Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 28.7s

Alternatives: 25

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = 7.733051506673494e+161
  2. Ec = -1.9132243378120024e-254
  3. Vef = -3.749410138337826e-159
  4. EDonor = -1.6473514668616318e-198
  5. mu = 2.0783962254908347e-256
  6. KbT = 2.2280354013583903e-193
  7. NaChar = -2.6783173812597487e+161
  8. Ev = -3.0905712143312193e+32
  9. EAccept = -1.2034417076194655e+235

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $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 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. Final simplification 100.0%

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

Alternative 2: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{if}\;NdChar \leq -1.7 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;NdChar \leq 5.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))
          (* NaChar (/ 1.0 (+ 1.0 (* (/ Ev KbT) (* (/ Ev KbT) 0.5))))))))
   (if (<= NdChar -1.7e+22)
     t_1
     (if (<= NdChar 1.75e-56)
       t_0
       (if (<= NdChar 5.8e+37)
         (+
          (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
          (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
         (if (<= NdChar 1.05e+162) t_0 t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	double t_1 = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -1.7e+22) {
		tmp = t_1;
	} else if (NdChar <= 1.75e-56) {
		tmp = t_0;
	} else if (NdChar <= 5.8e+37) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (NdChar <= 1.05e+162) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / (1.0d0 + exp((mu / kbt))))
    t_1 = (ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))) + (nachar * (1.0d0 / (1.0d0 + ((ev / kbt) * ((ev / kbt) * 0.5d0)))))
    if (ndchar <= (-1.7d+22)) then
        tmp = t_1
    else if (ndchar <= 1.75d-56) then
        tmp = t_0
    else if (ndchar <= 5.8d+37) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (ndchar <= 1.05d+162) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + Math.exp((mu / KbT))));
	double t_1 = (NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -1.7e+22) {
		tmp = t_1;
	} else if (NdChar <= 1.75e-56) {
		tmp = t_0;
	} else if (NdChar <= 5.8e+37) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (NdChar <= 1.05e+162) {
		tmp = t_0;
	} 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)))) + (NdChar / (1.0 + math.exp((mu / KbT))))
	t_1 = (NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))))
	tmp = 0
	if NdChar <= -1.7e+22:
		tmp = t_1
	elif NdChar <= 1.75e-56:
		tmp = t_0
	elif NdChar <= 5.8e+37:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif NdChar <= 1.05e+162:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT)))) + Float64(NaChar * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) * Float64(Float64(Ev / KbT) * 0.5))))))
	tmp = 0.0
	if (NdChar <= -1.7e+22)
		tmp = t_1;
	elseif (NdChar <= 1.75e-56)
		tmp = t_0;
	elseif (NdChar <= 5.8e+37)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (NdChar <= 1.05e+162)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	t_1 = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	tmp = 0.0;
	if (NdChar <= -1.7e+22)
		tmp = t_1;
	elseif (NdChar <= 1.75e-56)
		tmp = t_0;
	elseif (NdChar <= 5.8e+37)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (NdChar <= 1.05e+162)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NaChar / 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[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{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[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] * N[(N[(Ev / KbT), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.7e+22], t$95$1, If[LessEqual[NdChar, 1.75e-56], t$95$0, If[LessEqual[NdChar, 5.8e+37], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.05e+162], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.7e22 or 1.05e162 < 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 Ev around inf 76.6%

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

   5. Taylor expanded in Ev around 0 61.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 61.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 61.2%

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

   7. Simplified 61.2%

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

      1. div-inv 61.2%

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

      2. times-frac 68.5%

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

   9. Applied egg-rr 68.5%

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

       1. associate-+r+ 68.5%

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

       2. times-frac 61.2%

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

       3. unpow2 61.2%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 61.2%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 61.2%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 61.2%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 61.2%

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

       8. times-frac 68.5%

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

       9. associate-*r* 68.5%

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

   11. Simplified 68.5%

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

   12. Taylor expanded in Ev around inf 70.0%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   13. Step-by-step derivation

       1. *-commutative 70.0%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{{Ev}^{2}}{{KbT}^{2}} \cdot 0.5}} \]

       2. unpow2 70.0%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}} \cdot 0.5} \]

       3. unpow2 70.0%

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

       4. times-frac 77.3%

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

       5. associate-*r* 77.3%

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

   14. Simplified 77.3%

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

   if -1.7e22 < NdChar < 1.7499999999999999e-56 or 5.79999999999999957e37 < NdChar < 1.05e162

   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 78.5%

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

   if 1.7499999999999999e-56 < NdChar < 5.79999999999999957e37

   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 Ev around inf 78.8%

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

   5. Taylor expanded in EDonor around inf 71.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.7 \cdot 10^{+22}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{-56}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 5.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.05 \cdot 10^{+162}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \end{array} \]

Alternative 3: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_1 := t_0 + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{if}\;NdChar \leq -5.5 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{-56}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 4.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.15 \cdot 10^{+141}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT)))))
        (t_1
         (+ t_0 (* NaChar (/ 1.0 (+ 1.0 (* (/ Ev KbT) (* (/ Ev KbT) 0.5))))))))
   (if (<= NdChar -5.5e+21)
     t_1
     (if (<= NdChar 1.75e-56)
       (+
        (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
        (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
       (if (<= NdChar 4.1e+26)
         (+
          (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
          (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
         (if (<= NdChar 1.15e+141)
           (+ t_0 (/ NaChar (+ 1.0 (exp (/ EAccept 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 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_1 = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -5.5e+21) {
		tmp = t_1;
	} else if (NdChar <= 1.75e-56) {
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	} else if (NdChar <= 4.1e+26) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (NdChar <= 1.15e+141) {
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / 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 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_1 = t_0 + (nachar * (1.0d0 / (1.0d0 + ((ev / kbt) * ((ev / kbt) * 0.5d0)))))
    if (ndchar <= (-5.5d+21)) then
        tmp = t_1
    else if (ndchar <= 1.75d-56) then
        tmp = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / (1.0d0 + exp((mu / kbt))))
    else if (ndchar <= 4.1d+26) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (ndchar <= 1.15d+141) then
        tmp = t_0 + (nachar / (1.0d0 + exp((eaccept / 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 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_1 = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -5.5e+21) {
		tmp = t_1;
	} else if (NdChar <= 1.75e-56) {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else if (NdChar <= 4.1e+26) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (NdChar <= 1.15e+141) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	t_1 = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))))
	tmp = 0
	if NdChar <= -5.5e+21:
		tmp = t_1
	elif NdChar <= 1.75e-56:
		tmp = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + math.exp((mu / KbT))))
	elif NdChar <= 4.1e+26:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif NdChar <= 1.15e+141:
		tmp = t_0 + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	else:
		tmp = t_1
	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 * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) * Float64(Float64(Ev / KbT) * 0.5))))))
	tmp = 0.0
	if (NdChar <= -5.5e+21)
		tmp = t_1;
	elseif (NdChar <= 1.75e-56)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	elseif (NdChar <= 4.1e+26)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (NdChar <= 1.15e+141)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / 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 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	t_1 = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	tmp = 0.0;
	if (NdChar <= -5.5e+21)
		tmp = t_1;
	elseif (NdChar <= 1.75e-56)
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	elseif (NdChar <= 4.1e+26)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (NdChar <= 1.15e+141)
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / 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[(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 * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] * N[(N[(Ev / KbT), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -5.5e+21], t$95$1, If[LessEqual[NdChar, 1.75e-56], 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[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 4.1e+26], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.15e+141], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -5.5e21 or 1.1500000000000001e141 < 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 Ev around inf 75.4%

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

   5. Taylor expanded in Ev around 0 58.1%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 58.1%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 58.1%

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

   7. Simplified 58.1%

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

      1. div-inv 58.1%

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

      2. times-frac 65.0%

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

   9. Applied egg-rr 65.0%

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

       1. associate-+r+ 65.0%

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

       2. times-frac 58.1%

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

       3. unpow2 58.1%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 58.1%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 58.1%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 58.1%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 58.1%

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

       8. times-frac 65.0%

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

       9. associate-*r* 65.0%

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

   11. Simplified 65.0%

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

   12. Taylor expanded in Ev around inf 68.1%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   13. Step-by-step derivation

       1. *-commutative 68.1%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{{Ev}^{2}}{{KbT}^{2}} \cdot 0.5}} \]

       2. unpow2 68.1%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}} \cdot 0.5} \]

       3. unpow2 68.1%

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

       4. times-frac 75.0%

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

       5. associate-*r* 75.0%

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

   14. Simplified 75.0%

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

   if -5.5e21 < NdChar < 1.7499999999999999e-56

   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 78.2%

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

   if 1.7499999999999999e-56 < NdChar < 4.09999999999999983e26

   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 Ev around inf 75.7%

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

   5. Taylor expanded in EDonor around inf 75.7%

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

   if 4.09999999999999983e26 < NdChar < 1.1500000000000001e141

   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 EAccept around inf 84.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -5.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{elif}\;NdChar \leq 1.75 \cdot 10^{-56}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 4.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.15 \cdot 10^{+141}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \end{array} \]

Alternative 4: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_1 := \frac{Ev}{KbT} \cdot 0.5\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ t_3 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;mu \leq -1.1 \cdot 10^{+160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;mu \leq -6.2 \cdot 10^{+43}:\\ \;\;\;\;t_3 + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;mu \leq -9.5 \cdot 10^{-120}:\\ \;\;\;\;t_0 + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + \frac{Ev \cdot t_1}{KbT}\right)\right)}\\ \mathbf{elif}\;mu \leq 4.6 \cdot 10^{-264}:\\ \;\;\;\;t_3 + \frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(\frac{EDonor}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 1.06 \cdot 10^{+185}:\\ \;\;\;\;t_0 + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

