 + (nachar / (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 = NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	double t_2 = NdChar / (1.0 + Math.exp((mu / KbT)));
	double tmp;
	if (mu <= -0.37) {
		tmp = t_2 + (NaChar / (1.0 + Math.exp((((Vef + Ev) - mu) / KbT))));
	} else if (mu <= 1.42e-216) {
		tmp = t_0 + (NaChar / (1.0 + (mu * (((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu) + (-1.0 / KbT)))));
	} else if (mu <= 9.5e-150) {
		tmp = t_1 + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (mu <= 6.6e-117) {
		tmp = t_0 + (NaChar / (1.0 + ((1.0 + (Ev * ((1.0 / KbT) + ((EAccept / (KbT * Ev)) + (Vef / (KbT * Ev)))))) - (mu / KbT))));
	} else if (mu <= 31.5) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + t_1;
	} else if (mu <= 2.35e+173) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else {
		tmp = t_2 + (NaChar / (1.0 + Math.exp((mu / -KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))
	t_1 = NdChar / (1.0 + math.exp((EDonor / KbT)))
	t_2 = NdChar / (1.0 + math.exp((mu / KbT)))
	tmp = 0
	if mu <= -0.37:
		tmp = t_2 + (NaChar / (1.0 + math.exp((((Vef + Ev) - mu) / KbT))))
	elif mu <= 1.42e-216:
		tmp = t_0 + (NaChar / (1.0 + (mu * (((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu) + (-1.0 / KbT)))))
	elif mu <= 9.5e-150:
		tmp = t_1 + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif mu <= 6.6e-117:
		tmp = t_0 + (NaChar / (1.0 + ((1.0 + (Ev * ((1.0 / KbT) + ((EAccept / (KbT * Ev)) + (Vef / (KbT * Ev)))))) - (mu / KbT))))
	elif mu <= 31.5:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + t_1
	elif mu <= 2.35e+173:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	else:
		tmp = t_2 + (NaChar / (1.0 + math.exp((mu / -KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))))
	t_2 = Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT))))
	tmp = 0.0
	if (mu <= -0.37)
		tmp = Float64(t_2 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)))));
	elseif (mu <= 1.42e-216)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + Float64(mu * Float64(Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) / mu) + Float64(-1.0 / KbT))))));
	elseif (mu <= 9.5e-150)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	elseif (mu <= 6.6e-117)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Ev * Float64(Float64(1.0 / KbT) + Float64(Float64(EAccept / Float64(KbT * Ev)) + Float64(Vef / Float64(KbT * Ev)))))) - Float64(mu / KbT)))));
	elseif (mu <= 31.5)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + t_1);
	elseif (mu <= 2.35e+173)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))));
	else
		tmp = Float64(t_2 + Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	t_1 = NdChar / (1.0 + exp((EDonor / KbT)));
	t_2 = NdChar / (1.0 + exp((mu / KbT)));
	tmp = 0.0;
	if (mu <= -0.37)
		tmp = t_2 + (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT))));
	elseif (mu <= 1.42e-216)
		tmp = t_0 + (NaChar / (1.0 + (mu * (((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu) + (-1.0 / KbT)))));
	elseif (mu <= 9.5e-150)
		tmp = t_1 + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (mu <= 6.6e-117)
		tmp = t_0 + (NaChar / (1.0 + ((1.0 + (Ev * ((1.0 / KbT) + ((EAccept / (KbT * Ev)) + (Vef / (KbT * Ev)))))) - (mu / KbT))));
	elseif (mu <= 31.5)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + t_1;
	elseif (mu <= 2.35e+173)
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	else
		tmp = t_2 + (NaChar / (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[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -0.37], N[(t$95$2 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.42e-216], N[(t$95$0 + N[(NaChar / N[(1.0 + N[(mu * N[(N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 9.5e-150], N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 6.6e-117], N[(t$95$0 + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(Ev * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(EAccept / N[(KbT * Ev), $MachinePrecision]), $MachinePrecision] + N[(Vef / N[(KbT * Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 31.5], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[mu, 2.35e+173], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(NaChar / N[(1.0 + N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if mu < -0.37

   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%

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

   5. Taylor expanded in mu around inf 82.1%

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

   6. Taylor expanded in EAccept around 0 77.4%

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

   if -0.37 < mu < 1.42000000000000004e-216

   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%

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

   5. Taylor expanded in KbT around inf 69.0%

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

   6. Taylor expanded in mu around -inf 75.7%

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

   if 1.42000000000000004e-216 < mu < 9.50000000000000013e-150

   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%

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

   5. Taylor expanded in Ev around inf 70.1%

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

   6. Taylor expanded in EDonor around inf 70.1%

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

   if 9.50000000000000013e-150 < mu < 6.6000000000000003e-117

   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%

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

   5. Taylor expanded in KbT around inf 64.2%

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

   6. Taylor expanded in Ev around inf 64.4%

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

   if 6.6000000000000003e-117 < mu < 31.5

   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%

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

   5. Taylor expanded in EAccept around inf 74.8%

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

   6. Taylor expanded in EDonor around inf 61.3%

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

   if 31.5 < mu < 2.35000000000000007e173

   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%

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

   5. Taylor expanded in mu around inf 75.9%

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

   6. Taylor expanded in mu around 0 73.4%

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

   if 2.35000000000000007e173 < 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%

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

   5. Taylor expanded in mu around inf 90.2%

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

   6. Taylor expanded in mu around inf 90.2%

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

      1. associate-*r/ 90.2%

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

      2. mul-1-neg 90.2%

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

   8. Simplified 90.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\color{blue}{\frac{-mu}{KbT}}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -0.37:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.42 \cdot 10^{-216}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + mu \cdot \left(\frac{1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 9.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;mu \leq 6.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + Ev \cdot \left(\frac{1}{KbT} + \left(\frac{EAccept}{KbT \cdot Ev} + \frac{Vef}{KbT \cdot Ev}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 31.5:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.35 \cdot 10^{+173}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ t_1 := e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}\\ \mathbf{if}\;mu \leq -2.8 \cdot 10^{+219}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq 8.9 \cdot 10^{-218}:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - t\_1}\\ \mathbf{elif}\;mu \leq 1.65 \cdot 10^{-161}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;mu \leq 9.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{NdChar}{1 + t\_1} + \frac{NaChar}{1 + \left(\left(1 + Ev \cdot \left(\frac{1}{KbT} + \left(\frac{EAccept}{KbT \cdot Ev} + \frac{Vef}{KbT \cdot Ev}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 1.9 \cdot 10^{+172}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ mu (- KbT)))))))
        (t_1 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
   (if (<= mu -2.8e+219)
     t_0
     (if (<= mu 8.9e-218)
       (-
        (/
         NaChar
         (-
          (+ 2.0 (* EAccept (+ (/ 1.0 KbT) (/ Vef (* KbT EAccept)))))
          (/ mu KbT)))
        (/ NdChar (- -1.0 t_1)))
       (if (<= mu 1.65e-161)
         (+
          (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
          (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
         (if (<= mu 9.2e-48)
           (+
            (/ NdChar (+ 1.0 t_1))
            (/
             NaChar
             (+
              1.0
              (-
               (+
                1.0
                (*
                 Ev
                 (+
                  (/ 1.0 KbT)
                  (+ (/ EAccept (* KbT Ev)) (/ Vef (* KbT Ev))))))
               (/ mu KbT)))))
           (if (<= mu 1.9e+172)
             (+
              (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
              (/ NdChar (+ 1.0 (+ 1.0 (/ mu 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 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	double t_1 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double tmp;
	if (mu <= -2.8e+219) {
		tmp = t_0;
	} else if (mu <= 8.9e-218) {
		tmp = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_1));
	} else if (mu <= 1.65e-161) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (mu <= 9.2e-48) {
		tmp = (NdChar / (1.0 + t_1)) + (NaChar / (1.0 + ((1.0 + (Ev * ((1.0 / KbT) + ((EAccept / (KbT * Ev)) + (Vef / (KbT * Ev)))))) - (mu / KbT))));
	} else if (mu <= 1.9e+172) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / 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) :: t_1
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((mu / -kbt))))
    t_1 = exp(((edonor + (mu + (vef - ec))) / kbt))
    if (mu <= (-2.8d+219)) then
        tmp = t_0
    else if (mu <= 8.9d-218) then
        tmp = (nachar / ((2.0d0 + (eaccept * ((1.0d0 / kbt) + (vef / (kbt * eaccept))))) - (mu / kbt))) - (ndchar / ((-1.0d0) - t_1))
    else if (mu <= 1.65d-161) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (mu <= 9.2d-48) then
        tmp = (ndchar / (1.0d0 + t_1)) + (nachar / (1.0d0 + ((1.0d0 + (ev * ((1.0d0 / kbt) + ((eaccept / (kbt * ev)) + (vef / (kbt * ev)))))) - (mu / kbt))))
    else if (mu <= 1.9d+172) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (mu / 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 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((mu / -KbT))));
	double t_1 = Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double tmp;
	if (mu <= -2.8e+219) {
		tmp = t_0;
	} else if (mu <= 8.9e-218) {
		tmp = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_1));
	} else if (mu <= 1.65e-161) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (mu <= 9.2e-48) {
		tmp = (NdChar / (1.0 + t_1)) + (NaChar / (1.0 + ((1.0 + (Ev * ((1.0 / KbT) + ((EAccept / (KbT * Ev)) + (Vef / (KbT * Ev)))))) - (mu / KbT))));
	} else if (mu <= 1.9e+172) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((mu / -KbT))))
	t_1 = math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))
	tmp = 0
	if mu <= -2.8e+219:
		tmp = t_0
	elif mu <= 8.9e-218:
		tmp = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_1))
	elif mu <= 1.65e-161:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif mu <= 9.2e-48:
		tmp = (NdChar / (1.0 + t_1)) + (NaChar / (1.0 + ((1.0 + (Ev * ((1.0 / KbT) + ((EAccept / (KbT * Ev)) + (Vef / (KbT * Ev)))))) - (mu / KbT))))
	elif mu <= 1.9e+172:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	else:
		tmp = t_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(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT))))))
	t_1 = exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))
	tmp = 0.0
	if (mu <= -2.8e+219)
		tmp = t_0;
	elseif (mu <= 8.9e-218)
		tmp = Float64(Float64(NaChar / Float64(Float64(2.0 + Float64(EAccept * Float64(Float64(1.0 / KbT) + Float64(Vef / Float64(KbT * EAccept))))) - Float64(mu / KbT))) - Float64(NdChar / Float64(-1.0 - t_1)));
	elseif (mu <= 1.65e-161)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	elseif (mu <= 9.2e-48)
		tmp = Float64(Float64(NdChar / Float64(1.0 + t_1)) + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Ev * Float64(Float64(1.0 / KbT) + Float64(Float64(EAccept / Float64(KbT * Ev)) + Float64(Vef / Float64(KbT * Ev)))))) - Float64(mu / KbT)))));
	elseif (mu <= 1.9e+172)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / 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 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	t_1 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	tmp = 0.0;
	if (mu <= -2.8e+219)
		tmp = t_0;
	elseif (mu <= 8.9e-218)
		tmp = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_1));
	elseif (mu <= 1.65e-161)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (mu <= 9.2e-48)
		tmp = (NdChar / (1.0 + t_1)) + (NaChar / (1.0 + ((1.0 + (Ev * ((1.0 / KbT) + ((EAccept / (KbT * Ev)) + (Vef / (KbT * Ev)))))) - (mu / KbT))));
	elseif (mu <= 1.9e+172)
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / 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[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[mu, -2.8e+219], t$95$0, If[LessEqual[mu, 8.9e-218], N[(N[(NaChar / N[(N[(2.0 + N[(EAccept * N[(N[(1.0 / KbT), $MachinePrecision] + N[(Vef / N[(KbT * EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.65e-161], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 9.2e-48], N[(N[(NdChar / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(Ev * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(EAccept / N[(KbT * Ev), $MachinePrecision]), $MachinePrecision] + N[(Vef / N[(KbT * Ev), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.9e+172], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if mu < -2.80000000000000015e219 or 1.89999999999999985e172 < 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%

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

   5. Taylor expanded in mu around inf 91.4%

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

   6. Taylor expanded in mu around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. mul-1-neg 91.4%

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

   8. Simplified 91.4%

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

   if -2.80000000000000015e219 < mu < 8.8999999999999999e-218

   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%

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

   5. Taylor expanded in KbT around inf 67.4%

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

   6. Taylor expanded in EAccept around inf 71.2%

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

   7. Taylor expanded in Ev around 0 74.9%

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

   if 8.8999999999999999e-218 < mu < 1.6499999999999999e-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%

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

   5. Taylor expanded in Ev around inf 70.1%

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

   6. Taylor expanded in EDonor around inf 70.1%

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

   if 1.6499999999999999e-161 < mu < 9.2000000000000003e-48

   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%

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

   5. Taylor expanded in KbT around inf 62.8%

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

   6. Taylor expanded in Ev around inf 62.8%

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

   if 9.2000000000000003e-48 < mu < 1.89999999999999985e172

   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%

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

   5. Taylor expanded in mu around inf 74.0%

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

   6. Taylor expanded in mu around 0 70.5%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\left(1 + \frac{mu}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -2.8 \cdot 10^{+219}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;mu \leq 8.9 \cdot 10^{-218}:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.65 \cdot 10^{-161}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;mu \leq 9.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + Ev \cdot \left(\frac{1}{KbT} + \left(\frac{EAccept}{KbT \cdot Ev} + \frac{Vef}{KbT \cdot Ev}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 1.9 \cdot 10^{+172}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_2 := t\_1 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{if}\;Vef \leq -2.35 \cdot 10^{+39}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;Vef \leq 1.75 \cdot 10^{-28}:\\ \;\;\;\;t\_1 + t\_0\\ \mathbf{elif}\;Vef \leq 2.25 \cdot 10^{+102} \lor \neg \left(Vef \leq 2 \cdot 10^{+185}\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_2 (+ t_1 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))))
   (if (<= Vef -2.35e+39)
     t_2
     (if (<= Vef 1.75e-28)
       (+ t_1 t_0)
       (if (or (<= Vef 2.25e+102) (not (<= Vef 2e+185)))
         t_2
         (+ t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef Ev) 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 = NdChar / (1.0 + exp((mu / KbT)));
	double t_1 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_2 = t_1 + (NdChar / (1.0 + exp((Vef / KbT))));
	double tmp;
	if (Vef <= -2.35e+39) {
		tmp = t_2;
	} else if (Vef <= 1.75e-28) {
		tmp = t_1 + t_0;
	} else if ((Vef <= 2.25e+102) || !(Vef <= 2e+185)) {
		tmp = t_2;
	} else {
		tmp = t_0 + (NaChar / (1.0 + exp((((Vef + Ev) - 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) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp((mu / kbt)))
    t_1 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_2 = t_1 + (ndchar / (1.0d0 + exp((vef / kbt))))
    if (vef <= (-2.35d+39)) then
        tmp = t_2
    else if (vef <= 1.75d-28) then
        tmp = t_1 + t_0
    else if ((vef <= 2.25d+102) .or. (.not. (vef <= 2d+185))) then
        tmp = t_2
    else
        tmp = t_0 + (nachar / (1.0d0 + exp((((vef + ev) - 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 = NdChar / (1.0 + Math.exp((mu / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_2 = t_1 + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	double tmp;
	if (Vef <= -2.35e+39) {
		tmp = t_2;
	} else if (Vef <= 1.75e-28) {
		tmp = t_1 + t_0;
	} else if ((Vef <= 2.25e+102) || !(Vef <= 2e+185)) {
		tmp = t_2;
	} else {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((((Vef + Ev) - mu) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((mu / KbT)))
	t_1 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_2 = t_1 + (NdChar / (1.0 + math.exp((Vef / KbT))))
	tmp = 0
	if Vef <= -2.35e+39:
		tmp = t_2
	elif Vef <= 1.75e-28:
		tmp = t_1 + t_0
	elif (Vef <= 2.25e+102) or not (Vef <= 2e+185):
		tmp = t_2
	else:
		tmp = t_0 + (NaChar / (1.0 + math.exp((((Vef + Ev) - mu) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_2 = Float64(t_1 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))))
	tmp = 0.0
	if (Vef <= -2.35e+39)
		tmp = t_2;
	elseif (Vef <= 1.75e-28)
		tmp = Float64(t_1 + t_0);
	elseif ((Vef <= 2.25e+102) || !(Vef <= 2e+185))
		tmp = t_2;
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)))));
	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 / KbT)));
	t_1 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_2 = t_1 + (NdChar / (1.0 + exp((Vef / KbT))));
	tmp = 0.0;
	if (Vef <= -2.35e+39)
		tmp = t_2;
	elseif (Vef <= 1.75e-28)
		tmp = t_1 + t_0;
	elseif ((Vef <= 2.25e+102) || ~((Vef <= 2e+185)))
		tmp = t_2;
	else
		tmp = t_0 + (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT))));
	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[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -2.35e+39], t$95$2, If[LessEqual[Vef, 1.75e-28], N[(t$95$1 + t$95$0), $MachinePrecision], If[Or[LessEqual[Vef, 2.25e+102], N[Not[LessEqual[Vef, 2e+185]], $MachinePrecision]], t$95$2, N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -2.35e39 or 1.75e-28 < Vef < 2.2500000000000001e102 or 2e185 < 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%

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

   5. Taylor expanded in Vef around inf 87.8%

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

   if -2.35e39 < Vef < 1.75e-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%

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

   5. Taylor expanded in mu around inf 78.3%

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

   if 2.2500000000000001e102 < Vef < 2e185

   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%

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

   5. Taylor expanded in mu around inf 100.0%

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

   6. Taylor expanded in EAccept around 0 100.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -2.35 \cdot 10^{+39}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.75 \cdot 10^{-28}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 2.25 \cdot 10^{+102} \lor \neg \left(Vef \leq 2 \cdot 10^{+185}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -7.5 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 5.8 \cdot 10^{-226}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 5.6 \cdot 10^{-64}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1
         (-
          (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))
          (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))))
   (if (<= NdChar -7.5e-88)
     t_1
     (if (<= NdChar 5.8e-226)
       (+ t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
       (if (<= NdChar 5.6e-64)
         (+ t_0 (/ NdChar (+ 1.0 (exp (/ mu 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 + (EAccept - mu))) / KbT)));
	double t_1 = (NaChar / (1.0 + exp((EAccept / KbT)))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	double tmp;
	if (NdChar <= -7.5e-88) {
		tmp = t_1;
	} else if (NdChar <= 5.8e-226) {
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	} else if (NdChar <= 5.6e-64) {
		tmp = t_0 + (NdChar / (1.0 + exp((mu / 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 + (eaccept - mu))) / kbt)))
    t_1 = (nachar / (1.0d0 + exp((eaccept / kbt)))) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt))))
    if (ndchar <= (-7.5d-88)) then
        tmp = t_1
    else if (ndchar <= 5.8d-226) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((vef / kbt))))
    else if (ndchar <= 5.6d-64) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((mu / 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 + (EAccept - mu))) / KbT)));
	double t_1 = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	double tmp;
	if (NdChar <= -7.5e-88) {
		tmp = t_1;
	} else if (NdChar <= 5.8e-226) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	} else if (NdChar <= 5.6e-64) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((mu / 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 + (EAccept - mu))) / KbT)))
	t_1 = (NaChar / (1.0 + math.exp((EAccept / KbT)))) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	tmp = 0
	if NdChar <= -7.5e-88:
		tmp = t_1
	elif NdChar <= 5.8e-226:
		tmp = t_0 + (NdChar / (1.0 + math.exp((Vef / KbT))))
	elif NdChar <= 5.6e-64:
		tmp = t_0 + (NdChar / (1.0 + math.exp((mu / 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(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))))
	tmp = 0.0
	if (NdChar <= -7.5e-88)
		tmp = t_1;
	elseif (NdChar <= 5.8e-226)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	elseif (NdChar <= 5.6e-64)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / 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 + (EAccept - mu))) / KbT)));
	t_1 = (NaChar / (1.0 + exp((EAccept / KbT)))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	tmp = 0.0;
	if (NdChar <= -7.5e-88)
		tmp = t_1;
	elseif (NdChar <= 5.8e-226)
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	elseif (NdChar <= 5.6e-64)
		tmp = t_0 + (NdChar / (1.0 + exp((mu / 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[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -7.5e-88], t$95$1, If[LessEqual[NdChar, 5.8e-226], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 5.6e-64], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -7.50000000000000041e-88 or 5.60000000000000008e-64 < NdChar

