Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 32.1s

Alternatives: 27

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (pow E (/ (/ (+ (- mu Ec) (+ Vef EDonor)) KbT) 2.0))))
   (+
    (/ NdChar (+ 1.0 (* t_0 t_0)))
    (/ 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) {
	double t_0 = pow(((double) M_E), ((((mu - Ec) + (Vef + EDonor)) / KbT) / 2.0));
	return (NdChar / (1.0 + (t_0 * t_0))) + (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT))));
}

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

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.pow(math.e, ((((mu - Ec) + (Vef + EDonor)) / KbT) / 2.0))
	return (NdChar / (1.0 + (t_0 * t_0))) + (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT))))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(1) ^ Float64(Float64(Float64(Float64(mu - Ec) + Float64(Vef + EDonor)) / KbT) / 2.0)
	return Float64(Float64(NdChar / Float64(1.0 + Float64(t_0 * t_0))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 2.71828182845904523536 ^ ((((mu - Ec) + (Vef + EDonor)) / KbT) / 2.0);
	tmp = (NdChar / (1.0 + (t_0 * t_0))) + (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. associate--r- 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

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

   5. sub-neg 99.9%

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

   6. associate--l- 99.9%

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

   7. unsub-neg 99.9%

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

   8. +-commutative 99.9%

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

   9. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 100.0%

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

   3. exp-1-e 100.0%

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

   4. associate--r- 100.0%

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

5. Applied egg-rr 100.0%

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

   1. sqr-pow 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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 = (mu - Ec) + (Vef + EDonor);
	return (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (Math.pow(Math.E, ((t_0 / KbT) / 2.0)) * Math.exp((t_0 / (KbT * 2.0))))));
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (mu - Ec) + (Vef + EDonor)
	return (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (math.pow(math.e, ((t_0 / KbT) / 2.0)) * math.exp((t_0 / (KbT * 2.0))))))

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(mu - Ec) + Float64(Vef + EDonor))
	return Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64((exp(1) ^ Float64(Float64(t_0 / KbT) / 2.0)) * exp(Float64(t_0 / Float64(KbT * 2.0)))))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (mu - Ec) + (Vef + EDonor);
	tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + ((2.71828182845904523536 ^ ((t_0 / KbT) / 2.0)) * exp((t_0 / (KbT * 2.0))))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. associate--r- 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

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

   5. sub-neg 99.9%

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

   6. associate--l- 99.9%

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

   7. unsub-neg 99.9%

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

   8. +-commutative 99.9%

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

   9. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 100.0%

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

   3. exp-1-e 100.0%

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

   4. associate--r- 100.0%

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

5. Applied egg-rr 100.0%

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

   1. sqr-pow 100.0%

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

7. Applied egg-rr 100.0%

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

   1. pow-to-exp 100.0%

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

   2. e-exp-1 100.0%

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

   3. add-log-exp 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. associate-/l/ 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / Float64(1.0 + (exp(1) ^ Float64(Float64(Float64(mu - Ec) + Float64(Vef + EDonor)) / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (2.71828182845904523536 ^ (((mu - Ec) + (Vef + EDonor)) / KbT))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. associate--r- 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

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

   5. sub-neg 99.9%

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

   6. associate--l- 99.9%

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

   7. unsub-neg 99.9%

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

   8. +-commutative 99.9%

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

   9. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 100.0%

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

   3. exp-1-e 100.0%

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

   4. associate--r- 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 4: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := t_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{if}\;NdChar \leq -2 \cdot 10^{+135}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -2.3 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq 1.8 \cdot 10^{-187}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 1.4 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} - \frac{mu}{KbT}\right)}\\ \end{array} \end{array} \]

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 (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))))
   (if (<= NdChar -2e+135)
     (+
      (/ NdChar (+ 1.0 (exp (/ mu KbT))))
      (/ NaChar (+ 1.0 (exp (/ (- mu) KbT)))))
     (if (<= NdChar -2.3e-216)
       t_1
       (if (<= NdChar 1.8e-187)
         (+
          t_0
          (/ NdChar (+ 1.0 (+ 1.0 (/ (+ (+ mu Vef) (- EDonor Ec)) KbT)))))
         (if (<= NdChar 1.4e+183)
           t_1
           (+
            (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))
            (/ NaChar (+ 1.0 (- (/ Ev KbT) (/ mu KbT)))))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	double tmp;
	if (NdChar <= -2e+135) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-mu / KbT))));
	} else if (NdChar <= -2.3e-216) {
		tmp = t_1;
	} else if (NdChar <= 1.8e-187) {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	} else if (NdChar <= 1.4e+183) {
		tmp = t_1;
	} else {
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    if (ndchar <= (-2d+135)) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((-mu / kbt))))
    else if (ndchar <= (-2.3d-216)) then
        tmp = t_1
    else if (ndchar <= 1.8d-187) then
        tmp = t_0 + (ndchar / (1.0d0 + (1.0d0 + (((mu + vef) + (edonor - ec)) / kbt))))
    else if (ndchar <= 1.4d+183) then
        tmp = t_1
    else
        tmp = (ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))) + (nachar / (1.0d0 + ((ev / kbt) - (mu / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	double tmp;
	if (NdChar <= -2e+135) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((-mu / KbT))));
	} else if (NdChar <= -2.3e-216) {
		tmp = t_1;
	} else if (NdChar <= 1.8e-187) {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	} else if (NdChar <= 1.4e+183) {
		tmp = t_1;
	} else {
		tmp = (NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))
	t_1 = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	tmp = 0
	if NdChar <= -2e+135:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((-mu / KbT))))
	elif NdChar <= -2.3e-216:
		tmp = t_1
	elif NdChar <= 1.8e-187:
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))))
	elif NdChar <= 1.4e+183:
		tmp = t_1
	else:
		tmp = (NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT))))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))))
	tmp = 0.0
	if (NdChar <= -2e+135)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))));
	elseif (NdChar <= -2.3e-216)
		tmp = t_1;
	elseif (NdChar <= 1.8e-187)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(mu + Vef) + Float64(EDonor - Ec)) / KbT)))));
	elseif (NdChar <= 1.4e+183)
		tmp = t_1;
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(Ev / KbT) - Float64(mu / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	t_1 = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	tmp = 0.0;
	if (NdChar <= -2e+135)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-mu / KbT))));
	elseif (NdChar <= -2.3e-216)
		tmp = t_1;
	elseif (NdChar <= 1.8e-187)
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	elseif (NdChar <= 1.4e+183)
		tmp = t_1;
	else
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -2e+135], 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], If[LessEqual[NdChar, -2.3e-216], t$95$1, If[LessEqual[NdChar, 1.8e-187], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(1.0 + N[(N[(N[(mu + Vef), $MachinePrecision] + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.4e+183], t$95$1, N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -1.99999999999999992e135

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in mu around inf 76.7%

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

   5. Taylor expanded in mu around inf 70.7%

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

      1. neg-mul-1 70.7%

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

      2. distribute-neg-frac 70.7%

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

   7. Simplified 70.7%

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

   if -1.99999999999999992e135 < NdChar < -2.29999999999999997e-216 or 1.79999999999999997e-187 < NdChar < 1.40000000000000009e183

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in EDonor around inf 76.8%

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

   if -2.29999999999999997e-216 < NdChar < 1.79999999999999997e-187

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

      3. exp-1-e 100.0%

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

      4. associate--r- 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in KbT around inf 82.5%

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

      1. log-E 82.5%

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

      2. *-lft-identity 82.5%

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

      3. associate--l+ 82.5%

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

      4. associate--l+ 82.5%

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

      5. sub-neg 82.5%

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

      6. +-commutative 82.5%

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

      7. mul-1-neg 82.5%

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

      8. associate-+r+ 82.5%

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

      9. mul-1-neg 82.5%

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

      10. +-commutative 82.5%

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

      11. sub-neg 82.5%

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

   8. Simplified 82.5%

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

   if 1.40000000000000009e183 < NdChar

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 72.9%

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

      1. flip-+ 65.8%

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

   6. Applied egg-rr 65.8%

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

   7. Taylor expanded in Ev around inf 81.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2 \cdot 10^{+135}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -2.3 \cdot 10^{-216}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.8 \cdot 10^{-187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 1.4 \cdot 10^{+183}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} - \frac{mu}{KbT}\right)}\\ \end{array} \]

Alternative 5: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := t_0 + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \mathbf{if}\;Ec \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Ec \leq 2.55 \cdot 10^{-202}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;Ec \leq 1.6 \cdot 10^{+99}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 (+ t_0 (/ NdChar (+ 1.0 (exp (/ (- Ec) KbT)))))))
   (if (<= Ec -3.2e-9)
     t_1
     (if (<= Ec 2.55e-202)
       (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
       (if (<= Ec 1.6e+99)
         (+ t_0 (/ NdChar (+ 1.0 (exp (/ Vef 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 = t_0 + (NdChar / (1.0 + exp((-Ec / KbT))));
	double tmp;
	if (Ec <= -3.2e-9) {
		tmp = t_1;
	} else if (Ec <= 2.55e-202) {
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (Ec <= 1.6e+99) {
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / 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 = t_0 + (ndchar / (1.0d0 + exp((-ec / kbt))))
    if (ec <= (-3.2d-9)) then
        tmp = t_1
    else if (ec <= 2.55d-202) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (ec <= 1.6d+99) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((vef / 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 = t_0 + (NdChar / (1.0 + Math.exp((-Ec / KbT))));
	double tmp;
	if (Ec <= -3.2e-9) {
		tmp = t_1;
	} else if (Ec <= 2.55e-202) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (Ec <= 1.6e+99) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((Vef / 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 = t_0 + (NdChar / (1.0 + math.exp((-Ec / KbT))))
	tmp = 0
	if Ec <= -3.2e-9:
		tmp = t_1
	elif Ec <= 2.55e-202:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif Ec <= 1.6e+99:
		tmp = t_0 + (NdChar / (1.0 + math.exp((Vef / 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(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT))))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Ec) / KbT)))))
	tmp = 0.0
	if (Ec <= -3.2e-9)
		tmp = t_1;
	elseif (Ec <= 2.55e-202)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (Ec <= 1.6e+99)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / 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 = t_0 + (NdChar / (1.0 + exp((-Ec / KbT))));
	tmp = 0.0;
	if (Ec <= -3.2e-9)
		tmp = t_1;
	elseif (Ec <= 2.55e-202)
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (Ec <= 1.6e+99)
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / 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[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Ec, -3.2e-9], t$95$1, If[LessEqual[Ec, 2.55e-202], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ec, 1.6e+99], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if Ec < -3.20000000000000012e-9 or 1.6e99 < Ec

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in Ec around inf 87.5%

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

      1. associate-*r/ 87.5%

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

      2. mul-1-neg 87.5%

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

   6. Simplified 87.5%

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

   if -3.20000000000000012e-9 < Ec < 2.54999999999999998e-202

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in EDonor around inf 77.5%

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

   if 2.54999999999999998e-202 < Ec < 1.6e99

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in Vef around inf 84.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ec \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \mathbf{elif}\;Ec \leq 2.55 \cdot 10^{-202}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;Ec \leq 1.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \end{array} \]

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / 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 = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / 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 (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. neg-sub0 99.9%

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

   2. associate--r- 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

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

   5. sub-neg 99.9%

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

   6. associate--l- 99.9%

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

   7. unsub-neg 99.9%

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

   8. +-commutative 99.9%

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

   9. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 7: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;Vef \leq -1.6 \cdot 10^{+161} \lor \neg \left(Vef \leq 2 \cdot 10^{+32}\right):\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))))
   (if (or (<= Vef -1.6e+161) (not (<= Vef 2e+32)))
     (+ t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
     (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double tmp;
	if ((Vef <= -1.6e+161) || !(Vef <= 2e+32)) {
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	} else {
		tmp = t_0 + (NdChar / (1.0 + exp((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) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    if ((vef <= (-1.6d+161)) .or. (.not. (vef <= 2d+32))) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((vef / kbt))))
    else
        tmp = t_0 + (ndchar / (1.0d0 + exp((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 t_0 = NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double tmp;
	if ((Vef <= -1.6e+161) || !(Vef <= 2e+32)) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	} else {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	}
	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)))
	tmp = 0
	if (Vef <= -1.6e+161) or not (Vef <= 2e+32):
		tmp = t_0 + (NdChar / (1.0 + math.exp((Vef / KbT))))
	else:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT))))
	tmp = 0.0
	if ((Vef <= -1.6e+161) || !(Vef <= 2e+32))
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	tmp = 0.0;
	if ((Vef <= -1.6e+161) || ~((Vef <= 2e+32)))
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	else
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[Vef, -1.6e+161], N[Not[LessEqual[Vef, 2e+32]], $MachinePrecision]], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -1.60000000000000001e161 or 2.00000000000000011e32 < Vef

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in Vef around inf 89.8%

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

   if -1.60000000000000001e161 < Vef < 2.00000000000000011e32

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EDonor around inf 77.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.6 \cdot 10^{+161} \lor \neg \left(Vef \leq 2 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \end{array} \]

Alternative 8: 63.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -3.7 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq -1.6 \cdot 10^{+119}:\\ \;\;\;\;t_2 + \frac{KbT \cdot NaChar}{\left(Ev + \frac{EAccept \cdot EAccept}{EAccept - Vef}\right) - \left(mu + \frac{Vef \cdot Vef}{EAccept - Vef}\right)}\\ \mathbf{elif}\;NdChar \leq -4 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;NdChar \leq 0.028:\\ \;\;\;\;t_2 + \frac{NaChar}{1 + \left(\left(1 + \left(\left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{KbT - \left(\frac{EAccept \cdot EAccept}{KbT} + \left(\frac{EAccept}{KbT} + -1\right) \cdot \left(KbT \cdot \left(-1 - \frac{EAccept}{KbT}\right)\right)\right)}{Vef}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.4 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2 + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} - \frac{mu}{KbT}\right)}\\ \end{array} \end{array} \]