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 (* (/ Ev KbT) 0.5))
        (t_2
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- mu) KbT))))))
        (t_3 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))))
   (if (<= mu -1.1e+160)
     t_2
     (if (<= mu -6.2e+43)
       (+
        t_3
        (/
         NdChar
         (+ 1.0 (+ (/ mu KbT) (+ 1.0 (* 0.5 (/ (* mu mu) (* KbT KbT))))))))
       (if (<= mu -9.5e-120)
         (+
          t_0
          (* NaChar (/ 1.0 (+ 1.0 (+ (/ Ev KbT) (+ 1.0 (/ (* Ev t_1) KbT)))))))
         (if (<= mu 4.6e-264)
           (+
            t_3
            (/
             NdChar
             (+
              1.0
              (-
               (+ (/ mu KbT) (+ (/ EDonor KbT) (+ 1.0 (/ Vef KbT))))
               (/ Ec KbT)))))
           (if (<= mu 1.06e+185)
             (+ t_0 (* NaChar (/ 1.0 (+ 1.0 (* (/ Ev KbT) 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 = (Ev / KbT) * 0.5;
	double t_2 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-mu / KbT))));
	double t_3 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double tmp;
	if (mu <= -1.1e+160) {
		tmp = t_2;
	} else if (mu <= -6.2e+43) {
		tmp = t_3 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	} else if (mu <= -9.5e-120) {
		tmp = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + (1.0 + ((Ev * t_1) / KbT))))));
	} else if (mu <= 4.6e-264) {
		tmp = t_3 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))));
	} else if (mu <= 1.06e+185) {
		tmp = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * 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) :: t_3
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_1 = (ev / kbt) * 0.5d0
    t_2 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((-mu / kbt))))
    t_3 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    if (mu <= (-1.1d+160)) then
        tmp = t_2
    else if (mu <= (-6.2d+43)) then
        tmp = t_3 + (ndchar / (1.0d0 + ((mu / kbt) + (1.0d0 + (0.5d0 * ((mu * mu) / (kbt * kbt)))))))
    else if (mu <= (-9.5d-120)) then
        tmp = t_0 + (nachar * (1.0d0 / (1.0d0 + ((ev / kbt) + (1.0d0 + ((ev * t_1) / kbt))))))
    else if (mu <= 4.6d-264) then
        tmp = t_3 + (ndchar / (1.0d0 + (((mu / kbt) + ((edonor / kbt) + (1.0d0 + (vef / kbt)))) - (ec / kbt))))
    else if (mu <= 1.06d+185) then
        tmp = t_0 + (nachar * (1.0d0 / (1.0d0 + ((ev / kbt) * 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 = (Ev / KbT) * 0.5;
	double t_2 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((-mu / KbT))));
	double t_3 = NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double tmp;
	if (mu <= -1.1e+160) {
		tmp = t_2;
	} else if (mu <= -6.2e+43) {
		tmp = t_3 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	} else if (mu <= -9.5e-120) {
		tmp = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + (1.0 + ((Ev * t_1) / KbT))))));
	} else if (mu <= 4.6e-264) {
		tmp = t_3 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))));
	} else if (mu <= 1.06e+185) {
		tmp = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * 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 = (Ev / KbT) * 0.5
	t_2 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((-mu / KbT))))
	t_3 = NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))
	tmp = 0
	if mu <= -1.1e+160:
		tmp = t_2
	elif mu <= -6.2e+43:
		tmp = t_3 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))))
	elif mu <= -9.5e-120:
		tmp = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + (1.0 + ((Ev * t_1) / KbT))))))
	elif mu <= 4.6e-264:
		tmp = t_3 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))))
	elif mu <= 1.06e+185:
		tmp = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * 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(Float64(Ev / KbT) * 0.5)
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))))
	t_3 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT))))
	tmp = 0.0
	if (mu <= -1.1e+160)
		tmp = t_2;
	elseif (mu <= -6.2e+43)
		tmp = Float64(t_3 + Float64(NdChar / Float64(1.0 + Float64(Float64(mu / KbT) + Float64(1.0 + Float64(0.5 * Float64(Float64(mu * mu) / Float64(KbT * KbT))))))));
	elseif (mu <= -9.5e-120)
		tmp = Float64(t_0 + Float64(NaChar * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) + Float64(1.0 + Float64(Float64(Ev * t_1) / KbT)))))));
	elseif (mu <= 4.6e-264)
		tmp = Float64(t_3 + Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(mu / KbT) + Float64(Float64(EDonor / KbT) + Float64(1.0 + Float64(Vef / KbT)))) - Float64(Ec / KbT)))));
	elseif (mu <= 1.06e+185)
		tmp = Float64(t_0 + Float64(NaChar * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) * 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 = (Ev / KbT) * 0.5;
	t_2 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-mu / KbT))));
	t_3 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	tmp = 0.0;
	if (mu <= -1.1e+160)
		tmp = t_2;
	elseif (mu <= -6.2e+43)
		tmp = t_3 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	elseif (mu <= -9.5e-120)
		tmp = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + (1.0 + ((Ev * t_1) / KbT))))));
	elseif (mu <= 4.6e-264)
		tmp = t_3 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))));
	elseif (mu <= 1.06e+185)
		tmp = t_0 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * 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[(N[(Ev / KbT), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = 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$3 = 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[mu, -1.1e+160], t$95$2, If[LessEqual[mu, -6.2e+43], N[(t$95$3 + N[(NdChar / N[(1.0 + N[(N[(mu / KbT), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(N[(mu * mu), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, -9.5e-120], N[(t$95$0 + N[(NaChar * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] + N[(1.0 + N[(N[(Ev * t$95$1), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 4.6e-264], N[(t$95$3 + N[(NdChar / N[(1.0 + N[(N[(N[(mu / KbT), $MachinePrecision] + N[(N[(EDonor / KbT), $MachinePrecision] + N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.06e+185], N[(t$95$0 + N[(NaChar * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if mu < -1.09999999999999996e160 or 1.06000000000000004e185 < mu

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 88.9%

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

   5. Taylor expanded in mu around inf 87.0%

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

      1. neg-mul-1 87.0%

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

   7. Simplified 87.0%

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

   if -1.09999999999999996e160 < mu < -6.2000000000000003e43

   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 mu around inf 87.5%

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

   5. Taylor expanded in mu around 0 76.9%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{{mu}^{2}}{{KbT}^{2}}\right)\right)}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   6. Step-by-step derivation

      1. unpow2 76.9%

         \[\leadsto \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{mu \cdot mu}}{{KbT}^{2}}\right)\right)} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

      2. unpow2 76.9%

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

   7. Simplified 76.9%

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

   if -6.2000000000000003e43 < mu < -9.49999999999999937e-120

   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 Ev around inf 86.5%

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

   5. Taylor expanded in Ev around 0 48.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 48.7%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 48.7%

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

   7. Simplified 48.7%

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

      1. div-inv 48.7%

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

      2. times-frac 66.0%

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

   9. Applied egg-rr 66.0%

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

       1. associate-+r+ 66.0%

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

       2. times-frac 48.7%

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

       3. unpow2 48.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 48.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 48.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 48.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 48.7%

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

       8. times-frac 66.0%

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

       9. associate-*r* 66.0%

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

   11. Simplified 66.0%

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

       1. associate-*r/ 66.0%

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

       2. *-commutative 66.0%

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

   13. Applied egg-rr 66.0%

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

   if -9.49999999999999937e-120 < mu < 4.60000000000000023e-264

   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 75.3%

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

      1. associate-+r+ 32.3%

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

   6. Simplified 75.3%

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

   if 4.60000000000000023e-264 < mu < 1.06000000000000004e185

   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 Ev around inf 64.9%

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

   5. Taylor expanded in Ev around 0 57.7%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 57.7%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 57.7%

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

   7. Simplified 57.7%

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

      1. div-inv 57.7%

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

      2. times-frac 60.2%

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

   9. Applied egg-rr 60.2%

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

       1. associate-+r+ 60.2%

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

       2. times-frac 57.7%

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

       3. unpow2 57.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 57.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 57.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 57.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 57.7%

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

       8. times-frac 60.2%

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

       9. associate-*r* 60.2%

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

   11. Simplified 60.2%

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

   12. Taylor expanded in Ev around inf 57.5%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   13. Step-by-step derivation

       1. *-commutative 57.5%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{{Ev}^{2}}{{KbT}^{2}} \cdot 0.5}} \]

       2. unpow2 57.5%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}} \cdot 0.5} \]

       3. unpow2 57.5%

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

       4. times-frac 60.0%

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

       5. associate-*r* 60.0%

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

   14. Simplified 60.0%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;mu \leq -6.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;mu \leq -9.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + \frac{Ev \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}{KbT}\right)\right)}\\ \mathbf{elif}\;mu \leq 4.6 \cdot 10^{-264}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(\frac{EDonor}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;mu \leq 1.06 \cdot 10^{+185}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \end{array} \]

Alternative 5: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_2 := t_1 + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{if}\;NdChar \leq -1.8 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;NdChar \leq -4.4 \cdot 10^{-36}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq -4.8 \cdot 10^{-166}:\\ \;\;\;\;t_1 + \frac{NaChar}{1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}}\\ \mathbf{elif}\;NdChar \leq 1.7 \cdot 10^{-62}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(\frac{EDonor}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 2.05 \cdot 10^{+127}:\\ \;\;\;\;t_1 + \frac{NaChar}{\frac{EAccept}{KbT} + \left(2 + 0.5 \cdot \frac{EAccept \cdot EAccept}{KbT \cdot KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ 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 (/ 1.0 (+ 1.0 (* (/ Ev KbT) (* (/ Ev KbT) 0.5))))))))
   (if (<= NdChar -1.8e+22)
     t_2
     (if (<= NdChar -4.4e-36)
       (+
        t_0
        (/
         NdChar
         (+ 1.0 (+ (/ mu KbT) (+ 1.0 (* 0.5 (/ (* mu mu) (* KbT KbT))))))))
       (if (<= NdChar -4.8e-166)
         (+ t_1 (/ NaChar (+ 1.0 (* 0.5 (/ (* Ev Ev) (* KbT KbT))))))
         (if (<= NdChar 1.7e-62)
           (+
            t_0
            (/
             NdChar
             (+
              1.0
              (-
               (+ (/ mu KbT) (+ (/ EDonor KbT) (+ 1.0 (/ Vef KbT))))
               (/ Ec KbT)))))
           (if (<= NdChar 2.9e+27)
             (+
              (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
              (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
             (if (<= NdChar 2.05e+127)
               (+
                t_1
                (/
                 NaChar
                 (+
                  (/ EAccept KbT)
                  (+ 2.0 (* 0.5 (/ (* EAccept EAccept) (* KbT KbT)))))))
               t_2))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -1.8e+22) {
		tmp = t_2;
	} else if (NdChar <= -4.4e-36) {
		tmp = t_0 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	} else if (NdChar <= -4.8e-166) {
		tmp = t_1 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))));
	} else if (NdChar <= 1.7e-62) {
		tmp = t_0 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))));
	} else if (NdChar <= 2.9e+27) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (NdChar <= 2.05e+127) {
		tmp = t_1 + (NaChar / ((EAccept / KbT) + (2.0 + (0.5 * ((EAccept * EAccept) / (KbT * KbT))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 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 * (1.0d0 / (1.0d0 + ((ev / kbt) * ((ev / kbt) * 0.5d0)))))
    if (ndchar <= (-1.8d+22)) then
        tmp = t_2
    else if (ndchar <= (-4.4d-36)) then
        tmp = t_0 + (ndchar / (1.0d0 + ((mu / kbt) + (1.0d0 + (0.5d0 * ((mu * mu) / (kbt * kbt)))))))
    else if (ndchar <= (-4.8d-166)) then
        tmp = t_1 + (nachar / (1.0d0 + (0.5d0 * ((ev * ev) / (kbt * kbt)))))
    else if (ndchar <= 1.7d-62) then
        tmp = t_0 + (ndchar / (1.0d0 + (((mu / kbt) + ((edonor / kbt) + (1.0d0 + (vef / kbt)))) - (ec / kbt))))
    else if (ndchar <= 2.9d+27) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (ndchar <= 2.05d+127) then
        tmp = t_1 + (nachar / ((eaccept / kbt) + (2.0d0 + (0.5d0 * ((eaccept * eaccept) / (kbt * kbt))))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -1.8e+22) {
		tmp = t_2;
	} else if (NdChar <= -4.4e-36) {
		tmp = t_0 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	} else if (NdChar <= -4.8e-166) {
		tmp = t_1 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))));
	} else if (NdChar <= 1.7e-62) {
		tmp = t_0 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))));
	} else if (NdChar <= 2.9e+27) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (NdChar <= 2.05e+127) {
		tmp = t_1 + (NaChar / ((EAccept / KbT) + (2.0 + (0.5 * ((EAccept * EAccept) / (KbT * KbT))))));
	} 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))))
	tmp = 0
	if NdChar <= -1.8e+22:
		tmp = t_2
	elif NdChar <= -4.4e-36:
		tmp = t_0 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))))
	elif NdChar <= -4.8e-166:
		tmp = t_1 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))
	elif NdChar <= 1.7e-62:
		tmp = t_0 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))))
	elif NdChar <= 2.9e+27:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif NdChar <= 2.05e+127:
		tmp = t_1 + (NaChar / ((EAccept / KbT) + (2.0 + (0.5 * ((EAccept * EAccept) / (KbT * KbT))))))
	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 * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) * Float64(Float64(Ev / KbT) * 0.5))))))
	tmp = 0.0
	if (NdChar <= -1.8e+22)
		tmp = t_2;
	elseif (NdChar <= -4.4e-36)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Float64(mu / KbT) + Float64(1.0 + Float64(0.5 * Float64(Float64(mu * mu) / Float64(KbT * KbT))))))));
	elseif (NdChar <= -4.8e-166)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + Float64(0.5 * Float64(Float64(Ev * Ev) / Float64(KbT * KbT))))));
	elseif (NdChar <= 1.7e-62)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(mu / KbT) + Float64(Float64(EDonor / KbT) + Float64(1.0 + Float64(Vef / KbT)))) - Float64(Ec / KbT)))));
	elseif (NdChar <= 2.9e+27)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (NdChar <= 2.05e+127)
		tmp = Float64(t_1 + Float64(NaChar / Float64(Float64(EAccept / KbT) + Float64(2.0 + Float64(0.5 * Float64(Float64(EAccept * EAccept) / Float64(KbT * KbT)))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	tmp = 0.0;
	if (NdChar <= -1.8e+22)
		tmp = t_2;
	elseif (NdChar <= -4.4e-36)
		tmp = t_0 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	elseif (NdChar <= -4.8e-166)
		tmp = t_1 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))));
	elseif (NdChar <= 1.7e-62)
		tmp = t_0 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))));
	elseif (NdChar <= 2.9e+27)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (NdChar <= 2.05e+127)
		tmp = t_1 + (NaChar / ((EAccept / KbT) + (2.0 + (0.5 * ((EAccept * EAccept) / (KbT * KbT))))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(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 * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] * N[(N[(Ev / KbT), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.8e+22], t$95$2, If[LessEqual[NdChar, -4.4e-36], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(N[(mu / KbT), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(N[(mu * mu), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -4.8e-166], N[(t$95$1 + N[(NaChar / N[(1.0 + N[(0.5 * N[(N[(Ev * Ev), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.7e-62], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(N[(N[(mu / KbT), $MachinePrecision] + N[(N[(EDonor / KbT), $MachinePrecision] + N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.9e+27], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.05e+127], N[(t$95$1 + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + N[(2.0 + N[(0.5 * N[(N[(EAccept * EAccept), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if NdChar < -1.8e22 or 2.04999999999999991e127 < 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 Ev around inf 75.8%