   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%

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

   5. Taylor expanded in EAccept around inf 83.2%

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

   if -7.50000000000000041e-88 < NdChar < 5.80000000000000003e-226

   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%

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

   5. Taylor expanded in Vef around inf 84.6%

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

   if 5.80000000000000003e-226 < NdChar < 5.60000000000000008e-64

   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%

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

   5. Taylor expanded in mu around inf 89.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 5.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 5.6 \cdot 10^{-64}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -8.4 \cdot 10^{+77}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.25 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT))))))
   (if (<= mu -8.4e+77)
     (+ t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef Ev) mu) KbT)))))
     (if (<= mu 1.25e+187)
       (+
        (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
        (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
       (+ t_0 (/ NaChar (+ 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 = NdChar / (1.0 + exp((mu / KbT)));
	double tmp;
	if (mu <= -8.4e+77) {
		tmp = t_0 + (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT))));
	} else if (mu <= 1.25e+187) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + exp((Vef / KbT))));
	} else {
		tmp = t_0 + (NaChar / (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) :: tmp
    t_0 = ndchar / (1.0d0 + exp((mu / kbt)))
    if (mu <= (-8.4d+77)) then
        tmp = t_0 + (nachar / (1.0d0 + exp((((vef + ev) - mu) / kbt))))
    else if (mu <= 1.25d+187) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + exp((vef / kbt))))
    else
        tmp = t_0 + (nachar / (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 = NdChar / (1.0 + Math.exp((mu / KbT)));
	double tmp;
	if (mu <= -8.4e+77) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((((Vef + Ev) - mu) / KbT))));
	} else if (mu <= 1.25e+187) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((mu / -KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((mu / KbT)))
	tmp = 0
	if mu <= -8.4e+77:
		tmp = t_0 + (NaChar / (1.0 + math.exp((((Vef + Ev) - mu) / KbT))))
	elif mu <= 1.25e+187:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + math.exp((Vef / KbT))))
	else:
		tmp = t_0 + (NaChar / (1.0 + math.exp((mu / -KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT))))
	tmp = 0.0
	if (mu <= -8.4e+77)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)))));
	elseif (mu <= 1.25e+187)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT))))));
	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 / KbT)));
	tmp = 0.0;
	if (mu <= -8.4e+77)
		tmp = t_0 + (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT))));
	elseif (mu <= 1.25e+187)
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + exp((Vef / KbT))));
	else
		tmp = t_0 + (NaChar / (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[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -8.4e+77], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.25e+187], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -8.3999999999999995e77

   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%

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

   5. Taylor expanded in mu around inf 83.1%

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

   6. Taylor expanded in EAccept around 0 83.1%

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

   if -8.3999999999999995e77 < mu < 1.25e187

   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%

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

   5. Taylor expanded in Vef around inf 78.9%

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

   if 1.25e187 < 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%

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

   5. Taylor expanded in mu around inf 94.8%

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

   6. Taylor expanded in mu around inf 94.8%

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

      1. associate-*r/ 94.8%

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

      2. mul-1-neg 94.8%

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

   8. Simplified 94.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -8.4 \cdot 10^{+77}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.25 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp(((Vef + (Ev + (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[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 9: 63.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}\\ t_1 := \frac{NdChar}{1 + t\_0}\\ t_2 := t\_1 + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + Vef \cdot \left(\frac{1}{KbT} + \frac{Ev}{Vef \cdot KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ t_3 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{if}\;NaChar \leq -3 \cdot 10^{+60}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;NaChar \leq -1.05 \cdot 10^{-85}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NaChar \leq 2.5 \cdot 10^{-257}:\\ \;\;\;\;t\_1 + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - t\_0}\\ \mathbf{elif}\;NaChar \leq 3.1 \cdot 10^{+67} \lor \neg \left(NaChar \leq 8.5 \cdot 10^{+144}\right):\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))
        (t_1 (/ NdChar (+ 1.0 t_0)))
        (t_2
         (+
          t_1
          (/
           NaChar
           (+
            1.0
            (-
             (+
              1.0
              (+ (/ EAccept KbT) (* Vef (+ (/ 1.0 KbT) (/ Ev (* Vef KbT))))))
             (/ mu KbT))))))
        (t_3
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
          (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))))
   (if (<= NaChar -3e+60)
     t_3
     (if (<= NaChar -1.05e-85)
       t_2
       (if (<= NaChar 2.5e-257)
         (+ t_1 (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu))))))
         (if (<= NaChar 0.5)
           (-
            (/
             NaChar
             (-
              (+ 2.0 (* EAccept (+ (/ 1.0 KbT) (/ Vef (* KbT EAccept)))))
              (/ mu KbT)))
            (/ NdChar (- -1.0 t_0)))
           (if (or (<= NaChar 3.1e+67) (not (<= NaChar 8.5e+144)))
             t_3
             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 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = NdChar / (1.0 + t_0);
	double t_2 = t_1 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + (Vef * ((1.0 / KbT) + (Ev / (Vef * KbT)))))) - (mu / KbT))));
	double t_3 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -3e+60) {
		tmp = t_3;
	} else if (NaChar <= -1.05e-85) {
		tmp = t_2;
	} else if (NaChar <= 2.5e-257) {
		tmp = t_1 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} else if (NaChar <= 0.5) {
		tmp = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	} else if ((NaChar <= 3.1e+67) || !(NaChar <= 8.5e+144)) {
		tmp = t_3;
	} 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) :: t_3
    real(8) :: tmp
    t_0 = exp(((edonor + (mu + (vef - ec))) / kbt))
    t_1 = ndchar / (1.0d0 + t_0)
    t_2 = t_1 + (nachar / (1.0d0 + ((1.0d0 + ((eaccept / kbt) + (vef * ((1.0d0 / kbt) + (ev / (vef * kbt)))))) - (mu / kbt))))
    t_3 = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    if (nachar <= (-3d+60)) then
        tmp = t_3
    else if (nachar <= (-1.05d-85)) then
        tmp = t_2
    else if (nachar <= 2.5d-257) then
        tmp = t_1 + (kbt * (nachar / (eaccept + (ev + (vef - mu)))))
    else if (nachar <= 0.5d0) then
        tmp = (nachar / ((2.0d0 + (eaccept * ((1.0d0 / kbt) + (vef / (kbt * eaccept))))) - (mu / kbt))) - (ndchar / ((-1.0d0) - t_0))
    else if ((nachar <= 3.1d+67) .or. (.not. (nachar <= 8.5d+144))) then
        tmp = t_3
    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 = Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = NdChar / (1.0 + t_0);
	double t_2 = t_1 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + (Vef * ((1.0 / KbT) + (Ev / (Vef * KbT)))))) - (mu / KbT))));
	double t_3 = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -3e+60) {
		tmp = t_3;
	} else if (NaChar <= -1.05e-85) {
		tmp = t_2;
	} else if (NaChar <= 2.5e-257) {
		tmp = t_1 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} else if (NaChar <= 0.5) {
		tmp = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	} else if ((NaChar <= 3.1e+67) || !(NaChar <= 8.5e+144)) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))
	t_1 = NdChar / (1.0 + t_0)
	t_2 = t_1 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + (Vef * ((1.0 / KbT) + (Ev / (Vef * KbT)))))) - (mu / KbT))))
	t_3 = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	tmp = 0
	if NaChar <= -3e+60:
		tmp = t_3
	elif NaChar <= -1.05e-85:
		tmp = t_2
	elif NaChar <= 2.5e-257:
		tmp = t_1 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))))
	elif NaChar <= 0.5:
		tmp = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0))
	elif (NaChar <= 3.1e+67) or not (NaChar <= 8.5e+144):
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))
	t_1 = Float64(NdChar / Float64(1.0 + t_0))
	t_2 = Float64(t_1 + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Vef * Float64(Float64(1.0 / KbT) + Float64(Ev / Float64(Vef * KbT)))))) - Float64(mu / KbT)))))
	t_3 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))))
	tmp = 0.0
	if (NaChar <= -3e+60)
		tmp = t_3;
	elseif (NaChar <= -1.05e-85)
		tmp = t_2;
	elseif (NaChar <= 2.5e-257)
		tmp = Float64(t_1 + Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))));
	elseif (NaChar <= 0.5)
		tmp = Float64(Float64(NaChar / Float64(Float64(2.0 + Float64(EAccept * Float64(Float64(1.0 / KbT) + Float64(Vef / Float64(KbT * EAccept))))) - Float64(mu / KbT))) - Float64(NdChar / Float64(-1.0 - t_0)));
	elseif ((NaChar <= 3.1e+67) || !(NaChar <= 8.5e+144))
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	t_1 = NdChar / (1.0 + t_0);
	t_2 = t_1 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + (Vef * ((1.0 / KbT) + (Ev / (Vef * KbT)))))) - (mu / KbT))));
	t_3 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	tmp = 0.0;
	if (NaChar <= -3e+60)
		tmp = t_3;
	elseif (NaChar <= -1.05e-85)
		tmp = t_2;
	elseif (NaChar <= 2.5e-257)
		tmp = t_1 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	elseif (NaChar <= 0.5)
		tmp = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	elseif ((NaChar <= 3.1e+67) || ~((NaChar <= 8.5e+144)))
		tmp = t_3;
	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[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(Vef * N[(N[(1.0 / KbT), $MachinePrecision] + N[(Ev / N[(Vef * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -3e+60], t$95$3, If[LessEqual[NaChar, -1.05e-85], t$95$2, If[LessEqual[NaChar, 2.5e-257], N[(t$95$1 + N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 0.5], N[(N[(NaChar / N[(N[(2.0 + N[(EAccept * N[(N[(1.0 / KbT), $MachinePrecision] + N[(Vef / N[(KbT * EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[NaChar, 3.1e+67], N[Not[LessEqual[NaChar, 8.5e+144]], $MachinePrecision]], t$95$3, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -2.9999999999999998e60 or 0.5 < NaChar < 3.09999999999999996e67 or 8.4999999999999998e144 < 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%

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

   5. Taylor expanded in mu around inf 83.6%

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

   6. Taylor expanded in mu around 0 74.6%

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

   if -2.9999999999999998e60 < NaChar < -1.05e-85 or 3.09999999999999996e67 < NaChar < 8.4999999999999998e144

   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%

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

   5. Taylor expanded in KbT around inf 60.5%

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

   6. Taylor expanded in Vef around inf 66.1%

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

   if -1.05e-85 < NaChar < 2.49999999999999994e-257

   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%

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

   5. Taylor expanded in KbT around inf 72.4%

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

   6. Taylor expanded in KbT around 0 79.0%

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

      1. associate-/l* 80.5%

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

      2. associate--l+ 80.5%

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

      3. associate--l+ 80.5%

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

   8. Simplified 80.5%

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

   if 2.49999999999999994e-257 < NaChar < 0.5

   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%

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

   5. Taylor expanded in KbT around inf 78.5%

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

   6. Taylor expanded in EAccept around inf 78.4%

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

   7. Taylor expanded in Ev around 0 73.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -3 \cdot 10^{+60}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq -1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + Vef \cdot \left(\frac{1}{KbT} + \frac{Ev}{Vef \cdot KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 2.5 \cdot 10^{-257}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 3.1 \cdot 10^{+67} \lor \neg \left(NaChar \leq 8.5 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + Vef \cdot \left(\frac{1}{KbT} + \frac{Ev}{Vef \cdot KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}\\ t_1 := \frac{NdChar}{1 + t\_0}\\ t_2 := t\_1 + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} - Vef \cdot \left(\frac{-1}{KbT} - \frac{Ev}{Vef \cdot KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ t_3 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{if}\;NaChar \leq -6.5 \cdot 10^{+57}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;NaChar \leq -4.8 \cdot 10^{-85}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NaChar \leq 1.5 \cdot 10^{-258}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} - \frac{NdChar}{-1 - t\_0}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;t\_1 + \frac{NaChar}{1 + mu \cdot \left(\frac{1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 2.9 \cdot 10^{+67} \lor \neg \left(NaChar \leq 8.5 \cdot 10^{+144}\right):\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))
        (t_1 (/ NdChar (+ 1.0 t_0)))
        (t_2
         (+
          t_1
          (/
           NaChar
           (+
            1.0
            (-
             (+
              1.0
              (- (/ EAccept KbT) (* Vef (- (/ -1.0 KbT) (/ Ev (* Vef KbT))))))
             (/ mu KbT))))))
        (t_3
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
          (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))))
   (if (<= NaChar -6.5e+57)
     t_3
     (if (<= NaChar -4.8e-85)
       t_2
       (if (<= NaChar 1.5e-258)
         (-
          (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu)))))
          (/ NdChar (- -1.0 t_0)))
         (if (<= NaChar 0.5)
           (+
            t_1
            (/
             NaChar
             (+
              1.0
              (*
               mu
               (+
                (/ (+ 1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT)))) mu)
                (/ -1.0 KbT))))))
           (if (or (<= NaChar 2.9e+67) (not (<= NaChar 8.5e+144)))
             t_3
             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 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = NdChar / (1.0 + t_0);
	double t_2 = t_1 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) - (Vef * ((-1.0 / KbT) - (Ev / (Vef * KbT)))))) - (mu / KbT))));
	double t_3 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -6.5e+57) {
		tmp = t_3;
	} else if (NaChar <= -4.8e-85) {
		tmp = t_2;
	} else if (NaChar <= 1.5e-258) {
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - (NdChar / (-1.0 - t_0));
	} else if (NaChar <= 0.5) {
		tmp = t_1 + (NaChar / (1.0 + (mu * (((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu) + (-1.0 / KbT)))));
	} else if ((NaChar <= 2.9e+67) || !(NaChar <= 8.5e+144)) {
		tmp = t_3;
	} 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) :: t_3
    real(8) :: tmp
    t_0 = exp(((edonor + (mu + (vef - ec))) / kbt))
    t_1 = ndchar / (1.0d0 + t_0)
    t_2 = t_1 + (nachar / (1.0d0 + ((1.0d0 + ((eaccept / kbt) - (vef * (((-1.0d0) / kbt) - (ev / (vef * kbt)))))) - (mu / kbt))))
    t_3 = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    if (nachar <= (-6.5d+57)) then
        tmp = t_3
    else if (nachar <= (-4.8d-85)) then
        tmp = t_2
    else if (nachar <= 1.5d-258) then
        tmp = (kbt * (nachar / (eaccept + (ev + (vef - mu))))) - (ndchar / ((-1.0d0) - t_0))
    else if (nachar <= 0.5d0) then
        tmp = t_1 + (nachar / (1.0d0 + (mu * (((1.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) / mu) + ((-1.0d0) / kbt)))))
    else if ((nachar <= 2.9d+67) .or. (.not. (nachar <= 8.5d+144))) then
        tmp = t_3
    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 = Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = NdChar / (1.0 + t_0);
	double t_2 = t_1 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) - (Vef * ((-1.0 / KbT) - (Ev / (Vef * KbT)))))) - (mu / KbT))));
	double t_3 = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -6.5e+57) {
		tmp = t_3;
	} else if (NaChar <= -4.8e-85) {
		tmp = t_2;
	} else if (NaChar <= 1.5e-258) {
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - (NdChar / (-1.0 - t_0));
	} else if (NaChar <= 0.5) {
		tmp = t_1 + (NaChar / (1.0 + (mu * (((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu) + (-1.0 / KbT)))));
	} else if ((NaChar <= 2.9e+67) || !(NaChar <= 8.5e+144)) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))
	t_1 = NdChar / (1.0 + t_0)
	t_2 = t_1 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) - (Vef * ((-1.0 / KbT) - (Ev / (Vef * KbT)))))) - (mu / KbT))))
	t_3 = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	tmp = 0
	if NaChar <= -6.5e+57:
		tmp = t_3
	elif NaChar <= -4.8e-85:
		tmp = t_2
	elif NaChar <= 1.5e-258:
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - (NdChar / (-1.0 - t_0))
	elif NaChar <= 0.5:
		tmp = t_1 + (NaChar / (1.0 + (mu * (((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu) + (-1.0 / KbT)))))
	elif (NaChar <= 2.9e+67) or not (NaChar <= 8.5e+144):
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))
	t_1 = Float64(NdChar / Float64(1.0 + t_0))
	t_2 = Float64(t_1 + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) - Float64(Vef * Float64(Float64(-1.0 / KbT) - Float64(Ev / Float64(Vef * KbT)))))) - Float64(mu / KbT)))))
	t_3 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))))
	tmp = 0.0
	if (NaChar <= -6.5e+57)
		tmp = t_3;
	elseif (NaChar <= -4.8e-85)
		tmp = t_2;
	elseif (NaChar <= 1.5e-258)
		tmp = Float64(Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))) - Float64(NdChar / Float64(-1.0 - t_0)));
	elseif (NaChar <= 0.5)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + Float64(mu * Float64(Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) / mu) + Float64(-1.0 / KbT))))));
	elseif ((NaChar <= 2.9e+67) || !(NaChar <= 8.5e+144))
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	t_1 = NdChar / (1.0 + t_0);
	t_2 = t_1 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) - (Vef * ((-1.0 / KbT) - (Ev / (Vef * KbT)))))) - (mu / KbT))));
	t_3 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	tmp = 0.0;
	if (NaChar <= -6.5e+57)
		tmp = t_3;
	elseif (NaChar <= -4.8e-85)
		tmp = t_2;
	elseif (NaChar <= 1.5e-258)
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - (NdChar / (-1.0 - t_0));
	elseif (NaChar <= 0.5)
		tmp = t_1 + (NaChar / (1.0 + (mu * (((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) / mu) + (-1.0 / KbT)))));
	elseif ((NaChar <= 2.9e+67) || ~((NaChar <= 8.5e+144)))
		tmp = t_3;
	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[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] - N[(Vef * N[(N[(-1.0 / KbT), $MachinePrecision] - N[(Ev / N[(Vef * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -6.5e+57], t$95$3, If[LessEqual[NaChar, -4.8e-85], t$95$2, If[LessEqual[NaChar, 1.5e-258], N[(N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 0.5], N[(t$95$1 + N[(NaChar / N[(1.0 + N[(mu * N[(N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[NaChar, 2.9e+67], N[Not[LessEqual[NaChar, 8.5e+144]], $MachinePrecision]], t$95$3, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -6.4999999999999997e57 or 0.5 < NaChar < 2.90000000000000023e67 or 8.4999999999999998e144 < 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%