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))))
          (/ NdChar (+ 1.0 (+ 1.0 (/ (+ (+ mu Vef) (- EDonor Ec)) KbT))))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- mu) KbT))))))
        (t_2 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))))
   (if (<= NdChar -3.7e+148)
     t_1
     (if (<= NdChar -1.6e+119)
       (+
        t_2
        (/
         (* KbT NaChar)
         (-
          (+ Ev (/ (* EAccept EAccept) (- EAccept Vef)))
          (+ mu (/ (* Vef Vef) (- EAccept Vef))))))
       (if (<= NdChar -4e+33)
         t_1
         (if (<= NdChar 6.8e-18)
           t_0
           (if (<= NdChar 0.028)
             (+
              t_2
              (/
               NaChar
               (+
                1.0
                (-
                 (+
                  1.0
                  (+
                   (+ (/ Ev KbT) (/ EAccept KbT))
                   (+
                    (/ Vef KbT)
                    (/
                     (-
                      KbT
                      (+
                       (/ (* EAccept EAccept) KbT)
                       (*
                        (+ (/ EAccept KbT) -1.0)
                        (* KbT (- -1.0 (/ EAccept KbT))))))
                     Vef))))
                 (/ mu KbT)))))
             (if (<= NdChar 2.4e+72)
               t_0
               (+ t_2 (/ NaChar (+ 1.0 (- (/ Ev KbT) (/ mu KbT)))))))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	double t_1 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-mu / KbT))));
	double t_2 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double tmp;
	if (NdChar <= -3.7e+148) {
		tmp = t_1;
	} else if (NdChar <= -1.6e+119) {
		tmp = t_2 + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))));
	} else if (NdChar <= -4e+33) {
		tmp = t_1;
	} else if (NdChar <= 6.8e-18) {
		tmp = t_0;
	} else if (NdChar <= 0.028) {
		tmp = t_2 + (NaChar / (1.0 + ((1.0 + (((Ev / KbT) + (EAccept / KbT)) + ((Vef / KbT) + ((KbT - (((EAccept * EAccept) / KbT) + (((EAccept / KbT) + -1.0) * (KbT * (-1.0 - (EAccept / KbT)))))) / Vef)))) - (mu / KbT))));
	} else if (NdChar <= 2.4e+72) {
		tmp = t_0;
	} else {
		tmp = t_2 + (NaChar / (1.0 + ((Ev / KbT) - (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 = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (((mu + vef) + (edonor - ec)) / kbt))))
    t_1 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((-mu / kbt))))
    t_2 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    if (ndchar <= (-3.7d+148)) then
        tmp = t_1
    else if (ndchar <= (-1.6d+119)) then
        tmp = t_2 + ((kbt * nachar) / ((ev + ((eaccept * eaccept) / (eaccept - vef))) - (mu + ((vef * vef) / (eaccept - vef)))))
    else if (ndchar <= (-4d+33)) then
        tmp = t_1
    else if (ndchar <= 6.8d-18) then
        tmp = t_0
    else if (ndchar <= 0.028d0) then
        tmp = t_2 + (nachar / (1.0d0 + ((1.0d0 + (((ev / kbt) + (eaccept / kbt)) + ((vef / kbt) + ((kbt - (((eaccept * eaccept) / kbt) + (((eaccept / kbt) + (-1.0d0)) * (kbt * ((-1.0d0) - (eaccept / kbt)))))) / vef)))) - (mu / kbt))))
    else if (ndchar <= 2.4d+72) then
        tmp = t_0
    else
        tmp = t_2 + (nachar / (1.0d0 + ((ev / kbt) - (mu / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	double t_1 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((-mu / KbT))));
	double t_2 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double tmp;
	if (NdChar <= -3.7e+148) {
		tmp = t_1;
	} else if (NdChar <= -1.6e+119) {
		tmp = t_2 + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))));
	} else if (NdChar <= -4e+33) {
		tmp = t_1;
	} else if (NdChar <= 6.8e-18) {
		tmp = t_0;
	} else if (NdChar <= 0.028) {
		tmp = t_2 + (NaChar / (1.0 + ((1.0 + (((Ev / KbT) + (EAccept / KbT)) + ((Vef / KbT) + ((KbT - (((EAccept * EAccept) / KbT) + (((EAccept / KbT) + -1.0) * (KbT * (-1.0 - (EAccept / KbT)))))) / Vef)))) - (mu / KbT))));
	} else if (NdChar <= 2.4e+72) {
		tmp = t_0;
	} else {
		tmp = t_2 + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))))
	t_1 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((-mu / KbT))))
	t_2 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	tmp = 0
	if NdChar <= -3.7e+148:
		tmp = t_1
	elif NdChar <= -1.6e+119:
		tmp = t_2 + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))))
	elif NdChar <= -4e+33:
		tmp = t_1
	elif NdChar <= 6.8e-18:
		tmp = t_0
	elif NdChar <= 0.028:
		tmp = t_2 + (NaChar / (1.0 + ((1.0 + (((Ev / KbT) + (EAccept / KbT)) + ((Vef / KbT) + ((KbT - (((EAccept * EAccept) / KbT) + (((EAccept / KbT) + -1.0) * (KbT * (-1.0 - (EAccept / KbT)))))) / Vef)))) - (mu / KbT))))
	elif NdChar <= 2.4e+72:
		tmp = t_0
	else:
		tmp = t_2 + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(mu + Vef) + Float64(EDonor - Ec)) / KbT)))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))))
	t_2 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	tmp = 0.0
	if (NdChar <= -3.7e+148)
		tmp = t_1;
	elseif (NdChar <= -1.6e+119)
		tmp = Float64(t_2 + Float64(Float64(KbT * NaChar) / Float64(Float64(Ev + Float64(Float64(EAccept * EAccept) / Float64(EAccept - Vef))) - Float64(mu + Float64(Float64(Vef * Vef) / Float64(EAccept - Vef))))));
	elseif (NdChar <= -4e+33)
		tmp = t_1;
	elseif (NdChar <= 6.8e-18)
		tmp = t_0;
	elseif (NdChar <= 0.028)
		tmp = Float64(t_2 + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(Float64(Ev / KbT) + Float64(EAccept / KbT)) + Float64(Float64(Vef / KbT) + Float64(Float64(KbT - Float64(Float64(Float64(EAccept * EAccept) / KbT) + Float64(Float64(Float64(EAccept / KbT) + -1.0) * Float64(KbT * Float64(-1.0 - Float64(EAccept / KbT)))))) / Vef)))) - Float64(mu / KbT)))));
	elseif (NdChar <= 2.4e+72)
		tmp = t_0;
	else
		tmp = Float64(t_2 + Float64(NaChar / Float64(1.0 + Float64(Float64(Ev / KbT) - Float64(mu / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	t_1 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-mu / KbT))));
	t_2 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	tmp = 0.0;
	if (NdChar <= -3.7e+148)
		tmp = t_1;
	elseif (NdChar <= -1.6e+119)
		tmp = t_2 + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))));
	elseif (NdChar <= -4e+33)
		tmp = t_1;
	elseif (NdChar <= 6.8e-18)
		tmp = t_0;
	elseif (NdChar <= 0.028)
		tmp = t_2 + (NaChar / (1.0 + ((1.0 + (((Ev / KbT) + (EAccept / KbT)) + ((Vef / KbT) + ((KbT - (((EAccept * EAccept) / KbT) + (((EAccept / KbT) + -1.0) * (KbT * (-1.0 - (EAccept / KbT)))))) / Vef)))) - (mu / KbT))));
	elseif (NdChar <= 2.4e+72)
		tmp = t_0;
	else
		tmp = t_2 + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	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[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(N[(N[(mu + Vef), $MachinePrecision] + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$2 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -3.7e+148], t$95$1, If[LessEqual[NdChar, -1.6e+119], N[(t$95$2 + N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(Ev + N[(N[(EAccept * EAccept), $MachinePrecision] / N[(EAccept - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu + N[(N[(Vef * Vef), $MachinePrecision] / N[(EAccept - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -4e+33], t$95$1, If[LessEqual[NdChar, 6.8e-18], t$95$0, If[LessEqual[NdChar, 0.028], N[(t$95$2 + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(N[(Ev / KbT), $MachinePrecision] + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(N[(KbT - N[(N[(N[(EAccept * EAccept), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(EAccept / KbT), $MachinePrecision] + -1.0), $MachinePrecision] * N[(KbT * N[(-1.0 - N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.4e+72], t$95$0, N[(t$95$2 + N[(NaChar / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -3.7000000000000002e148 or -1.59999999999999995e119 < NdChar < -3.9999999999999998e33

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in mu around inf 75.8%

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

   5. Taylor expanded in mu around inf 68.5%

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

      1. neg-mul-1 68.5%

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

      2. distribute-neg-frac 68.5%

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

   7. Simplified 68.5%

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

   if -3.7000000000000002e148 < NdChar < -1.59999999999999995e119

   1. Initial program 99.5%

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

      1. neg-sub0 99.5%

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

      2. associate--r- 99.5%

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

      3. +-commutative 99.5%

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

      4. neg-sub0 99.5%

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

      5. sub-neg 99.5%

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

      6. associate--l- 99.5%

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

      7. unsub-neg 99.5%

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

      8. +-commutative 99.5%

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

      9. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in KbT around inf 87.5%

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

      1. flip-+ 50.0%

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in KbT around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. +-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. +-commutative 99.5%

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

      5. unpow2 99.5%

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

   9. Simplified 99.5%

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

   if -3.9999999999999998e33 < NdChar < 6.80000000000000002e-18 or 0.0280000000000000006 < NdChar < 2.4000000000000001e72

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

      3. exp-1-e 100.0%

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

      4. associate--r- 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in KbT around inf 71.2%

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

      1. log-E 71.2%

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

      2. *-lft-identity 71.2%

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

      3. associate--l+ 71.2%

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

      4. associate--l+ 71.2%

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

      5. sub-neg 71.2%

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

      6. +-commutative 71.2%

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

      7. mul-1-neg 71.2%

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

      8. associate-+r+ 71.2%

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

      9. mul-1-neg 71.2%

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

      10. +-commutative 71.2%

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

      11. sub-neg 71.2%

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

   8. Simplified 71.2%

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

   if 6.80000000000000002e-18 < NdChar < 0.0280000000000000006

   1. Initial program 99.0%

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

      1. neg-sub0 99.0%

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

      2. associate--r- 99.0%

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

      3. +-commutative 99.0%

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

      4. neg-sub0 99.0%

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

      5. sub-neg 99.0%

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

      6. associate--l- 99.0%

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

      7. unsub-neg 99.0%

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

      8. +-commutative 99.0%

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

      9. associate-+l+ 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in KbT around inf 71.6%

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

      1. flip-+ 57.3%

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in Vef around -inf 71.3%

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

      1. associate-+r+ 71.3%

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

      2. mul-1-neg 71.3%

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

      3. unsub-neg 71.3%

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

   9. Simplified 71.3%

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

   if 2.4000000000000001e72 < NdChar

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 60.9%

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

      1. flip-+ 55.1%

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

   6. Applied egg-rr 55.1%

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

   7. Taylor expanded in Ev around inf 67.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -3.7 \cdot 10^{+148}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -1.6 \cdot 10^{+119}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{KbT \cdot NaChar}{\left(Ev + \frac{EAccept \cdot EAccept}{EAccept - Vef}\right) - \left(mu + \frac{Vef \cdot Vef}{EAccept - Vef}\right)}\\ \mathbf{elif}\;NdChar \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 0.028:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{KbT - \left(\frac{EAccept \cdot EAccept}{KbT} + \left(\frac{EAccept}{KbT} + -1\right) \cdot \left(KbT \cdot \left(-1 - \frac{EAccept}{KbT}\right)\right)\right)}{Vef}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} - \frac{mu}{KbT}\right)}\\ \end{array} \]