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

   5. Taylor expanded in Ev around 0 58.8%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 58.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 58.8%

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

   7. Simplified 58.8%

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

      1. div-inv 58.8%

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

      2. times-frac 65.6%

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

   9. Applied egg-rr 65.6%

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

       1. associate-+r+ 65.6%

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

       2. times-frac 58.8%

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

       3. unpow2 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 58.8%

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

       8. times-frac 65.6%

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

       9. associate-*r* 65.6%

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

   11. Simplified 65.6%

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

   12. Taylor expanded in Ev around inf 68.6%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   13. Step-by-step derivation

       1. *-commutative 68.6%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{{Ev}^{2}}{{KbT}^{2}} \cdot 0.5}} \]

       2. unpow2 68.6%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}} \cdot 0.5} \]

       3. unpow2 68.6%

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

       4. times-frac 75.4%

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

       5. associate-*r* 75.4%

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

   14. Simplified 75.4%

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

   if -1.8e22 < NdChar < -4.3999999999999999e-36

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 88.1%

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

   5. Taylor expanded in mu around 0 75.0%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{{mu}^{2}}{{KbT}^{2}}\right)\right)}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   6. Step-by-step derivation

      1. unpow2 75.0%

         \[\leadsto \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{mu \cdot mu}}{{KbT}^{2}}\right)\right)} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

      2. unpow2 75.0%

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

   7. Simplified 75.0%

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

   if -4.3999999999999999e-36 < NdChar < -4.7999999999999997e-166

   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 Ev around inf 57.2%

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

   5. Taylor expanded in Ev around 0 72.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 72.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in Ev around inf 80.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   9. Step-by-step derivation

      1. unpow2 80.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}} \]

      2. unpow2 80.0%

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

   10. Simplified 80.0%

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

   if -4.7999999999999997e-166 < NdChar < 1.69999999999999994e-62

   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 70.1%

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

      1. associate-+r+ 25.5%

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

   6. Simplified 70.1%

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

   if 1.69999999999999994e-62 < NdChar < 2.9000000000000001e27

   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 Ev around inf 78.5%

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

   5. Taylor expanded in EDonor around inf 72.9%

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

   if 2.9000000000000001e27 < NdChar < 2.04999999999999991e127

   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 EAccept around inf 82.3%

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

   5. Taylor expanded in EAccept around 0 63.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\frac{EAccept}{KbT} + \left(2 + 0.5 \cdot \frac{{EAccept}^{2}}{{KbT}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. unpow2 63.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + \left(2 + 0.5 \cdot \frac{\color{blue}{EAccept \cdot EAccept}}{{KbT}^{2}}\right)} \]

      2. unpow2 63.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + \left(2 + 0.5 \cdot \frac{EAccept \cdot EAccept}{\color{blue}{KbT \cdot KbT}}\right)} \]

   7. Simplified 63.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\frac{EAccept}{KbT} + \left(2 + 0.5 \cdot \frac{EAccept \cdot EAccept}{KbT \cdot KbT}\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{elif}\;NdChar \leq -4.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq -4.8 \cdot 10^{-166}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}}\\ \mathbf{elif}\;NdChar \leq 1.7 \cdot 10^{-62}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(\frac{EDonor}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 2.05 \cdot 10^{+127}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + \left(2 + 0.5 \cdot \frac{EAccept \cdot EAccept}{KbT \cdot KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \end{array} \]

Alternative 6: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ \mathbf{if}\;EAccept \leq 3.5 \cdot 10^{-48}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;EAccept \leq 5 \cdot 10^{+145}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \end{array} \]

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 (<= EAccept 3.5e-48)
     (+ t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
     (if (<= EAccept 5e+145)
       (+ t_0 (/ NaChar (+ 1.0 (exp (/ Vef KbT)))))
       (+ t_0 (/ NaChar (+ 1.0 (exp (/ EAccept 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 (EAccept <= 3.5e-48) {
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (EAccept <= 5e+145) {
		tmp = t_0 + (NaChar / (1.0 + exp((Vef / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / 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 (eaccept <= 3.5d-48) then
        tmp = t_0 + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (eaccept <= 5d+145) then
        tmp = t_0 + (nachar / (1.0d0 + exp((vef / kbt))))
    else
        tmp = t_0 + (nachar / (1.0d0 + exp((eaccept / 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 (EAccept <= 3.5e-48) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (EAccept <= 5e+145) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((Vef / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((EAccept / 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 EAccept <= 3.5e-48:
		tmp = t_0 + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif EAccept <= 5e+145:
		tmp = t_0 + (NaChar / (1.0 + math.exp((Vef / KbT))))
	else:
		tmp = t_0 + (NaChar / (1.0 + math.exp((EAccept / 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 (EAccept <= 3.5e-48)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	elseif (EAccept <= 5e+145)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / 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 (EAccept <= 3.5e-48)
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (EAccept <= 5e+145)
		tmp = t_0 + (NaChar / (1.0 + exp((Vef / KbT))));
	else
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / 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[EAccept, 3.5e-48], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EAccept, 5e+145], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if EAccept < 3.49999999999999991e-48

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 Ev around inf 68.6%

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

   if 3.49999999999999991e-48 < EAccept < 4.99999999999999967e145

   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 73.2%

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

   if 4.99999999999999967e145 < EAccept

   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 EAccept around inf 84.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq 3.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;EAccept \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \]

Alternative 7: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ \mathbf{if}\;EAccept \leq 1.8 \cdot 10^{+83}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \end{array} \]

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 (<= EAccept 1.8e+83)
     (+ t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
     (+ t_0 (/ NaChar (+ 1.0 (exp (/ EAccept 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 (EAccept <= 1.8e+83) {
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / 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 (eaccept <= 1.8d+83) then
        tmp = t_0 + (nachar / (1.0d0 + exp((ev / kbt))))
    else
        tmp = t_0 + (nachar / (1.0d0 + exp((eaccept / 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 (EAccept <= 1.8e+83) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((EAccept / 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 EAccept <= 1.8e+83:
		tmp = t_0 + (NaChar / (1.0 + math.exp((Ev / KbT))))
	else:
		tmp = t_0 + (NaChar / (1.0 + math.exp((EAccept / 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 (EAccept <= 1.8e+83)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / 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 (EAccept <= 1.8e+83)
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	else
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / 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[EAccept, 1.8e+83], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if EAccept < 1.7999999999999999e83

   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 Ev around inf 68.7%

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

   if 1.7999999999999999e83 < EAccept

   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 EAccept around inf 77.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq 1.8 \cdot 10^{+83}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \]

Alternative 8: 61.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := t_0 + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_3 := t_2 + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{if}\;NdChar \leq -2.7 \cdot 10^{+25}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;NdChar \leq -4.7 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq -4.5 \cdot 10^{-198}:\\ \;\;\;\;t_2 + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq 6.2 \cdot 10^{-68}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 4.8 \cdot 10^{+128}:\\ \;\;\;\;t_2 + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{elif}\;NdChar \leq 1.02 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