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

   5. Taylor expanded in mu around inf 83.6%

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

   6. Taylor expanded in mu around 0 74.6%

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

   if -6.4999999999999997e57 < NaChar < -4.8000000000000001e-85 or 2.90000000000000023e67 < NaChar < 8.4999999999999998e144

   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%

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

   5. Taylor expanded in KbT around inf 60.5%

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

   6. Taylor expanded in Vef around inf 66.1%

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

   if -4.8000000000000001e-85 < NaChar < 1.5000000000000001e-258

   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%

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

   5. Taylor expanded in KbT around inf 72.4%

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

   6. Taylor expanded in KbT around 0 79.0%

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

      1. associate-/l* 80.5%

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

      2. associate--l+ 80.5%

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

      3. associate--l+ 80.5%

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

   8. Simplified 80.5%

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

   if 1.5000000000000001e-258 < NaChar < 0.5

   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%

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

   5. Taylor expanded in KbT around inf 78.5%

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

   6. Taylor expanded in mu around -inf 80.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -6.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq -4.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} - Vef \cdot \left(\frac{-1}{KbT} - \frac{Ev}{Vef \cdot KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 1.5 \cdot 10^{-258}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + mu \cdot \left(\frac{1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)}{mu} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 2.9 \cdot 10^{+67} \lor \neg \left(NaChar \leq 8.5 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} - Vef \cdot \left(\frac{-1}{KbT} - \frac{Ev}{Vef \cdot KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}\\ t_1 := \frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - t\_0}\\ t_2 := \frac{NdChar}{1 + t\_0}\\ t_3 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{if}\;NaChar \leq -2.3 \cdot 10^{+60}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;NaChar \leq -1.75 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 2 \cdot 10^{-258}:\\ \;\;\;\;t\_2 + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 1.7 \cdot 10^{+69}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;NaChar \leq 1.25 \cdot 10^{+145}:\\ \;\;\;\;t\_2 + \frac{1}{\frac{2 + \left(\frac{EAccept}{KbT} + \left(\frac{Ev}{KbT} + \left(\frac{Vef}{KbT} - \frac{mu}{KbT}\right)\right)\right)}{NaChar}}\\ \mathbf{elif}\;NaChar \leq 1.9 \cdot 10^{+163}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))
        (t_1
         (-
          (/
           NaChar
           (-
            (+ 2.0 (* EAccept (+ (/ 1.0 KbT) (/ Vef (* KbT EAccept)))))
            (/ mu KbT)))
          (/ NdChar (- -1.0 t_0))))
        (t_2 (/ NdChar (+ 1.0 t_0)))
        (t_3
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
          (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))))
   (if (<= NaChar -2.3e+60)
     t_3
     (if (<= NaChar -1.75e-87)
       t_1
       (if (<= NaChar 2e-258)
         (+ t_2 (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu))))))
         (if (<= NaChar 0.5)
           t_1
           (if (<= NaChar 1.7e+69)
             t_3
             (if (<= NaChar 1.25e+145)
               (+
                t_2
                (/
                 1.0
                 (/
                  (+
                   2.0
                   (+
                    (/ EAccept KbT)
                    (+ (/ Ev KbT) (- (/ Vef KbT) (/ mu KbT)))))
                  NaChar)))
               (if (<= NaChar 1.9e+163)
                 (+
                  (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
                  (/ NdChar (+ 2.0 (/ EDonor KbT))))
                 t_3)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	double t_2 = NdChar / (1.0 + t_0);
	double t_3 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -2.3e+60) {
		tmp = t_3;
	} else if (NaChar <= -1.75e-87) {
		tmp = t_1;
	} else if (NaChar <= 2e-258) {
		tmp = t_2 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} else if (NaChar <= 0.5) {
		tmp = t_1;
	} else if (NaChar <= 1.7e+69) {
		tmp = t_3;
	} else if (NaChar <= 1.25e+145) {
		tmp = t_2 + (1.0 / ((2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef / KbT) - (mu / KbT))))) / NaChar));
	} else if (NaChar <= 1.9e+163) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = exp(((edonor + (mu + (vef - ec))) / kbt))
    t_1 = (nachar / ((2.0d0 + (eaccept * ((1.0d0 / kbt) + (vef / (kbt * eaccept))))) - (mu / kbt))) - (ndchar / ((-1.0d0) - t_0))
    t_2 = ndchar / (1.0d0 + t_0)
    t_3 = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    if (nachar <= (-2.3d+60)) then
        tmp = t_3
    else if (nachar <= (-1.75d-87)) then
        tmp = t_1
    else if (nachar <= 2d-258) then
        tmp = t_2 + (kbt * (nachar / (eaccept + (ev + (vef - mu)))))
    else if (nachar <= 0.5d0) then
        tmp = t_1
    else if (nachar <= 1.7d+69) then
        tmp = t_3
    else if (nachar <= 1.25d+145) then
        tmp = t_2 + (1.0d0 / ((2.0d0 + ((eaccept / kbt) + ((ev / kbt) + ((vef / kbt) - (mu / kbt))))) / nachar))
    else if (nachar <= 1.9d+163) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (2.0d0 + (edonor / kbt)))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	double t_2 = NdChar / (1.0 + t_0);
	double t_3 = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -2.3e+60) {
		tmp = t_3;
	} else if (NaChar <= -1.75e-87) {
		tmp = t_1;
	} else if (NaChar <= 2e-258) {
		tmp = t_2 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} else if (NaChar <= 0.5) {
		tmp = t_1;
	} else if (NaChar <= 1.7e+69) {
		tmp = t_3;
	} else if (NaChar <= 1.25e+145) {
		tmp = t_2 + (1.0 / ((2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef / KbT) - (mu / KbT))))) / NaChar));
	} else if (NaChar <= 1.9e+163) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))
	t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0))
	t_2 = NdChar / (1.0 + t_0)
	t_3 = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	tmp = 0
	if NaChar <= -2.3e+60:
		tmp = t_3
	elif NaChar <= -1.75e-87:
		tmp = t_1
	elif NaChar <= 2e-258:
		tmp = t_2 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))))
	elif NaChar <= 0.5:
		tmp = t_1
	elif NaChar <= 1.7e+69:
		tmp = t_3
	elif NaChar <= 1.25e+145:
		tmp = t_2 + (1.0 / ((2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef / KbT) - (mu / KbT))))) / NaChar))
	elif NaChar <= 1.9e+163:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)))
	else:
		tmp = t_3
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))
	t_1 = Float64(Float64(NaChar / Float64(Float64(2.0 + Float64(EAccept * Float64(Float64(1.0 / KbT) + Float64(Vef / Float64(KbT * EAccept))))) - Float64(mu / KbT))) - Float64(NdChar / Float64(-1.0 - t_0)))
	t_2 = Float64(NdChar / Float64(1.0 + t_0))
	t_3 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))))
	tmp = 0.0
	if (NaChar <= -2.3e+60)
		tmp = t_3;
	elseif (NaChar <= -1.75e-87)
		tmp = t_1;
	elseif (NaChar <= 2e-258)
		tmp = Float64(t_2 + Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))));
	elseif (NaChar <= 0.5)
		tmp = t_1;
	elseif (NaChar <= 1.7e+69)
		tmp = t_3;
	elseif (NaChar <= 1.25e+145)
		tmp = Float64(t_2 + Float64(1.0 / Float64(Float64(2.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Ev / KbT) + Float64(Float64(Vef / KbT) - Float64(mu / KbT))))) / NaChar)));
	elseif (NaChar <= 1.9e+163)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(2.0 + Float64(EDonor / KbT))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	t_2 = NdChar / (1.0 + t_0);
	t_3 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	tmp = 0.0;
	if (NaChar <= -2.3e+60)
		tmp = t_3;
	elseif (NaChar <= -1.75e-87)
		tmp = t_1;
	elseif (NaChar <= 2e-258)
		tmp = t_2 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	elseif (NaChar <= 0.5)
		tmp = t_1;
	elseif (NaChar <= 1.7e+69)
		tmp = t_3;
	elseif (NaChar <= 1.25e+145)
		tmp = t_2 + (1.0 / ((2.0 + ((EAccept / KbT) + ((Ev / KbT) + ((Vef / KbT) - (mu / KbT))))) / NaChar));
	elseif (NaChar <= 1.9e+163)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[(2.0 + N[(EAccept * N[(N[(1.0 / KbT), $MachinePrecision] + N[(Vef / N[(KbT * EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -2.3e+60], t$95$3, If[LessEqual[NaChar, -1.75e-87], t$95$1, If[LessEqual[NaChar, 2e-258], N[(t$95$2 + N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 0.5], t$95$1, If[LessEqual[NaChar, 1.7e+69], t$95$3, If[LessEqual[NaChar, 1.25e+145], N[(t$95$2 + N[(1.0 / N[(N[(2.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Ev / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.9e+163], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NaChar < -2.30000000000000017e60 or 0.5 < NaChar < 1.69999999999999993e69 or 1.90000000000000004e163 < 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%

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

   5. Taylor expanded in mu around inf 84.7%

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

   6. Taylor expanded in mu around 0 76.0%

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

   if -2.30000000000000017e60 < NaChar < -1.75000000000000006e-87 or 1.99999999999999991e-258 < NaChar < 0.5

   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%

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

   5. Taylor expanded in KbT around inf 68.5%

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

   6. Taylor expanded in EAccept around inf 71.9%

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

   7. Taylor expanded in Ev around 0 71.4%

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

   if -1.75000000000000006e-87 < NaChar < 1.99999999999999991e-258

   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%

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

   5. Taylor expanded in KbT around inf 72.4%

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

   6. Taylor expanded in KbT around 0 79.0%

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

      1. associate-/l* 80.5%

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

      2. associate--l+ 80.5%

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

      3. associate--l+ 80.5%

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

   8. Simplified 80.5%

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

   if 1.69999999999999993e69 < NaChar < 1.24999999999999992e145

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in KbT around inf 70.6%

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

      1. clear-num 70.6%

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

      2. inv-pow 70.6%

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

      3. associate--l+ 70.6%

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

   7. Applied egg-rr 70.6%

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

      1. unpow-1 70.6%

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

      2. associate-+r+ 70.6%

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

      3. metadata-eval 70.6%

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

      4. associate--l+ 70.6%

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

      5. associate--l+ 70.6%

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

   9. Simplified 70.6%

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

   if 1.24999999999999992e145 < NaChar < 1.90000000000000004e163

   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%

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

   5. Taylor expanded in Ev around inf 59.4%

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

   6. Taylor expanded in EDonor around inf 56.4%

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

   7. Taylor expanded in EDonor around 0 56.4%

      \[\leadsto \frac{NdChar}{\color{blue}{2 + \frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.3 \cdot 10^{+60}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq -1.75 \cdot 10^{-87}:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 2 \cdot 10^{-258}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.7 \cdot 10^{+69}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 1.25 \cdot 10^{+145}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{1}{\frac{2 + \left(\frac{EAccept}{KbT} + \left(\frac{Ev}{KbT} + \left(\frac{Vef}{KbT} - \frac{mu}{KbT}\right)\right)\right)}{NaChar}}\\ \mathbf{elif}\;NaChar \leq 1.9 \cdot 10^{+163}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}\\ t_1 := \frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - t\_0}\\ t_2 := \frac{NdChar}{1 + t\_0}\\ t_3 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{if}\;NaChar \leq -1.55 \cdot 10^{+59}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;NaChar \leq -4.4 \cdot 10^{-84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 5 \cdot 10^{-258}:\\ \;\;\;\;t\_2 + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 2.6 \cdot 10^{+69}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;NaChar \leq 1.6 \cdot 10^{+146}:\\ \;\;\;\;t\_2 + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 1.9 \cdot 10^{+163}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))
        (t_1
         (-
          (/
           NaChar
           (-
            (+ 2.0 (* EAccept (+ (/ 1.0 KbT) (/ Vef (* KbT EAccept)))))
            (/ mu KbT)))
          (/ NdChar (- -1.0 t_0))))
        (t_2 (/ NdChar (+ 1.0 t_0)))
        (t_3
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
          (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))))
   (if (<= NaChar -1.55e+59)
     t_3
     (if (<= NaChar -4.4e-84)
       t_1
       (if (<= NaChar 5e-258)
         (+ t_2 (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu))))))
         (if (<= NaChar 0.5)
           t_1
           (if (<= NaChar 2.6e+69)
             t_3
             (if (<= NaChar 1.6e+146)
               (+
                t_2
                (/
                 NaChar
                 (+
                  1.0
                  (-
                   (+ 1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
                   (/ mu KbT)))))
               (if (<= NaChar 1.9e+163)
                 (+
                  (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
                  (/ NdChar (+ 2.0 (/ EDonor KbT))))
                 t_3)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	double t_2 = NdChar / (1.0 + t_0);
	double t_3 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -1.55e+59) {
		tmp = t_3;
	} else if (NaChar <= -4.4e-84) {
		tmp = t_1;
	} else if (NaChar <= 5e-258) {
		tmp = t_2 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} else if (NaChar <= 0.5) {
		tmp = t_1;
	} else if (NaChar <= 2.6e+69) {
		tmp = t_3;
	} else if (NaChar <= 1.6e+146) {
		tmp = t_2 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	} else if (NaChar <= 1.9e+163) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = exp(((edonor + (mu + (vef - ec))) / kbt))
    t_1 = (nachar / ((2.0d0 + (eaccept * ((1.0d0 / kbt) + (vef / (kbt * eaccept))))) - (mu / kbt))) - (ndchar / ((-1.0d0) - t_0))
    t_2 = ndchar / (1.0d0 + t_0)
    t_3 = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    if (nachar <= (-1.55d+59)) then
        tmp = t_3
    else if (nachar <= (-4.4d-84)) then
        tmp = t_1
    else if (nachar <= 5d-258) then
        tmp = t_2 + (kbt * (nachar / (eaccept + (ev + (vef - mu)))))
    else if (nachar <= 0.5d0) then
        tmp = t_1
    else if (nachar <= 2.6d+69) then
        tmp = t_3
    else if (nachar <= 1.6d+146) then
        tmp = t_2 + (nachar / (1.0d0 + ((1.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt))))
    else if (nachar <= 1.9d+163) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (2.0d0 + (edonor / kbt)))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	double t_2 = NdChar / (1.0 + t_0);
	double t_3 = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -1.55e+59) {
		tmp = t_3;
	} else if (NaChar <= -4.4e-84) {
		tmp = t_1;
	} else if (NaChar <= 5e-258) {
		tmp = t_2 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} else if (NaChar <= 0.5) {
		tmp = t_1;
	} else if (NaChar <= 2.6e+69) {
		tmp = t_3;
	} else if (NaChar <= 1.6e+146) {
		tmp = t_2 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	} else if (NaChar <= 1.9e+163) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))
	t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0))
	t_2 = NdChar / (1.0 + t_0)
	t_3 = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	tmp = 0
	if NaChar <= -1.55e+59:
		tmp = t_3
	elif NaChar <= -4.4e-84:
		tmp = t_1
	elif NaChar <= 5e-258:
		tmp = t_2 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))))
	elif NaChar <= 0.5:
		tmp = t_1
	elif NaChar <= 2.6e+69:
		tmp = t_3
	elif NaChar <= 1.6e+146:
		tmp = t_2 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))))
	elif NaChar <= 1.9e+163:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)))
	else:
		tmp = t_3
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))
	t_1 = Float64(Float64(NaChar / Float64(Float64(2.0 + Float64(EAccept * Float64(Float64(1.0 / KbT) + Float64(Vef / Float64(KbT * EAccept))))) - Float64(mu / KbT))) - Float64(NdChar / Float64(-1.0 - t_0)))
	t_2 = Float64(NdChar / Float64(1.0 + t_0))
	t_3 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))))
	tmp = 0.0
	if (NaChar <= -1.55e+59)
		tmp = t_3;
	elseif (NaChar <= -4.4e-84)
		tmp = t_1;
	elseif (NaChar <= 5e-258)
		tmp = Float64(t_2 + Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))));
	elseif (NaChar <= 0.5)
		tmp = t_1;
	elseif (NaChar <= 2.6e+69)
		tmp = t_3;
	elseif (NaChar <= 1.6e+146)
		tmp = Float64(t_2 + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT)))));
	elseif (NaChar <= 1.9e+163)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(2.0 + Float64(EDonor / KbT))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	t_2 = NdChar / (1.0 + t_0);
	t_3 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	tmp = 0.0;
	if (NaChar <= -1.55e+59)
		tmp = t_3;
	elseif (NaChar <= -4.4e-84)
		tmp = t_1;
	elseif (NaChar <= 5e-258)
		tmp = t_2 + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	elseif (NaChar <= 0.5)
		tmp = t_1;
	elseif (NaChar <= 2.6e+69)
		tmp = t_3;
	elseif (NaChar <= 1.6e+146)
		tmp = t_2 + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	elseif (NaChar <= 1.9e+163)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[(2.0 + N[(EAccept * N[(N[(1.0 / KbT), $MachinePrecision] + N[(Vef / N[(KbT * EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.55e+59], t$95$3, If[LessEqual[NaChar, -4.4e-84], t$95$1, If[LessEqual[NaChar, 5e-258], N[(t$95$2 + N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 0.5], t$95$1, If[LessEqual[NaChar, 2.6e+69], t$95$3, If[LessEqual[NaChar, 1.6e+146], N[(t$95$2 + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.9e+163], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NaChar < -1.55000000000000007e59 or 0.5 < NaChar < 2.6000000000000002e69 or 1.90000000000000004e163 < 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%

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

   5. Taylor expanded in mu around inf 84.7%

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

   6. Taylor expanded in mu around 0 76.0%

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

   if -1.55000000000000007e59 < NaChar < -4.3999999999999998e-84 or 4.9999999999999999e-258 < NaChar < 0.5