Alternative 9: 62.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{if}\;NdChar \leq -1.3 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq -8.6 \cdot 10^{+18}:\\ \;\;\;\;t_0 + \frac{KbT \cdot NaChar}{\left(Ev + \frac{EAccept \cdot EAccept}{EAccept - Vef}\right) - \left(mu + \frac{Vef \cdot Vef}{EAccept - Vef}\right)}\\ \mathbf{elif}\;NdChar \leq 1.65 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq 12:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + \left(\left(1 + \left(\left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{KbT - \left(\frac{EAccept \cdot EAccept}{KbT} + \left(\frac{EAccept}{KbT} + -1\right) \cdot \left(KbT \cdot \left(-1 - \frac{EAccept}{KbT}\right)\right)\right)}{Vef}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} - \frac{mu}{KbT}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT)))))
        (t_1
         (+
          (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
          (/ NdChar (+ 1.0 (+ 1.0 (/ (+ (+ mu Vef) (- EDonor Ec)) KbT)))))))
   (if (<= NdChar -1.3e+153)
     t_1
     (if (<= NdChar -8.6e+18)
       (+
        t_0
        (/
         (* KbT NaChar)
         (-
          (+ Ev (/ (* EAccept EAccept) (- EAccept Vef)))
          (+ mu (/ (* Vef Vef) (- EAccept Vef))))))
       (if (<= NdChar 1.65e-17)
         t_1
         (if (<= NdChar 12.0)
           (+
            t_0
            (/
             NaChar
             (+
              1.0
              (-
               (+
                1.0
                (+
                 (+ (/ Ev KbT) (/ EAccept KbT))
                 (+
                  (/ Vef KbT)
                  (/
                   (-
                    KbT
                    (+
                     (/ (* EAccept EAccept) KbT)
                     (*
                      (+ (/ EAccept KbT) -1.0)
                      (* KbT (- -1.0 (/ EAccept KbT))))))
                   Vef))))
               (/ mu KbT)))))
           (if (<= NdChar 1.75e+74)
             t_1
             (+ t_0 (/ NaChar (+ 1.0 (- (/ Ev KbT) (/ 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 + ((Vef + EDonor) - Ec)) / KbT)));
	double t_1 = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	double tmp;
	if (NdChar <= -1.3e+153) {
		tmp = t_1;
	} else if (NdChar <= -8.6e+18) {
		tmp = t_0 + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))));
	} else if (NdChar <= 1.65e-17) {
		tmp = t_1;
	} else if (NdChar <= 12.0) {
		tmp = t_0 + (NaChar / (1.0 + ((1.0 + (((Ev / KbT) + (EAccept / KbT)) + ((Vef / KbT) + ((KbT - (((EAccept * EAccept) / KbT) + (((EAccept / KbT) + -1.0) * (KbT * (-1.0 - (EAccept / KbT)))))) / Vef)))) - (mu / KbT))));
	} else if (NdChar <= 1.75e+74) {
		tmp = t_1;
	} else {
		tmp = t_0 + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_1 = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (((mu + vef) + (edonor - ec)) / kbt))))
    if (ndchar <= (-1.3d+153)) then
        tmp = t_1
    else if (ndchar <= (-8.6d+18)) then
        tmp = t_0 + ((kbt * nachar) / ((ev + ((eaccept * eaccept) / (eaccept - vef))) - (mu + ((vef * vef) / (eaccept - vef)))))
    else if (ndchar <= 1.65d-17) then
        tmp = t_1
    else if (ndchar <= 12.0d0) then
        tmp = t_0 + (nachar / (1.0d0 + ((1.0d0 + (((ev / kbt) + (eaccept / kbt)) + ((vef / kbt) + ((kbt - (((eaccept * eaccept) / kbt) + (((eaccept / kbt) + (-1.0d0)) * (kbt * ((-1.0d0) - (eaccept / kbt)))))) / vef)))) - (mu / kbt))))
    else if (ndchar <= 1.75d+74) then
        tmp = t_1
    else
        tmp = t_0 + (nachar / (1.0d0 + ((ev / kbt) - (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 + ((Vef + EDonor) - Ec)) / KbT)));
	double t_1 = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	double tmp;
	if (NdChar <= -1.3e+153) {
		tmp = t_1;
	} else if (NdChar <= -8.6e+18) {
		tmp = t_0 + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))));
	} else if (NdChar <= 1.65e-17) {
		tmp = t_1;
	} else if (NdChar <= 12.0) {
		tmp = t_0 + (NaChar / (1.0 + ((1.0 + (((Ev / KbT) + (EAccept / KbT)) + ((Vef / KbT) + ((KbT - (((EAccept * EAccept) / KbT) + (((EAccept / KbT) + -1.0) * (KbT * (-1.0 - (EAccept / KbT)))))) / Vef)))) - (mu / KbT))));
	} else if (NdChar <= 1.75e+74) {
		tmp = t_1;
	} else {
		tmp = t_0 + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	t_1 = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))))
	tmp = 0
	if NdChar <= -1.3e+153:
		tmp = t_1
	elif NdChar <= -8.6e+18:
		tmp = t_0 + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))))
	elif NdChar <= 1.65e-17:
		tmp = t_1
	elif NdChar <= 12.0:
		tmp = t_0 + (NaChar / (1.0 + ((1.0 + (((Ev / KbT) + (EAccept / KbT)) + ((Vef / KbT) + ((KbT - (((EAccept * EAccept) / KbT) + (((EAccept / KbT) + -1.0) * (KbT * (-1.0 - (EAccept / KbT)))))) / Vef)))) - (mu / KbT))))
	elif NdChar <= 1.75e+74:
		tmp = t_1
	else:
		tmp = t_0 + (NaChar / (1.0 + ((Ev / KbT) - (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(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(mu + Vef) + Float64(EDonor - Ec)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -1.3e+153)
		tmp = t_1;
	elseif (NdChar <= -8.6e+18)
		tmp = Float64(t_0 + Float64(Float64(KbT * NaChar) / Float64(Float64(Ev + Float64(Float64(EAccept * EAccept) / Float64(EAccept - Vef))) - Float64(mu + Float64(Float64(Vef * Vef) / Float64(EAccept - Vef))))));
	elseif (NdChar <= 1.65e-17)
		tmp = t_1;
	elseif (NdChar <= 12.0)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(Float64(Ev / KbT) + Float64(EAccept / KbT)) + Float64(Float64(Vef / KbT) + Float64(Float64(KbT - Float64(Float64(Float64(EAccept * EAccept) / KbT) + Float64(Float64(Float64(EAccept / KbT) + -1.0) * Float64(KbT * Float64(-1.0 - Float64(EAccept / KbT)))))) / Vef)))) - Float64(mu / KbT)))));
	elseif (NdChar <= 1.75e+74)
		tmp = t_1;
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + Float64(Float64(Ev / KbT) - Float64(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 + ((Vef + EDonor) - Ec)) / KbT)));
	t_1 = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -1.3e+153)
		tmp = t_1;
	elseif (NdChar <= -8.6e+18)
		tmp = t_0 + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))));
	elseif (NdChar <= 1.65e-17)
		tmp = t_1;
	elseif (NdChar <= 12.0)
		tmp = t_0 + (NaChar / (1.0 + ((1.0 + (((Ev / KbT) + (EAccept / KbT)) + ((Vef / KbT) + ((KbT - (((EAccept * EAccept) / KbT) + (((EAccept / KbT) + -1.0) * (KbT * (-1.0 - (EAccept / KbT)))))) / Vef)))) - (mu / KbT))));
	elseif (NdChar <= 1.75e+74)
		tmp = t_1;
	else
		tmp = t_0 + (NaChar / (1.0 + ((Ev / KbT) - (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[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(N[(N[(mu + Vef), $MachinePrecision] + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.3e+153], t$95$1, If[LessEqual[NdChar, -8.6e+18], N[(t$95$0 + N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(Ev + N[(N[(EAccept * EAccept), $MachinePrecision] / N[(EAccept - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu + N[(N[(Vef * Vef), $MachinePrecision] / N[(EAccept - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.65e-17], t$95$1, If[LessEqual[NdChar, 12.0], N[(t$95$0 + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(N[(Ev / KbT), $MachinePrecision] + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(N[(KbT - N[(N[(N[(EAccept * EAccept), $MachinePrecision] / KbT), $MachinePrecision] + N[(N[(N[(EAccept / KbT), $MachinePrecision] + -1.0), $MachinePrecision] * N[(KbT * N[(-1.0 - N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.75e+74], t$95$1, N[(t$95$0 + N[(NaChar / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -1.2999999999999999e153 or -8.6e18 < NdChar < 1.65e-17 or 12 < NdChar < 1.75000000000000007e74

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

      3. exp-1-e 100.0%

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

      4. associate--r- 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in KbT around inf 69.7%

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

      1. log-E 69.7%

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

      2. *-lft-identity 69.7%

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

      3. associate--l+ 69.7%

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

      4. associate--l+ 69.7%

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

      5. sub-neg 69.7%

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

      6. +-commutative 69.7%

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

      7. mul-1-neg 69.7%

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

      8. associate-+r+ 69.7%

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

      9. mul-1-neg 69.7%

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

      10. +-commutative 69.7%

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

      11. sub-neg 69.7%

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

   8. Simplified 69.7%

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

   if -1.2999999999999999e153 < NdChar < -8.6e18

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 65.4%

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

      1. flip-+ 47.8%

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

   6. Applied egg-rr 47.8%

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

   7. Taylor expanded in KbT around 0 65.3%

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

      1. *-commutative 65.3%

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

      2. +-commutative 65.3%

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

      3. unpow2 65.3%

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

      4. +-commutative 65.3%

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

      5. unpow2 65.3%

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

   9. Simplified 65.3%

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

   if 1.65e-17 < NdChar < 12

   1. Initial program 99.0%

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

      1. neg-sub0 99.0%

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

      2. associate--r- 99.0%

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

      3. +-commutative 99.0%

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

      4. neg-sub0 99.0%

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

      5. sub-neg 99.0%

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

      6. associate--l- 99.0%

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

      7. unsub-neg 99.0%

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

      8. +-commutative 99.0%

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

      9. associate-+l+ 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in KbT around inf 71.6%

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

      1. flip-+ 57.3%

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in Vef around -inf 71.3%

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

      1. associate-+r+ 71.3%

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

      2. mul-1-neg 71.3%

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

      3. unsub-neg 71.3%

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

   9. Simplified 71.3%

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

   if 1.75000000000000007e74 < NdChar

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 60.9%

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

      1. flip-+ 55.1%

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

   6. Applied egg-rr 55.1%

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

   7. Taylor expanded in Ev around inf 67.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq -8.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{KbT \cdot NaChar}{\left(Ev + \frac{EAccept \cdot EAccept}{EAccept - Vef}\right) - \left(mu + \frac{Vef \cdot Vef}{EAccept - Vef}\right)}\\ \mathbf{elif}\;NdChar \leq 1.65 \cdot 10^{-17}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 12:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\left(\frac{Ev}{KbT} + \frac{EAccept}{KbT}\right) + \left(\frac{Vef}{KbT} + \frac{KbT - \left(\frac{EAccept \cdot EAccept}{KbT} + \left(\frac{EAccept}{KbT} + -1\right) \cdot \left(KbT \cdot \left(-1 - \frac{EAccept}{KbT}\right)\right)\right)}{Vef}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{+74}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} - \frac{mu}{KbT}\right)}\\ \end{array} \]

Alternative 10: 64.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -5800 \lor \neg \left(NaChar \leq 4.2 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{KbT \cdot NaChar}{\left(Ev + \frac{EAccept \cdot EAccept}{EAccept - Vef}\right) - \left(mu + \frac{Vef \cdot Vef}{EAccept - Vef}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -5800.0) (not (<= NaChar 4.2e-75)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
    (/ NdChar (+ 1.0 (+ 1.0 (/ (+ (+ mu Vef) (- EDonor Ec)) KbT)))))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))
    (/
     (* KbT NaChar)
     (-
      (+ Ev (/ (* EAccept EAccept) (- EAccept Vef)))
      (+ mu (/ (* Vef Vef) (- EAccept Vef))))))))

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 <= -5800.0) || !(NaChar <= 4.2e-75)) {
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	} else {
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))));
	}
	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 <= (-5800.0d0)) .or. (.not. (nachar <= 4.2d-75))) then
        tmp = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (((mu + vef) + (edonor - ec)) / kbt))))
    else
        tmp = (ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))) + ((kbt * nachar) / ((ev + ((eaccept * eaccept) / (eaccept - vef))) - (mu + ((vef * vef) / (eaccept - vef)))))
    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 <= -5800.0) || !(NaChar <= 4.2e-75)) {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	} else {
		tmp = (NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -5800.0) or not (NaChar <= 4.2e-75):
		tmp = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))))
	else:
		tmp = (NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -5800.0) || !(NaChar <= 4.2e-75))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(mu + Vef) + Float64(EDonor - Ec)) / KbT)))));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT)))) + Float64(Float64(KbT * NaChar) / Float64(Float64(Ev + Float64(Float64(EAccept * EAccept) / Float64(EAccept - Vef))) - Float64(mu + Float64(Float64(Vef * Vef) / Float64(EAccept - Vef))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -5800.0) || ~((NaChar <= 4.2e-75)))
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	else
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + ((KbT * NaChar) / ((Ev + ((EAccept * EAccept) / (EAccept - Vef))) - (mu + ((Vef * Vef) / (EAccept - Vef)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -5800.0], N[Not[LessEqual[NaChar, 4.2e-75]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(N[(N[(mu + Vef), $MachinePrecision] + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(KbT * NaChar), $MachinePrecision] / N[(N[(Ev + N[(N[(EAccept * EAccept), $MachinePrecision] / N[(EAccept - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu + N[(N[(Vef * Vef), $MachinePrecision] / N[(EAccept - Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -5800 or 4.2000000000000002e-75 < NaChar

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

      3. exp-1-e 100.0%

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

      4. associate--r- 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in KbT around inf 66.0%

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

      1. log-E 66.0%

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

      2. *-lft-identity 66.0%

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

      3. associate--l+ 66.0%

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

      4. associate--l+ 66.0%

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

      5. sub-neg 66.0%

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

      6. +-commutative 66.0%

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

      7. mul-1-neg 66.0%

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

      8. associate-+r+ 66.0%

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

      9. mul-1-neg 66.0%

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

      10. +-commutative 66.0%

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

      11. sub-neg 66.0%

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

   8. Simplified 66.0%

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

   if -5800 < NaChar < 4.2000000000000002e-75

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 65.0%

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

      1. flip-+ 58.3%

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

   6. Applied egg-rr 58.3%

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

   7. Taylor expanded in KbT around 0 63.7%

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

      1. *-commutative 63.7%

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

      2. +-commutative 63.7%

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

      3. unpow2 63.7%

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

      4. +-commutative 63.7%

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

      5. unpow2 63.7%

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

   9. Simplified 63.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -5800 \lor \neg \left(NaChar \leq 4.2 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{KbT \cdot NaChar}{\left(Ev + \frac{EAccept \cdot EAccept}{EAccept - Vef}\right) - \left(mu + \frac{Vef \cdot Vef}{EAccept - Vef}\right)}\\ \end{array} \]

Alternative 11: 61.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;NaChar \leq -3 \cdot 10^{+163}:\\ \;\;\;\;t_0 + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;NaChar \leq -8.8 \cdot 10^{+141} \lor \neg \left(NaChar \leq -0.52\right) \land NaChar \leq 4.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} - \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \end{array} \end{array} \]

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))))))
   (if (<= NaChar -3e+163)
     (+ t_0 (/ NdChar (+ 2.0 (/ EDonor KbT))))
     (if (or (<= NaChar -8.8e+141)
             (and (not (<= NaChar -0.52)) (<= NaChar 4.3e-75)))
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))
        (/ NaChar (+ 1.0 (- (/ Ev KbT) (/ mu KbT)))))
       (+ t_0 (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double tmp;
	if (NaChar <= -3e+163) {
		tmp = t_0 + (NdChar / (2.0 + (EDonor / KbT)));
	} else if ((NaChar <= -8.8e+141) || (!(NaChar <= -0.52) && (NaChar <= 4.3e-75))) {
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	} else {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (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 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    if (nachar <= (-3d+163)) then
        tmp = t_0 + (ndchar / (2.0d0 + (edonor / kbt)))
    else if ((nachar <= (-8.8d+141)) .or. (.not. (nachar <= (-0.52d0))) .and. (nachar <= 4.3d-75)) then
        tmp = (ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))) + (nachar / (1.0d0 + ((ev / kbt) - (mu / kbt))))
    else
        tmp = t_0 + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double tmp;
	if (NaChar <= -3e+163) {
		tmp = t_0 + (NdChar / (2.0 + (EDonor / KbT)));
	} else if ((NaChar <= -8.8e+141) || (!(NaChar <= -0.52) && (NaChar <= 4.3e-75))) {
		tmp = (NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	} else {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))
	tmp = 0
	if NaChar <= -3e+163:
		tmp = t_0 + (NdChar / (2.0 + (EDonor / KbT)))
	elif (NaChar <= -8.8e+141) or (not (NaChar <= -0.52) and (NaChar <= 4.3e-75)):
		tmp = (NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))))
	else:
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT))))
	tmp = 0.0
	if (NaChar <= -3e+163)
		tmp = Float64(t_0 + Float64(NdChar / Float64(2.0 + Float64(EDonor / KbT))));
	elseif ((NaChar <= -8.8e+141) || (!(NaChar <= -0.52) && (NaChar <= 4.3e-75)))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(Ev / KbT) - Float64(mu / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	tmp = 0.0;
	if (NaChar <= -3e+163)
		tmp = t_0 + (NdChar / (2.0 + (EDonor / KbT)));
	elseif ((NaChar <= -8.8e+141) || (~((NaChar <= -0.52)) && (NaChar <= 4.3e-75)))
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	else
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -3e+163], N[(t$95$0 + N[(NdChar / N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[NaChar, -8.8e+141], And[N[Not[LessEqual[NaChar, -0.52]], $MachinePrecision], LessEqual[NaChar, 4.3e-75]]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -3.00000000000000013e163

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in EDonor around inf 73.2%

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

   5. Taylor expanded in EDonor around 0 66.8%

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

   if -3.00000000000000013e163 < NaChar < -8.8e141 or -0.52000000000000002 < NaChar < 4.2999999999999999e-75

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 65.6%

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

      1. flip-+ 58.4%

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

   6. Applied egg-rr 58.4%

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

   7. Taylor expanded in Ev around inf 66.4%

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

   if -8.8e141 < NaChar < -0.52000000000000002 or 4.2999999999999999e-75 < NaChar

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in mu around inf 70.6%

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

   5. Taylor expanded in mu around 0 61.5%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -3 \cdot 10^{+163}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;NaChar \leq -8.8 \cdot 10^{+141} \lor \neg \left(NaChar \leq -0.52\right) \land NaChar \leq 4.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} - \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \end{array} \]