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 (+ (/ mu KbT) (+ 1.0 (* 0.5 (/ (* mu mu) (* KbT KbT)))))))))
        (t_2 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT)))))
        (t_3
         (+ t_2 (* NaChar (/ 1.0 (+ 1.0 (* (/ Ev KbT) (* (/ Ev KbT) 0.5))))))))
   (if (<= NdChar -2.7e+25)
     t_3
     (if (<= NdChar -4.7e-39)
       t_1
       (if (<= NdChar -4.5e-198)
         (+
          t_2
          (/
           NaChar
           (+ 1.0 (+ (/ Ev KbT) (+ 1.0 (* 0.5 (/ (* Ev Ev) (* KbT KbT))))))))
         (if (<= NdChar 6.2e-68)
           (+ t_0 (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))
           (if (<= NdChar 4.8e+128)
             (+ t_2 (/ NaChar (+ (/ EAccept KbT) 2.0)))
             (if (<= NdChar 1.02e+176) t_1 t_3))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	double t_2 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_3 = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -2.7e+25) {
		tmp = t_3;
	} else if (NdChar <= -4.7e-39) {
		tmp = t_1;
	} else if (NdChar <= -4.5e-198) {
		tmp = t_2 + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))));
	} else if (NdChar <= 6.2e-68) {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else if (NdChar <= 4.8e+128) {
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0));
	} else if (NdChar <= 1.02e+176) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + ((mu / kbt) + (1.0d0 + (0.5d0 * ((mu * mu) / (kbt * kbt)))))))
    t_2 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_3 = t_2 + (nachar * (1.0d0 / (1.0d0 + ((ev / kbt) * ((ev / kbt) * 0.5d0)))))
    if (ndchar <= (-2.7d+25)) then
        tmp = t_3
    else if (ndchar <= (-4.7d-39)) then
        tmp = t_1
    else if (ndchar <= (-4.5d-198)) then
        tmp = t_2 + (nachar / (1.0d0 + ((ev / kbt) + (1.0d0 + (0.5d0 * ((ev * ev) / (kbt * kbt)))))))
    else if (ndchar <= 6.2d-68) then
        tmp = t_0 + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    else if (ndchar <= 4.8d+128) then
        tmp = t_2 + (nachar / ((eaccept / kbt) + 2.0d0))
    else if (ndchar <= 1.02d+176) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	double t_2 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_3 = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -2.7e+25) {
		tmp = t_3;
	} else if (NdChar <= -4.7e-39) {
		tmp = t_1;
	} else if (NdChar <= -4.5e-198) {
		tmp = t_2 + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))));
	} else if (NdChar <= 6.2e-68) {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else if (NdChar <= 4.8e+128) {
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0));
	} else if (NdChar <= 1.02e+176) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))))
	t_2 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	t_3 = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))))
	tmp = 0
	if NdChar <= -2.7e+25:
		tmp = t_3
	elif NdChar <= -4.7e-39:
		tmp = t_1
	elif NdChar <= -4.5e-198:
		tmp = t_2 + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))))
	elif NdChar <= 6.2e-68:
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	elif NdChar <= 4.8e+128:
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0))
	elif NdChar <= 1.02e+176:
		tmp = t_1
	else:
		tmp = t_3
	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 + Float64(Float64(mu / KbT) + Float64(1.0 + Float64(0.5 * Float64(Float64(mu * mu) / Float64(KbT * KbT))))))))
	t_2 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	t_3 = Float64(t_2 + Float64(NaChar * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) * Float64(Float64(Ev / KbT) * 0.5))))))
	tmp = 0.0
	if (NdChar <= -2.7e+25)
		tmp = t_3;
	elseif (NdChar <= -4.7e-39)
		tmp = t_1;
	elseif (NdChar <= -4.5e-198)
		tmp = Float64(t_2 + Float64(NaChar / Float64(1.0 + Float64(Float64(Ev / KbT) + Float64(1.0 + Float64(0.5 * Float64(Float64(Ev * Ev) / Float64(KbT * KbT))))))));
	elseif (NdChar <= 6.2e-68)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))));
	elseif (NdChar <= 4.8e+128)
		tmp = Float64(t_2 + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	elseif (NdChar <= 1.02e+176)
		tmp = t_1;
	else
		tmp = t_3;
	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 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	t_2 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	t_3 = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	tmp = 0.0;
	if (NdChar <= -2.7e+25)
		tmp = t_3;
	elseif (NdChar <= -4.7e-39)
		tmp = t_1;
	elseif (NdChar <= -4.5e-198)
		tmp = t_2 + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))));
	elseif (NdChar <= 6.2e-68)
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	elseif (NdChar <= 4.8e+128)
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0));
	elseif (NdChar <= 1.02e+176)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(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[(N[(mu / KbT), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(N[(mu * mu), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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]}, Block[{t$95$3 = N[(t$95$2 + N[(NaChar * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] * N[(N[(Ev / KbT), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -2.7e+25], t$95$3, If[LessEqual[NdChar, -4.7e-39], t$95$1, If[LessEqual[NdChar, -4.5e-198], N[(t$95$2 + N[(NaChar / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(N[(Ev * Ev), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 6.2e-68], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 4.8e+128], N[(t$95$2 + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.02e+176], t$95$1, t$95$3]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -2.7e25 or 1.02000000000000001e176 < 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 Ev around inf 77.4%

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

   5. Taylor expanded in Ev around 0 62.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 62.4%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 62.4%

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

   7. Simplified 62.4%

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

      1. div-inv 62.4%

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

      2. times-frac 69.8%

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

   9. Applied egg-rr 69.8%

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

       1. associate-+r+ 69.8%

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

       2. times-frac 62.4%

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

       3. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 62.4%

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

       8. times-frac 69.8%

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

       9. associate-*r* 69.8%

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

   11. Simplified 69.8%

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

   12. Taylor expanded in Ev around inf 70.7%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   13. Step-by-step derivation

       1. *-commutative 70.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{{Ev}^{2}}{{KbT}^{2}} \cdot 0.5}} \]

       2. unpow2 70.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}} \cdot 0.5} \]

       3. unpow2 70.7%

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

       4. times-frac 77.3%

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

       5. associate-*r* 77.3%

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

   14. Simplified 77.3%

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

   if -2.7e25 < NdChar < -4.7000000000000002e-39 or 4.8000000000000004e128 < NdChar < 1.02000000000000001e176

   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 86.6%

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

   5. Taylor expanded in mu around 0 79.6%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{{mu}^{2}}{{KbT}^{2}}\right)\right)}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   6. Step-by-step derivation

      1. unpow2 79.6%

         \[\leadsto \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{mu \cdot mu}}{{KbT}^{2}}\right)\right)} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

      2. unpow2 79.6%

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

   7. Simplified 79.6%

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

   if -4.7000000000000002e-39 < NdChar < -4.4999999999999998e-198

   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 Ev around inf 54.7%

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

   5. Taylor expanded in Ev around 0 63.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 63.4%

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

   7. Simplified 63.4%

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

   if -4.4999999999999998e-198 < NdChar < 6.1999999999999999e-68

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 79.8%

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

   5. Taylor expanded in mu around 0 65.7%

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

   if 6.1999999999999999e-68 < NdChar < 4.8000000000000004e128

   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 EAccept around inf 80.8%

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

   5. Taylor expanded in EAccept around 0 68.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\frac{EAccept}{KbT} + 2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.7 \cdot 10^{+25}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{elif}\;NdChar \leq -4.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq -4.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq 6.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 4.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{elif}\;NdChar \leq 1.02 \cdot 10^{+176}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \end{array} \]

Alternative 9: 61.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := t_0 + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_3 := 1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)\\ t_4 := t_2 + NaChar \cdot \frac{1}{t_3}\\ \mathbf{if}\;NdChar \leq -1.8 \cdot 10^{+22}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;NdChar \leq -5.8 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;t_2 + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + t_3\right)}\\ \mathbf{elif}\;NdChar \leq 1.5 \cdot 10^{-68}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 4.4 \cdot 10^{+128}:\\ \;\;\;\;t_2 + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{elif}\;NdChar \leq 1.02 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \end{array} \]

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 (+ (/ mu KbT) (+ 1.0 (* 0.5 (/ (* mu mu) (* KbT KbT)))))))))
        (t_2 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT)))))
        (t_3 (+ 1.0 (* (/ Ev KbT) (* (/ Ev KbT) 0.5))))
        (t_4 (+ t_2 (* NaChar (/ 1.0 t_3)))))
   (if (<= NdChar -1.8e+22)
     t_4
     (if (<= NdChar -5.8e-41)
       t_1
       (if (<= NdChar -6.5e-201)
         (+ t_2 (* NaChar (/ 1.0 (+ 1.0 (+ (/ Ev KbT) t_3)))))
         (if (<= NdChar 1.5e-68)
           (+ t_0 (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))
           (if (<= NdChar 4.4e+128)
             (+ t_2 (/ NaChar (+ (/ EAccept KbT) 2.0)))
             (if (<= NdChar 1.02e+176) t_1 t_4))))))))

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 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	double t_2 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_3 = 1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5));
	double t_4 = t_2 + (NaChar * (1.0 / t_3));
	double tmp;
	if (NdChar <= -1.8e+22) {
		tmp = t_4;
	} else if (NdChar <= -5.8e-41) {
		tmp = t_1;
	} else if (NdChar <= -6.5e-201) {
		tmp = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + t_3))));
	} else if (NdChar <= 1.5e-68) {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else if (NdChar <= 4.4e+128) {
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0));
	} else if (NdChar <= 1.02e+176) {
		tmp = t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + ((mu / kbt) + (1.0d0 + (0.5d0 * ((mu * mu) / (kbt * kbt)))))))
    t_2 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_3 = 1.0d0 + ((ev / kbt) * ((ev / kbt) * 0.5d0))
    t_4 = t_2 + (nachar * (1.0d0 / t_3))
    if (ndchar <= (-1.8d+22)) then
        tmp = t_4
    else if (ndchar <= (-5.8d-41)) then
        tmp = t_1
    else if (ndchar <= (-6.5d-201)) then
        tmp = t_2 + (nachar * (1.0d0 / (1.0d0 + ((ev / kbt) + t_3))))
    else if (ndchar <= 1.5d-68) then
        tmp = t_0 + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    else if (ndchar <= 4.4d+128) then
        tmp = t_2 + (nachar / ((eaccept / kbt) + 2.0d0))
    else if (ndchar <= 1.02d+176) then
        tmp = t_1
    else
        tmp = t_4
    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 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	double t_2 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_3 = 1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5));
	double t_4 = t_2 + (NaChar * (1.0 / t_3));
	double tmp;
	if (NdChar <= -1.8e+22) {
		tmp = t_4;
	} else if (NdChar <= -5.8e-41) {
		tmp = t_1;
	} else if (NdChar <= -6.5e-201) {
		tmp = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + t_3))));
	} else if (NdChar <= 1.5e-68) {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else if (NdChar <= 4.4e+128) {
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0));
	} else if (NdChar <= 1.02e+176) {
		tmp = t_1;
	} else {
		tmp = t_4;
	}
	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 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))))
	t_2 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	t_3 = 1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5))
	t_4 = t_2 + (NaChar * (1.0 / t_3))
	tmp = 0
	if NdChar <= -1.8e+22:
		tmp = t_4
	elif NdChar <= -5.8e-41:
		tmp = t_1
	elif NdChar <= -6.5e-201:
		tmp = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + t_3))))
	elif NdChar <= 1.5e-68:
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	elif NdChar <= 4.4e+128:
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0))
	elif NdChar <= 1.02e+176:
		tmp = t_1
	else:
		tmp = t_4
	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 + Float64(Float64(mu / KbT) + Float64(1.0 + Float64(0.5 * Float64(Float64(mu * mu) / Float64(KbT * KbT))))))))
	t_2 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	t_3 = Float64(1.0 + Float64(Float64(Ev / KbT) * Float64(Float64(Ev / KbT) * 0.5)))
	t_4 = Float64(t_2 + Float64(NaChar * Float64(1.0 / t_3)))
	tmp = 0.0
	if (NdChar <= -1.8e+22)
		tmp = t_4;
	elseif (NdChar <= -5.8e-41)
		tmp = t_1;
	elseif (NdChar <= -6.5e-201)
		tmp = Float64(t_2 + Float64(NaChar * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) + t_3)))));
	elseif (NdChar <= 1.5e-68)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))));
	elseif (NdChar <= 4.4e+128)
		tmp = Float64(t_2 + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	elseif (NdChar <= 1.02e+176)
		tmp = t_1;
	else
		tmp = t_4;
	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 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	t_2 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	t_3 = 1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5));
	t_4 = t_2 + (NaChar * (1.0 / t_3));
	tmp = 0.0;
	if (NdChar <= -1.8e+22)
		tmp = t_4;
	elseif (NdChar <= -5.8e-41)
		tmp = t_1;
	elseif (NdChar <= -6.5e-201)
		tmp = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + t_3))));
	elseif (NdChar <= 1.5e-68)
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	elseif (NdChar <= 4.4e+128)
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0));
	elseif (NdChar <= 1.02e+176)
		tmp = t_1;
	else
		tmp = t_4;
	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[(N[(mu / KbT), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(N[(mu * mu), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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]}, Block[{t$95$3 = N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] * N[(N[(Ev / KbT), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[(NaChar * N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.8e+22], t$95$4, If[LessEqual[NdChar, -5.8e-41], t$95$1, If[LessEqual[NdChar, -6.5e-201], N[(t$95$2 + N[(NaChar * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.5e-68], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 4.4e+128], N[(t$95$2 + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.02e+176], t$95$1, t$95$4]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -1.8e22 or 1.02000000000000001e176 < 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 Ev around inf 77.4%