   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%

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

   5. Taylor expanded in KbT around inf 68.5%

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

   6. Taylor expanded in EAccept around inf 71.9%

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

   7. Taylor expanded in Ev around 0 71.4%

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

   if -4.3999999999999998e-84 < NaChar < 4.9999999999999999e-258

   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%

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

   5. Taylor expanded in KbT around inf 72.4%

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

   6. Taylor expanded in KbT around 0 79.0%

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

      1. associate-/l* 80.5%

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

      2. associate--l+ 80.5%

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

      3. associate--l+ 80.5%

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

   8. Simplified 80.5%

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

   if 2.6000000000000002e69 < NaChar < 1.6e146

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in KbT around inf 70.6%

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

   if 1.6e146 < NaChar < 1.90000000000000004e163

   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%

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

   5. Taylor expanded in Ev around inf 59.4%

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

   6. Taylor expanded in EDonor around inf 56.4%

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

   7. Taylor expanded in EDonor around 0 56.4%

      \[\leadsto \frac{NdChar}{\color{blue}{2 + \frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.55 \cdot 10^{+59}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq -4.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 5 \cdot 10^{-258}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 2.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 1.6 \cdot 10^{+146}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 1.9 \cdot 10^{+163}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}\\ t_1 := \frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - t\_0}\\ t_2 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{if}\;NaChar \leq -5.6 \cdot 10^{+59}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NaChar \leq -2.6 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 10^{-257}:\\ \;\;\;\;\frac{NdChar}{1 + t\_0} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 2.6 \cdot 10^{+69}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NaChar \leq 1.15 \cdot 10^{+146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 1.9 \cdot 10^{+163}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))
        (t_1
         (-
          (/
           NaChar
           (-
            (+ 2.0 (* EAccept (+ (/ 1.0 KbT) (/ Vef (* KbT EAccept)))))
            (/ mu KbT)))
          (/ NdChar (- -1.0 t_0))))
        (t_2
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
          (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))))
   (if (<= NaChar -5.6e+59)
     t_2
     (if (<= NaChar -2.6e-85)
       t_1
       (if (<= NaChar 1e-257)
         (+
          (/ NdChar (+ 1.0 t_0))
          (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu))))))
         (if (<= NaChar 0.5)
           t_1
           (if (<= NaChar 2.6e+69)
             t_2
             (if (<= NaChar 1.15e+146)
               t_1
               (if (<= NaChar 1.9e+163)
                 (+
                  (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
                  (/ NdChar (+ 2.0 (/ EDonor 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 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	double t_2 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -5.6e+59) {
		tmp = t_2;
	} else if (NaChar <= -2.6e-85) {
		tmp = t_1;
	} else if (NaChar <= 1e-257) {
		tmp = (NdChar / (1.0 + t_0)) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} else if (NaChar <= 0.5) {
		tmp = t_1;
	} else if (NaChar <= 2.6e+69) {
		tmp = t_2;
	} else if (NaChar <= 1.15e+146) {
		tmp = t_1;
	} else if (NaChar <= 1.9e+163) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / 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 = exp(((edonor + (mu + (vef - ec))) / kbt))
    t_1 = (nachar / ((2.0d0 + (eaccept * ((1.0d0 / kbt) + (vef / (kbt * eaccept))))) - (mu / kbt))) - (ndchar / ((-1.0d0) - t_0))
    t_2 = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    if (nachar <= (-5.6d+59)) then
        tmp = t_2
    else if (nachar <= (-2.6d-85)) then
        tmp = t_1
    else if (nachar <= 1d-257) then
        tmp = (ndchar / (1.0d0 + t_0)) + (kbt * (nachar / (eaccept + (ev + (vef - mu)))))
    else if (nachar <= 0.5d0) then
        tmp = t_1
    else if (nachar <= 2.6d+69) then
        tmp = t_2
    else if (nachar <= 1.15d+146) then
        tmp = t_1
    else if (nachar <= 1.9d+163) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (2.0d0 + (edonor / 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 = Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	double t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	double t_2 = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -5.6e+59) {
		tmp = t_2;
	} else if (NaChar <= -2.6e-85) {
		tmp = t_1;
	} else if (NaChar <= 1e-257) {
		tmp = (NdChar / (1.0 + t_0)) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} else if (NaChar <= 0.5) {
		tmp = t_1;
	} else if (NaChar <= 2.6e+69) {
		tmp = t_2;
	} else if (NaChar <= 1.15e+146) {
		tmp = t_1;
	} else if (NaChar <= 1.9e+163) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))
	t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0))
	t_2 = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	tmp = 0
	if NaChar <= -5.6e+59:
		tmp = t_2
	elif NaChar <= -2.6e-85:
		tmp = t_1
	elif NaChar <= 1e-257:
		tmp = (NdChar / (1.0 + t_0)) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))))
	elif NaChar <= 0.5:
		tmp = t_1
	elif NaChar <= 2.6e+69:
		tmp = t_2
	elif NaChar <= 1.15e+146:
		tmp = t_1
	elif NaChar <= 1.9e+163:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)))
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))
	t_1 = Float64(Float64(NaChar / Float64(Float64(2.0 + Float64(EAccept * Float64(Float64(1.0 / KbT) + Float64(Vef / Float64(KbT * EAccept))))) - Float64(mu / KbT))) - Float64(NdChar / Float64(-1.0 - t_0)))
	t_2 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))))
	tmp = 0.0
	if (NaChar <= -5.6e+59)
		tmp = t_2;
	elseif (NaChar <= -2.6e-85)
		tmp = t_1;
	elseif (NaChar <= 1e-257)
		tmp = Float64(Float64(NdChar / Float64(1.0 + t_0)) + Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))));
	elseif (NaChar <= 0.5)
		tmp = t_1;
	elseif (NaChar <= 2.6e+69)
		tmp = t_2;
	elseif (NaChar <= 1.15e+146)
		tmp = t_1;
	elseif (NaChar <= 1.9e+163)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(2.0 + Float64(EDonor / 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 = exp(((EDonor + (mu + (Vef - Ec))) / KbT));
	t_1 = (NaChar / ((2.0 + (EAccept * ((1.0 / KbT) + (Vef / (KbT * EAccept))))) - (mu / KbT))) - (NdChar / (-1.0 - t_0));
	t_2 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	tmp = 0.0;
	if (NaChar <= -5.6e+59)
		tmp = t_2;
	elseif (NaChar <= -2.6e-85)
		tmp = t_1;
	elseif (NaChar <= 1e-257)
		tmp = (NdChar / (1.0 + t_0)) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	elseif (NaChar <= 0.5)
		tmp = t_1;
	elseif (NaChar <= 2.6e+69)
		tmp = t_2;
	elseif (NaChar <= 1.15e+146)
		tmp = t_1;
	elseif (NaChar <= 1.9e+163)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / 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[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(N[(2.0 + N[(EAccept * N[(N[(1.0 / KbT), $MachinePrecision] + N[(Vef / N[(KbT * EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -5.6e+59], t$95$2, If[LessEqual[NaChar, -2.6e-85], t$95$1, If[LessEqual[NaChar, 1e-257], N[(N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 0.5], t$95$1, If[LessEqual[NaChar, 2.6e+69], t$95$2, If[LessEqual[NaChar, 1.15e+146], t$95$1, If[LessEqual[NaChar, 1.9e+163], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -5.5999999999999996e59 or 0.5 < NaChar < 2.6000000000000002e69 or 1.90000000000000004e163 < 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%

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

   5. Taylor expanded in mu around inf 84.7%

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

   6. Taylor expanded in mu around 0 76.0%

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

   if -5.5999999999999996e59 < NaChar < -2.60000000000000011e-85 or 9.9999999999999998e-258 < NaChar < 0.5 or 2.6000000000000002e69 < NaChar < 1.15e146

   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%

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

   5. Taylor expanded in KbT around inf 68.9%

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

   6. Taylor expanded in EAccept around inf 71.7%

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

   7. Taylor expanded in Ev around 0 70.2%

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

   if -2.60000000000000011e-85 < NaChar < 9.9999999999999998e-258

   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%

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

   5. Taylor expanded in KbT around inf 72.4%

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

   6. Taylor expanded in KbT around 0 79.0%

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

      1. associate-/l* 80.5%

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

      2. associate--l+ 80.5%

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

      3. associate--l+ 80.5%

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

   8. Simplified 80.5%

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

   if 1.15e146 < NaChar < 1.90000000000000004e163

   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%

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

   5. Taylor expanded in Ev around inf 59.4%

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

   6. Taylor expanded in EDonor around inf 56.4%

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

   7. Taylor expanded in EDonor around 0 56.4%

      \[\leadsto \frac{NdChar}{\color{blue}{2 + \frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -5.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq -2.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 10^{-257}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 2.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq 1.15 \cdot 10^{+146}:\\ \;\;\;\;\frac{NaChar}{\left(2 + EAccept \cdot \left(\frac{1}{KbT} + \frac{Vef}{KbT \cdot EAccept}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.9 \cdot 10^{+163}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{if}\;NaChar \leq -5 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq -1.15 \cdot 10^{+19}:\\ \;\;\;\;\frac{NaChar}{\left(2 + \left(\frac{EAccept}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}} - t\_0\\ \mathbf{elif}\;NaChar \leq -2.9 \cdot 10^{-27} \lor \neg \left(NaChar \leq 0.5\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
        (t_1
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
          (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))))
   (if (<= NaChar -5e+60)
     t_1
     (if (<= NaChar -1.15e+19)
       (-
        (/ NaChar (- (+ 2.0 (+ (/ EAccept KbT) (/ Vef KbT))) (/ mu KbT)))
        t_0)
       (if (or (<= NaChar -2.9e-27) (not (<= NaChar 0.5)))
         t_1
         (- (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu))))) t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -5e+60) {
		tmp = t_1;
	} else if (NaChar <= -1.15e+19) {
		tmp = (NaChar / ((2.0 + ((EAccept / KbT) + (Vef / KbT))) - (mu / KbT))) - t_0;
	} else if ((NaChar <= -2.9e-27) || !(NaChar <= 0.5)) {
		tmp = t_1;
	} else {
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - 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 = ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt)))
    t_1 = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    if (nachar <= (-5d+60)) then
        tmp = t_1
    else if (nachar <= (-1.15d+19)) then
        tmp = (nachar / ((2.0d0 + ((eaccept / kbt) + (vef / kbt))) - (mu / kbt))) - t_0
    else if ((nachar <= (-2.9d-27)) .or. (.not. (nachar <= 0.5d0))) then
        tmp = t_1
    else
        tmp = (kbt * (nachar / (eaccept + (ev + (vef - mu))))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	double tmp;
	if (NaChar <= -5e+60) {
		tmp = t_1;
	} else if (NaChar <= -1.15e+19) {
		tmp = (NaChar / ((2.0 + ((EAccept / KbT) + (Vef / KbT))) - (mu / KbT))) - t_0;
	} else if ((NaChar <= -2.9e-27) || !(NaChar <= 0.5)) {
		tmp = t_1;
	} else {
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))
	t_1 = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	tmp = 0
	if NaChar <= -5e+60:
		tmp = t_1
	elif NaChar <= -1.15e+19:
		tmp = (NaChar / ((2.0 + ((EAccept / KbT) + (Vef / KbT))) - (mu / KbT))) - t_0
	elif (NaChar <= -2.9e-27) or not (NaChar <= 0.5):
		tmp = t_1
	else:
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))))
	tmp = 0.0
	if (NaChar <= -5e+60)
		tmp = t_1;
	elseif (NaChar <= -1.15e+19)
		tmp = Float64(Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(EAccept / KbT) + Float64(Vef / KbT))) - Float64(mu / KbT))) - t_0);
	elseif ((NaChar <= -2.9e-27) || !(NaChar <= 0.5))
		tmp = t_1;
	else
		tmp = Float64(Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))) - t_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(((EDonor + (mu + (Vef - Ec))) / KbT)));
	t_1 = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	tmp = 0.0;
	if (NaChar <= -5e+60)
		tmp = t_1;
	elseif (NaChar <= -1.15e+19)
		tmp = (NaChar / ((2.0 + ((EAccept / KbT) + (Vef / KbT))) - (mu / KbT))) - t_0;
	elseif ((NaChar <= -2.9e-27) || ~((NaChar <= 0.5)))
		tmp = t_1;
	else
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -5e+60], t$95$1, If[LessEqual[NaChar, -1.15e+19], N[(N[(NaChar / N[(N[(2.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[Or[LessEqual[NaChar, -2.9e-27], N[Not[LessEqual[NaChar, 0.5]], $MachinePrecision]], t$95$1, N[(N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -4.99999999999999975e60 or -1.15e19 < NaChar < -2.90000000000000004e-27 or 0.5 < 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%

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

   5. Taylor expanded in mu around inf 78.1%

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

   6. Taylor expanded in mu around 0 68.7%

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

   if -4.99999999999999975e60 < NaChar < -1.15e19

   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%

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

   5. Taylor expanded in KbT around inf 73.1%

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

   6. Taylor expanded in Ev around 0 82.2%

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

   if -2.90000000000000004e-27 < NaChar < 0.5

   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%

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

   5. Taylor expanded in KbT around inf 74.9%

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

   6. Taylor expanded in KbT around 0 77.4%

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

      1. associate-/l* 79.0%

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

      2. associate--l+ 79.0%

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

      3. associate--l+ 79.0%

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

   8. Simplified 79.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -5 \cdot 10^{+60}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NaChar \leq -1.15 \cdot 10^{+19}:\\ \;\;\;\;\frac{NaChar}{\left(2 + \left(\frac{EAccept}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{mu}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq -2.9 \cdot 10^{-27} \lor \neg \left(NaChar \leq 0.5\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ t_1 := NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_2 := NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{if}\;NaChar \leq -3.1 \cdot 10^{+55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NaChar \leq -5.2 \cdot 10^{-233}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq -3.5 \cdot 10^{-299}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
          (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu)))))))
        (t_1
         (-
          (* NaChar 0.5)
          (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))))
        (t_2
         (+
          (* NdChar 0.5)
          (/
           1.0
           (/ (+ 1.0 (exp (/ (- Vef (- (- mu Ev) EAccept)) KbT))) NaChar)))))
   (if (<= NaChar -3.1e+55)
     t_2
     (if (<= NaChar -5.2e-233)
       t_1
       (if (<= NaChar -3.5e-299)
         t_0
         (if (<= NaChar 1.3e-68) t_1 (if (<= NaChar 0.5) 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((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	double t_1 = (NaChar * 0.5) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	double t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	double tmp;
	if (NaChar <= -3.1e+55) {
		tmp = t_2;
	} else if (NaChar <= -5.2e-233) {
		tmp = t_1;
	} else if (NaChar <= -3.5e-299) {
		tmp = t_0;
	} else if (NaChar <= 1.3e-68) {
		tmp = t_1;
	} else if (NaChar <= 0.5) {
		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((vef / kbt)))) + (kbt * (nachar / (eaccept + (ev + (vef - mu)))))
    t_1 = (nachar * 0.5d0) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt))))
    t_2 = (ndchar * 0.5d0) + (1.0d0 / ((1.0d0 + exp(((vef - ((mu - ev) - eaccept)) / kbt))) / nachar))
    if (nachar <= (-3.1d+55)) then
        tmp = t_2
    else if (nachar <= (-5.2d-233)) then
        tmp = t_1
    else if (nachar <= (-3.5d-299)) then
        tmp = t_0
    else if (nachar <= 1.3d-68) then
        tmp = t_1
    else if (nachar <= 0.5d0) 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((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	double t_1 = (NaChar * 0.5) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	double t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + Math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	double tmp;
	if (NaChar <= -3.1e+55) {
		tmp = t_2;
	} else if (NaChar <= -5.2e-233) {
		tmp = t_1;
	} else if (NaChar <= -3.5e-299) {
		tmp = t_0;
	} else if (NaChar <= 1.3e-68) {
		tmp = t_1;
	} else if (NaChar <= 0.5) {
		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((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))))
	t_1 = (NaChar * 0.5) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar))
	tmp = 0
	if NaChar <= -3.1e+55:
		tmp = t_2
	elif NaChar <= -5.2e-233:
		tmp = t_1
	elif NaChar <= -3.5e-299:
		tmp = t_0
	elif NaChar <= 1.3e-68:
		tmp = t_1
	elif NaChar <= 0.5:
		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(Vef / KbT)))) + Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))))
	t_1 = Float64(Float64(NaChar * 0.5) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))))
	t_2 = Float64(Float64(NdChar * 0.5) + Float64(1.0 / Float64(Float64(1.0 + exp(Float64(Float64(Vef - Float64(Float64(mu - Ev) - EAccept)) / KbT))) / NaChar)))
	tmp = 0.0
	if (NaChar <= -3.1e+55)
		tmp = t_2;
	elseif (NaChar <= -5.2e-233)
		tmp = t_1;
	elseif (NaChar <= -3.5e-299)
		tmp = t_0;
	elseif (NaChar <= 1.3e-68)
		tmp = t_1;
	elseif (NaChar <= 0.5)
		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((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	t_1 = (NaChar * 0.5) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	tmp = 0.0;
	if (NaChar <= -3.1e+55)
		tmp = t_2;
	elseif (NaChar <= -5.2e-233)
		tmp = t_1;
	elseif (NaChar <= -3.5e-299)
		tmp = t_0;
	elseif (NaChar <= 1.3e-68)
		tmp = t_1;
	elseif (NaChar <= 0.5)
		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[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar * 0.5), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar * 0.5), $MachinePrecision] + N[(1.0 / N[(N[(1.0 + N[Exp[N[(N[(Vef - N[(N[(mu - Ev), $MachinePrecision] - EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -3.1e+55], t$95$2, If[LessEqual[NaChar, -5.2e-233], t$95$1, If[LessEqual[NaChar, -3.5e-299], t$95$0, If[LessEqual[NaChar, 1.3e-68], t$95$1, If[LessEqual[NaChar, 0.5], t$95$0, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -3.09999999999999994e55 or 0.5 < 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%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+l- 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in KbT around inf 67.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{1}{\frac{1 + e^{\frac{Vef + \left(EAccept - \left(mu - Ev\right)\right)}{KbT}}}{NaChar}} \]
   10. Step-by-step derivation

       1. *-commutative 10.0%

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

   11. Simplified 67.1%

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

   if -3.09999999999999994e55 < NaChar < -5.1999999999999996e-233 or -3.49999999999999991e-299 < NaChar < 1.2999999999999999e-68

   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%

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

   5. Taylor expanded in KbT around inf 60.8%

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

   if -5.1999999999999996e-233 < NaChar < -3.49999999999999991e-299 or 1.2999999999999999e-68 < NaChar < 0.5