Alternative 12: 65.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -0.05 \lor \neg \left(NaChar \leq 1.85 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} - \frac{mu}{KbT}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -0.05) (not (<= NaChar 1.85e-75)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
    (/ NdChar (+ 1.0 (+ 1.0 (/ (+ (+ mu Vef) (- EDonor Ec)) KbT)))))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))
    (/ NaChar (+ 1.0 (- (/ Ev KbT) (/ mu 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 <= -0.05) || !(NaChar <= 1.85e-75)) {
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	} else {
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (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) :: tmp
    if ((nachar <= (-0.05d0)) .or. (.not. (nachar <= 1.85d-75))) then
        tmp = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / (1.0d0 + (1.0d0 + (((mu + vef) + (edonor - ec)) / kbt))))
    else
        tmp = (ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))) + (nachar / (1.0d0 + ((ev / kbt) - (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 tmp;
	if ((NaChar <= -0.05) || !(NaChar <= 1.85e-75)) {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	} else {
		tmp = (NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -0.05) or not (NaChar <= 1.85e-75):
		tmp = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))))
	else:
		tmp = (NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -0.05) || !(NaChar <= 1.85e-75))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(mu + Vef) + Float64(EDonor - Ec)) / KbT)))));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(Ev / KbT) - Float64(mu / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -0.05) || ~((NaChar <= 1.85e-75)))
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + (1.0 + (((mu + Vef) + (EDonor - Ec)) / KbT))));
	else
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) - (mu / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -0.05], N[Not[LessEqual[NaChar, 1.85e-75]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(1.0 + N[(N[(N[(mu + Vef), $MachinePrecision] + N[(EDonor - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -0.050000000000000003 or 1.85000000000000012e-75 < NaChar

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

      3. exp-1-e 100.0%

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

      4. associate--r- 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in KbT around inf 65.8%

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

      1. log-E 65.8%

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

      2. *-lft-identity 65.8%

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

      3. associate--l+ 65.8%

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

      4. associate--l+ 65.8%

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

      5. sub-neg 65.8%

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

      6. +-commutative 65.8%

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

      7. mul-1-neg 65.8%

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

      8. associate-+r+ 65.8%

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

      9. mul-1-neg 65.8%

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

      10. +-commutative 65.8%

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

      11. sub-neg 65.8%

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

   8. Simplified 65.8%

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

   if -0.050000000000000003 < NaChar < 1.85000000000000012e-75

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 66.1%

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

      1. flip-+ 58.4%

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

   6. Applied egg-rr 58.4%

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

   7. Taylor expanded in Ev around inf 66.0%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -0.05 \lor \neg \left(NaChar \leq 1.85 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{\left(mu + Vef\right) + \left(EDonor - Ec\right)}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} - \frac{mu}{KbT}\right)}\\ \end{array} \]

Alternative 13: 49.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_2 := t_1 + NaChar \cdot 0.5\\ \mathbf{if}\;KbT \leq -4.5 \cdot 10^{+156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;KbT \leq -7.8 \cdot 10^{+78}:\\ \;\;\;\;t_0 + -0.25 \cdot \frac{Vef}{\frac{KbT}{NdChar}}\\ \mathbf{elif}\;KbT \leq -2.1 \cdot 10^{-111}:\\ \;\;\;\;t_0 + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 4.8 \cdot 10^{-182}:\\ \;\;\;\;t_1 + \frac{NaChar}{\frac{Vef}{KbT}}\\ \mathbf{elif}\;KbT \leq 8.6 \cdot 10^{-8}:\\ \;\;\;\;t_1 + KbT \cdot \frac{NaChar}{EAccept}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT)))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT)))))
        (t_2 (+ t_1 (* NaChar 0.5))))
   (if (<= KbT -4.5e+156)
     t_2
     (if (<= KbT -7.8e+78)
       (+ t_0 (* -0.25 (/ Vef (/ KbT NdChar))))
       (if (<= KbT -2.1e-111)
         (+ t_0 (* NdChar 0.5))
         (if (<= KbT 4.8e-182)
           (+ t_1 (/ NaChar (/ Vef KbT)))
           (if (<= KbT 8.6e-8) (+ t_1 (* KbT (/ NaChar EAccept))) t_2)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_2 = t_1 + (NaChar * 0.5);
	double tmp;
	if (KbT <= -4.5e+156) {
		tmp = t_2;
	} else if (KbT <= -7.8e+78) {
		tmp = t_0 + (-0.25 * (Vef / (KbT / NdChar)));
	} else if (KbT <= -2.1e-111) {
		tmp = t_0 + (NdChar * 0.5);
	} else if (KbT <= 4.8e-182) {
		tmp = t_1 + (NaChar / (Vef / KbT));
	} else if (KbT <= 8.6e-8) {
		tmp = t_1 + (KbT * (NaChar / EAccept));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    t_1 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_2 = t_1 + (nachar * 0.5d0)
    if (kbt <= (-4.5d+156)) then
        tmp = t_2
    else if (kbt <= (-7.8d+78)) then
        tmp = t_0 + ((-0.25d0) * (vef / (kbt / ndchar)))
    else if (kbt <= (-2.1d-111)) then
        tmp = t_0 + (ndchar * 0.5d0)
    else if (kbt <= 4.8d-182) then
        tmp = t_1 + (nachar / (vef / kbt))
    else if (kbt <= 8.6d-8) then
        tmp = t_1 + (kbt * (nachar / eaccept))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_2 = t_1 + (NaChar * 0.5);
	double tmp;
	if (KbT <= -4.5e+156) {
		tmp = t_2;
	} else if (KbT <= -7.8e+78) {
		tmp = t_0 + (-0.25 * (Vef / (KbT / NdChar)));
	} else if (KbT <= -2.1e-111) {
		tmp = t_0 + (NdChar * 0.5);
	} else if (KbT <= 4.8e-182) {
		tmp = t_1 + (NaChar / (Vef / KbT));
	} else if (KbT <= 8.6e-8) {
		tmp = t_1 + (KbT * (NaChar / EAccept));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))
	t_1 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	t_2 = t_1 + (NaChar * 0.5)
	tmp = 0
	if KbT <= -4.5e+156:
		tmp = t_2
	elif KbT <= -7.8e+78:
		tmp = t_0 + (-0.25 * (Vef / (KbT / NdChar)))
	elif KbT <= -2.1e-111:
		tmp = t_0 + (NdChar * 0.5)
	elif KbT <= 4.8e-182:
		tmp = t_1 + (NaChar / (Vef / KbT))
	elif KbT <= 8.6e-8:
		tmp = t_1 + (KbT * (NaChar / EAccept))
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT))))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	t_2 = Float64(t_1 + Float64(NaChar * 0.5))
	tmp = 0.0
	if (KbT <= -4.5e+156)
		tmp = t_2;
	elseif (KbT <= -7.8e+78)
		tmp = Float64(t_0 + Float64(-0.25 * Float64(Vef / Float64(KbT / NdChar))));
	elseif (KbT <= -2.1e-111)
		tmp = Float64(t_0 + Float64(NdChar * 0.5));
	elseif (KbT <= 4.8e-182)
		tmp = Float64(t_1 + Float64(NaChar / Float64(Vef / KbT)));
	elseif (KbT <= 8.6e-8)
		tmp = Float64(t_1 + Float64(KbT * Float64(NaChar / EAccept)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	t_1 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	t_2 = t_1 + (NaChar * 0.5);
	tmp = 0.0;
	if (KbT <= -4.5e+156)
		tmp = t_2;
	elseif (KbT <= -7.8e+78)
		tmp = t_0 + (-0.25 * (Vef / (KbT / NdChar)));
	elseif (KbT <= -2.1e-111)
		tmp = t_0 + (NdChar * 0.5);
	elseif (KbT <= 4.8e-182)
		tmp = t_1 + (NaChar / (Vef / KbT));
	elseif (KbT <= 8.6e-8)
		tmp = t_1 + (KbT * (NaChar / EAccept));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -4.5e+156], t$95$2, If[LessEqual[KbT, -7.8e+78], N[(t$95$0 + N[(-0.25 * N[(Vef / N[(KbT / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -2.1e-111], N[(t$95$0 + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4.8e-182], N[(t$95$1 + N[(NaChar / N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 8.6e-8], N[(t$95$1 + N[(KbT * N[(NaChar / EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if KbT < -4.50000000000000031e156 or 8.6000000000000002e-8 < KbT

   1. Initial program 99.8%

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

      1. neg-sub0 99.8%

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

      2. associate--r- 99.8%

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

      3. +-commutative 99.8%

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

      4. neg-sub0 99.8%

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

      5. sub-neg 99.8%

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

      6. associate--l- 99.8%

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

      7. unsub-neg 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in KbT around inf 65.7%

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

   5. Taylor expanded in KbT around inf 65.7%

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

   if -4.50000000000000031e156 < KbT < -7.8000000000000008e78

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 33.4%

      \[\leadsto \color{blue}{\left(-0.25 \cdot \frac{NdChar \cdot \left(\left(Vef + \left(mu + EDonor\right)\right) - Ec\right)}{KbT} + 0.5 \cdot NdChar\right)} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   5. Taylor expanded in Vef around inf 37.8%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{Vef \cdot NdChar}{KbT}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   6. Step-by-step derivation

      1. associate-/l* 54.0%

         \[\leadsto -0.25 \cdot \color{blue}{\frac{Vef}{\frac{KbT}{NdChar}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   7. Simplified 54.0%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{Vef}{\frac{KbT}{NdChar}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   if -7.8000000000000008e78 < KbT < -2.0999999999999999e-111

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 42.8%

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

      1. *-commutative 8.1%

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

   6. Simplified 42.8%

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

   if -2.0999999999999999e-111 < KbT < 4.7999999999999997e-182

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 43.3%

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

   5. Taylor expanded in Vef around inf 52.9%

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

      1. associate-/l* 59.1%

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

   7. Simplified 59.1%

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

   if 4.7999999999999997e-182 < KbT < 8.6000000000000002e-8

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 44.7%

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

   5. Taylor expanded in EAccept around inf 40.2%

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

      1. associate-/l* 40.0%

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

   7. Simplified 40.0%

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

      1. associate-/r/ 40.1%

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

   9. Applied egg-rr 40.1%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq -7.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + -0.25 \cdot \frac{Vef}{\frac{KbT}{NdChar}}\\ \mathbf{elif}\;KbT \leq -2.1 \cdot 10^{-111}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 4.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{Vef}{KbT}}\\ \mathbf{elif}\;KbT \leq 8.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 14: 49.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_2 := t_1 + NaChar \cdot 0.5\\ \mathbf{if}\;KbT \leq -2.5 \cdot 10^{+157}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;KbT \leq -1.28 \cdot 10^{+79}:\\ \;\;\;\;t_0 + -0.25 \cdot \frac{NdChar \cdot mu}{KbT}\\ \mathbf{elif}\;KbT \leq -3.2 \cdot 10^{-111}:\\ \;\;\;\;t_0 + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 2.8 \cdot 10^{-180}:\\ \;\;\;\;t_1 + \frac{NaChar}{\frac{Vef}{KbT}}\\ \mathbf{elif}\;KbT \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;t_1 + KbT \cdot \frac{NaChar}{EAccept}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT)))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT)))))
        (t_2 (+ t_1 (* NaChar 0.5))))
   (if (<= KbT -2.5e+157)
     t_2
     (if (<= KbT -1.28e+79)
       (+ t_0 (* -0.25 (/ (* NdChar mu) KbT)))
       (if (<= KbT -3.2e-111)
         (+ t_0 (* NdChar 0.5))
         (if (<= KbT 2.8e-180)
           (+ t_1 (/ NaChar (/ Vef KbT)))
           (if (<= KbT 2.4e-5) (+ t_1 (* KbT (/ NaChar EAccept))) t_2)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_2 = t_1 + (NaChar * 0.5);
	double tmp;
	if (KbT <= -2.5e+157) {
		tmp = t_2;
	} else if (KbT <= -1.28e+79) {
		tmp = t_0 + (-0.25 * ((NdChar * mu) / KbT));
	} else if (KbT <= -3.2e-111) {
		tmp = t_0 + (NdChar * 0.5);
	} else if (KbT <= 2.8e-180) {
		tmp = t_1 + (NaChar / (Vef / KbT));
	} else if (KbT <= 2.4e-5) {
		tmp = t_1 + (KbT * (NaChar / EAccept));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    t_1 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_2 = t_1 + (nachar * 0.5d0)
    if (kbt <= (-2.5d+157)) then
        tmp = t_2
    else if (kbt <= (-1.28d+79)) then
        tmp = t_0 + ((-0.25d0) * ((ndchar * mu) / kbt))
    else if (kbt <= (-3.2d-111)) then
        tmp = t_0 + (ndchar * 0.5d0)
    else if (kbt <= 2.8d-180) then
        tmp = t_1 + (nachar / (vef / kbt))
    else if (kbt <= 2.4d-5) then
        tmp = t_1 + (kbt * (nachar / eaccept))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_2 = t_1 + (NaChar * 0.5);
	double tmp;
	if (KbT <= -2.5e+157) {
		tmp = t_2;
	} else if (KbT <= -1.28e+79) {
		tmp = t_0 + (-0.25 * ((NdChar * mu) / KbT));
	} else if (KbT <= -3.2e-111) {
		tmp = t_0 + (NdChar * 0.5);
	} else if (KbT <= 2.8e-180) {
		tmp = t_1 + (NaChar / (Vef / KbT));
	} else if (KbT <= 2.4e-5) {
		tmp = t_1 + (KbT * (NaChar / EAccept));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))
	t_1 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	t_2 = t_1 + (NaChar * 0.5)
	tmp = 0
	if KbT <= -2.5e+157:
		tmp = t_2
	elif KbT <= -1.28e+79:
		tmp = t_0 + (-0.25 * ((NdChar * mu) / KbT))
	elif KbT <= -3.2e-111:
		tmp = t_0 + (NdChar * 0.5)
	elif KbT <= 2.8e-180:
		tmp = t_1 + (NaChar / (Vef / KbT))
	elif KbT <= 2.4e-5:
		tmp = t_1 + (KbT * (NaChar / EAccept))
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT))))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	t_2 = Float64(t_1 + Float64(NaChar * 0.5))
	tmp = 0.0
	if (KbT <= -2.5e+157)
		tmp = t_2;
	elseif (KbT <= -1.28e+79)
		tmp = Float64(t_0 + Float64(-0.25 * Float64(Float64(NdChar * mu) / KbT)));
	elseif (KbT <= -3.2e-111)
		tmp = Float64(t_0 + Float64(NdChar * 0.5));
	elseif (KbT <= 2.8e-180)
		tmp = Float64(t_1 + Float64(NaChar / Float64(Vef / KbT)));
	elseif (KbT <= 2.4e-5)
		tmp = Float64(t_1 + Float64(KbT * Float64(NaChar / EAccept)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	t_1 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	t_2 = t_1 + (NaChar * 0.5);
	tmp = 0.0;
	if (KbT <= -2.5e+157)
		tmp = t_2;
	elseif (KbT <= -1.28e+79)
		tmp = t_0 + (-0.25 * ((NdChar * mu) / KbT));
	elseif (KbT <= -3.2e-111)
		tmp = t_0 + (NdChar * 0.5);
	elseif (KbT <= 2.8e-180)
		tmp = t_1 + (NaChar / (Vef / KbT));
	elseif (KbT <= 2.4e-5)
		tmp = t_1 + (KbT * (NaChar / EAccept));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -2.5e+157], t$95$2, If[LessEqual[KbT, -1.28e+79], N[(t$95$0 + N[(-0.25 * N[(N[(NdChar * mu), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -3.2e-111], N[(t$95$0 + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.8e-180], N[(t$95$1 + N[(NaChar / N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.4e-5], N[(t$95$1 + N[(KbT * N[(NaChar / EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if KbT < -2.49999999999999988e157 or 2.4000000000000001e-5 < KbT