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

   5. Taylor expanded in Ev around 0 62.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 62.4%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 62.4%

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

   7. Simplified 62.4%

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

      1. div-inv 62.4%

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

      2. times-frac 69.8%

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

   9. Applied egg-rr 69.8%

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

       1. associate-+r+ 69.8%

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

       2. times-frac 62.4%

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

       3. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 62.4%

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

       8. times-frac 69.8%

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

       9. associate-*r* 69.8%

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

   11. Simplified 69.8%

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

   12. Taylor expanded in Ev around inf 70.7%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   13. Step-by-step derivation

       1. *-commutative 70.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{{Ev}^{2}}{{KbT}^{2}} \cdot 0.5}} \]

       2. unpow2 70.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}} \cdot 0.5} \]

       3. unpow2 70.7%

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

       4. times-frac 77.3%

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

       5. associate-*r* 77.3%

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

   14. Simplified 77.3%

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

   if -1.8e22 < NdChar < -5.79999999999999955e-41 or 4.40000000000000033e128 < NdChar < 1.02000000000000001e176

   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 86.6%

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

   5. Taylor expanded in mu around 0 79.6%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{{mu}^{2}}{{KbT}^{2}}\right)\right)}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   6. Step-by-step derivation

      1. unpow2 79.6%

         \[\leadsto \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{mu \cdot mu}}{{KbT}^{2}}\right)\right)} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

      2. unpow2 79.6%

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

   7. Simplified 79.6%

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

   if -5.79999999999999955e-41 < NdChar < -6.49999999999999974e-201

   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 Ev around inf 54.7%

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

   5. Taylor expanded in Ev around 0 63.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 63.4%

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

   7. Simplified 63.4%

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

      1. div-inv 63.4%

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

      2. times-frac 64.0%

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

   9. Applied egg-rr 64.0%

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

       1. associate-+r+ 64.0%

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

       2. times-frac 63.4%

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

       3. unpow2 63.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 63.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 63.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 63.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 63.4%

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

       8. times-frac 64.0%

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

       9. associate-*r* 64.0%

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

   11. Simplified 64.0%

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

   if -6.49999999999999974e-201 < NdChar < 1.5e-68

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 79.8%

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

   5. Taylor expanded in mu around 0 65.7%

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

   if 1.5e-68 < NdChar < 4.40000000000000033e128

   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 EAccept around inf 80.8%

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

   5. Taylor expanded in EAccept around 0 68.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\frac{EAccept}{KbT} + 2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{elif}\;NdChar \leq -5.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq -6.5 \cdot 10^{-201}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)\right)\right)}\\ \mathbf{elif}\;NdChar \leq 1.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 4.4 \cdot 10^{+128}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{elif}\;NdChar \leq 1.02 \cdot 10^{+176}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \end{array} \]

Alternative 10: 61.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{Ev}{KbT} \cdot 0.5\\ t_1 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_2 := t_1 + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ t_3 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_4 := t_3 + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot t_0}\\ \mathbf{if}\;NdChar \leq -2.7 \cdot 10^{+25}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;NdChar \leq -4 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;NdChar \leq -9.5 \cdot 10^{-198}:\\ \;\;\;\;t_3 + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + \frac{Ev \cdot t_0}{KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;t_1 + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 6.6 \cdot 10^{+127}:\\ \;\;\;\;t_3 + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{elif}\;NdChar \leq 1.02 \cdot 10^{+176}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* (/ Ev KbT) 0.5))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT)))))
        (t_2
         (+
          t_1
          (/
           NdChar
           (+ 1.0 (+ (/ mu KbT) (+ 1.0 (* 0.5 (/ (* mu mu) (* KbT KbT)))))))))
        (t_3 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT)))))
        (t_4 (+ t_3 (* NaChar (/ 1.0 (+ 1.0 (* (/ Ev KbT) t_0)))))))
   (if (<= NdChar -2.7e+25)
     t_4
     (if (<= NdChar -4e-44)
       t_2
       (if (<= NdChar -9.5e-198)
         (+
          t_3
          (* NaChar (/ 1.0 (+ 1.0 (+ (/ Ev KbT) (+ 1.0 (/ (* Ev t_0) KbT)))))))
         (if (<= NdChar 6.5e-71)
           (+ t_1 (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))
           (if (<= NdChar 6.6e+127)
             (+ t_3 (/ NaChar (+ (/ EAccept KbT) 2.0)))
             (if (<= NdChar 1.02e+176) t_2 t_4))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (Ev / KbT) * 0.5;
	double t_1 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_2 = t_1 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	double t_3 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_4 = t_3 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * t_0))));
	double tmp;
	if (NdChar <= -2.7e+25) {
		tmp = t_4;
	} else if (NdChar <= -4e-44) {
		tmp = t_2;
	} else if (NdChar <= -9.5e-198) {
		tmp = t_3 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + (1.0 + ((Ev * t_0) / KbT))))));
	} else if (NdChar <= 6.5e-71) {
		tmp = t_1 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else if (NdChar <= 6.6e+127) {
		tmp = t_3 + (NaChar / ((EAccept / KbT) + 2.0));
	} else if (NdChar <= 1.02e+176) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (ev / kbt) * 0.5d0
    t_1 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    t_2 = t_1 + (ndchar / (1.0d0 + ((mu / kbt) + (1.0d0 + (0.5d0 * ((mu * mu) / (kbt * kbt)))))))
    t_3 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_4 = t_3 + (nachar * (1.0d0 / (1.0d0 + ((ev / kbt) * t_0))))
    if (ndchar <= (-2.7d+25)) then
        tmp = t_4
    else if (ndchar <= (-4d-44)) then
        tmp = t_2
    else if (ndchar <= (-9.5d-198)) then
        tmp = t_3 + (nachar * (1.0d0 / (1.0d0 + ((ev / kbt) + (1.0d0 + ((ev * t_0) / kbt))))))
    else if (ndchar <= 6.5d-71) then
        tmp = t_1 + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    else if (ndchar <= 6.6d+127) then
        tmp = t_3 + (nachar / ((eaccept / kbt) + 2.0d0))
    else if (ndchar <= 1.02d+176) then
        tmp = t_2
    else
        tmp = t_4
    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 = (Ev / KbT) * 0.5;
	double t_1 = NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_2 = t_1 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	double t_3 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_4 = t_3 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * t_0))));
	double tmp;
	if (NdChar <= -2.7e+25) {
		tmp = t_4;
	} else if (NdChar <= -4e-44) {
		tmp = t_2;
	} else if (NdChar <= -9.5e-198) {
		tmp = t_3 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + (1.0 + ((Ev * t_0) / KbT))))));
	} else if (NdChar <= 6.5e-71) {
		tmp = t_1 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else if (NdChar <= 6.6e+127) {
		tmp = t_3 + (NaChar / ((EAccept / KbT) + 2.0));
	} else if (NdChar <= 1.02e+176) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (Ev / KbT) * 0.5
	t_1 = NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))
	t_2 = t_1 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))))
	t_3 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	t_4 = t_3 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * t_0))))
	tmp = 0
	if NdChar <= -2.7e+25:
		tmp = t_4
	elif NdChar <= -4e-44:
		tmp = t_2
	elif NdChar <= -9.5e-198:
		tmp = t_3 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + (1.0 + ((Ev * t_0) / KbT))))))
	elif NdChar <= 6.5e-71:
		tmp = t_1 + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	elif NdChar <= 6.6e+127:
		tmp = t_3 + (NaChar / ((EAccept / KbT) + 2.0))
	elif NdChar <= 1.02e+176:
		tmp = t_2
	else:
		tmp = t_4
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(Ev / KbT) * 0.5)
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT))))
	t_2 = Float64(t_1 + Float64(NdChar / Float64(1.0 + Float64(Float64(mu / KbT) + Float64(1.0 + Float64(0.5 * Float64(Float64(mu * mu) / Float64(KbT * KbT))))))))
	t_3 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	t_4 = Float64(t_3 + Float64(NaChar * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) * t_0)))))
	tmp = 0.0
	if (NdChar <= -2.7e+25)
		tmp = t_4;
	elseif (NdChar <= -4e-44)
		tmp = t_2;
	elseif (NdChar <= -9.5e-198)
		tmp = Float64(t_3 + Float64(NaChar * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) + Float64(1.0 + Float64(Float64(Ev * t_0) / KbT)))))));
	elseif (NdChar <= 6.5e-71)
		tmp = Float64(t_1 + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))));
	elseif (NdChar <= 6.6e+127)
		tmp = Float64(t_3 + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	elseif (NdChar <= 1.02e+176)
		tmp = t_2;
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (Ev / KbT) * 0.5;
	t_1 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	t_2 = t_1 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	t_3 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	t_4 = t_3 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * t_0))));
	tmp = 0.0;
	if (NdChar <= -2.7e+25)
		tmp = t_4;
	elseif (NdChar <= -4e-44)
		tmp = t_2;
	elseif (NdChar <= -9.5e-198)
		tmp = t_3 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) + (1.0 + ((Ev * t_0) / KbT))))));
	elseif (NdChar <= 6.5e-71)
		tmp = t_1 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	elseif (NdChar <= 6.6e+127)
		tmp = t_3 + (NaChar / ((EAccept / KbT) + 2.0));
	elseif (NdChar <= 1.02e+176)
		tmp = t_2;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(Ev / KbT), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = 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$2 = N[(t$95$1 + N[(NdChar / N[(1.0 + N[(N[(mu / KbT), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(N[(mu * mu), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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$4 = N[(t$95$3 + N[(NaChar * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -2.7e+25], t$95$4, If[LessEqual[NdChar, -4e-44], t$95$2, If[LessEqual[NdChar, -9.5e-198], N[(t$95$3 + N[(NaChar * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] + N[(1.0 + N[(N[(Ev * t$95$0), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 6.5e-71], N[(t$95$1 + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 6.6e+127], N[(t$95$3 + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.02e+176], t$95$2, t$95$4]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -2.7e25 or 1.02000000000000001e176 < 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 Ev around inf 77.4%