   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%

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

   5. Taylor expanded in KbT around inf 79.9%

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

   6. Taylor expanded in KbT around 0 82.5%

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

      1. associate-/l* 87.3%

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

      2. associate--l+ 87.3%

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

      3. associate--l+ 87.3%

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

   8. Simplified 87.3%

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

   9. Taylor expanded in Vef around inf 73.4%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -3.1 \cdot 10^{+55}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{elif}\;NaChar \leq -5.2 \cdot 10^{-233}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq -3.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{elif}\;NaChar \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := NaChar \cdot 0.5 - t\_0\\ t_2 := NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{if}\;NaChar \leq -2 \cdot 10^{+56}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NaChar \leq -3.8 \cdot 10^{-196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 2.9 \cdot 10^{-280}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{EAccept} - t\_0\\ \mathbf{elif}\;NaChar \leq 1.65 \cdot 10^{-68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
        (t_1 (- (* NaChar 0.5) t_0))
        (t_2
         (+
          (* NdChar 0.5)
          (/
           1.0
           (/ (+ 1.0 (exp (/ (- Vef (- (- mu Ev) EAccept)) KbT))) NaChar)))))
   (if (<= NaChar -2e+56)
     t_2
     (if (<= NaChar -3.8e-196)
       t_1
       (if (<= NaChar 2.9e-280)
         (- (* KbT (/ NaChar EAccept)) t_0)
         (if (<= NaChar 1.65e-68)
           t_1
           (if (<= NaChar 0.5)
             (+
              (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
              (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu))))))
             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(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar * 0.5) - t_0;
	double t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	double tmp;
	if (NaChar <= -2e+56) {
		tmp = t_2;
	} else if (NaChar <= -3.8e-196) {
		tmp = t_1;
	} else if (NaChar <= 2.9e-280) {
		tmp = (KbT * (NaChar / EAccept)) - t_0;
	} else if (NaChar <= 1.65e-68) {
		tmp = t_1;
	} else if (NaChar <= 0.5) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} 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(((edonor + (mu + (vef - ec))) / kbt)))
    t_1 = (nachar * 0.5d0) - t_0
    t_2 = (ndchar * 0.5d0) + (1.0d0 / ((1.0d0 + exp(((vef - ((mu - ev) - eaccept)) / kbt))) / nachar))
    if (nachar <= (-2d+56)) then
        tmp = t_2
    else if (nachar <= (-3.8d-196)) then
        tmp = t_1
    else if (nachar <= 2.9d-280) then
        tmp = (kbt * (nachar / eaccept)) - t_0
    else if (nachar <= 1.65d-68) then
        tmp = t_1
    else if (nachar <= 0.5d0) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (kbt * (nachar / (eaccept + (ev + (vef - mu)))))
    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(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar * 0.5) - t_0;
	double t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + Math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	double tmp;
	if (NaChar <= -2e+56) {
		tmp = t_2;
	} else if (NaChar <= -3.8e-196) {
		tmp = t_1;
	} else if (NaChar <= 2.9e-280) {
		tmp = (KbT * (NaChar / EAccept)) - t_0;
	} else if (NaChar <= 1.65e-68) {
		tmp = t_1;
	} else if (NaChar <= 0.5) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} 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(((EDonor + (mu + (Vef - Ec))) / KbT)))
	t_1 = (NaChar * 0.5) - t_0
	t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar))
	tmp = 0
	if NaChar <= -2e+56:
		tmp = t_2
	elif NaChar <= -3.8e-196:
		tmp = t_1
	elif NaChar <= 2.9e-280:
		tmp = (KbT * (NaChar / EAccept)) - t_0
	elif NaChar <= 1.65e-68:
		tmp = t_1
	elif NaChar <= 0.5:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))))
	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(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	t_1 = Float64(Float64(NaChar * 0.5) - t_0)
	t_2 = Float64(Float64(NdChar * 0.5) + Float64(1.0 / Float64(Float64(1.0 + exp(Float64(Float64(Vef - Float64(Float64(mu - Ev) - EAccept)) / KbT))) / NaChar)))
	tmp = 0.0
	if (NaChar <= -2e+56)
		tmp = t_2;
	elseif (NaChar <= -3.8e-196)
		tmp = t_1;
	elseif (NaChar <= 2.9e-280)
		tmp = Float64(Float64(KbT * Float64(NaChar / EAccept)) - t_0);
	elseif (NaChar <= 1.65e-68)
		tmp = t_1;
	elseif (NaChar <= 0.5)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))));
	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(((EDonor + (mu + (Vef - Ec))) / KbT)));
	t_1 = (NaChar * 0.5) - t_0;
	t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	tmp = 0.0;
	if (NaChar <= -2e+56)
		tmp = t_2;
	elseif (NaChar <= -3.8e-196)
		tmp = t_1;
	elseif (NaChar <= 2.9e-280)
		tmp = (KbT * (NaChar / EAccept)) - t_0;
	elseif (NaChar <= 1.65e-68)
		tmp = t_1;
	elseif (NaChar <= 0.5)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	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[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar * 0.5), $MachinePrecision] + N[(1.0 / N[(N[(1.0 + N[Exp[N[(N[(Vef - N[(N[(mu - Ev), $MachinePrecision] - EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -2e+56], t$95$2, If[LessEqual[NaChar, -3.8e-196], t$95$1, If[LessEqual[NaChar, 2.9e-280], N[(N[(KbT * N[(NaChar / EAccept), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[NaChar, 1.65e-68], t$95$1, If[LessEqual[NaChar, 0.5], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -2.00000000000000018e56 or 0.5 < 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%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+l- 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in KbT around inf 67.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{1}{\frac{1 + e^{\frac{Vef + \left(EAccept - \left(mu - Ev\right)\right)}{KbT}}}{NaChar}} \]
   10. Step-by-step derivation

       1. *-commutative 10.0%

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

   11. Simplified 67.1%

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

   if -2.00000000000000018e56 < NaChar < -3.8000000000000001e-196 or 2.9e-280 < NaChar < 1.6499999999999999e-68

   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%

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

   5. Taylor expanded in KbT around inf 61.3%

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

   if -3.8000000000000001e-196 < NaChar < 2.9e-280

   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%

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

   5. Taylor expanded in KbT around inf 74.7%

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

   6. Taylor expanded in KbT around 0 82.8%

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

      1. associate-/l* 85.2%

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

      2. associate--l+ 85.2%

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

      3. associate--l+ 85.2%

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

   8. Simplified 85.2%

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

   9. Taylor expanded in EAccept around inf 70.1%

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

       1. associate-/l* 66.8%

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

   11. Simplified 66.8%

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

   if 1.6499999999999999e-68 < NaChar < 0.5

   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%

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

   5. Taylor expanded in KbT around inf 79.2%

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

   6. Taylor expanded in KbT around 0 79.2%

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

      1. associate-/l* 84.5%

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

      2. associate--l+ 84.5%

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

      3. associate--l+ 84.5%

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

   8. Simplified 84.5%

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

   9. Taylor expanded in Vef around inf 69.3%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2 \cdot 10^{+56}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{elif}\;NaChar \leq -3.8 \cdot 10^{-196}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 2.9 \cdot 10^{-280}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{EAccept} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.65 \cdot 10^{-68}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := NaChar \cdot 0.5 - t\_0\\ t_2 := NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{if}\;NaChar \leq -1.1 \cdot 10^{+56}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NaChar \leq -4.1 \cdot 10^{-196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 6.8 \cdot 10^{-137}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{Vef} - t\_0\\ \mathbf{elif}\;NaChar \leq 2.2 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
        (t_1 (- (* NaChar 0.5) t_0))
        (t_2
         (+
          (* NdChar 0.5)
          (/
           1.0
           (/ (+ 1.0 (exp (/ (- Vef (- (- mu Ev) EAccept)) KbT))) NaChar)))))
   (if (<= NaChar -1.1e+56)
     t_2
     (if (<= NaChar -4.1e-196)
       t_1
       (if (<= NaChar 6.8e-137)
         (- (* KbT (/ NaChar Vef)) t_0)
         (if (<= NaChar 2.2e-69)
           t_1
           (if (<= NaChar 5.5e-6)
             (+
              (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
              (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu))))))
             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(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar * 0.5) - t_0;
	double t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	double tmp;
	if (NaChar <= -1.1e+56) {
		tmp = t_2;
	} else if (NaChar <= -4.1e-196) {
		tmp = t_1;
	} else if (NaChar <= 6.8e-137) {
		tmp = (KbT * (NaChar / Vef)) - t_0;
	} else if (NaChar <= 2.2e-69) {
		tmp = t_1;
	} else if (NaChar <= 5.5e-6) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} 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(((edonor + (mu + (vef - ec))) / kbt)))
    t_1 = (nachar * 0.5d0) - t_0
    t_2 = (ndchar * 0.5d0) + (1.0d0 / ((1.0d0 + exp(((vef - ((mu - ev) - eaccept)) / kbt))) / nachar))
    if (nachar <= (-1.1d+56)) then
        tmp = t_2
    else if (nachar <= (-4.1d-196)) then
        tmp = t_1
    else if (nachar <= 6.8d-137) then
        tmp = (kbt * (nachar / vef)) - t_0
    else if (nachar <= 2.2d-69) then
        tmp = t_1
    else if (nachar <= 5.5d-6) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (kbt * (nachar / (eaccept + (ev + (vef - mu)))))
    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(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = (NaChar * 0.5) - t_0;
	double t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + Math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	double tmp;
	if (NaChar <= -1.1e+56) {
		tmp = t_2;
	} else if (NaChar <= -4.1e-196) {
		tmp = t_1;
	} else if (NaChar <= 6.8e-137) {
		tmp = (KbT * (NaChar / Vef)) - t_0;
	} else if (NaChar <= 2.2e-69) {
		tmp = t_1;
	} else if (NaChar <= 5.5e-6) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} 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(((EDonor + (mu + (Vef - Ec))) / KbT)))
	t_1 = (NaChar * 0.5) - t_0
	t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar))
	tmp = 0
	if NaChar <= -1.1e+56:
		tmp = t_2
	elif NaChar <= -4.1e-196:
		tmp = t_1
	elif NaChar <= 6.8e-137:
		tmp = (KbT * (NaChar / Vef)) - t_0
	elif NaChar <= 2.2e-69:
		tmp = t_1
	elif NaChar <= 5.5e-6:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))))
	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(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	t_1 = Float64(Float64(NaChar * 0.5) - t_0)
	t_2 = Float64(Float64(NdChar * 0.5) + Float64(1.0 / Float64(Float64(1.0 + exp(Float64(Float64(Vef - Float64(Float64(mu - Ev) - EAccept)) / KbT))) / NaChar)))
	tmp = 0.0
	if (NaChar <= -1.1e+56)
		tmp = t_2;
	elseif (NaChar <= -4.1e-196)
		tmp = t_1;
	elseif (NaChar <= 6.8e-137)
		tmp = Float64(Float64(KbT * Float64(NaChar / Vef)) - t_0);
	elseif (NaChar <= 2.2e-69)
		tmp = t_1;
	elseif (NaChar <= 5.5e-6)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))));
	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(((EDonor + (mu + (Vef - Ec))) / KbT)));
	t_1 = (NaChar * 0.5) - t_0;
	t_2 = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	tmp = 0.0;
	if (NaChar <= -1.1e+56)
		tmp = t_2;
	elseif (NaChar <= -4.1e-196)
		tmp = t_1;
	elseif (NaChar <= 6.8e-137)
		tmp = (KbT * (NaChar / Vef)) - t_0;
	elseif (NaChar <= 2.2e-69)
		tmp = t_1;
	elseif (NaChar <= 5.5e-6)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	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[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar * 0.5), $MachinePrecision] + N[(1.0 / N[(N[(1.0 + N[Exp[N[(N[(Vef - N[(N[(mu - Ev), $MachinePrecision] - EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.1e+56], t$95$2, If[LessEqual[NaChar, -4.1e-196], t$95$1, If[LessEqual[NaChar, 6.8e-137], N[(N[(KbT * N[(NaChar / Vef), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[NaChar, 2.2e-69], t$95$1, If[LessEqual[NaChar, 5.5e-6], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -1.10000000000000008e56 or 5.4999999999999999e-6 < 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%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+l- 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in KbT around inf 66.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{1}{\frac{1 + e^{\frac{Vef + \left(EAccept - \left(mu - Ev\right)\right)}{KbT}}}{NaChar}} \]
   10. Step-by-step derivation

       1. *-commutative 10.0%

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

   11. Simplified 66.1%

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

   if -1.10000000000000008e56 < NaChar < -4.10000000000000021e-196 or 6.80000000000000028e-137 < NaChar < 2.2e-69

   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%

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

   5. Taylor expanded in KbT around inf 57.8%

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

   if -4.10000000000000021e-196 < NaChar < 6.80000000000000028e-137

   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%

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

   5. Taylor expanded in KbT around inf 78.4%

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

   6. Taylor expanded in EAccept around inf 80.0%

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

   7. Taylor expanded in Vef around inf 73.8%

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

      1. associate-/l* 73.6%

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

   9. Simplified 73.6%

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

   if 2.2e-69 < NaChar < 5.4999999999999999e-6

   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%

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

   5. Taylor expanded in KbT around inf 76.6%

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

   6. Taylor expanded in KbT around 0 76.6%

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

      1. associate-/l* 82.6%

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

      2. associate--l+ 82.6%

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

      3. associate--l+ 82.6%

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

   8. Simplified 82.6%

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

   9. Taylor expanded in Vef around inf 70.5%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.1 \cdot 10^{+56}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{elif}\;NaChar \leq -4.1 \cdot 10^{-196}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 6.8 \cdot 10^{-137}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{Vef} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 2.2 \cdot 10^{-69}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{if}\;NaChar \leq -4.1 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq -1.35 \cdot 10^{-280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq -1.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \mathbf{elif}\;NaChar \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (-
          (* NaChar 0.5)
          (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))))
        (t_1
         (+
          (* NdChar 0.5)
          (/
           1.0
           (/ (+ 1.0 (exp (/ (- Vef (- (- mu Ev) EAccept)) KbT))) NaChar)))))
   (if (<= NaChar -4.1e+55)
     t_1
     (if (<= NaChar -1.35e-280)
       t_0
       (if (<= NaChar -1.5e-294)
         (/ (* KbT NaChar) (- (+ EAccept (+ Vef Ev)) mu))
         (if (<= NaChar 5e-7) t_0 t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar * 0.5) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	double t_1 = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	double tmp;
	if (NaChar <= -4.1e+55) {
		tmp = t_1;
	} else if (NaChar <= -1.35e-280) {
		tmp = t_0;
	} else if (NaChar <= -1.5e-294) {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	} else if (NaChar <= 5e-7) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar * 0.5d0) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt))))
    t_1 = (ndchar * 0.5d0) + (1.0d0 / ((1.0d0 + exp(((vef - ((mu - ev) - eaccept)) / kbt))) / nachar))
    if (nachar <= (-4.1d+55)) then
        tmp = t_1
    else if (nachar <= (-1.35d-280)) then
        tmp = t_0
    else if (nachar <= (-1.5d-294)) then
        tmp = (kbt * nachar) / ((eaccept + (vef + ev)) - mu)
    else if (nachar <= 5d-7) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar * 0.5) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	double t_1 = (NdChar * 0.5) + (1.0 / ((1.0 + Math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	double tmp;
	if (NaChar <= -4.1e+55) {
		tmp = t_1;
	} else if (NaChar <= -1.35e-280) {
		tmp = t_0;
	} else if (NaChar <= -1.5e-294) {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	} else if (NaChar <= 5e-7) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar * 0.5) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	t_1 = (NdChar * 0.5) + (1.0 / ((1.0 + math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar))
	tmp = 0
	if NaChar <= -4.1e+55:
		tmp = t_1
	elif NaChar <= -1.35e-280:
		tmp = t_0
	elif NaChar <= -1.5e-294:
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu)
	elif NaChar <= 5e-7:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar * 0.5) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))))
	t_1 = Float64(Float64(NdChar * 0.5) + Float64(1.0 / Float64(Float64(1.0 + exp(Float64(Float64(Vef - Float64(Float64(mu - Ev) - EAccept)) / KbT))) / NaChar)))
	tmp = 0.0
	if (NaChar <= -4.1e+55)
		tmp = t_1;
	elseif (NaChar <= -1.35e-280)
		tmp = t_0;
	elseif (NaChar <= -1.5e-294)
		tmp = Float64(Float64(KbT * NaChar) / Float64(Float64(EAccept + Float64(Vef + Ev)) - mu));
	elseif (NaChar <= 5e-7)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar * 0.5) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	t_1 = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	tmp = 0.0;
	if (NaChar <= -4.1e+55)
		tmp = t_1;
	elseif (NaChar <= -1.35e-280)
		tmp = t_0;
	elseif (NaChar <= -1.5e-294)
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	elseif (NaChar <= 5e-7)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar * 0.5), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar * 0.5), $MachinePrecision] + N[(1.0 / N[(N[(1.0 + N[Exp[N[(N[(Vef - N[(N[(mu - Ev), $MachinePrecision] - EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -4.1e+55], t$95$1, If[LessEqual[NaChar, -1.35e-280], t$95$0, If[LessEqual[NaChar, -1.5e-294], N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 5e-7], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -4.09999999999999981e55 or 4.99999999999999977e-7 < 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%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+l- 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in KbT around inf 66.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{1}{\frac{1 + e^{\frac{Vef + \left(EAccept - \left(mu - Ev\right)\right)}{KbT}}}{NaChar}} \]
   10. Step-by-step derivation

       1. *-commutative 10.0%

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

   11. Simplified 66.1%

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

   if -4.09999999999999981e55 < NaChar < -1.34999999999999992e-280 or -1.4999999999999999e-294 < NaChar < 4.99999999999999977e-7

   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%

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

   5. Taylor expanded in KbT around inf 59.0%

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

   if -1.34999999999999992e-280 < NaChar < -1.4999999999999999e-294

   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%

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

   5. Taylor expanded in KbT around inf 100.0%

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

   6. Taylor expanded in KbT around 0 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

      3. associate--l+ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in KbT around inf 3.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 3.1%

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

   11. Simplified 3.1%

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

   12. Taylor expanded in NdChar around 0 100.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -4.1 \cdot 10^{+55}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{elif}\;NaChar \leq -1.35 \cdot 10^{-280}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq -1.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \mathbf{elif}\;NaChar \leq 5 \cdot 10^{-7}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 40.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \mathbf{if}\;mu \leq -1.65 \cdot 10^{+71}:\\ \;\;\;\;t\_1 + t\_0\\ \mathbf{elif}\;mu \leq 6.5 \cdot 10^{-215}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;mu \leq 6.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;mu \leq 2.8 \cdot 10^{-104}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;mu \leq 2.85 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 2.0 (/ Ev KbT))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
        (t_2 (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) t_0)))
   (if (<= mu -1.65e+71)
     (+ t_1 t_0)
     (if (<= mu 6.5e-215)
       t_2
       (if (<= mu 6.4e-157)
         (+
          (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
          (/ NdChar (+ 2.0 (/ EDonor KbT))))
         (if (<= mu 2.8e-104)
           t_2
           (if (<= mu 2.85e+187)
             (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))
             (+ t_1 (* 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 t_0 = NaChar / (2.0 + (Ev / KbT));
	double t_1 = NdChar / (1.0 + exp((mu / KbT)));
	double t_2 = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	double tmp;
	if (mu <= -1.65e+71) {
		tmp = t_1 + t_0;
	} else if (mu <= 6.5e-215) {
		tmp = t_2;
	} else if (mu <= 6.4e-157) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	} else if (mu <= 2.8e-104) {
		tmp = t_2;
	} else if (mu <= 2.85e+187) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_1 + (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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (2.0d0 + (ev / kbt))
    t_1 = ndchar / (1.0d0 + exp((mu / kbt)))
    t_2 = (ndchar / (1.0d0 + exp((vef / kbt)))) + t_0
    if (mu <= (-1.65d+71)) then
        tmp = t_1 + t_0
    else if (mu <= 6.5d-215) then
        tmp = t_2
    else if (mu <= 6.4d-157) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (2.0d0 + (edonor / kbt)))
    else if (mu <= 2.8d-104) then
        tmp = t_2
    else if (mu <= 2.85d+187) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = t_1 + (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 t_0 = NaChar / (2.0 + (Ev / KbT));
	double t_1 = NdChar / (1.0 + Math.exp((mu / KbT)));
	double t_2 = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	double tmp;
	if (mu <= -1.65e+71) {
		tmp = t_1 + t_0;
	} else if (mu <= 6.5e-215) {
		tmp = t_2;
	} else if (mu <= 6.4e-157) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	} else if (mu <= 2.8e-104) {
		tmp = t_2;
	} else if (mu <= 2.85e+187) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_1 + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (2.0 + (Ev / KbT))
	t_1 = NdChar / (1.0 + math.exp((mu / KbT)))
	t_2 = (NdChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	tmp = 0
	if mu <= -1.65e+71:
		tmp = t_1 + t_0
	elif mu <= 6.5e-215:
		tmp = t_2
	elif mu <= 6.4e-157:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)))
	elif mu <= 2.8e-104:
		tmp = t_2
	elif mu <= 2.85e+187:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	else:
		tmp = t_1 + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(2.0 + Float64(Ev / KbT)))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT))))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0)
	tmp = 0.0
	if (mu <= -1.65e+71)
		tmp = Float64(t_1 + t_0);
	elseif (mu <= 6.5e-215)
		tmp = t_2;
	elseif (mu <= 6.4e-157)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(2.0 + Float64(EDonor / KbT))));
	elseif (mu <= 2.8e-104)
		tmp = t_2;
	elseif (mu <= 2.85e+187)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(t_1 + Float64(NaChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (2.0 + (Ev / KbT));
	t_1 = NdChar / (1.0 + exp((mu / KbT)));
	t_2 = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	tmp = 0.0;
	if (mu <= -1.65e+71)
		tmp = t_1 + t_0;
	elseif (mu <= 6.5e-215)
		tmp = t_2;
	elseif (mu <= 6.4e-157)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	elseif (mu <= 2.8e-104)
		tmp = t_2;
	elseif (mu <= 2.85e+187)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	else
		tmp = t_1 + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[mu, -1.65e+71], N[(t$95$1 + t$95$0), $MachinePrecision], If[LessEqual[mu, 6.5e-215], t$95$2, If[LessEqual[mu, 6.4e-157], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2.8e-104], t$95$2, If[LessEqual[mu, 2.85e+187], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if mu < -1.6499999999999999e71