   1. Initial program 99.8%

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

      1. neg-sub0 99.8%

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

      2. associate--r- 99.8%

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

      3. +-commutative 99.8%

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

      4. neg-sub0 99.8%

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

      5. sub-neg 99.8%

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

      6. associate--l- 99.8%

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

      7. unsub-neg 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in KbT around inf 65.7%

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

   5. Taylor expanded in KbT around inf 65.7%

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

   if -2.49999999999999988e157 < KbT < -1.27999999999999998e79

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 33.4%

      \[\leadsto \color{blue}{\left(-0.25 \cdot \frac{NdChar \cdot \left(\left(Vef + \left(mu + EDonor\right)\right) - Ec\right)}{KbT} + 0.5 \cdot NdChar\right)} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   5. Taylor expanded in mu around inf 55.7%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{mu \cdot NdChar}{KbT}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   if -1.27999999999999998e79 < KbT < -3.1999999999999998e-111

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 42.8%

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

      1. *-commutative 8.1%

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

   6. Simplified 42.8%

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

   if -3.1999999999999998e-111 < KbT < 2.79999999999999997e-180

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 43.3%

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

   5. Taylor expanded in Vef around inf 52.9%

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

      1. associate-/l* 59.1%

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

   7. Simplified 59.1%

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

   if 2.79999999999999997e-180 < KbT < 2.4000000000000001e-5

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 44.7%

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

   5. Taylor expanded in EAccept around inf 40.2%

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

      1. associate-/l* 40.0%

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

   7. Simplified 40.0%

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

      1. associate-/r/ 40.1%

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

   9. Applied egg-rr 40.1%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq -1.28 \cdot 10^{+79}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + -0.25 \cdot \frac{NdChar \cdot mu}{KbT}\\ \mathbf{elif}\;KbT \leq -3.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 2.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{Vef}{KbT}}\\ \mathbf{elif}\;KbT \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 15: 48.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_2 := t_1 + NaChar \cdot 0.5\\ \mathbf{if}\;KbT \leq -4.8 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;KbT \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;t_0 + 0.25 \cdot \frac{NdChar}{\frac{KbT}{Ec}}\\ \mathbf{elif}\;KbT \leq -1.5 \cdot 10^{-111}:\\ \;\;\;\;t_0 + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 4.7 \cdot 10^{-182}:\\ \;\;\;\;t_1 + \frac{NaChar}{\frac{Vef}{KbT}}\\ \mathbf{elif}\;KbT \leq 0.00048:\\ \;\;\;\;t_1 + KbT \cdot \frac{NaChar}{EAccept}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT)))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT)))))
        (t_2 (+ t_1 (* NaChar 0.5))))
   (if (<= KbT -4.8e+151)
     t_2
     (if (<= KbT -6.4e-9)
       (+ t_0 (* 0.25 (/ NdChar (/ KbT Ec))))
       (if (<= KbT -1.5e-111)
         (+ t_0 (* NdChar 0.5))
         (if (<= KbT 4.7e-182)
           (+ t_1 (/ NaChar (/ Vef KbT)))
           (if (<= KbT 0.00048) (+ t_1 (* KbT (/ NaChar EAccept))) t_2)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_2 = t_1 + (NaChar * 0.5);
	double tmp;
	if (KbT <= -4.8e+151) {
		tmp = t_2;
	} else if (KbT <= -6.4e-9) {
		tmp = t_0 + (0.25 * (NdChar / (KbT / Ec)));
	} else if (KbT <= -1.5e-111) {
		tmp = t_0 + (NdChar * 0.5);
	} else if (KbT <= 4.7e-182) {
		tmp = t_1 + (NaChar / (Vef / KbT));
	} else if (KbT <= 0.00048) {
		tmp = t_1 + (KbT * (NaChar / EAccept));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    t_1 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_2 = t_1 + (nachar * 0.5d0)
    if (kbt <= (-4.8d+151)) then
        tmp = t_2
    else if (kbt <= (-6.4d-9)) then
        tmp = t_0 + (0.25d0 * (ndchar / (kbt / ec)))
    else if (kbt <= (-1.5d-111)) then
        tmp = t_0 + (ndchar * 0.5d0)
    else if (kbt <= 4.7d-182) then
        tmp = t_1 + (nachar / (vef / kbt))
    else if (kbt <= 0.00048d0) then
        tmp = t_1 + (kbt * (nachar / eaccept))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_2 = t_1 + (NaChar * 0.5);
	double tmp;
	if (KbT <= -4.8e+151) {
		tmp = t_2;
	} else if (KbT <= -6.4e-9) {
		tmp = t_0 + (0.25 * (NdChar / (KbT / Ec)));
	} else if (KbT <= -1.5e-111) {
		tmp = t_0 + (NdChar * 0.5);
	} else if (KbT <= 4.7e-182) {
		tmp = t_1 + (NaChar / (Vef / KbT));
	} else if (KbT <= 0.00048) {
		tmp = t_1 + (KbT * (NaChar / EAccept));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))
	t_1 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	t_2 = t_1 + (NaChar * 0.5)
	tmp = 0
	if KbT <= -4.8e+151:
		tmp = t_2
	elif KbT <= -6.4e-9:
		tmp = t_0 + (0.25 * (NdChar / (KbT / Ec)))
	elif KbT <= -1.5e-111:
		tmp = t_0 + (NdChar * 0.5)
	elif KbT <= 4.7e-182:
		tmp = t_1 + (NaChar / (Vef / KbT))
	elif KbT <= 0.00048:
		tmp = t_1 + (KbT * (NaChar / EAccept))
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT))))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	t_2 = Float64(t_1 + Float64(NaChar * 0.5))
	tmp = 0.0
	if (KbT <= -4.8e+151)
		tmp = t_2;
	elseif (KbT <= -6.4e-9)
		tmp = Float64(t_0 + Float64(0.25 * Float64(NdChar / Float64(KbT / Ec))));
	elseif (KbT <= -1.5e-111)
		tmp = Float64(t_0 + Float64(NdChar * 0.5));
	elseif (KbT <= 4.7e-182)
		tmp = Float64(t_1 + Float64(NaChar / Float64(Vef / KbT)));
	elseif (KbT <= 0.00048)
		tmp = Float64(t_1 + Float64(KbT * Float64(NaChar / EAccept)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	t_1 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	t_2 = t_1 + (NaChar * 0.5);
	tmp = 0.0;
	if (KbT <= -4.8e+151)
		tmp = t_2;
	elseif (KbT <= -6.4e-9)
		tmp = t_0 + (0.25 * (NdChar / (KbT / Ec)));
	elseif (KbT <= -1.5e-111)
		tmp = t_0 + (NdChar * 0.5);
	elseif (KbT <= 4.7e-182)
		tmp = t_1 + (NaChar / (Vef / KbT));
	elseif (KbT <= 0.00048)
		tmp = t_1 + (KbT * (NaChar / EAccept));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -4.8e+151], t$95$2, If[LessEqual[KbT, -6.4e-9], N[(t$95$0 + N[(0.25 * N[(NdChar / N[(KbT / Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -1.5e-111], N[(t$95$0 + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4.7e-182], N[(t$95$1 + N[(NaChar / N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 0.00048], N[(t$95$1 + N[(KbT * N[(NaChar / EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if KbT < -4.8000000000000002e151 or 4.80000000000000012e-4 < KbT

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 66.4%

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

   5. Taylor expanded in KbT around inf 66.4%

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

   if -4.8000000000000002e151 < KbT < -6.40000000000000023e-9

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 29.6%

      \[\leadsto \color{blue}{\left(-0.25 \cdot \frac{NdChar \cdot \left(\left(Vef + \left(mu + EDonor\right)\right) - Ec\right)}{KbT} + 0.5 \cdot NdChar\right)} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   5. Taylor expanded in Ec around inf 50.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{NdChar \cdot Ec}{KbT}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   6. Step-by-step derivation

      1. associate-/l* 50.2%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{NdChar}{\frac{KbT}{Ec}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   7. Simplified 50.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{NdChar}{\frac{KbT}{Ec}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   if -6.40000000000000023e-9 < KbT < -1.50000000000000004e-111

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 47.2%

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

      1. *-commutative 6.6%

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

   6. Simplified 47.2%

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

   if -1.50000000000000004e-111 < KbT < 4.7e-182

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 43.3%

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

   5. Taylor expanded in Vef around inf 52.9%

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

      1. associate-/l* 59.1%

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

   7. Simplified 59.1%

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

   if 4.7e-182 < KbT < 4.80000000000000012e-4

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 44.7%

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

   5. Taylor expanded in EAccept around inf 40.2%

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

      1. associate-/l* 40.0%

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

   7. Simplified 40.0%

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

      1. associate-/r/ 40.1%

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

   9. Applied egg-rr 40.1%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + 0.25 \cdot \frac{NdChar}{\frac{KbT}{Ec}}\\ \mathbf{elif}\;KbT \leq -1.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 4.7 \cdot 10^{-182}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{Vef}{KbT}}\\ \mathbf{elif}\;KbT \leq 0.00048:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 16: 56.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -6.6 \cdot 10^{+133}:\\ \;\;\;\;t_0 + NaChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq -1750000000:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 8.2 \cdot 10^{+182}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{NaChar}{\frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))))
   (if (<= NdChar -6.6e+133)
     (+ t_0 (* NaChar 0.5))
     (if (<= NdChar -1750000000.0)
       (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
       (if (<= NdChar 8.2e+182)
         (+
          (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
          (/ NdChar (+ 2.0 (/ EDonor KbT))))
         (- t_0 (/ NaChar (/ 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 + ((Vef + EDonor) - Ec)) / KbT)));
	double tmp;
	if (NdChar <= -6.6e+133) {
		tmp = t_0 + (NaChar * 0.5);
	} else if (NdChar <= -1750000000.0) {
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	} else if (NdChar <= 8.2e+182) {
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	} else {
		tmp = t_0 - (NaChar / (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 + ((vef + edonor) - ec)) / kbt)))
    if (ndchar <= (-6.6d+133)) then
        tmp = t_0 + (nachar * 0.5d0)
    else if (ndchar <= (-1750000000.0d0)) then
        tmp = ndchar / (1.0d0 + exp((edonor / kbt)))
    else if (ndchar <= 8.2d+182) then
        tmp = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / (2.0d0 + (edonor / kbt)))
    else
        tmp = t_0 - (nachar / (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 + ((Vef + EDonor) - Ec)) / KbT)));
	double tmp;
	if (NdChar <= -6.6e+133) {
		tmp = t_0 + (NaChar * 0.5);
	} else if (NdChar <= -1750000000.0) {
		tmp = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	} else if (NdChar <= 8.2e+182) {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	} else {
		tmp = t_0 - (NaChar / (mu / KbT));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	tmp = 0
	if NdChar <= -6.6e+133:
		tmp = t_0 + (NaChar * 0.5)
	elif NdChar <= -1750000000.0:
		tmp = NdChar / (1.0 + math.exp((EDonor / KbT)))
	elif NdChar <= 8.2e+182:
		tmp = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)))
	else:
		tmp = t_0 - (NaChar / (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(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	tmp = 0.0
	if (NdChar <= -6.6e+133)
		tmp = Float64(t_0 + Float64(NaChar * 0.5));
	elseif (NdChar <= -1750000000.0)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))));
	elseif (NdChar <= 8.2e+182)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / Float64(2.0 + Float64(EDonor / KbT))));
	else
		tmp = Float64(t_0 - Float64(NaChar / Float64(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 + ((Vef + EDonor) - Ec)) / KbT)));
	tmp = 0.0;
	if (NdChar <= -6.6e+133)
		tmp = t_0 + (NaChar * 0.5);
	elseif (NdChar <= -1750000000.0)
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	elseif (NdChar <= 8.2e+182)
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (2.0 + (EDonor / KbT)));
	else
		tmp = t_0 - (NaChar / (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[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -6.6e+133], N[(t$95$0 + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -1750000000.0], N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 8.2e+182], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(NaChar / N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -6.6e133

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. associate--r- 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. unsub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 56.3%

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

   5. Taylor expanded in KbT around inf 62.9%

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

   if -6.6e133 < NdChar < -1.75e9

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. associate--r- 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 61.0%

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

   5. Taylor expanded in EAccept around inf 49.8%

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

      1. associate-/l* 49.0%

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

   7. Simplified 49.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   8. Taylor expanded in EDonor around inf 37.7%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   9. Taylor expanded in NdChar around inf 61.7%

      \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}} \]

   if -1.75e9 < NdChar < 8.20000000000000006e182

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in EDonor around inf 73.6%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   5. Taylor expanded in EDonor around 0 57.9%

      \[\leadsto \frac{NdChar}{\color{blue}{2 + \frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   if 8.20000000000000006e182 < NdChar

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 72.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in mu around inf 56.5%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{-1 \cdot \frac{NaChar \cdot KbT}{mu}} \]
   6. Step-by-step derivation

      1. mul-1-neg 56.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\left(-\frac{NaChar \cdot KbT}{mu}\right)} \]

      2. associate-/l* 63.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \left(-\color{blue}{\frac{NaChar}{\frac{mu}{KbT}}}\right) \]

      3. distribute-neg-frac 63.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{-NaChar}{\frac{mu}{KbT}}} \]

   7. Simplified 63.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{-NaChar}{\frac{mu}{KbT}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -6.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq -1750000000:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 8.2 \cdot 10^{+182}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} - \frac{NaChar}{\frac{mu}{KbT}}\\ \end{array} \]