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

   5. Taylor expanded in Ev around 0 62.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 62.4%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 62.4%

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

   7. Simplified 62.4%

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

      1. div-inv 62.4%

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

      2. times-frac 69.8%

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

   9. Applied egg-rr 69.8%

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

       1. associate-+r+ 69.8%

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

       2. times-frac 62.4%

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

       3. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 62.4%

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

       8. times-frac 69.8%

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

       9. associate-*r* 69.8%

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

   11. Simplified 69.8%

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

   12. Taylor expanded in Ev around inf 70.7%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   13. Step-by-step derivation

       1. *-commutative 70.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{{Ev}^{2}}{{KbT}^{2}} \cdot 0.5}} \]

       2. unpow2 70.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}} \cdot 0.5} \]

       3. unpow2 70.7%

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

       4. times-frac 77.3%

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

       5. associate-*r* 77.3%

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

   14. Simplified 77.3%

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

   if -2.7e25 < NdChar < -3.99999999999999981e-44 or 6.59999999999999953e127 < NdChar < 1.02000000000000001e176

   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 86.6%

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

   5. Taylor expanded in mu around 0 79.6%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{{mu}^{2}}{{KbT}^{2}}\right)\right)}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   6. Step-by-step derivation

      1. unpow2 79.6%

         \[\leadsto \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{mu \cdot mu}}{{KbT}^{2}}\right)\right)} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

      2. unpow2 79.6%

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

   7. Simplified 79.6%

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

   if -3.99999999999999981e-44 < NdChar < -9.4999999999999997e-198

   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 Ev around inf 54.7%

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

   5. Taylor expanded in Ev around 0 63.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 63.4%

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

   7. Simplified 63.4%

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

      1. div-inv 63.4%

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

      2. times-frac 64.0%

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

   9. Applied egg-rr 64.0%

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

       1. associate-+r+ 64.0%

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

       2. times-frac 63.4%

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

       3. unpow2 63.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 63.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 63.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 63.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 63.4%

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

       8. times-frac 64.0%

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

       9. associate-*r* 64.0%

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

   11. Simplified 64.0%

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

       1. associate-*r/ 64.0%

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

       2. *-commutative 64.0%

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

   13. Applied egg-rr 64.0%

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

   if -9.4999999999999997e-198 < NdChar < 6.50000000000000005e-71

   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 79.8%

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

   5. Taylor expanded in mu around 0 65.7%

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

   if 6.50000000000000005e-71 < NdChar < 6.59999999999999953e127

   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 EAccept around inf 80.8%

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

   5. Taylor expanded in EAccept around 0 68.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\frac{EAccept}{KbT} + 2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.7 \cdot 10^{+25}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{elif}\;NdChar \leq -4 \cdot 10^{-44}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq -9.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + \frac{Ev \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}{KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 6.6 \cdot 10^{+127}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{elif}\;NdChar \leq 1.02 \cdot 10^{+176}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \end{array} \]

Alternative 11: 62.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := t_0 + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ t_2 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_3 := t_2 + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{if}\;NdChar \leq -7 \cdot 10^{+23}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;NdChar \leq -1.2 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq -1.6 \cdot 10^{-166}:\\ \;\;\;\;t_2 + \frac{NaChar}{1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}}\\ \mathbf{elif}\;NdChar \leq 9.2 \cdot 10^{-62}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(\frac{EDonor}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 4.8 \cdot 10^{+128}:\\ \;\;\;\;t_2 + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{elif}\;NdChar \leq 1.02 \cdot 10^{+176}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

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 (+ (/ mu KbT) (+ 1.0 (* 0.5 (/ (* mu mu) (* KbT KbT)))))))))
        (t_2 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT)))))
        (t_3
         (+ t_2 (* NaChar (/ 1.0 (+ 1.0 (* (/ Ev KbT) (* (/ Ev KbT) 0.5))))))))
   (if (<= NdChar -7e+23)
     t_3
     (if (<= NdChar -1.2e-36)
       t_1
       (if (<= NdChar -1.6e-166)
         (+ t_2 (/ NaChar (+ 1.0 (* 0.5 (/ (* Ev Ev) (* KbT KbT))))))
         (if (<= NdChar 9.2e-62)
           (+
            t_0
            (/
             NdChar
             (+
              1.0
              (-
               (+ (/ mu KbT) (+ (/ EDonor KbT) (+ 1.0 (/ Vef KbT))))
               (/ Ec KbT)))))
           (if (<= NdChar 4.8e+128)
             (+ t_2 (/ NaChar (+ (/ EAccept KbT) 2.0)))
             (if (<= NdChar 1.02e+176) t_1 t_3))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	double t_2 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_3 = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -7e+23) {
		tmp = t_3;
	} else if (NdChar <= -1.2e-36) {
		tmp = t_1;
	} else if (NdChar <= -1.6e-166) {
		tmp = t_2 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))));
	} else if (NdChar <= 9.2e-62) {
		tmp = t_0 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))));
	} else if (NdChar <= 4.8e+128) {
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0));
	} else if (NdChar <= 1.02e+176) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + ((mu / kbt) + (1.0d0 + (0.5d0 * ((mu * mu) / (kbt * kbt)))))))
    t_2 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    t_3 = t_2 + (nachar * (1.0d0 / (1.0d0 + ((ev / kbt) * ((ev / kbt) * 0.5d0)))))
    if (ndchar <= (-7d+23)) then
        tmp = t_3
    else if (ndchar <= (-1.2d-36)) then
        tmp = t_1
    else if (ndchar <= (-1.6d-166)) then
        tmp = t_2 + (nachar / (1.0d0 + (0.5d0 * ((ev * ev) / (kbt * kbt)))))
    else if (ndchar <= 9.2d-62) then
        tmp = t_0 + (ndchar / (1.0d0 + (((mu / kbt) + ((edonor / kbt) + (1.0d0 + (vef / kbt)))) - (ec / kbt))))
    else if (ndchar <= 4.8d+128) then
        tmp = t_2 + (nachar / ((eaccept / kbt) + 2.0d0))
    else if (ndchar <= 1.02d+176) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	double t_2 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double t_3 = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -7e+23) {
		tmp = t_3;
	} else if (NdChar <= -1.2e-36) {
		tmp = t_1;
	} else if (NdChar <= -1.6e-166) {
		tmp = t_2 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))));
	} else if (NdChar <= 9.2e-62) {
		tmp = t_0 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))));
	} else if (NdChar <= 4.8e+128) {
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0));
	} else if (NdChar <= 1.02e+176) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))))
	t_2 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	t_3 = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))))
	tmp = 0
	if NdChar <= -7e+23:
		tmp = t_3
	elif NdChar <= -1.2e-36:
		tmp = t_1
	elif NdChar <= -1.6e-166:
		tmp = t_2 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))
	elif NdChar <= 9.2e-62:
		tmp = t_0 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))))
	elif NdChar <= 4.8e+128:
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0))
	elif NdChar <= 1.02e+176:
		tmp = t_1
	else:
		tmp = t_3
	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 + Float64(Float64(mu / KbT) + Float64(1.0 + Float64(0.5 * Float64(Float64(mu * mu) / Float64(KbT * KbT))))))))
	t_2 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	t_3 = Float64(t_2 + Float64(NaChar * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) * Float64(Float64(Ev / KbT) * 0.5))))))
	tmp = 0.0
	if (NdChar <= -7e+23)
		tmp = t_3;
	elseif (NdChar <= -1.2e-36)
		tmp = t_1;
	elseif (NdChar <= -1.6e-166)
		tmp = Float64(t_2 + Float64(NaChar / Float64(1.0 + Float64(0.5 * Float64(Float64(Ev * Ev) / Float64(KbT * KbT))))));
	elseif (NdChar <= 9.2e-62)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(mu / KbT) + Float64(Float64(EDonor / KbT) + Float64(1.0 + Float64(Vef / KbT)))) - Float64(Ec / KbT)))));
	elseif (NdChar <= 4.8e+128)
		tmp = Float64(t_2 + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	elseif (NdChar <= 1.02e+176)
		tmp = t_1;
	else
		tmp = t_3;
	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 + ((mu / KbT) + (1.0 + (0.5 * ((mu * mu) / (KbT * KbT)))))));
	t_2 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	t_3 = t_2 + (NaChar * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	tmp = 0.0;
	if (NdChar <= -7e+23)
		tmp = t_3;
	elseif (NdChar <= -1.2e-36)
		tmp = t_1;
	elseif (NdChar <= -1.6e-166)
		tmp = t_2 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))));
	elseif (NdChar <= 9.2e-62)
		tmp = t_0 + (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT))));
	elseif (NdChar <= 4.8e+128)
		tmp = t_2 + (NaChar / ((EAccept / KbT) + 2.0));
	elseif (NdChar <= 1.02e+176)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(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[(N[(mu / KbT), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(N[(mu * mu), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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]}, Block[{t$95$3 = N[(t$95$2 + N[(NaChar * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] * N[(N[(Ev / KbT), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -7e+23], t$95$3, If[LessEqual[NdChar, -1.2e-36], t$95$1, If[LessEqual[NdChar, -1.6e-166], N[(t$95$2 + N[(NaChar / N[(1.0 + N[(0.5 * N[(N[(Ev * Ev), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 9.2e-62], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(N[(N[(mu / KbT), $MachinePrecision] + N[(N[(EDonor / KbT), $MachinePrecision] + N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 4.8e+128], N[(t$95$2 + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.02e+176], t$95$1, t$95$3]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -7.0000000000000004e23 or 1.02000000000000001e176 < 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 Ev around inf 77.4%

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

   5. Taylor expanded in Ev around 0 62.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 62.4%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 62.4%

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

   7. Simplified 62.4%

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

      1. div-inv 62.4%

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

      2. times-frac 69.8%

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

   9. Applied egg-rr 69.8%

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

       1. associate-+r+ 69.8%

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

       2. times-frac 62.4%

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

       3. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 62.4%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 62.4%

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

       8. times-frac 69.8%

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

       9. associate-*r* 69.8%

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

   11. Simplified 69.8%

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

   12. Taylor expanded in Ev around inf 70.7%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   13. Step-by-step derivation

       1. *-commutative 70.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{{Ev}^{2}}{{KbT}^{2}} \cdot 0.5}} \]

       2. unpow2 70.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}} \cdot 0.5} \]

       3. unpow2 70.7%

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

       4. times-frac 77.3%

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

       5. associate-*r* 77.3%

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

   14. Simplified 77.3%

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

   if -7.0000000000000004e23 < NdChar < -1.2e-36 or 4.8000000000000004e128 < NdChar < 1.02000000000000001e176

   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 86.1%

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

   5. Taylor expanded in mu around 0 78.8%

      \[\leadsto \frac{NdChar}{1 + \color{blue}{\left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{{mu}^{2}}{{KbT}^{2}}\right)\right)}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]
   6. Step-by-step derivation

      1. unpow2 78.8%

         \[\leadsto \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{mu \cdot mu}}{{KbT}^{2}}\right)\right)} + \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} \]