   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%

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

   5. Taylor expanded in Ev around inf 68.5%

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

   6. Taylor expanded in Ev around 0 63.5%

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

   7. Taylor expanded in mu around inf 49.0%

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

   if -1.6499999999999999e71 < mu < 6.4999999999999999e-215 or 6.40000000000000041e-157 < mu < 2.8e-104

   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%

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

   5. Taylor expanded in Ev around inf 76.3%

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

   6. Taylor expanded in Ev around 0 61.5%

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

   7. Taylor expanded in Vef around inf 51.3%

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

   if 6.4999999999999999e-215 < mu < 6.40000000000000041e-157

   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%

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

   5. Taylor expanded in Ev around inf 70.1%

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

   6. Taylor expanded in EDonor around inf 70.1%

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

   7. Taylor expanded in EDonor around 0 61.9%

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

   if 2.8e-104 < mu < 2.8500000000000002e187

   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%

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

   5. Taylor expanded in EAccept around inf 67.6%

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

   6. Taylor expanded in KbT around inf 44.5%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 13.7%

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

   8. Simplified 44.5%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]

   if 2.8500000000000002e187 < 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%

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

   5. Taylor expanded in KbT around inf 69.5%

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

   6. Taylor expanded in mu around inf 64.3%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu}}{KbT}}} + 0.5 \cdot NaChar \]
3. Recombined 5 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.65 \cdot 10^{+71}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;mu \leq 6.5 \cdot 10^{-215}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;mu \leq 6.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;mu \leq 2.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;mu \leq 2.85 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 40.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -5.2 \cdot 10^{+71}:\\ \;\;\;\;t\_1 + t\_0\\ \mathbf{elif}\;mu \leq -9.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \mathbf{elif}\;mu \leq 5.2 \cdot 10^{-114}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{Ec}{-KbT}}}\\ \mathbf{elif}\;mu \leq 1.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;mu \leq 1.66 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 2.0 (/ Ev KbT))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ mu KbT))))))
   (if (<= mu -5.2e+71)
     (+ t_1 t_0)
     (if (<= mu -9.2e-296)
       (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) t_0)
       (if (<= mu 5.2e-114)
         (+ t_0 (/ NdChar (+ 1.0 (exp (/ Ec (- KbT))))))
         (if (<= mu 1.2e-68)
           (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (* NaChar 0.5))
           (if (<= mu 1.66e+187)
             (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))
             (+ t_1 (* 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 t_0 = NaChar / (2.0 + (Ev / KbT));
	double t_1 = NdChar / (1.0 + exp((mu / KbT)));
	double tmp;
	if (mu <= -5.2e+71) {
		tmp = t_1 + t_0;
	} else if (mu <= -9.2e-296) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	} else if (mu <= 5.2e-114) {
		tmp = t_0 + (NdChar / (1.0 + exp((Ec / -KbT))));
	} else if (mu <= 1.2e-68) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else if (mu <= 1.66e+187) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_1 + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (2.0d0 + (ev / kbt))
    t_1 = ndchar / (1.0d0 + exp((mu / kbt)))
    if (mu <= (-5.2d+71)) then
        tmp = t_1 + t_0
    else if (mu <= (-9.2d-296)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + t_0
    else if (mu <= 5.2d-114) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((ec / -kbt))))
    else if (mu <= 1.2d-68) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar * 0.5d0)
    else if (mu <= 1.66d+187) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = t_1 + (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 t_0 = NaChar / (2.0 + (Ev / KbT));
	double t_1 = NdChar / (1.0 + Math.exp((mu / KbT)));
	double tmp;
	if (mu <= -5.2e+71) {
		tmp = t_1 + t_0;
	} else if (mu <= -9.2e-296) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	} else if (mu <= 5.2e-114) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((Ec / -KbT))));
	} else if (mu <= 1.2e-68) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else if (mu <= 1.66e+187) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_1 + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (2.0 + (Ev / KbT))
	t_1 = NdChar / (1.0 + math.exp((mu / KbT)))
	tmp = 0
	if mu <= -5.2e+71:
		tmp = t_1 + t_0
	elif mu <= -9.2e-296:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	elif mu <= 5.2e-114:
		tmp = t_0 + (NdChar / (1.0 + math.exp((Ec / -KbT))))
	elif mu <= 1.2e-68:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	elif mu <= 1.66e+187:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	else:
		tmp = t_1 + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(2.0 + Float64(Ev / KbT)))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT))))
	tmp = 0.0
	if (mu <= -5.2e+71)
		tmp = Float64(t_1 + t_0);
	elseif (mu <= -9.2e-296)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0);
	elseif (mu <= 5.2e-114)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Ec / Float64(-KbT))))));
	elseif (mu <= 1.2e-68)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar * 0.5));
	elseif (mu <= 1.66e+187)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(t_1 + Float64(NaChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (2.0 + (Ev / KbT));
	t_1 = NdChar / (1.0 + exp((mu / KbT)));
	tmp = 0.0;
	if (mu <= -5.2e+71)
		tmp = t_1 + t_0;
	elseif (mu <= -9.2e-296)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	elseif (mu <= 5.2e-114)
		tmp = t_0 + (NdChar / (1.0 + exp((Ec / -KbT))));
	elseif (mu <= 1.2e-68)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	elseif (mu <= 1.66e+187)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	else
		tmp = t_1 + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -5.2e+71], N[(t$95$1 + t$95$0), $MachinePrecision], If[LessEqual[mu, -9.2e-296], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[mu, 5.2e-114], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Ec / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.2e-68], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.66e+187], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if mu < -5.19999999999999983e71

   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%

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

   5. Taylor expanded in Ev around inf 68.5%

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

   6. Taylor expanded in Ev around 0 63.5%

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

   7. Taylor expanded in mu around inf 49.0%

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

   if -5.19999999999999983e71 < mu < -9.20000000000000016e-296

   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%

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

   5. Taylor expanded in Ev around inf 74.7%

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

   6. Taylor expanded in Ev around 0 59.8%

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

   7. Taylor expanded in Vef around inf 53.4%

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

   if -9.20000000000000016e-296 < mu < 5.20000000000000026e-114

   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%

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

   5. Taylor expanded in Ev around inf 77.5%

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

   6. Taylor expanded in Ev around 0 59.6%

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

   7. Taylor expanded in Ec around inf 52.1%

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

      1. mul-1-neg 43.8%

         \[\leadsto \frac{1}{\frac{1 + e^{\color{blue}{-\frac{Ec}{KbT}}}}{NdChar}} + 0.5 \cdot NaChar \]

      2. distribute-neg-frac 43.8%

         \[\leadsto \frac{1}{\frac{1 + e^{\color{blue}{\frac{-Ec}{KbT}}}}{NdChar}} + 0.5 \cdot NaChar \]

   9. Simplified 52.1%

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

   if 5.20000000000000026e-114 < mu < 1.19999999999999996e-68

   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%

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

   5. Taylor expanded in KbT around inf 64.4%

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

   6. Taylor expanded in EDonor around inf 64.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor}}{KbT}}} + 0.5 \cdot NaChar \]

   if 1.19999999999999996e-68 < mu < 1.6600000000000001e187

   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%

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

   5. Taylor expanded in EAccept around inf 64.4%

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

   6. Taylor expanded in KbT around inf 41.9%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 14.0%

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

   8. Simplified 41.9%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]

   if 1.6600000000000001e187 < 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%

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

   5. Taylor expanded in KbT around inf 69.5%

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

   6. Taylor expanded in mu around inf 64.3%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu}}{KbT}}} + 0.5 \cdot NaChar \]
3. Recombined 6 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -5.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;mu \leq -9.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;mu \leq 5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} + \frac{NdChar}{1 + e^{\frac{Ec}{-KbT}}}\\ \mathbf{elif}\;mu \leq 1.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;mu \leq 1.66 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 62.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -4.8 \cdot 10^{-31} \lor \neg \left(NaChar \leq 0.5\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -4.8e-31) (not (<= NaChar 0.5)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
    (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))
   (-
    (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu)))))
    (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -4.8e-31) || !(NaChar <= 0.5)) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else {
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / 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) :: tmp
    if ((nachar <= (-4.8d-31)) .or. (.not. (nachar <= 0.5d0))) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    else
        tmp = (kbt * (nachar / (eaccept + (ev + (vef - mu))))) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / 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 tmp;
	if ((NaChar <= -4.8e-31) || !(NaChar <= 0.5)) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else {
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -4.8e-31) or not (NaChar <= 0.5):
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	else:
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -4.8e-31) || !(NaChar <= 0.5))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))));
	else
		tmp = Float64(Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -4.8e-31) || ~((NaChar <= 0.5)))
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	else
		tmp = (KbT * (NaChar / (EAccept + (Ev + (Vef - mu))))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -4.8e-31], N[Not[LessEqual[NaChar, 0.5]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -4.8e-31 or 0.5 < 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%

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

   5. Taylor expanded in mu around inf 77.7%

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

   6. Taylor expanded in mu around 0 66.1%

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

   if -4.8e-31 < NaChar < 0.5

   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%

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

   5. Taylor expanded in KbT around inf 74.9%

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

   6. Taylor expanded in KbT around 0 77.4%

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

      1. associate-/l* 79.0%

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

      2. associate--l+ 79.0%

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

      3. associate--l+ 79.0%

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

   8. Simplified 79.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -4.8 \cdot 10^{-31} \lor \neg \left(NaChar \leq 0.5\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 51.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{if}\;NaChar \leq -2.4 \cdot 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq 3.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{EAccept} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (* NdChar 0.5)
          (/
           1.0
           (/ (+ 1.0 (exp (/ (- Vef (- (- mu Ev) EAccept)) KbT))) NaChar)))))
   (if (<= NaChar -2.4e-30)
     t_0
     (if (<= NaChar 3.8e-145)
       (-
        (/ (* KbT NaChar) EAccept)
        (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
       (if (<= NaChar 0.5)
         (+
          (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
          (* KbT (/ NaChar (+ EAccept (+ Ev (- Vef mu))))))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	double tmp;
	if (NaChar <= -2.4e-30) {
		tmp = t_0;
	} else if (NaChar <= 3.8e-145) {
		tmp = ((KbT * NaChar) / EAccept) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	} else if (NaChar <= 0.5) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar * 0.5d0) + (1.0d0 / ((1.0d0 + exp(((vef - ((mu - ev) - eaccept)) / kbt))) / nachar))
    if (nachar <= (-2.4d-30)) then
        tmp = t_0
    else if (nachar <= 3.8d-145) then
        tmp = ((kbt * nachar) / eaccept) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt))))
    else if (nachar <= 0.5d0) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (kbt * (nachar / (eaccept + (ev + (vef - mu)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar * 0.5) + (1.0 / ((1.0 + Math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	double tmp;
	if (NaChar <= -2.4e-30) {
		tmp = t_0;
	} else if (NaChar <= 3.8e-145) {
		tmp = ((KbT * NaChar) / EAccept) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	} else if (NaChar <= 0.5) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar * 0.5) + (1.0 / ((1.0 + math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar))
	tmp = 0
	if NaChar <= -2.4e-30:
		tmp = t_0
	elif NaChar <= 3.8e-145:
		tmp = ((KbT * NaChar) / EAccept) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	elif NaChar <= 0.5:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar * 0.5) + Float64(1.0 / Float64(Float64(1.0 + exp(Float64(Float64(Vef - Float64(Float64(mu - Ev) - EAccept)) / KbT))) / NaChar)))
	tmp = 0.0
	if (NaChar <= -2.4e-30)
		tmp = t_0;
	elseif (NaChar <= 3.8e-145)
		tmp = Float64(Float64(Float64(KbT * NaChar) / EAccept) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))));
	elseif (NaChar <= 0.5)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(KbT * Float64(NaChar / Float64(EAccept + Float64(Ev + Float64(Vef - mu))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	tmp = 0.0;
	if (NaChar <= -2.4e-30)
		tmp = t_0;
	elseif (NaChar <= 3.8e-145)
		tmp = ((KbT * NaChar) / EAccept) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	elseif (NaChar <= 0.5)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (KbT * (NaChar / (EAccept + (Ev + (Vef - mu)))));
	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[(NdChar * 0.5), $MachinePrecision] + N[(1.0 / N[(N[(1.0 + N[Exp[N[(N[(Vef - N[(N[(mu - Ev), $MachinePrecision] - EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -2.4e-30], t$95$0, If[LessEqual[NaChar, 3.8e-145], N[(N[(N[(KbT * NaChar), $MachinePrecision] / EAccept), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 0.5], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(KbT * N[(NaChar / N[(EAccept + N[(Ev + N[(Vef - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -2.39999999999999985e-30 or 0.5 < 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%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+l- 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in KbT around inf 62.3%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{1}{\frac{1 + e^{\frac{Vef + \left(EAccept - \left(mu - Ev\right)\right)}{KbT}}}{NaChar}} \]
   10. Step-by-step derivation

       1. *-commutative 10.2%

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

   11. Simplified 62.3%

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

   if -2.39999999999999985e-30 < NaChar < 3.8000000000000002e-145

   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%

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

   5. Taylor expanded in KbT around inf 74.4%

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

   6. Taylor expanded in EAccept around inf 66.5%

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

   if 3.8000000000000002e-145 < NaChar < 0.5

   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%

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

   5. Taylor expanded in KbT around inf 76.2%

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

   6. Taylor expanded in KbT around 0 66.3%

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

      1. associate-/l* 69.3%

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

      2. associate--l+ 69.3%

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

      3. associate--l+ 69.3%

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

   8. Simplified 69.3%

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

   9. Taylor expanded in Vef around inf 57.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.4 \cdot 10^{-30}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{elif}\;NaChar \leq 3.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{EAccept} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 0.5:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 62.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -1.06 \cdot 10^{-29} \lor \neg \left(NaChar \leq 0.5\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -1.06e-29) (not (<= NaChar 0.5)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
    (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))
   (-
    (/ NaChar (+ 2.0 (/ Ev KbT)))
    (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -1.06e-29) || !(NaChar <= 0.5)) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else {
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / 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) :: tmp
    if ((nachar <= (-1.06d-29)) .or. (.not. (nachar <= 0.5d0))) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    else
        tmp = (nachar / (2.0d0 + (ev / kbt))) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / 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 tmp;
	if ((NaChar <= -1.06e-29) || !(NaChar <= 0.5)) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else {
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -1.06e-29) or not (NaChar <= 0.5):
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	else:
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -1.06e-29) || !(NaChar <= 0.5))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))));
	else
		tmp = Float64(Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -1.06e-29) || ~((NaChar <= 0.5)))
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	else
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -1.06e-29], N[Not[LessEqual[NaChar, 0.5]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -1.05999999999999995e-29 or 0.5 < 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%

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

   5. Taylor expanded in mu around inf 77.7%

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

   6. Taylor expanded in mu around 0 66.1%

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

   if -1.05999999999999995e-29 < NaChar < 0.5

   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%

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

   5. Taylor expanded in Ev around inf 77.3%

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

   6. Taylor expanded in Ev around 0 70.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.06 \cdot 10^{-29} \lor \neg \left(NaChar \leq 0.5\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 39.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ t_1 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{if}\;NaChar \leq -8.2 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq -2.8 \cdot 10^{-276}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq -5.2 \cdot 10^{-295}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \mathbf{elif}\;NaChar \leq 8.6 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (* NaChar 0.5)))
        (t_1 (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))))
   (if (<= NaChar -8.2e-22)
     t_1
     (if (<= NaChar -2.8e-276)
       t_0
       (if (<= NaChar -5.2e-295)
         (/ (* KbT NaChar) (- (+ EAccept (+ Vef Ev)) mu))
         (if (<= NaChar 8.6e-123) t_0 t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	double t_1 = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	double tmp;
	if (NaChar <= -8.2e-22) {
		tmp = t_1;
	} else if (NaChar <= -2.8e-276) {
		tmp = t_0;
	} else if (NaChar <= -5.2e-295) {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	} else if (NaChar <= 8.6e-123) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar * 0.5d0)
    t_1 = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    if (nachar <= (-8.2d-22)) then
        tmp = t_1
    else if (nachar <= (-2.8d-276)) then
        tmp = t_0
    else if (nachar <= (-5.2d-295)) then
        tmp = (kbt * nachar) / ((eaccept + (vef + ev)) - mu)
    else if (nachar <= 8.6d-123) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	double t_1 = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	double tmp;
	if (NaChar <= -8.2e-22) {
		tmp = t_1;
	} else if (NaChar <= -2.8e-276) {
		tmp = t_0;
	} else if (NaChar <= -5.2e-295) {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	} else if (NaChar <= 8.6e-123) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	t_1 = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	tmp = 0
	if NaChar <= -8.2e-22:
		tmp = t_1
	elif NaChar <= -2.8e-276:
		tmp = t_0
	elif NaChar <= -5.2e-295:
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu)
	elif NaChar <= 8.6e-123:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar * 0.5))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5))
	tmp = 0.0
	if (NaChar <= -8.2e-22)
		tmp = t_1;
	elseif (NaChar <= -2.8e-276)
		tmp = t_0;
	elseif (NaChar <= -5.2e-295)
		tmp = Float64(Float64(KbT * NaChar) / Float64(Float64(EAccept + Float64(Vef + Ev)) - mu));
	elseif (NaChar <= 8.6e-123)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	t_1 = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	tmp = 0.0;
	if (NaChar <= -8.2e-22)
		tmp = t_1;
	elseif (NaChar <= -2.8e-276)
		tmp = t_0;
	elseif (NaChar <= -5.2e-295)
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	elseif (NaChar <= 8.6e-123)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -8.2e-22], t$95$1, If[LessEqual[NaChar, -2.8e-276], t$95$0, If[LessEqual[NaChar, -5.2e-295], N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 8.6e-123], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -8.1999999999999999e-22 or 8.60000000000000064e-123 < 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%

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

   5. Taylor expanded in EAccept around inf 70.2%

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

   6. Taylor expanded in KbT around inf 46.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 13.5%

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

   8. Simplified 46.1%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]

   if -8.1999999999999999e-22 < NaChar < -2.79999999999999986e-276 or -5.1999999999999997e-295 < NaChar < 8.60000000000000064e-123

   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%

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

   5. Taylor expanded in KbT around inf 62.4%

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

   6. Taylor expanded in EDonor around inf 42.3%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor}}{KbT}}} + 0.5 \cdot NaChar \]

   if -2.79999999999999986e-276 < NaChar < -5.1999999999999997e-295

   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%

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

   5. Taylor expanded in KbT around inf 100.0%

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

   6. Taylor expanded in KbT around 0 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

      3. associate--l+ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in KbT around inf 5.4%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 5.4%