Alternative 17: 56.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_1 := t_0 + NaChar \cdot 0.5\\ t_2 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{if}\;NaChar \leq -400:\\ \;\;\;\;t_2\\ \mathbf{elif}\;NaChar \leq -2.9 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NaChar \leq 2.6 \cdot 10^{-305}:\\ \;\;\;\;t_0 + KbT \cdot \frac{NaChar}{EAccept}\\ \mathbf{elif}\;NaChar \leq 2.7 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT)))))
        (t_1 (+ t_0 (* NaChar 0.5)))
        (t_2
         (+
          (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
          (* NdChar 0.5))))
   (if (<= NaChar -400.0)
     t_2
     (if (<= NaChar -2.9e-244)
       t_1
       (if (<= NaChar 2.6e-305)
         (+ t_0 (* KbT (/ NaChar EAccept)))
         (if (<= NaChar 2.7e-52) t_1 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_1 = t_0 + (NaChar * 0.5);
	double t_2 = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	double tmp;
	if (NaChar <= -400.0) {
		tmp = t_2;
	} else if (NaChar <= -2.9e-244) {
		tmp = t_1;
	} else if (NaChar <= 2.6e-305) {
		tmp = t_0 + (KbT * (NaChar / EAccept));
	} else if (NaChar <= 2.7e-52) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_1 = t_0 + (nachar * 0.5d0)
    t_2 = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar * 0.5d0)
    if (nachar <= (-400.0d0)) then
        tmp = t_2
    else if (nachar <= (-2.9d-244)) then
        tmp = t_1
    else if (nachar <= 2.6d-305) then
        tmp = t_0 + (kbt * (nachar / eaccept))
    else if (nachar <= 2.7d-52) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_1 = t_0 + (NaChar * 0.5);
	double t_2 = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	double tmp;
	if (NaChar <= -400.0) {
		tmp = t_2;
	} else if (NaChar <= -2.9e-244) {
		tmp = t_1;
	} else if (NaChar <= 2.6e-305) {
		tmp = t_0 + (KbT * (NaChar / EAccept));
	} else if (NaChar <= 2.7e-52) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	t_1 = t_0 + (NaChar * 0.5)
	t_2 = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5)
	tmp = 0
	if NaChar <= -400.0:
		tmp = t_2
	elif NaChar <= -2.9e-244:
		tmp = t_1
	elif NaChar <= 2.6e-305:
		tmp = t_0 + (KbT * (NaChar / EAccept))
	elif NaChar <= 2.7e-52:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	t_1 = Float64(t_0 + Float64(NaChar * 0.5))
	t_2 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar * 0.5))
	tmp = 0.0
	if (NaChar <= -400.0)
		tmp = t_2;
	elseif (NaChar <= -2.9e-244)
		tmp = t_1;
	elseif (NaChar <= 2.6e-305)
		tmp = Float64(t_0 + Float64(KbT * Float64(NaChar / EAccept)));
	elseif (NaChar <= 2.7e-52)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	t_1 = t_0 + (NaChar * 0.5);
	t_2 = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	tmp = 0.0;
	if (NaChar <= -400.0)
		tmp = t_2;
	elseif (NaChar <= -2.9e-244)
		tmp = t_1;
	elseif (NaChar <= 2.6e-305)
		tmp = t_0 + (KbT * (NaChar / EAccept));
	elseif (NaChar <= 2.7e-52)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -400.0], t$95$2, If[LessEqual[NaChar, -2.9e-244], t$95$1, If[LessEqual[NaChar, 2.6e-305], N[(t$95$0 + N[(KbT * N[(NaChar / EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 2.7e-52], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -400 or 2.70000000000000009e-52 < NaChar

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 52.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.7%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 52.1%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   if -400 < NaChar < -2.89999999999999996e-244 or 2.6000000000000002e-305 < NaChar < 2.70000000000000009e-52

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 65.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in KbT around inf 58.6%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{0.5 \cdot NaChar} \]

   if -2.89999999999999996e-244 < NaChar < 2.6000000000000002e-305

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 76.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in EAccept around inf 86.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar \cdot KbT}{EAccept}} \]
   6. Step-by-step derivation

      1. associate-/l* 76.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   7. Simplified 76.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{EAccept} \cdot KbT} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{EAccept} \cdot KbT} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -400:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq -2.9 \cdot 10^{-244}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq 2.6 \cdot 10^{-305}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept}\\ \mathbf{elif}\;NaChar \leq 2.7 \cdot 10^{-52}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \]

Alternative 18: 50.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ \mathbf{if}\;KbT \leq -4.9 \cdot 10^{-111}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 8.6 \cdot 10^{-181}:\\ \;\;\;\;t_0 + \frac{NaChar}{\frac{Vef}{KbT}}\\ \mathbf{elif}\;KbT \leq 0.00014:\\ \;\;\;\;t_0 + KbT \cdot \frac{NaChar}{EAccept}\\ \mathbf{else}:\\ \;\;\;\;t_0 + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))))
   (if (<= KbT -4.9e-111)
     (+
      (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
      (* NdChar 0.5))
     (if (<= KbT 8.6e-181)
       (+ t_0 (/ NaChar (/ Vef KbT)))
       (if (<= KbT 0.00014)
         (+ t_0 (* KbT (/ NaChar EAccept)))
         (+ t_0 (* 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 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double tmp;
	if (KbT <= -4.9e-111) {
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	} else if (KbT <= 8.6e-181) {
		tmp = t_0 + (NaChar / (Vef / KbT));
	} else if (KbT <= 0.00014) {
		tmp = t_0 + (KbT * (NaChar / EAccept));
	} else {
		tmp = t_0 + (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) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    if (kbt <= (-4.9d-111)) then
        tmp = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar * 0.5d0)
    else if (kbt <= 8.6d-181) then
        tmp = t_0 + (nachar / (vef / kbt))
    else if (kbt <= 0.00014d0) then
        tmp = t_0 + (kbt * (nachar / eaccept))
    else
        tmp = t_0 + (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 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double tmp;
	if (KbT <= -4.9e-111) {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	} else if (KbT <= 8.6e-181) {
		tmp = t_0 + (NaChar / (Vef / KbT));
	} else if (KbT <= 0.00014) {
		tmp = t_0 + (KbT * (NaChar / EAccept));
	} else {
		tmp = t_0 + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	tmp = 0
	if KbT <= -4.9e-111:
		tmp = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5)
	elif KbT <= 8.6e-181:
		tmp = t_0 + (NaChar / (Vef / KbT))
	elif KbT <= 0.00014:
		tmp = t_0 + (KbT * (NaChar / EAccept))
	else:
		tmp = t_0 + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	tmp = 0.0
	if (KbT <= -4.9e-111)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar * 0.5));
	elseif (KbT <= 8.6e-181)
		tmp = Float64(t_0 + Float64(NaChar / Float64(Vef / KbT)));
	elseif (KbT <= 0.00014)
		tmp = Float64(t_0 + Float64(KbT * Float64(NaChar / EAccept)));
	else
		tmp = Float64(t_0 + Float64(NaChar * 0.5));
	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 + ((Vef + EDonor) - Ec)) / KbT)));
	tmp = 0.0;
	if (KbT <= -4.9e-111)
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	elseif (KbT <= 8.6e-181)
		tmp = t_0 + (NaChar / (Vef / KbT));
	elseif (KbT <= 0.00014)
		tmp = t_0 + (KbT * (NaChar / EAccept));
	else
		tmp = t_0 + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -4.9e-111], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 8.6e-181], N[(t$95$0 + N[(NaChar / N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 0.00014], N[(t$95$0 + N[(KbT * N[(NaChar / EAccept), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -4.90000000000000019e-111

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 53.3%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 13.8%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 53.3%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   if -4.90000000000000019e-111 < KbT < 8.6e-181

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 43.3%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in Vef around inf 52.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar \cdot KbT}{Vef}} \]
   6. Step-by-step derivation

      1. associate-/l* 59.1%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{Vef}{KbT}}} \]

   7. Simplified 59.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{Vef}{KbT}}} \]

   if 8.6e-181 < KbT < 1.3999999999999999e-4

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 44.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in EAccept around inf 40.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar \cdot KbT}{EAccept}} \]
   6. Step-by-step derivation

      1. associate-/l* 40.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   7. Simplified 40.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 40.1%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{EAccept} \cdot KbT} \]

   9. Applied egg-rr 40.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{EAccept} \cdot KbT} \]

   if 1.3999999999999999e-4 < KbT

   1. Initial program 99.8%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 58.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in KbT around inf 57.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{0.5 \cdot NaChar} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.9 \cdot 10^{-111}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 8.6 \cdot 10^{-181}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{Vef}{KbT}}\\ \mathbf{elif}\;KbT \leq 0.00014:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + KbT \cdot \frac{NaChar}{EAccept}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 19: 40.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{Vef}{KbT}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ t_2 := NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{if}\;KbT \leq -9.8 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;KbT \leq -2.5 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;KbT \leq -2.5 \cdot 10^{-98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;KbT \leq 3.85 \cdot 10^{-168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;KbT \leq 2.1 \cdot 10^{-63}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;KbT \leq 2.1 \cdot 10^{+219}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(1 + \left(\frac{EDonor}{KbT} + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right) \cdot t_0}{\frac{EAccept}{KbT} + t_0}\right) - \frac{mu}{KbT}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ Vef KbT)))
        (t_1 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
        (t_2 (+ (* NdChar 0.5) (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))))
   (if (<= KbT -9.8e+42)
     t_2
     (if (<= KbT -2.5e-37)
       t_1
       (if (<= KbT -2.5e-98)
         t_2
         (if (<= KbT 3.85e-168)
           t_1
           (if (<= KbT 2.1e-63)
             (+ (* NdChar 0.5) (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
             (if (<= KbT 2.1e+219)
               t_1
               (+
                (/
                 NdChar
                 (+
                  1.0
                  (-
                   (+ (/ mu KbT) (+ 1.0 (+ (/ EDonor KbT) (/ Vef KbT))))
                   (/ Ec KbT))))
                (/
                 NaChar
                 (+
                  1.0
                  (-
                   (+
                    (/ Ev KbT)
                    (/
                     (+
                      (* (/ EAccept KbT) (/ EAccept KbT))
                      (* (+ 1.0 (/ Vef KbT)) t_0))
                     (+ (/ EAccept KbT) t_0)))
                   (/ 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 = -1.0 - (Vef / KbT);
	double t_1 = NdChar / (1.0 + exp((EDonor / KbT)));
	double t_2 = (NdChar * 0.5) + (NaChar / (1.0 + exp((EAccept / KbT))));
	double tmp;
	if (KbT <= -9.8e+42) {
		tmp = t_2;
	} else if (KbT <= -2.5e-37) {
		tmp = t_1;
	} else if (KbT <= -2.5e-98) {
		tmp = t_2;
	} else if (KbT <= 3.85e-168) {
		tmp = t_1;
	} else if (KbT <= 2.1e-63) {
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (KbT <= 2.1e+219) {
		tmp = t_1;
	} else {
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_0)) / ((EAccept / KbT) + t_0))) - (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 = (-1.0d0) - (vef / kbt)
    t_1 = ndchar / (1.0d0 + exp((edonor / kbt)))
    t_2 = (ndchar * 0.5d0) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    if (kbt <= (-9.8d+42)) then
        tmp = t_2
    else if (kbt <= (-2.5d-37)) then
        tmp = t_1
    else if (kbt <= (-2.5d-98)) then
        tmp = t_2
    else if (kbt <= 3.85d-168) then
        tmp = t_1
    else if (kbt <= 2.1d-63) then
        tmp = (ndchar * 0.5d0) + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (kbt <= 2.1d+219) then
        tmp = t_1
    else
        tmp = (ndchar / (1.0d0 + (((mu / kbt) + (1.0d0 + ((edonor / kbt) + (vef / kbt)))) - (ec / kbt)))) + (nachar / (1.0d0 + (((ev / kbt) + ((((eaccept / kbt) * (eaccept / kbt)) + ((1.0d0 + (vef / kbt)) * t_0)) / ((eaccept / kbt) + t_0))) - (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 = -1.0 - (Vef / KbT);
	double t_1 = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	double t_2 = (NdChar * 0.5) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	double tmp;
	if (KbT <= -9.8e+42) {
		tmp = t_2;
	} else if (KbT <= -2.5e-37) {
		tmp = t_1;
	} else if (KbT <= -2.5e-98) {
		tmp = t_2;
	} else if (KbT <= 3.85e-168) {
		tmp = t_1;
	} else if (KbT <= 2.1e-63) {
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (KbT <= 2.1e+219) {
		tmp = t_1;
	} else {
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_0)) / ((EAccept / KbT) + t_0))) - (mu / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = -1.0 - (Vef / KbT)
	t_1 = NdChar / (1.0 + math.exp((EDonor / KbT)))
	t_2 = (NdChar * 0.5) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	tmp = 0
	if KbT <= -9.8e+42:
		tmp = t_2
	elif KbT <= -2.5e-37:
		tmp = t_1
	elif KbT <= -2.5e-98:
		tmp = t_2
	elif KbT <= 3.85e-168:
		tmp = t_1
	elif KbT <= 2.1e-63:
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif KbT <= 2.1e+219:
		tmp = t_1
	else:
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_0)) / ((EAccept / KbT) + t_0))) - (mu / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(-1.0 - Float64(Vef / KbT))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))))
	t_2 = Float64(Float64(NdChar * 0.5) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))))
	tmp = 0.0
	if (KbT <= -9.8e+42)
		tmp = t_2;
	elseif (KbT <= -2.5e-37)
		tmp = t_1;
	elseif (KbT <= -2.5e-98)
		tmp = t_2;
	elseif (KbT <= 3.85e-168)
		tmp = t_1;
	elseif (KbT <= 2.1e-63)
		tmp = Float64(Float64(NdChar * 0.5) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	elseif (KbT <= 2.1e+219)
		tmp = t_1;
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(mu / KbT) + Float64(1.0 + Float64(Float64(EDonor / KbT) + Float64(Vef / KbT)))) - Float64(Ec / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(Float64(Ev / KbT) + Float64(Float64(Float64(Float64(EAccept / KbT) * Float64(EAccept / KbT)) + Float64(Float64(1.0 + Float64(Vef / KbT)) * t_0)) / Float64(Float64(EAccept / KbT) + t_0))) - Float64(mu / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = -1.0 - (Vef / KbT);
	t_1 = NdChar / (1.0 + exp((EDonor / KbT)));
	t_2 = (NdChar * 0.5) + (NaChar / (1.0 + exp((EAccept / KbT))));
	tmp = 0.0;
	if (KbT <= -9.8e+42)
		tmp = t_2;
	elseif (KbT <= -2.5e-37)
		tmp = t_1;
	elseif (KbT <= -2.5e-98)
		tmp = t_2;
	elseif (KbT <= 3.85e-168)
		tmp = t_1;
	elseif (KbT <= 2.1e-63)
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (KbT <= 2.1e+219)
		tmp = t_1;
	else
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_0)) / ((EAccept / KbT) + t_0))) - (mu / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(-1.0 - N[(Vef / KbT), $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[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -9.8e+42], t$95$2, If[LessEqual[KbT, -2.5e-37], t$95$1, If[LessEqual[KbT, -2.5e-98], t$95$2, If[LessEqual[KbT, 3.85e-168], t$95$1, If[LessEqual[KbT, 2.1e-63], N[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.1e+219], t$95$1, N[(N[(NdChar / N[(1.0 + N[(N[(N[(mu / KbT), $MachinePrecision] + N[(1.0 + N[(N[(EDonor / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(N[(Ev / KbT), $MachinePrecision] + N[(N[(N[(N[(EAccept / KbT), $MachinePrecision] * N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(EAccept / KbT), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -9.8000000000000004e42 or -2.4999999999999999e-37 < KbT < -2.50000000000000009e-98