      2. unpow2 78.8%

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

   7. Simplified 78.8%

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

   if -1.2e-36 < NdChar < -1.6e-166

   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 Ev around inf 57.2%

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

   5. Taylor expanded in Ev around 0 72.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 72.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in Ev around inf 80.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   9. Step-by-step derivation

      1. unpow2 80.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}} \]

      2. unpow2 80.0%

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

   10. Simplified 80.0%

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

   if -1.6e-166 < NdChar < 9.20000000000000002e-62

   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 70.1%

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

      1. associate-+r+ 25.5%

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

   6. Simplified 70.1%

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

   if 9.20000000000000002e-62 < NdChar < 4.8000000000000004e128

   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 EAccept around inf 80.8%

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

   5. Taylor expanded in EAccept around 0 68.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{\color{blue}{\frac{EAccept}{KbT} + 2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -7 \cdot 10^{+23}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{elif}\;NdChar \leq -1.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq -1.6 \cdot 10^{-166}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}}\\ \mathbf{elif}\;NdChar \leq 9.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(\frac{EDonor}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 4.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{elif}\;NdChar \leq 1.02 \cdot 10^{+176}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(\frac{mu}{KbT} + \left(1 + 0.5 \cdot \frac{mu \cdot mu}{KbT \cdot KbT}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \end{array} \]

Alternative 12: 63.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_2 := t_1 + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{if}\;NdChar \leq -1.25 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;NdChar \leq -8.2 \cdot 10^{-40}:\\ \;\;\;\;t_0 + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq -7 \cdot 10^{-201}:\\ \;\;\;\;t_1 + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq 1.05 \cdot 10^{-75}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ 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 (/ 1.0 (+ 1.0 (* (/ Ev KbT) (* (/ Ev KbT) 0.5))))))))
   (if (<= NdChar -1.25e-10)
     t_2
     (if (<= NdChar -8.2e-40)
       (+ t_0 (/ NdChar 2.0))
       (if (<= NdChar -7e-201)
         (+
          t_1
          (/
           NaChar
           (+ 1.0 (+ (/ Ev KbT) (+ 1.0 (* 0.5 (/ (* Ev Ev) (* KbT KbT))))))))
         (if (<= NdChar 1.05e-75)
           (+ t_0 (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))
           t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -1.25e-10) {
		tmp = t_2;
	} else if (NdChar <= -8.2e-40) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (NdChar <= -7e-201) {
		tmp = t_1 + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))));
	} else if (NdChar <= 1.05e-75) {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 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 * (1.0d0 / (1.0d0 + ((ev / kbt) * ((ev / kbt) * 0.5d0)))))
    if (ndchar <= (-1.25d-10)) then
        tmp = t_2
    else if (ndchar <= (-8.2d-40)) then
        tmp = t_0 + (ndchar / 2.0d0)
    else if (ndchar <= (-7d-201)) then
        tmp = t_1 + (nachar / (1.0d0 + ((ev / kbt) + (1.0d0 + (0.5d0 * ((ev * ev) / (kbt * kbt)))))))
    else if (ndchar <= 1.05d-75) then
        tmp = t_0 + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -1.25e-10) {
		tmp = t_2;
	} else if (NdChar <= -8.2e-40) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (NdChar <= -7e-201) {
		tmp = t_1 + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))));
	} else if (NdChar <= 1.05e-75) {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))))
	tmp = 0
	if NdChar <= -1.25e-10:
		tmp = t_2
	elif NdChar <= -8.2e-40:
		tmp = t_0 + (NdChar / 2.0)
	elif NdChar <= -7e-201:
		tmp = t_1 + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))))
	elif NdChar <= 1.05e-75:
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	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 * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) * Float64(Float64(Ev / KbT) * 0.5))))))
	tmp = 0.0
	if (NdChar <= -1.25e-10)
		tmp = t_2;
	elseif (NdChar <= -8.2e-40)
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	elseif (NdChar <= -7e-201)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + Float64(Float64(Ev / KbT) + Float64(1.0 + Float64(0.5 * Float64(Float64(Ev * Ev) / Float64(KbT * KbT))))))));
	elseif (NdChar <= 1.05e-75)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	tmp = 0.0;
	if (NdChar <= -1.25e-10)
		tmp = t_2;
	elseif (NdChar <= -8.2e-40)
		tmp = t_0 + (NdChar / 2.0);
	elseif (NdChar <= -7e-201)
		tmp = t_1 + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))));
	elseif (NdChar <= 1.05e-75)
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(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 * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] * N[(N[(Ev / KbT), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.25e-10], t$95$2, If[LessEqual[NdChar, -8.2e-40], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -7e-201], N[(t$95$1 + N[(NaChar / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(N[(Ev * Ev), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.05e-75], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -1.25000000000000008e-10 or 1.0500000000000001e-75 < 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 Ev around inf 75.2%

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

   5. Taylor expanded in Ev around 0 58.8%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 58.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 58.8%

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

   7. Simplified 58.8%

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

      1. div-inv 58.8%

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

      2. times-frac 65.5%

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

   9. Applied egg-rr 65.5%

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

       1. associate-+r+ 65.5%

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

       2. times-frac 58.8%

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

       3. unpow2 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 58.8%

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

       8. times-frac 65.5%

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

       9. associate-*r* 65.5%

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

   11. Simplified 65.5%

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

   12. Taylor expanded in Ev around inf 64.7%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   13. Step-by-step derivation

       1. *-commutative 64.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{{Ev}^{2}}{{KbT}^{2}} \cdot 0.5}} \]

       2. unpow2 64.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}} \cdot 0.5} \]

       3. unpow2 64.7%

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

       4. times-frac 71.5%

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

       5. associate-*r* 71.5%

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

   14. Simplified 71.5%

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

   if -1.25000000000000008e-10 < NdChar < -8.19999999999999926e-40

   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 79.7%

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

   if -8.19999999999999926e-40 < NdChar < -7.00000000000000016e-201

   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 Ev around inf 54.7%

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

   5. Taylor expanded in Ev around 0 63.4%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 63.4%

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

   7. Simplified 63.4%

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

   if -7.00000000000000016e-201 < NdChar < 1.0500000000000001e-75

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 79.8%

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

   5. Taylor expanded in mu around 0 65.7%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.25 \cdot 10^{-10}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{elif}\;NdChar \leq -8.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq -7 \cdot 10^{-201}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}\right)\right)}\\ \mathbf{elif}\;NdChar \leq 1.05 \cdot 10^{-75}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \end{array} \]

Alternative 13: 63.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ t_2 := t_1 + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{if}\;NdChar \leq -3.2 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;NdChar \leq -9.5 \cdot 10^{-40}:\\ \;\;\;\;t_0 + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq -3.2 \cdot 10^{-188}:\\ \;\;\;\;t_1 + \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.7 \cdot 10^{-76}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ 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 (/ 1.0 (+ 1.0 (* (/ Ev KbT) (* (/ Ev KbT) 0.5))))))))
   (if (<= NdChar -3.2e-12)
     t_2
     (if (<= NdChar -9.5e-40)
       (+ t_0 (/ NdChar 2.0))
       (if (<= NdChar -3.2e-188)
         (+ t_1 (/ NaChar (+ 1.0 (+ 1.0 (/ Ev KbT)))))
         (if (<= NdChar 2.7e-76)
           (+ t_0 (/ NdChar (+ 1.0 (+ 1.0 (/ mu KbT)))))
           t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -3.2e-12) {
		tmp = t_2;
	} else if (NdChar <= -9.5e-40) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (NdChar <= -3.2e-188) {
		tmp = t_1 + (NaChar / (1.0 + (1.0 + (Ev / KbT))));
	} else if (NdChar <= 2.7e-76) {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 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 * (1.0d0 / (1.0d0 + ((ev / kbt) * ((ev / kbt) * 0.5d0)))))
    if (ndchar <= (-3.2d-12)) then
        tmp = t_2
    else if (ndchar <= (-9.5d-40)) then
        tmp = t_0 + (ndchar / 2.0d0)
    else if (ndchar <= (-3.2d-188)) then
        tmp = t_1 + (nachar / (1.0d0 + (1.0d0 + (ev / kbt))))
    else if (ndchar <= 2.7d-76) then
        tmp = t_0 + (ndchar / (1.0d0 + (1.0d0 + (mu / kbt))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	double tmp;
	if (NdChar <= -3.2e-12) {
		tmp = t_2;
	} else if (NdChar <= -9.5e-40) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (NdChar <= -3.2e-188) {
		tmp = t_1 + (NaChar / (1.0 + (1.0 + (Ev / KbT))));
	} else if (NdChar <= 2.7e-76) {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	} 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))))
	tmp = 0
	if NdChar <= -3.2e-12:
		tmp = t_2
	elif NdChar <= -9.5e-40:
		tmp = t_0 + (NdChar / 2.0)
	elif NdChar <= -3.2e-188:
		tmp = t_1 + (NaChar / (1.0 + (1.0 + (Ev / KbT))))
	elif NdChar <= 2.7e-76:
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))))
	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 * Float64(1.0 / Float64(1.0 + Float64(Float64(Ev / KbT) * Float64(Float64(Ev / KbT) * 0.5))))))
	tmp = 0.0
	if (NdChar <= -3.2e-12)
		tmp = t_2;
	elseif (NdChar <= -9.5e-40)
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	elseif (NdChar <= -3.2e-188)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev / KbT)))));
	elseif (NdChar <= 2.7e-76)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(mu / KbT)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 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 * (1.0 / (1.0 + ((Ev / KbT) * ((Ev / KbT) * 0.5)))));
	tmp = 0.0;
	if (NdChar <= -3.2e-12)
		tmp = t_2;
	elseif (NdChar <= -9.5e-40)
		tmp = t_0 + (NdChar / 2.0);
	elseif (NdChar <= -3.2e-188)
		tmp = t_1 + (NaChar / (1.0 + (1.0 + (Ev / KbT))));
	elseif (NdChar <= 2.7e-76)
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (mu / KbT))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(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 * N[(1.0 / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] * N[(N[(Ev / KbT), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -3.2e-12], t$95$2, If[LessEqual[NdChar, -9.5e-40], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -3.2e-188], N[(t$95$1 + N[(NaChar / N[(1.0 + N[(1.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.7e-76], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -3.2000000000000001e-12 or 2.7e-76 < 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 Ev around inf 75.2%

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

   5. Taylor expanded in Ev around 0 58.8%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 58.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 58.8%

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

   7. Simplified 58.8%

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

      1. div-inv 58.8%

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

      2. times-frac 65.5%

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

   9. Applied egg-rr 65.5%

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

       1. associate-+r+ 65.5%

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

       2. times-frac 58.8%

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

       3. unpow2 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{\color{blue}{{Ev}^{2}}}{KbT \cdot KbT}\right)} \]

       4. unpow2 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\left(\frac{Ev}{KbT} + 1\right) + 0.5 \cdot \frac{{Ev}^{2}}{\color{blue}{{KbT}^{2}}}\right)} \]

       5. associate-+r+ 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]

       6. unpow2 58.8%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

       7. unpow2 58.8%

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

       8. times-frac 65.5%

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

       9. associate-*r* 65.5%

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

   11. Simplified 65.5%

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

   12. Taylor expanded in Ev around inf 64.7%

       \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   13. Step-by-step derivation