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

   11. Simplified 5.4%

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

   12. Taylor expanded in NdChar around 0 86.0%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -8.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq -2.8 \cdot 10^{-276}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq -5.2 \cdot 10^{-295}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \mathbf{elif}\;NaChar \leq 8.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 39.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{if}\;NaChar \leq -1.62 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NaChar \leq -2.8 \cdot 10^{-276}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq -1.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \mathbf{elif}\;NaChar \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))))
   (if (<= NaChar -1.62e-24)
     t_0
     (if (<= NaChar -2.8e-276)
       (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (* NaChar 0.5))
       (if (<= NaChar -1.5e-294)
         (/ (* KbT NaChar) (- (+ EAccept (+ Vef Ev)) mu))
         (if (<= NaChar 2.4e+23)
           (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) (* NaChar 0.5))
           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((EAccept / KbT)))) + (NdChar * 0.5);
	double tmp;
	if (NaChar <= -1.62e-24) {
		tmp = t_0;
	} else if (NaChar <= -2.8e-276) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else if (NaChar <= -1.5e-294) {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	} else if (NaChar <= 2.4e+23) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar * 0.5);
	} 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((eaccept / kbt)))) + (ndchar * 0.5d0)
    if (nachar <= (-1.62d-24)) then
        tmp = t_0
    else if (nachar <= (-2.8d-276)) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar * 0.5d0)
    else if (nachar <= (-1.5d-294)) then
        tmp = (kbt * nachar) / ((eaccept + (vef + ev)) - mu)
    else if (nachar <= 2.4d+23) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar * 0.5d0)
    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((EAccept / KbT)))) + (NdChar * 0.5);
	double tmp;
	if (NaChar <= -1.62e-24) {
		tmp = t_0;
	} else if (NaChar <= -2.8e-276) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else if (NaChar <= -1.5e-294) {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	} else if (NaChar <= 2.4e+23) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar * 0.5);
	} 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((EAccept / KbT)))) + (NdChar * 0.5)
	tmp = 0
	if NaChar <= -1.62e-24:
		tmp = t_0
	elif NaChar <= -2.8e-276:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	elif NaChar <= -1.5e-294:
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu)
	elif NaChar <= 2.4e+23:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar * 0.5)
	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(EAccept / KbT)))) + Float64(NdChar * 0.5))
	tmp = 0.0
	if (NaChar <= -1.62e-24)
		tmp = t_0;
	elseif (NaChar <= -2.8e-276)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar * 0.5));
	elseif (NaChar <= -1.5e-294)
		tmp = Float64(Float64(KbT * NaChar) / Float64(Float64(EAccept + Float64(Vef + Ev)) - mu));
	elseif (NaChar <= 2.4e+23)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar * 0.5));
	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((EAccept / KbT)))) + (NdChar * 0.5);
	tmp = 0.0;
	if (NaChar <= -1.62e-24)
		tmp = t_0;
	elseif (NaChar <= -2.8e-276)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	elseif (NaChar <= -1.5e-294)
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	elseif (NaChar <= 2.4e+23)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar * 0.5);
	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[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -1.62e-24], t$95$0, If[LessEqual[NaChar, -2.8e-276], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, -1.5e-294], N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 2.4e+23], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -1.62e-24 or 2.4e23 < 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%

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

   5. Taylor expanded in EAccept around inf 70.1%

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

   6. Taylor expanded in KbT around inf 47.7%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 9.9%

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

   8. Simplified 47.7%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]

   if -1.62e-24 < NaChar < -2.79999999999999986e-276

   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%

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

   5. Taylor expanded in KbT around inf 58.1%

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

   6. Taylor expanded in EDonor around inf 43.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{EDonor}}{KbT}}} + 0.5 \cdot NaChar \]

   if -2.79999999999999986e-276 < NaChar < -1.4999999999999999e-294

   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%

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

   5. Taylor expanded in KbT around inf 100.0%

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

   6. Taylor expanded in KbT around 0 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

      3. associate--l+ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in KbT around inf 5.4%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 5.4%

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

   11. Simplified 5.4%

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

   12. Taylor expanded in NdChar around 0 86.0%

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

   if -1.4999999999999999e-294 < NaChar < 2.4e23

   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%

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

   5. Taylor expanded in KbT around inf 61.4%

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

   6. Taylor expanded in Vef around inf 44.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{Vef}}{KbT}}} + 0.5 \cdot NaChar \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.62 \cdot 10^{-24}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq -2.8 \cdot 10^{-276}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq -1.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \mathbf{elif}\;NaChar \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 61.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -2.8 \cdot 10^{+55} \lor \neg \left(NaChar \leq 0.5\right):\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -2.8e+55) (not (<= NaChar 0.5)))
   (+
    (* NdChar 0.5)
    (/ 1.0 (/ (+ 1.0 (exp (/ (- Vef (- (- mu Ev) EAccept)) KbT))) NaChar)))
   (-
    (/ NaChar (+ 2.0 (/ Ev KbT)))
    (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -2.8e+55) || !(NaChar <= 0.5)) {
		tmp = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	} else {
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / 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) :: tmp
    if ((nachar <= (-2.8d+55)) .or. (.not. (nachar <= 0.5d0))) then
        tmp = (ndchar * 0.5d0) + (1.0d0 / ((1.0d0 + exp(((vef - ((mu - ev) - eaccept)) / kbt))) / nachar))
    else
        tmp = (nachar / (2.0d0 + (ev / kbt))) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / 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 tmp;
	if ((NaChar <= -2.8e+55) || !(NaChar <= 0.5)) {
		tmp = (NdChar * 0.5) + (1.0 / ((1.0 + Math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	} else {
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -2.8e+55) or not (NaChar <= 0.5):
		tmp = (NdChar * 0.5) + (1.0 / ((1.0 + math.exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar))
	else:
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -2.8e+55) || !(NaChar <= 0.5))
		tmp = Float64(Float64(NdChar * 0.5) + Float64(1.0 / Float64(Float64(1.0 + exp(Float64(Float64(Vef - Float64(Float64(mu - Ev) - EAccept)) / KbT))) / NaChar)));
	else
		tmp = Float64(Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -2.8e+55) || ~((NaChar <= 0.5)))
		tmp = (NdChar * 0.5) + (1.0 / ((1.0 + exp(((Vef - ((mu - Ev) - EAccept)) / KbT))) / NaChar));
	else
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -2.8e+55], N[Not[LessEqual[NaChar, 0.5]], $MachinePrecision]], N[(N[(NdChar * 0.5), $MachinePrecision] + N[(1.0 / N[(N[(1.0 + N[Exp[N[(N[(Vef - N[(N[(mu - Ev), $MachinePrecision] - EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / NaChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -2.8000000000000001e55 or 0.5 < 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%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+l- 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in KbT around inf 67.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{1}{\frac{1 + e^{\frac{Vef + \left(EAccept - \left(mu - Ev\right)\right)}{KbT}}}{NaChar}} \]
   10. Step-by-step derivation

       1. *-commutative 10.0%

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

   11. Simplified 67.1%

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

   if -2.8000000000000001e55 < NaChar < 0.5

   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%

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

   5. Taylor expanded in Ev around inf 75.5%

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

   6. Taylor expanded in Ev around 0 66.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.8 \cdot 10^{+55} \lor \neg \left(NaChar \leq 0.5\right):\\ \;\;\;\;NdChar \cdot 0.5 + \frac{1}{\frac{1 + e^{\frac{Vef - \left(\left(mu - Ev\right) - EAccept\right)}{KbT}}}{NaChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 51.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.6 \cdot 10^{-160} \lor \neg \left(NdChar \leq 1.8 \cdot 10^{-69}\right):\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -2.6e-160) (not (<= NdChar 1.8e-69)))
   (-
    (* NaChar 0.5)
    (/ NdChar (- -1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar (+ 2.0 (/ EDonor KbT))))))

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 <= -2.6e-160) || !(NdChar <= 1.8e-69)) {
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	} else {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / 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) :: tmp
    if ((ndchar <= (-2.6d-160)) .or. (.not. (ndchar <= 1.8d-69))) then
        tmp = (nachar * 0.5d0) - (ndchar / ((-1.0d0) - exp(((edonor + (mu + (vef - ec))) / kbt))))
    else
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (2.0d0 + (edonor / 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 tmp;
	if ((NdChar <= -2.6e-160) || !(NdChar <= 1.8e-69)) {
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -2.6e-160) or not (NdChar <= 1.8e-69):
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - math.exp(((EDonor + (mu + (Vef - Ec))) / KbT))))
	else:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -2.6e-160) || !(NdChar <= 1.8e-69))
		tmp = Float64(Float64(NaChar * 0.5) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(2.0 + Float64(EDonor / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -2.6e-160) || ~((NdChar <= 1.8e-69)))
		tmp = (NaChar * 0.5) - (NdChar / (-1.0 - exp(((EDonor + (mu + (Vef - Ec))) / KbT))));
	else
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -2.6e-160], N[Not[LessEqual[NdChar, 1.8e-69]], $MachinePrecision]], N[(N[(NaChar * 0.5), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.60000000000000003e-160 or 1.80000000000000009e-69 < NdChar

   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%

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

   5. Taylor expanded in KbT around inf 60.7%

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

   if -2.60000000000000003e-160 < NdChar < 1.80000000000000009e-69

   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%

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

   5. Taylor expanded in Ev around inf 57.3%

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

   6. Taylor expanded in EDonor around inf 50.1%

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

   7. Taylor expanded in EDonor around 0 44.3%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.6 \cdot 10^{-160} \lor \neg \left(NdChar \leq 1.8 \cdot 10^{-69}\right):\\ \;\;\;\;NaChar \cdot 0.5 - \frac{NdChar}{-1 - e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 39.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{if}\;mu \leq -4.8 \cdot 10^{+73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq -4.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;mu \leq 1.1 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* NaChar 0.5))))
   (if (<= mu -4.8e+73)
     t_0
     (if (<= mu -4.1e-57)
       (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) (* NaChar 0.5))
       (if (<= mu 1.1e+187)
         (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (mu <= -4.8e+73) {
		tmp = t_0;
	} else if (mu <= -4.1e-57) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar * 0.5);
	} else if (mu <= 1.1e+187) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar * 0.5d0)
    if (mu <= (-4.8d+73)) then
        tmp = t_0
    else if (mu <= (-4.1d-57)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar * 0.5d0)
    else if (mu <= 1.1d+187) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (mu <= -4.8e+73) {
		tmp = t_0;
	} else if (mu <= -4.1e-57) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar * 0.5);
	} else if (mu <= 1.1e+187) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar * 0.5)
	tmp = 0
	if mu <= -4.8e+73:
		tmp = t_0
	elif mu <= -4.1e-57:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar * 0.5)
	elif mu <= 1.1e+187:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	else:
		tmp = t_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(mu / KbT)))) + Float64(NaChar * 0.5))
	tmp = 0.0
	if (mu <= -4.8e+73)
		tmp = t_0;
	elseif (mu <= -4.1e-57)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar * 0.5));
	elseif (mu <= 1.1e+187)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	tmp = 0.0;
	if (mu <= -4.8e+73)
		tmp = t_0;
	elseif (mu <= -4.1e-57)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar * 0.5);
	elseif (mu <= 1.1e+187)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	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[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -4.8e+73], t$95$0, If[LessEqual[mu, -4.1e-57], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.1e+187], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -4.80000000000000004e73 or 1.0999999999999999e187 < 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%

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

   5. Taylor expanded in KbT around inf 62.0%

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

   6. Taylor expanded in mu around inf 52.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu}}{KbT}}} + 0.5 \cdot NaChar \]

   if -4.80000000000000004e73 < mu < -4.1000000000000001e-57

   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%

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

   5. Taylor expanded in KbT around inf 51.0%

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

   6. Taylor expanded in Vef around inf 47.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{Vef}}{KbT}}} + 0.5 \cdot NaChar \]

   if -4.1000000000000001e-57 < mu < 1.0999999999999999e187

   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%

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

   5. Taylor expanded in EAccept around inf 72.0%

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

   6. Taylor expanded in KbT around inf 43.0%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 14.8%

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

   8. Simplified 43.0%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -4.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;mu \leq -4.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;mu \leq 1.1 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 40.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{if}\;mu \leq -3.25 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq 2.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.1 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* NaChar 0.5))))
   (if (<= mu -3.25e+112)
     t_0
     (if (<= mu 2.2e-176)
       (-
        (/ NaChar (+ 2.0 (/ Ev KbT)))
        (/ NdChar (- -1.0 (exp (/ EDonor KbT)))))
       (if (<= mu 1.1e+187)
         (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (mu <= -3.25e+112) {
		tmp = t_0;
	} else if (mu <= 2.2e-176) {
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - exp((EDonor / KbT))));
	} else if (mu <= 1.1e+187) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar * 0.5d0)
    if (mu <= (-3.25d+112)) then
        tmp = t_0
    else if (mu <= 2.2d-176) then
        tmp = (nachar / (2.0d0 + (ev / kbt))) - (ndchar / ((-1.0d0) - exp((edonor / kbt))))
    else if (mu <= 1.1d+187) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (mu <= -3.25e+112) {
		tmp = t_0;
	} else if (mu <= 2.2e-176) {
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - Math.exp((EDonor / KbT))));
	} else if (mu <= 1.1e+187) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar * 0.5)
	tmp = 0
	if mu <= -3.25e+112:
		tmp = t_0
	elif mu <= 2.2e-176:
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - math.exp((EDonor / KbT))))
	elif mu <= 1.1e+187:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	else:
		tmp = t_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(mu / KbT)))) + Float64(NaChar * 0.5))
	tmp = 0.0
	if (mu <= -3.25e+112)
		tmp = t_0;
	elseif (mu <= 2.2e-176)
		tmp = Float64(Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))) - Float64(NdChar / Float64(-1.0 - exp(Float64(EDonor / KbT)))));
	elseif (mu <= 1.1e+187)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	tmp = 0.0;
	if (mu <= -3.25e+112)
		tmp = t_0;
	elseif (mu <= 2.2e-176)
		tmp = (NaChar / (2.0 + (Ev / KbT))) - (NdChar / (-1.0 - exp((EDonor / KbT))));
	elseif (mu <= 1.1e+187)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	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[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -3.25e+112], t$95$0, If[LessEqual[mu, 2.2e-176], N[(N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.1e+187], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -3.2499999999999999e112 or 1.0999999999999999e187 < 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%

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

   5. Taylor expanded in KbT around inf 64.0%

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

   6. Taylor expanded in mu around inf 56.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu}}{KbT}}} + 0.5 \cdot NaChar \]

   if -3.2499999999999999e112 < mu < 2.1999999999999999e-176

   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%

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

   5. Taylor expanded in Ev around inf 76.3%

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

   6. Taylor expanded in Ev around 0 61.9%

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

   7. Taylor expanded in EDonor around inf 50.9%

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

   if 2.1999999999999999e-176 < mu < 1.0999999999999999e187

   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%

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

   5. Taylor expanded in EAccept around inf 64.8%

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

   6. Taylor expanded in KbT around inf 42.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 13.7%

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

   8. Simplified 42.1%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -3.25 \cdot 10^{+112}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;mu \leq 2.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.1 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 39.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{if}\;mu \leq -3.5 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq -1.85 \cdot 10^{-301}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;mu \leq 1.12 \cdot 10^{+188}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* NaChar 0.5))))
   (if (<= mu -3.5e+112)
     t_0
     (if (<= mu -1.85e-301)
       (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) (/ NaChar (+ 2.0 (/ Ev KbT))))
       (if (<= mu 1.12e+188)
         (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (mu <= -3.5e+112) {
		tmp = t_0;
	} else if (mu <= -1.85e-301) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (2.0 + (Ev / KbT)));
	} else if (mu <= 1.12e+188) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar * 0.5d0)
    if (mu <= (-3.5d+112)) then
        tmp = t_0
    else if (mu <= (-1.85d-301)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / (2.0d0 + (ev / kbt)))
    else if (mu <= 1.12d+188) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (mu <= -3.5e+112) {
		tmp = t_0;
	} else if (mu <= -1.85e-301) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / (2.0 + (Ev / KbT)));
	} else if (mu <= 1.12e+188) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar * 0.5)
	tmp = 0
	if mu <= -3.5e+112:
		tmp = t_0
	elif mu <= -1.85e-301:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / (2.0 + (Ev / KbT)))
	elif mu <= 1.12e+188:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	else:
		tmp = t_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(mu / KbT)))) + Float64(NaChar * 0.5))
	tmp = 0.0
	if (mu <= -3.5e+112)
		tmp = t_0;
	elseif (mu <= -1.85e-301)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))));
	elseif (mu <= 1.12e+188)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	tmp = 0.0;
	if (mu <= -3.5e+112)
		tmp = t_0;
	elseif (mu <= -1.85e-301)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (2.0 + (Ev / KbT)));
	elseif (mu <= 1.12e+188)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	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[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -3.5e+112], t$95$0, If[LessEqual[mu, -1.85e-301], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.12e+188], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -3.49999999999999997e112 or 1.11999999999999996e188 < 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%

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

   5. Taylor expanded in KbT around inf 64.0%

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

   6. Taylor expanded in mu around inf 56.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu}}{KbT}}} + 0.5 \cdot NaChar \]

   if -3.49999999999999997e112 < mu < -1.8499999999999999e-301

   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%

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

   5. Taylor expanded in Ev around inf 74.2%

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

   6. Taylor expanded in Ev around 0 60.7%

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

   7. Taylor expanded in Vef around inf 51.9%

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

   if -1.8499999999999999e-301 < mu < 1.11999999999999996e188

   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%

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

   5. Taylor expanded in EAccept around inf 68.2%

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

   6. Taylor expanded in KbT around inf 42.7%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 13.9%

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

   8. Simplified 42.7%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -3.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;mu \leq -1.85 \cdot 10^{-301}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;mu \leq 1.12 \cdot 10^{+188}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 39.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -1.6 \cdot 10^{+71}:\\ \;\;\;\;t\_1 + t\_0\\ \mathbf{elif}\;mu \leq -4 \cdot 10^{-301}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \mathbf{elif}\;mu \leq 3.9 \cdot 10^{+189}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 2.0 (/ Ev KbT))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ mu KbT))))))
   (if (<= mu -1.6e+71)
     (+ t_1 t_0)
     (if (<= mu -4e-301)
       (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) t_0)
       (if (<= mu 3.9e+189)
         (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))
         (+ t_1 (* 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 t_0 = NaChar / (2.0 + (Ev / KbT));
	double t_1 = NdChar / (1.0 + exp((mu / KbT)));
	double tmp;
	if (mu <= -1.6e+71) {
		tmp = t_1 + t_0;
	} else if (mu <= -4e-301) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	} else if (mu <= 3.9e+189) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_1 + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (2.0d0 + (ev / kbt))
    t_1 = ndchar / (1.0d0 + exp((mu / kbt)))
    if (mu <= (-1.6d+71)) then
        tmp = t_1 + t_0
    else if (mu <= (-4d-301)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + t_0
    else if (mu <= 3.9d+189) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = t_1 + (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 t_0 = NaChar / (2.0 + (Ev / KbT));
	double t_1 = NdChar / (1.0 + Math.exp((mu / KbT)));
	double tmp;
	if (mu <= -1.6e+71) {
		tmp = t_1 + t_0;
	} else if (mu <= -4e-301) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	} else if (mu <= 3.9e+189) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = t_1 + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (2.0 + (Ev / KbT))
	t_1 = NdChar / (1.0 + math.exp((mu / KbT)))
	tmp = 0
	if mu <= -1.6e+71:
		tmp = t_1 + t_0
	elif mu <= -4e-301:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	elif mu <= 3.9e+189:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	else:
		tmp = t_1 + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(2.0 + Float64(Ev / KbT)))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT))))
	tmp = 0.0
	if (mu <= -1.6e+71)
		tmp = Float64(t_1 + t_0);
	elseif (mu <= -4e-301)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0);
	elseif (mu <= 3.9e+189)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(t_1 + Float64(NaChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (2.0 + (Ev / KbT));
	t_1 = NdChar / (1.0 + exp((mu / KbT)));
	tmp = 0.0;
	if (mu <= -1.6e+71)
		tmp = t_1 + t_0;
	elseif (mu <= -4e-301)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	elseif (mu <= 3.9e+189)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	else
		tmp = t_1 + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -1.6e+71], N[(t$95$1 + t$95$0), $MachinePrecision], If[LessEqual[mu, -4e-301], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[mu, 3.9e+189], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -1.60000000000000012e71