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 60.6%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 16.4%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 60.6%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   7. Taylor expanded in EAccept around inf 48.9%

      \[\leadsto NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   if -9.8000000000000004e42 < KbT < -2.4999999999999999e-37 or -2.50000000000000009e-98 < KbT < 3.84999999999999985e-168 or 2.1e-63 < KbT < 2.09999999999999988e219

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 46.8%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in EAccept around inf 35.3%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar \cdot KbT}{EAccept}} \]
   6. Step-by-step derivation

      1. associate-/l* 35.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   7. Simplified 35.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   8. Taylor expanded in EDonor around inf 21.5%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   9. Taylor expanded in NdChar around inf 43.5%

      \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}} \]

   if 3.84999999999999985e-168 < KbT < 2.1e-63

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 34.2%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 5.7%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 34.2%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   7. Taylor expanded in Ev around inf 21.7%

      \[\leadsto NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   if 2.09999999999999988e219 < KbT

   1. Initial program 99.5%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 96.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]
   5. Step-by-step derivation

      1. flip-+ 96.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \color{blue}{\frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right) \cdot \left(1 + \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right)}}\right) - \frac{mu}{KbT}\right)} \]

   6. Applied egg-rr 96.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \color{blue}{\frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right) \cdot \left(1 + \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right)}}\right) - \frac{mu}{KbT}\right)} \]

   7. Taylor expanded in KbT around inf 85.5%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\left(\left(\frac{mu}{KbT} + \left(1 + \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right) \cdot \left(1 + \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right)}\right) - \frac{mu}{KbT}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -9.8 \cdot 10^{+42}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq -2.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;KbT \leq -2.5 \cdot 10^{-98}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 3.85 \cdot 10^{-168}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;KbT \leq 2.1 \cdot 10^{-63}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;KbT \leq 2.1 \cdot 10^{+219}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(1 + \left(\frac{EDonor}{KbT} + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right) \cdot \left(-1 - \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} + \left(-1 - \frac{Vef}{KbT}\right)}\right) - \frac{mu}{KbT}\right)}\\ \end{array} \]

Alternative 20: 50.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -30000 \lor \neg \left(NaChar \leq 6 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -30000.0) (not (<= NaChar 6e-76)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
    (* NdChar 0.5))
   (/ NdChar (+ 1.0 (exp (/ 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 ((NaChar <= -30000.0) || !(NaChar <= 6e-76)) {
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = NdChar / (1.0 + exp((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 ((nachar <= (-30000.0d0)) .or. (.not. (nachar <= 6d-76))) then
        tmp = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = ndchar / (1.0d0 + exp((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 ((NaChar <= -30000.0) || !(NaChar <= 6e-76)) {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -30000.0) or not (NaChar <= 6e-76):
		tmp = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5)
	else:
		tmp = NdChar / (1.0 + math.exp((EDonor / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -30000.0) || !(NaChar <= 6e-76))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(NdChar / Float64(1.0 + exp(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 ((NaChar <= -30000.0) || ~((NaChar <= 6e-76)))
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	else
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -30000.0], N[Not[LessEqual[NaChar, 6e-76]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -3e4 or 6.00000000000000048e-76 < NaChar

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 53.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 8.4%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 53.1%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   if -3e4 < NaChar < 6.00000000000000048e-76

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 65.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in EAccept around inf 45.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar \cdot KbT}{EAccept}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   7. Simplified 42.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   8. Taylor expanded in EDonor around inf 28.6%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   9. Taylor expanded in NdChar around inf 51.4%

      \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -30000 \lor \neg \left(NaChar \leq 6 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \end{array} \]

Alternative 21: 56.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -21 \lor \neg \left(NaChar \leq 1.6 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -21.0) (not (<= NaChar 1.6e-51)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
    (* NdChar 0.5))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))
    (* NaChar 0.5))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -21.0) || !(NaChar <= 1.6e-51)) {
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((nachar <= (-21.0d0)) .or. (.not. (nachar <= 1.6d-51))) then
        tmp = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = (ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))) + (nachar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -21.0) || !(NaChar <= 1.6e-51)) {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -21.0) or not (NaChar <= 1.6e-51):
		tmp = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5)
	else:
		tmp = (NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -21.0) || !(NaChar <= 1.6e-51))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT)))) + Float64(NaChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -21.0) || ~((NaChar <= 1.6e-51)))
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar * 0.5);
	else
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -21.0], N[Not[LessEqual[NaChar, 1.6e-51]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -21 or 1.6e-51 < NaChar

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 52.1%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.7%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 52.1%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   if -21 < NaChar < 1.6e-51

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 66.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in KbT around inf 58.6%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{0.5 \cdot NaChar} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -21 \lor \neg \left(NaChar \leq 1.6 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 22: 38.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EDonor \leq -5 \cdot 10^{+248}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;EDonor \leq -3.2 \cdot 10^{-146} \lor \neg \left(EDonor \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EDonor -5e+248)
   (+ (* NdChar 0.5) (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
   (if (or (<= EDonor -3.2e-146) (not (<= EDonor 5e-10)))
     (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
     (+ (* NdChar 0.5) (/ NaChar (+ 1.0 (exp (/ Vef 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 (EDonor <= -5e+248) {
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if ((EDonor <= -3.2e-146) || !(EDonor <= 5e-10)) {
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	} else {
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + exp((Vef / 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 (edonor <= (-5d+248)) then
        tmp = (ndchar * 0.5d0) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else if ((edonor <= (-3.2d-146)) .or. (.not. (edonor <= 5d-10))) then
        tmp = ndchar / (1.0d0 + exp((edonor / kbt)))
    else
        tmp = (ndchar * 0.5d0) + (nachar / (1.0d0 + exp((vef / 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 (EDonor <= -5e+248) {
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else if ((EDonor <= -3.2e-146) || !(EDonor <= 5e-10)) {
		tmp = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	} else {
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + Math.exp((Vef / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EDonor <= -5e+248:
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	elif (EDonor <= -3.2e-146) or not (EDonor <= 5e-10):
		tmp = NdChar / (1.0 + math.exp((EDonor / KbT)))
	else:
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + math.exp((Vef / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EDonor <= -5e+248)
		tmp = Float64(Float64(NdChar * 0.5) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif ((EDonor <= -3.2e-146) || !(EDonor <= 5e-10))
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))));
	else
		tmp = Float64(Float64(NdChar * 0.5) + Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (EDonor <= -5e+248)
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + exp((EAccept / KbT))));
	elseif ((EDonor <= -3.2e-146) || ~((EDonor <= 5e-10)))
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	else
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + exp((Vef / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EDonor, -5e+248], N[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[EDonor, -3.2e-146], N[Not[LessEqual[EDonor, 5e-10]], $MachinePrecision]], N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if EDonor < -4.9999999999999996e248

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 62.9%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 5.2%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 62.9%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   7. Taylor expanded in EAccept around inf 47.9%

      \[\leadsto NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   if -4.9999999999999996e248 < EDonor < -3.1999999999999999e-146 or 5.00000000000000031e-10 < EDonor

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 52.6%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in EAccept around inf 29.8%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar \cdot KbT}{EAccept}} \]
   6. Step-by-step derivation

      1. associate-/l* 29.7%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   7. Simplified 29.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   8. Taylor expanded in EDonor around inf 21.5%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   9. Taylor expanded in NdChar around inf 48.5%

      \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}} \]

   if -3.1999999999999999e-146 < EDonor < 5.00000000000000031e-10

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 47.3%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 13.9%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 47.3%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   7. Taylor expanded in Vef around inf 37.8%

      \[\leadsto NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{\color{blue}{Vef}}{KbT}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -5 \cdot 10^{+248}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;EDonor \leq -3.2 \cdot 10^{-146} \lor \neg \left(EDonor \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]

Alternative 23: 38.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EDonor \leq -6 \cdot 10^{+248}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;EDonor \leq -1.35 \cdot 10^{-110} \lor \neg \left(EDonor \leq 7.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EDonor -6e+248)
   (+ (* NdChar 0.5) (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
   (if (or (<= EDonor -1.35e-110) (not (<= EDonor 7.5e-9)))
     (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
     (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (/ NaChar 2.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EDonor <= -6e+248) {
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if ((EDonor <= -1.35e-110) || !(EDonor <= 7.5e-9)) {
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	} else {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (edonor <= (-6d+248)) then
        tmp = (ndchar * 0.5d0) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else if ((edonor <= (-1.35d-110)) .or. (.not. (edonor <= 7.5d-9))) then
        tmp = ndchar / (1.0d0 + exp((edonor / kbt)))
    else
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EDonor <= -6e+248) {
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else if ((EDonor <= -1.35e-110) || !(EDonor <= 7.5e-9)) {
		tmp = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	} else {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EDonor <= -6e+248:
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	elif (EDonor <= -1.35e-110) or not (EDonor <= 7.5e-9):
		tmp = NdChar / (1.0 + math.exp((EDonor / KbT)))
	else:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EDonor <= -6e+248)
		tmp = Float64(Float64(NdChar * 0.5) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif ((EDonor <= -1.35e-110) || !(EDonor <= 7.5e-9))
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (EDonor <= -6e+248)
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + exp((EAccept / KbT))));
	elseif ((EDonor <= -1.35e-110) || ~((EDonor <= 7.5e-9)))
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	else
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EDonor, -6e+248], N[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[EDonor, -1.35e-110], N[Not[LessEqual[EDonor, 7.5e-9]], $MachinePrecision]], N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if EDonor < -6e248

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 62.9%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 5.2%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 62.9%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   7. Taylor expanded in EAccept around inf 47.9%

      \[\leadsto NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   if -6e248 < EDonor < -1.3499999999999999e-110 or 7.49999999999999933e-9 < EDonor

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 52.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in EAccept around inf 29.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar \cdot KbT}{EAccept}} \]
   6. Step-by-step derivation

      1. associate-/l* 29.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   7. Simplified 29.5%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   8. Taylor expanded in EDonor around inf 21.8%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   9. Taylor expanded in NdChar around inf 49.2%

      \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}} \]

   if -1.3499999999999999e-110 < EDonor < 7.49999999999999933e-9

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in mu around inf 70.1%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{mu}{KbT}}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   5. Taylor expanded in KbT around inf 38.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{\color{blue}{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -6 \cdot 10^{+248}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;EDonor \leq -1.35 \cdot 10^{-110} \lor \neg \left(EDonor \leq 7.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \]

Alternative 24: 40.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ t_1 := NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ t_2 := -1 - \frac{Vef}{KbT}\\ \mathbf{if}\;KbT \leq -1.65 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;KbT \leq -2.85 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;KbT \leq -1.45 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;KbT \leq 5.8 \cdot 10^{+219}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(1 + \left(\frac{EDonor}{KbT} + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right) \cdot t_2}{\frac{EAccept}{KbT} + t_2}\right) - \frac{mu}{KbT}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
        (t_1 (+ (* NdChar 0.5) (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))))
        (t_2 (- -1.0 (/ Vef KbT))))
   (if (<= KbT -1.65e+38)
     t_1
     (if (<= KbT -2.85e-37)
       t_0
       (if (<= KbT -1.45e-98)
         t_1
         (if (<= KbT 5.8e+219)
           t_0
           (+
            (/
             NdChar
             (+
              1.0
              (-
               (+ (/ mu KbT) (+ 1.0 (+ (/ EDonor KbT) (/ Vef KbT))))
               (/ Ec KbT))))
            (/
             NaChar
             (+
              1.0
              (-
               (+
                (/ Ev KbT)
                (/
                 (+
                  (* (/ EAccept KbT) (/ EAccept KbT))
                  (* (+ 1.0 (/ Vef KbT)) t_2))
                 (+ (/ EAccept KbT) t_2)))
               (/ 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 / KbT)));
	double t_1 = (NdChar * 0.5) + (NaChar / (1.0 + exp((EAccept / KbT))));
	double t_2 = -1.0 - (Vef / KbT);
	double tmp;
	if (KbT <= -1.65e+38) {
		tmp = t_1;
	} else if (KbT <= -2.85e-37) {
		tmp = t_0;
	} else if (KbT <= -1.45e-98) {
		tmp = t_1;
	} else if (KbT <= 5.8e+219) {
		tmp = t_0;
	} else {
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_2)) / ((EAccept / KbT) + t_2))) - (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 / kbt)))
    t_1 = (ndchar * 0.5d0) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    t_2 = (-1.0d0) - (vef / kbt)
    if (kbt <= (-1.65d+38)) then
        tmp = t_1
    else if (kbt <= (-2.85d-37)) then
        tmp = t_0
    else if (kbt <= (-1.45d-98)) then
        tmp = t_1
    else if (kbt <= 5.8d+219) then
        tmp = t_0
    else
        tmp = (ndchar / (1.0d0 + (((mu / kbt) + (1.0d0 + ((edonor / kbt) + (vef / kbt)))) - (ec / kbt)))) + (nachar / (1.0d0 + (((ev / kbt) + ((((eaccept / kbt) * (eaccept / kbt)) + ((1.0d0 + (vef / kbt)) * t_2)) / ((eaccept / kbt) + t_2))) - (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 / KbT)));
	double t_1 = (NdChar * 0.5) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	double t_2 = -1.0 - (Vef / KbT);
	double tmp;
	if (KbT <= -1.65e+38) {
		tmp = t_1;
	} else if (KbT <= -2.85e-37) {
		tmp = t_0;
	} else if (KbT <= -1.45e-98) {
		tmp = t_1;
	} else if (KbT <= 5.8e+219) {
		tmp = t_0;
	} else {
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_2)) / ((EAccept / KbT) + t_2))) - (mu / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp((EDonor / KbT)))
	t_1 = (NdChar * 0.5) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	t_2 = -1.0 - (Vef / KbT)
	tmp = 0
	if KbT <= -1.65e+38:
		tmp = t_1
	elif KbT <= -2.85e-37:
		tmp = t_0
	elif KbT <= -1.45e-98:
		tmp = t_1
	elif KbT <= 5.8e+219:
		tmp = t_0
	else:
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_2)) / ((EAccept / KbT) + t_2))) - (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(EDonor / KbT))))
	t_1 = Float64(Float64(NdChar * 0.5) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))))
	t_2 = Float64(-1.0 - Float64(Vef / KbT))
	tmp = 0.0
	if (KbT <= -1.65e+38)
		tmp = t_1;
	elseif (KbT <= -2.85e-37)
		tmp = t_0;
	elseif (KbT <= -1.45e-98)
		tmp = t_1;
	elseif (KbT <= 5.8e+219)
		tmp = t_0;
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(mu / KbT) + Float64(1.0 + Float64(Float64(EDonor / KbT) + Float64(Vef / KbT)))) - Float64(Ec / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(Float64(Ev / KbT) + Float64(Float64(Float64(Float64(EAccept / KbT) * Float64(EAccept / KbT)) + Float64(Float64(1.0 + Float64(Vef / KbT)) * t_2)) / Float64(Float64(EAccept / KbT) + t_2))) - Float64(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((EDonor / KbT)));
	t_1 = (NdChar * 0.5) + (NaChar / (1.0 + exp((EAccept / KbT))));
	t_2 = -1.0 - (Vef / KbT);
	tmp = 0.0;
	if (KbT <= -1.65e+38)
		tmp = t_1;
	elseif (KbT <= -2.85e-37)
		tmp = t_0;
	elseif (KbT <= -1.45e-98)
		tmp = t_1;
	elseif (KbT <= 5.8e+219)
		tmp = t_0;
	else
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_2)) / ((EAccept / KbT) + t_2))) - (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[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.65e+38], t$95$1, If[LessEqual[KbT, -2.85e-37], t$95$0, If[LessEqual[KbT, -1.45e-98], t$95$1, If[LessEqual[KbT, 5.8e+219], t$95$0, N[(N[(NdChar / N[(1.0 + N[(N[(N[(mu / KbT), $MachinePrecision] + N[(1.0 + N[(N[(EDonor / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(N[(Ev / KbT), $MachinePrecision] + N[(N[(N[(N[(EAccept / KbT), $MachinePrecision] * N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(EAccept / KbT), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.65e38 or -2.84999999999999987e-37 < KbT < -1.45e-98

   1. Initial program 99.9%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.9%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 60.6%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 16.4%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 60.6%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   7. Taylor expanded in EAccept around inf 48.9%

      \[\leadsto NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   if -1.65e38 < KbT < -2.84999999999999987e-37 or -1.45e-98 < KbT < 5.79999999999999958e219

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 44.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in EAccept around inf 34.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar \cdot KbT}{EAccept}} \]
   6. Step-by-step derivation

      1. associate-/l* 34.1%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   7. Simplified 34.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   8. Taylor expanded in EDonor around inf 19.3%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   9. Taylor expanded in NdChar around inf 39.1%

      \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}} \]

   if 5.79999999999999958e219 < KbT

   1. Initial program 99.5%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 96.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]
   5. Step-by-step derivation

      1. flip-+ 96.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \color{blue}{\frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right) \cdot \left(1 + \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right)}}\right) - \frac{mu}{KbT}\right)} \]

   6. Applied egg-rr 96.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \color{blue}{\frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right) \cdot \left(1 + \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right)}}\right) - \frac{mu}{KbT}\right)} \]

   7. Taylor expanded in KbT around inf 85.5%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\left(\left(\frac{mu}{KbT} + \left(1 + \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right) \cdot \left(1 + \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right)}\right) - \frac{mu}{KbT}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.65 \cdot 10^{+38}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq -2.85 \cdot 10^{-37}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;KbT \leq -1.45 \cdot 10^{-98}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 5.8 \cdot 10^{+219}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(1 + \left(\frac{EDonor}{KbT} + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right) \cdot \left(-1 - \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} + \left(-1 - \frac{Vef}{KbT}\right)}\right) - \frac{mu}{KbT}\right)}\\ \end{array} \]

Alternative 25: 40.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{Vef}{KbT}\\ \mathbf{if}\;KbT \leq -6 \cdot 10^{+163}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 5.8 \cdot 10^{+219}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(1 + \left(\frac{EDonor}{KbT} + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right) \cdot t_0}{\frac{EAccept}{KbT} + t_0}\right) - \frac{mu}{KbT}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ Vef KbT))))
   (if (<= KbT -6e+163)
     (+ (* NdChar 0.5) (/ NaChar 2.0))
     (if (<= KbT 5.8e+219)
       (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
       (+
        (/
         NdChar
         (+
          1.0
          (-
           (+ (/ mu KbT) (+ 1.0 (+ (/ EDonor KbT) (/ Vef KbT))))
           (/ Ec KbT))))
        (/
         NaChar
         (+
          1.0
          (-
           (+
            (/ Ev KbT)
            (/
             (+
              (* (/ EAccept KbT) (/ EAccept KbT))
              (* (+ 1.0 (/ Vef KbT)) t_0))
             (+ (/ EAccept KbT) t_0)))
           (/ 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 = -1.0 - (Vef / KbT);
	double tmp;
	if (KbT <= -6e+163) {
		tmp = (NdChar * 0.5) + (NaChar / 2.0);
	} else if (KbT <= 5.8e+219) {
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	} else {
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_0)) / ((EAccept / KbT) + t_0))) - (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 = (-1.0d0) - (vef / kbt)
    if (kbt <= (-6d+163)) then
        tmp = (ndchar * 0.5d0) + (nachar / 2.0d0)
    else if (kbt <= 5.8d+219) then
        tmp = ndchar / (1.0d0 + exp((edonor / kbt)))
    else
        tmp = (ndchar / (1.0d0 + (((mu / kbt) + (1.0d0 + ((edonor / kbt) + (vef / kbt)))) - (ec / kbt)))) + (nachar / (1.0d0 + (((ev / kbt) + ((((eaccept / kbt) * (eaccept / kbt)) + ((1.0d0 + (vef / kbt)) * t_0)) / ((eaccept / kbt) + t_0))) - (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 = -1.0 - (Vef / KbT);
	double tmp;
	if (KbT <= -6e+163) {
		tmp = (NdChar * 0.5) + (NaChar / 2.0);
	} else if (KbT <= 5.8e+219) {
		tmp = NdChar / (1.0 + Math.exp((EDonor / KbT)));
	} else {
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_0)) / ((EAccept / KbT) + t_0))) - (mu / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = -1.0 - (Vef / KbT)
	tmp = 0
	if KbT <= -6e+163:
		tmp = (NdChar * 0.5) + (NaChar / 2.0)
	elif KbT <= 5.8e+219:
		tmp = NdChar / (1.0 + math.exp((EDonor / KbT)))
	else:
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_0)) / ((EAccept / KbT) + t_0))) - (mu / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(-1.0 - Float64(Vef / KbT))
	tmp = 0.0
	if (KbT <= -6e+163)
		tmp = Float64(Float64(NdChar * 0.5) + Float64(NaChar / 2.0));
	elseif (KbT <= 5.8e+219)
		tmp = Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT))));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(mu / KbT) + Float64(1.0 + Float64(Float64(EDonor / KbT) + Float64(Vef / KbT)))) - Float64(Ec / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(Float64(Ev / KbT) + Float64(Float64(Float64(Float64(EAccept / KbT) * Float64(EAccept / KbT)) + Float64(Float64(1.0 + Float64(Vef / KbT)) * t_0)) / Float64(Float64(EAccept / KbT) + t_0))) - Float64(mu / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = -1.0 - (Vef / KbT);
	tmp = 0.0;
	if (KbT <= -6e+163)
		tmp = (NdChar * 0.5) + (NaChar / 2.0);
	elseif (KbT <= 5.8e+219)
		tmp = NdChar / (1.0 + exp((EDonor / KbT)));
	else
		tmp = (NdChar / (1.0 + (((mu / KbT) + (1.0 + ((EDonor / KbT) + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + (((Ev / KbT) + ((((EAccept / KbT) * (EAccept / KbT)) + ((1.0 + (Vef / KbT)) * t_0)) / ((EAccept / KbT) + t_0))) - (mu / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(-1.0 - N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -6e+163], N[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 5.8e+219], N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[(N[(N[(mu / KbT), $MachinePrecision] + N[(1.0 + N[(N[(EDonor / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(N[(Ev / KbT), $MachinePrecision] + N[(N[(N[(N[(EAccept / KbT), $MachinePrecision] * N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(EAccept / KbT), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -6.00000000000000027e163

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 78.9%

      \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   5. Step-by-step derivation

      1. *-commutative 27.0%

         \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   6. Simplified 78.9%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

   7. Taylor expanded in KbT around inf 70.1%

      \[\leadsto NdChar \cdot 0.5 + \frac{NaChar}{\color{blue}{2}} \]

   if -6.00000000000000027e163 < KbT < 5.79999999999999958e219

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 100.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 42.8%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

   5. Taylor expanded in EAccept around inf 31.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar \cdot KbT}{EAccept}} \]
   6. Step-by-step derivation

      1. associate-/l* 31.7%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   7. Simplified 31.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

   8. Taylor expanded in EDonor around inf 17.9%

      \[\leadsto \frac{NdChar}{1 + e^{\color{blue}{\frac{EDonor}{KbT}}}} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

   9. Taylor expanded in NdChar around inf 35.7%

      \[\leadsto \color{blue}{\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}} \]

   if 5.79999999999999958e219 < KbT

   1. Initial program 99.5%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   2. Step-by-step derivation

      1. neg-sub0 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      2. associate--r- 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      4. neg-sub0 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      5. sub-neg 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      6. associate--l- 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

      7. unsub-neg 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

      8. +-commutative 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

      9. associate-+l+ 99.5%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

   4. Taylor expanded in KbT around inf 96.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]
   5. Step-by-step derivation

      1. flip-+ 96.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \color{blue}{\frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right) \cdot \left(1 + \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right)}}\right) - \frac{mu}{KbT}\right)} \]

   6. Applied egg-rr 96.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \color{blue}{\frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right) \cdot \left(1 + \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right)}}\right) - \frac{mu}{KbT}\right)} \]

   7. Taylor expanded in KbT around inf 85.5%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\left(\left(\frac{mu}{KbT} + \left(1 + \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right) \cdot \left(1 + \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} - \left(1 + \frac{Vef}{KbT}\right)}\right) - \frac{mu}{KbT}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6 \cdot 10^{+163}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 5.8 \cdot 10^{+219}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(1 + \left(\frac{EDonor}{KbT} + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 + \left(\left(\frac{Ev}{KbT} + \frac{\frac{EAccept}{KbT} \cdot \frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right) \cdot \left(-1 - \frac{Vef}{KbT}\right)}{\frac{EAccept}{KbT} + \left(-1 - \frac{Vef}{KbT}\right)}\right) - \frac{mu}{KbT}\right)}\\ \end{array} \]

Alternative 26: 27.8% accurate, 32.7× speedup

Math

\[\begin{array}{l} \\ NdChar \cdot 0.5 + \frac{NaChar}{2} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+ (* NdChar 0.5) (/ NaChar 2.0)))

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 / 2.0);
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (ndchar * 0.5d0) + (nachar / 2.0d0)
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar * 0.5) + (NaChar / 2.0);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar * 0.5) + (NaChar / 2.0)

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar * 0.5) + Float64(NaChar / 2.0))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar * 0.5) + (NaChar / 2.0);
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
2. Step-by-step derivation

   1. neg-sub0 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   2. associate--r- 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. +-commutative 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   4. neg-sub0 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   5. sub-neg 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   6. associate--l- 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   7. unsub-neg 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

   8. +-commutative 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

   9. associate-+l+ 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

4. Taylor expanded in KbT around inf 44.2%

   \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
5. Step-by-step derivation

   1. *-commutative 11.9%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

6. Simplified 44.2%

   \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

7. Taylor expanded in KbT around inf 26.5%

   \[\leadsto NdChar \cdot 0.5 + \frac{NaChar}{\color{blue}{2}} \]

8. Final simplification 26.5%

   \[\leadsto NdChar \cdot 0.5 + \frac{NaChar}{2} \]

Alternative 27: 7.9% accurate, 45.8× speedup

Math

\[\begin{array}{l} \\ \frac{KbT \cdot NaChar}{EAccept} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (/ (* KbT NaChar) EAccept))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (KbT * NaChar) / EAccept;
}

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 = (kbt * nachar) / eaccept
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 (KbT * NaChar) / EAccept;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (KbT * NaChar) / EAccept

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(KbT * NaChar) / EAccept)
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (KbT * NaChar) / EAccept;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(KbT * NaChar), $MachinePrecision] / EAccept), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
2. Step-by-step derivation

   1. neg-sub0 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{0 - \left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   2. associate--r- 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{\left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right) + mu}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. +-commutative 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu + \left(0 - \left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   4. neg-sub0 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu + \color{blue}{\left(-\left(\left(Ec - Vef\right) - EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   5. sub-neg 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{\color{blue}{mu - \left(\left(Ec - Vef\right) - EDonor\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   6. associate--l- 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \color{blue}{\left(Ec - \left(Vef + EDonor\right)\right)}}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   7. unsub-neg 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(\left(Ev + Vef\right) + EAccept\right) - mu}}{KbT}}} \]

   8. +-commutative 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\color{blue}{\left(Vef + Ev\right)} + EAccept\right) - mu}{KbT}}} \]

   9. associate-+l+ 99.9%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\color{blue}{\left(Vef + \left(Ev + EAccept\right)\right)} - mu}{KbT}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}} \]

4. Taylor expanded in KbT around inf 50.4%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\left(\frac{Ev}{KbT} + \left(\frac{EAccept}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}} \]

5. Taylor expanded in EAccept around inf 30.7%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar \cdot KbT}{EAccept}} \]
6. Step-by-step derivation

   1. associate-/l* 30.5%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

7. Simplified 30.5%

   \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \color{blue}{\frac{NaChar}{\frac{EAccept}{KbT}}} \]

8. Taylor expanded in KbT around inf 11.9%

   \[\leadsto \color{blue}{0.5 \cdot NdChar} + \frac{NaChar}{\frac{EAccept}{KbT}} \]
9. Step-by-step derivation

   1. *-commutative 11.9%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

10. Simplified 11.9%

    \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{\frac{EAccept}{KbT}} \]

11. Taylor expanded in NdChar around 0 7.0%

    \[\leadsto \color{blue}{\frac{NaChar \cdot KbT}{EAccept}} \]

12. Final simplification 7.0%

    \[\leadsto \frac{KbT \cdot NaChar}{EAccept} \]

Reproduce

herbie shell --seed 2023217 
(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))))))