   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%

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

   5. Taylor expanded in Ev around inf 68.5%

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

   6. Taylor expanded in Ev around 0 63.5%

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

   7. Taylor expanded in mu around inf 49.0%

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

   if -1.60000000000000012e71 < mu < -4.00000000000000027e-301

   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%

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

   5. Taylor expanded in Ev around inf 74.2%

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

   6. Taylor expanded in Ev around 0 59.6%

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

   7. Taylor expanded in Vef around inf 53.4%

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

   if -4.00000000000000027e-301 < mu < 3.9e189

   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%

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

   5. Taylor expanded in EAccept around inf 68.2%

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

   6. Taylor expanded in KbT around inf 42.7%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 13.9%

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

   8. Simplified 42.7%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]

   if 3.9e189 < 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%

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

   5. Taylor expanded in KbT around inf 69.5%

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

   6. Taylor expanded in mu around inf 64.3%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu}}{KbT}}} + 0.5 \cdot NaChar \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;mu \leq -4 \cdot 10^{-301}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2 + \frac{Ev}{KbT}}\\ \mathbf{elif}\;mu \leq 3.9 \cdot 10^{+189}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 32: 33.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -5.7 \cdot 10^{-200} \lor \neg \left(NaChar \leq 1.65 \cdot 10^{-142}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -5.7e-200) (not (<= NaChar 1.65e-142)))
   (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))
   (/ (* KbT NaChar) (- (+ EAccept (+ Vef Ev)) mu))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -5.7e-200) || !(NaChar <= 1.65e-142)) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	}
	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 ((nachar <= (-5.7d-200)) .or. (.not. (nachar <= 1.65d-142))) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = (kbt * nachar) / ((eaccept + (vef + ev)) - mu)
    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 ((NaChar <= -5.7e-200) || !(NaChar <= 1.65e-142)) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -5.7e-200) or not (NaChar <= 1.65e-142):
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	else:
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -5.7e-200) || !(NaChar <= 1.65e-142))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(Float64(KbT * NaChar) / Float64(Float64(EAccept + Float64(Vef + Ev)) - mu));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -5.7e-200) || ~((NaChar <= 1.65e-142)))
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	else
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -5.7e-200], N[Not[LessEqual[NaChar, 1.65e-142]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -5.69999999999999949e-200 or 1.6499999999999998e-142 < 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%

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

   5. Taylor expanded in EAccept around inf 71.2%

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

   6. Taylor expanded in KbT around inf 44.0%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 14.8%

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

   8. Simplified 44.0%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]

   if -5.69999999999999949e-200 < NaChar < 1.6499999999999998e-142

   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%

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

   5. Taylor expanded in KbT around inf 78.1%

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

   6. Taylor expanded in KbT around 0 84.8%

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

      1. associate-/l* 86.3%

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

      2. associate--l+ 86.3%

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

      3. associate--l+ 86.3%

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

   8. Simplified 86.3%

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

   9. Taylor expanded in KbT around inf 18.8%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in NdChar around 0 34.6%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -5.7 \cdot 10^{-200} \lor \neg \left(NaChar \leq 1.65 \cdot 10^{-142}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \end{array} \]
5. Add Preprocessing

Alternative 33: 37.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -5.4 \cdot 10^{+172}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -5.4e+172)
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (* NdChar 0.5))
   (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 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 <= -5.4e+172) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 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 <= (-5.4d+172)) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 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 <= -5.4e+172) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -5.4e+172:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar * 0.5)
	else:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -5.4e+172)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 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 <= -5.4e+172)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar * 0.5);
	else
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -5.4e+172], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Ev < -5.4e172

   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%

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

   5. Taylor expanded in Ev around inf 93.3%

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

   6. Taylor expanded in KbT around inf 53.5%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 25.4%

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

   8. Simplified 53.5%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} \]

   if -5.4e172 < Ev

   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%

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

   5. Taylor expanded in EAccept around inf 72.5%

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

   6. Taylor expanded in KbT around inf 37.6%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} \]
   7. Step-by-step derivation

      1. *-commutative 14.5%

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

   8. Simplified 37.6%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -5.4 \cdot 10^{+172}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 34: 29.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4 \cdot 10^{+46} \lor \neg \left(KbT \leq 4.1 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= KbT -4e+46) (not (<= KbT 4.1e+38)))
   (+
    (/
     NaChar
     (+
      1.0
      (- (+ 1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT)))) (/ mu KbT))))
    (* NdChar 0.5))
   (/ (* KbT NaChar) (- (+ EAccept (+ Vef Ev)) mu))))

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 <= -4e+46) || !(KbT <= 4.1e+38)) {
		tmp = (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	}
	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 <= (-4d+46)) .or. (.not. (kbt <= 4.1d+38))) then
        tmp = (nachar / (1.0d0 + ((1.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = (kbt * nachar) / ((eaccept + (vef + ev)) - mu)
    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 <= -4e+46) || !(KbT <= 4.1e+38)) {
		tmp = (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (KbT <= -4e+46) or not (KbT <= 4.1e+38):
		tmp = (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)))) + (NdChar * 0.5)
	else:
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((KbT <= -4e+46) || !(KbT <= 4.1e+38))
		tmp = Float64(Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(Float64(KbT * NaChar) / Float64(Float64(EAccept + Float64(Vef + Ev)) - mu));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((KbT <= -4e+46) || ~((KbT <= 4.1e+38)))
		tmp = (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)))) + (NdChar * 0.5);
	else
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[KbT, -4e+46], N[Not[LessEqual[KbT, 4.1e+38]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -4e46 or 4.1000000000000003e38 < 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%

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

   5. Taylor expanded in KbT around inf 71.7%

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

   6. Taylor expanded in KbT around inf 55.3%

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

      1. *-commutative 25.4%

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

   8. Simplified 55.3%

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

   if -4e46 < KbT < 4.1000000000000003e38

   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%

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

   5. Taylor expanded in KbT around inf 49.4%

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

   6. Taylor expanded in KbT around 0 56.0%

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

      1. associate-/l* 56.0%

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

      2. associate--l+ 56.0%

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

      3. associate--l+ 56.0%

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

   8. Simplified 56.0%

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

   9. Taylor expanded in KbT around inf 9.6%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 9.6%

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

   11. Simplified 9.6%

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

   12. Taylor expanded in NdChar around 0 22.8%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4 \cdot 10^{+46} \lor \neg \left(KbT \leq 4.1 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \end{array} \]
5. Add Preprocessing

Alternative 35: 29.3% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 4.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{1}{\frac{Ec}{NdChar \cdot KbT} - \frac{2}{NdChar}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4.4e+46)
   (+ (/ NaChar (+ 2.0 (/ Ev KbT))) (* NdChar 0.5))
   (if (<= KbT 4.1e+38)
     (/ (* KbT NaChar) (- (+ EAccept (+ Vef Ev)) mu))
     (- (* NaChar 0.5) (/ 1.0 (- (/ Ec (* NdChar KbT)) (/ 2.0 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 (KbT <= -4.4e+46) {
		tmp = (NaChar / (2.0 + (Ev / KbT))) + (NdChar * 0.5);
	} else if (KbT <= 4.1e+38) {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	} else {
		tmp = (NaChar * 0.5) - (1.0 / ((Ec / (NdChar * KbT)) - (2.0 / 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 (kbt <= (-4.4d+46)) then
        tmp = (nachar / (2.0d0 + (ev / kbt))) + (ndchar * 0.5d0)
    else if (kbt <= 4.1d+38) then
        tmp = (kbt * nachar) / ((eaccept + (vef + ev)) - mu)
    else
        tmp = (nachar * 0.5d0) - (1.0d0 / ((ec / (ndchar * kbt)) - (2.0d0 / 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 (KbT <= -4.4e+46) {
		tmp = (NaChar / (2.0 + (Ev / KbT))) + (NdChar * 0.5);
	} else if (KbT <= 4.1e+38) {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	} else {
		tmp = (NaChar * 0.5) - (1.0 / ((Ec / (NdChar * KbT)) - (2.0 / NdChar)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -4.4e+46:
		tmp = (NaChar / (2.0 + (Ev / KbT))) + (NdChar * 0.5)
	elif KbT <= 4.1e+38:
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu)
	else:
		tmp = (NaChar * 0.5) - (1.0 / ((Ec / (NdChar * KbT)) - (2.0 / NdChar)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4.4e+46)
		tmp = Float64(Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))) + Float64(NdChar * 0.5));
	elseif (KbT <= 4.1e+38)
		tmp = Float64(Float64(KbT * NaChar) / Float64(Float64(EAccept + Float64(Vef + Ev)) - mu));
	else
		tmp = Float64(Float64(NaChar * 0.5) - Float64(1.0 / Float64(Float64(Ec / Float64(NdChar * KbT)) - Float64(2.0 / NdChar))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -4.4e+46)
		tmp = (NaChar / (2.0 + (Ev / KbT))) + (NdChar * 0.5);
	elseif (KbT <= 4.1e+38)
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	else
		tmp = (NaChar * 0.5) - (1.0 / ((Ec / (NdChar * KbT)) - (2.0 / NdChar)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4.4e+46], N[(N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4.1e+38], N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar * 0.5), $MachinePrecision] - N[(1.0 / N[(N[(Ec / N[(NdChar * KbT), $MachinePrecision]), $MachinePrecision] - N[(2.0 / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.4000000000000001e46

   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%

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

   5. Taylor expanded in Ev around inf 76.9%

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

   6. Taylor expanded in Ev around 0 65.4%

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

   7. Taylor expanded in KbT around inf 51.9%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{2 + \frac{Ev}{KbT}} \]
   8. Step-by-step derivation

      1. *-commutative 21.4%

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

   9. Simplified 51.9%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{2 + \frac{Ev}{KbT}} \]

   if -4.4000000000000001e46 < KbT < 4.1000000000000003e38

   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%

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

   5. Taylor expanded in KbT around inf 49.4%

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

   6. Taylor expanded in KbT around 0 56.0%

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

      1. associate-/l* 56.0%

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

      2. associate--l+ 56.0%

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

      3. associate--l+ 56.0%

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

   8. Simplified 56.0%

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

   9. Taylor expanded in KbT around inf 9.6%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 9.6%

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

   11. Simplified 9.6%

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

   12. Taylor expanded in NdChar around 0 22.8%

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

   if 4.1000000000000003e38 < 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%

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

   5. Taylor expanded in KbT around inf 73.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \color{blue}{0.5 \cdot NaChar} \]
   6. Step-by-step derivation

      1. clear-num 73.2%

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

      2. inv-pow 73.2%

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

      3. associate-+r+ 73.2%

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

   7. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{{\left(\frac{1 + e^{\frac{\left(EDonor + mu\right) + \left(Vef - Ec\right)}{KbT}}}{NdChar}\right)}^{-1}} + 0.5 \cdot NaChar \]
   8. Step-by-step derivation

      1. unpow-1 73.2%

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

      2. associate-+l+ 73.2%

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

      3. associate-+r- 73.2%

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

      4. +-commutative 73.2%

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

      5. sub-neg 73.2%

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

      6. associate-+r+ 73.2%

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

      7. mul-1-neg 73.2%

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

      8. mul-1-neg 73.2%

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

      9. associate-+r+ 73.2%

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

      10. sub-neg 73.2%

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

      11. +-commutative 73.2%

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

      12. associate-+r- 73.2%

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

   9. Simplified 73.2%

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

   10. Taylor expanded in Ec around inf 58.7%

       \[\leadsto \frac{1}{\frac{1 + e^{\color{blue}{-1 \cdot \frac{Ec}{KbT}}}}{NdChar}} + 0.5 \cdot NaChar \]
   11. Step-by-step derivation

       1. mul-1-neg 58.7%

          \[\leadsto \frac{1}{\frac{1 + e^{\color{blue}{-\frac{Ec}{KbT}}}}{NdChar}} + 0.5 \cdot NaChar \]

       2. distribute-neg-frac 58.7%

          \[\leadsto \frac{1}{\frac{1 + e^{\color{blue}{\frac{-Ec}{KbT}}}}{NdChar}} + 0.5 \cdot NaChar \]

   12. Simplified 58.7%

       \[\leadsto \frac{1}{\frac{1 + e^{\color{blue}{\frac{-Ec}{KbT}}}}{NdChar}} + 0.5 \cdot NaChar \]

   13. Taylor expanded in Ec around 0 55.6%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{Ec}{KbT \cdot NdChar} + 2 \cdot \frac{1}{NdChar}}} + 0.5 \cdot NaChar \]
   14. Step-by-step derivation

       1. +-commutative 55.6%

          \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{1}{NdChar} + -1 \cdot \frac{Ec}{KbT \cdot NdChar}}} + 0.5 \cdot NaChar \]

       2. mul-1-neg 55.6%

          \[\leadsto \frac{1}{2 \cdot \frac{1}{NdChar} + \color{blue}{\left(-\frac{Ec}{KbT \cdot NdChar}\right)}} + 0.5 \cdot NaChar \]

       3. unsub-neg 55.6%

          \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{1}{NdChar} - \frac{Ec}{KbT \cdot NdChar}}} + 0.5 \cdot NaChar \]

       4. associate-*r/ 55.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot 1}{NdChar}} - \frac{Ec}{KbT \cdot NdChar}} + 0.5 \cdot NaChar \]

       5. metadata-eval 55.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{2}}{NdChar} - \frac{Ec}{KbT \cdot NdChar}} + 0.5 \cdot NaChar \]

   15. Simplified 55.6%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 4.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 - \frac{1}{\frac{Ec}{NdChar \cdot KbT} - \frac{2}{NdChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 36: 29.3% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -1.7e+51)
   (+ (/ NaChar (+ 2.0 (/ Ev KbT))) (* NdChar 0.5))
   (if (<= KbT 3.3e+38)
     (/ (* KbT NaChar) (- (+ EAccept (+ Vef Ev)) mu))
     (+ (* NdChar 0.5) (* NaChar 0.5)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.7e+51) {
		tmp = (NaChar / (2.0 + (Ev / KbT))) + (NdChar * 0.5);
	} else if (KbT <= 3.3e+38) {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	} else {
		tmp = (NdChar * 0.5) + (NaChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-1.7d+51)) then
        tmp = (nachar / (2.0d0 + (ev / kbt))) + (ndchar * 0.5d0)
    else if (kbt <= 3.3d+38) then
        tmp = (kbt * nachar) / ((eaccept + (vef + ev)) - mu)
    else
        tmp = (ndchar * 0.5d0) + (nachar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.7e+51) {
		tmp = (NaChar / (2.0 + (Ev / KbT))) + (NdChar * 0.5);
	} else if (KbT <= 3.3e+38) {
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	} else {
		tmp = (NdChar * 0.5) + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -1.7e+51:
		tmp = (NaChar / (2.0 + (Ev / KbT))) + (NdChar * 0.5)
	elif KbT <= 3.3e+38:
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu)
	else:
		tmp = (NdChar * 0.5) + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -1.7e+51)
		tmp = Float64(Float64(NaChar / Float64(2.0 + Float64(Ev / KbT))) + Float64(NdChar * 0.5));
	elseif (KbT <= 3.3e+38)
		tmp = Float64(Float64(KbT * NaChar) / Float64(Float64(EAccept + Float64(Vef + Ev)) - mu));
	else
		tmp = Float64(Float64(NdChar * 0.5) + Float64(NaChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -1.7e+51)
		tmp = (NaChar / (2.0 + (Ev / KbT))) + (NdChar * 0.5);
	elseif (KbT <= 3.3e+38)
		tmp = (KbT * NaChar) / ((EAccept + (Vef + Ev)) - mu);
	else
		tmp = (NdChar * 0.5) + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -1.7e+51], N[(N[(NaChar / N[(2.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 3.3e+38], N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.69999999999999992e51

   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%

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

   5. Taylor expanded in Ev around inf 76.9%

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

   6. Taylor expanded in Ev around 0 65.4%

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

   7. Taylor expanded in KbT around inf 51.9%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{2 + \frac{Ev}{KbT}} \]
   8. Step-by-step derivation

      1. *-commutative 21.4%

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

   9. Simplified 51.9%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{2 + \frac{Ev}{KbT}} \]

   if -1.69999999999999992e51 < KbT < 3.2999999999999999e38

   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%

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

   5. Taylor expanded in KbT around inf 49.4%

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

   6. Taylor expanded in KbT around 0 56.0%

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

      1. associate-/l* 56.0%

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

      2. associate--l+ 56.0%

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

      3. associate--l+ 56.0%

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

   8. Simplified 56.0%

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

   9. Taylor expanded in KbT around inf 9.6%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 9.6%

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

   11. Simplified 9.6%

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

   12. Taylor expanded in NdChar around 0 22.8%

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

   if 3.2999999999999999e38 < 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%

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

   5. Taylor expanded in KbT around inf 73.2%

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

   6. Taylor expanded in KbT around inf 55.5%

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

      1. *-commutative 29.4%

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

   8. Simplified 55.5%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Ev}{KbT}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{KbT \cdot NaChar}{\left(EAccept + \left(Vef + Ev\right)\right) - mu}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 37: 27.7% accurate, 32.7× speedup

Math

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

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+ (* NdChar 0.5) (* 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 (NdChar * 0.5) + (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 = (ndchar * 0.5d0) + (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 (NdChar * 0.5) + (NaChar * 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in KbT around inf 48.8%

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

6. Taylor expanded in KbT around inf 28.4%

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

   1. *-commutative 15.8%

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

8. Simplified 28.4%

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

9. Final simplification 28.4%

   \[\leadsto NdChar \cdot 0.5 + NaChar \cdot 0.5 \]
10. Add Preprocessing

Alternative 38: 18.4% accurate, 76.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in KbT around inf 58.1%

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

6. Taylor expanded in KbT around 0 48.9%

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

   1. associate-/l* 50.2%

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

   2. associate--l+ 50.2%

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

   3. associate--l+ 50.2%

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

8. Simplified 50.2%

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

9. Taylor expanded in KbT around inf 15.8%

   \[\leadsto \color{blue}{0.5 \cdot NdChar} + KbT \cdot \frac{NaChar}{EAccept + \left(Ev + \left(Vef - mu\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 15.8%

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

11. Simplified 15.8%

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

12. Taylor expanded in NdChar around inf 20.4%

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

13. Final simplification 20.4%

    \[\leadsto NdChar \cdot 0.5 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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))))))