       1. *-commutative 64.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \color{blue}{\frac{{Ev}^{2}}{{KbT}^{2}} \cdot 0.5}} \]

       2. unpow2 64.7%

          \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}} \cdot 0.5} \]

       3. unpow2 64.7%

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

       4. times-frac 71.5%

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

       5. associate-*r* 71.5%

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

   14. Simplified 71.5%

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

   if -3.2000000000000001e-12 < NdChar < -9.5000000000000006e-40

   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 79.7%

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

   if -9.5000000000000006e-40 < NdChar < -3.20000000000000022e-188

   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 Ev around inf 47.1%

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

   5. Taylor expanded in Ev around 0 58.1%

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

   if -3.20000000000000022e-188 < NdChar < 2.7e-76

   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 79.5%

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

   5. Taylor expanded in mu around 0 66.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -3.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \mathbf{elif}\;NdChar \leq -9.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq -3.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot \frac{1}{1 + \frac{Ev}{KbT} \cdot \left(\frac{Ev}{KbT} \cdot 0.5\right)}\\ \end{array} \]

Alternative 14: 60.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)}\\ \mathbf{if}\;NdChar \leq -6500000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;NdChar \leq -5.4 \cdot 10^{-62}:\\ \;\;\;\;t_0 + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq -3.5 \cdot 10^{-188} \lor \neg \left(NdChar \leq 2.7 \cdot 10^{-67}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \end{array} \end{array} \]

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)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))
          (/ NaChar (+ 1.0 (+ 1.0 (/ Ev KbT)))))))
   (if (<= NdChar -6500000000000.0)
     t_1
     (if (<= NdChar -5.4e-62)
       (+ t_0 (/ NdChar 2.0))
       (if (or (<= NdChar -3.5e-188) (not (<= NdChar 2.7e-67)))
         t_1
         (+ 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 t_1 = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + (1.0 + (Ev / KbT))));
	double tmp;
	if (NdChar <= -6500000000000.0) {
		tmp = t_1;
	} else if (NdChar <= -5.4e-62) {
		tmp = t_0 + (NdChar / 2.0);
	} else if ((NdChar <= -3.5e-188) || !(NdChar <= 2.7e-67)) {
		tmp = t_1;
	} 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) :: t_1
    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)))) + (nachar / (1.0d0 + (1.0d0 + (ev / kbt))))
    if (ndchar <= (-6500000000000.0d0)) then
        tmp = t_1
    else if (ndchar <= (-5.4d-62)) then
        tmp = t_0 + (ndchar / 2.0d0)
    else if ((ndchar <= (-3.5d-188)) .or. (.not. (ndchar <= 2.7d-67))) then
        tmp = t_1
    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 t_1 = (NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + (1.0 + (Ev / KbT))));
	double tmp;
	if (NdChar <= -6500000000000.0) {
		tmp = t_1;
	} else if (NdChar <= -5.4e-62) {
		tmp = t_0 + (NdChar / 2.0);
	} else if ((NdChar <= -3.5e-188) || !(NdChar <= 2.7e-67)) {
		tmp = t_1;
	} 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)))
	t_1 = (NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + (1.0 + (Ev / KbT))))
	tmp = 0
	if NdChar <= -6500000000000.0:
		tmp = t_1
	elif NdChar <= -5.4e-62:
		tmp = t_0 + (NdChar / 2.0)
	elif (NdChar <= -3.5e-188) or not (NdChar <= 2.7e-67):
		tmp = t_1
	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))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev / KbT)))))
	tmp = 0.0
	if (NdChar <= -6500000000000.0)
		tmp = t_1;
	elseif (NdChar <= -5.4e-62)
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	elseif ((NdChar <= -3.5e-188) || !(NdChar <= 2.7e-67))
		tmp = t_1;
	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)));
	t_1 = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / (1.0 + (1.0 + (Ev / KbT))));
	tmp = 0.0;
	if (NdChar <= -6500000000000.0)
		tmp = t_1;
	elseif (NdChar <= -5.4e-62)
		tmp = t_0 + (NdChar / 2.0);
	elseif ((NdChar <= -3.5e-188) || ~((NdChar <= 2.7e-67)))
		tmp = t_1;
	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]}, Block[{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[(1.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -6500000000000.0], t$95$1, If[LessEqual[NdChar, -5.4e-62], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[NdChar, -3.5e-188], N[Not[LessEqual[NdChar, 2.7e-67]], $MachinePrecision]], t$95$1, 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 NdChar < -6.5e12 or -5.40000000000000039e-62 < NdChar < -3.5e-188 or 2.70000000000000016e-67 < 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 Ev around inf 73.3%

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

   5. Taylor expanded in Ev around 0 64.6%

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

   if -6.5e12 < NdChar < -5.40000000000000039e-62

   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%

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

   if -3.5e-188 < NdChar < 2.70000000000000016e-67

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 79.5%

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

   5. Taylor expanded in mu around 0 66.0%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -6500000000000:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq -5.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq -3.5 \cdot 10^{-188} \lor \neg \left(NdChar \leq 2.7 \cdot 10^{-67}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \end{array} \]

Alternative 15: 59.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t_1 + \frac{NaChar}{1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}}\\ \mathbf{elif}\;NdChar \leq -7.8 \cdot 10^{-44}:\\ \;\;\;\;t_0 + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq -3.5 \cdot 10^{-188} \lor \neg \left(NdChar \leq 1.05 \cdot 10^{-68}\right):\\ \;\;\;\;t_1 + \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \end{array} \end{array} \]

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)))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))))
   (if (<= NdChar -2e+14)
     (+ t_1 (/ NaChar (+ 1.0 (* 0.5 (/ (* Ev Ev) (* KbT KbT))))))
     (if (<= NdChar -7.8e-44)
       (+ t_0 (/ NdChar 2.0))
       (if (or (<= NdChar -3.5e-188) (not (<= NdChar 1.05e-68)))
         (+ t_1 (/ NaChar (+ 1.0 (+ 1.0 (/ Ev 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 t_1 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double tmp;
	if (NdChar <= -2e+14) {
		tmp = t_1 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))));
	} else if (NdChar <= -7.8e-44) {
		tmp = t_0 + (NdChar / 2.0);
	} else if ((NdChar <= -3.5e-188) || !(NdChar <= 1.05e-68)) {
		tmp = t_1 + (NaChar / (1.0 + (1.0 + (Ev / 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) :: t_1
    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)))
    if (ndchar <= (-2d+14)) then
        tmp = t_1 + (nachar / (1.0d0 + (0.5d0 * ((ev * ev) / (kbt * kbt)))))
    else if (ndchar <= (-7.8d-44)) then
        tmp = t_0 + (ndchar / 2.0d0)
    else if ((ndchar <= (-3.5d-188)) .or. (.not. (ndchar <= 1.05d-68))) then
        tmp = t_1 + (nachar / (1.0d0 + (1.0d0 + (ev / 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 t_1 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double tmp;
	if (NdChar <= -2e+14) {
		tmp = t_1 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))));
	} else if (NdChar <= -7.8e-44) {
		tmp = t_0 + (NdChar / 2.0);
	} else if ((NdChar <= -3.5e-188) || !(NdChar <= 1.05e-68)) {
		tmp = t_1 + (NaChar / (1.0 + (1.0 + (Ev / 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)))
	t_1 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	tmp = 0
	if NdChar <= -2e+14:
		tmp = t_1 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))
	elif NdChar <= -7.8e-44:
		tmp = t_0 + (NdChar / 2.0)
	elif (NdChar <= -3.5e-188) or not (NdChar <= 1.05e-68):
		tmp = t_1 + (NaChar / (1.0 + (1.0 + (Ev / 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))))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	tmp = 0.0
	if (NdChar <= -2e+14)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + Float64(0.5 * Float64(Float64(Ev * Ev) / Float64(KbT * KbT))))));
	elseif (NdChar <= -7.8e-44)
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	elseif ((NdChar <= -3.5e-188) || !(NdChar <= 1.05e-68))
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev / 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)));
	t_1 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	tmp = 0.0;
	if (NdChar <= -2e+14)
		tmp = t_1 + (NaChar / (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))));
	elseif (NdChar <= -7.8e-44)
		tmp = t_0 + (NdChar / 2.0);
	elseif ((NdChar <= -3.5e-188) || ~((NdChar <= 1.05e-68)))
		tmp = t_1 + (NaChar / (1.0 + (1.0 + (Ev / 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]}, 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]}, If[LessEqual[NdChar, -2e+14], N[(t$95$1 + N[(NaChar / N[(1.0 + N[(0.5 * N[(N[(Ev * Ev), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -7.8e-44], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[NdChar, -3.5e-188], N[Not[LessEqual[NdChar, 1.05e-68]], $MachinePrecision]], N[(t$95$1 + N[(NaChar / N[(1.0 + N[(1.0 + N[(Ev / 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 4 regimes

2. if NdChar < -2e14

   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 Ev around inf 72.3%

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

   5. Taylor expanded in Ev around 0 61.8%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 61.8%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 61.8%

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

   7. Simplified 61.8%

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

   8. Taylor expanded in Ev around inf 68.0%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}}} \]
   9. Step-by-step derivation

      1. unpow2 68.0%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}} \]

      2. unpow2 68.0%

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

   10. Simplified 68.0%

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

   if -2e14 < NdChar < -7.8000000000000004e-44

   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%

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

   if -7.8000000000000004e-44 < NdChar < -3.5e-188 or 1.05000000000000004e-68 < 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 Ev around inf 73.1%

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

   5. Taylor expanded in Ev around 0 66.0%

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

   if -3.5e-188 < NdChar < 1.05000000000000004e-68

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 79.5%

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

   5. Taylor expanded in mu around 0 66.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}}\\ \mathbf{elif}\;NdChar \leq -7.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq -3.5 \cdot 10^{-188} \lor \neg \left(NdChar \leq 1.05 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{mu}{KbT}\right)}\\ \end{array} \]

Alternative 16: 60.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -2.2 \cdot 10^{+40} \lor \neg \left(NaChar \leq 1.3 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -2.2e+40) (not (<= NaChar 1.3e+25)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
    (/ NdChar 2.0))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))
    (/ NaChar (+ (/ EAccept 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 tmp;
	if ((NaChar <= -2.2e+40) || !(NaChar <= 1.3e+25)) {
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / ((EAccept / KbT) + 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 ((nachar <= (-2.2d+40)) .or. (.not. (nachar <= 1.3d+25))) then
        tmp = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / 2.0d0)
    else
        tmp = (ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))) + (nachar / ((eaccept / kbt) + 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 ((NaChar <= -2.2e+40) || !(NaChar <= 1.3e+25)) {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -2.2e+40) or not (NaChar <= 1.3e+25):
		tmp = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0)
	else:
		tmp = (NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -2.2e+40) || !(NaChar <= 1.3e+25))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / 2.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT)))) + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -2.2e+40) || ~((NaChar <= 1.3e+25)))
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0);
	else
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -2.2e+40], N[Not[LessEqual[NaChar, 1.3e+25]], $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 / 2.0), $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[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -2.1999999999999999e40 or 1.2999999999999999e25 < 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 54.6%

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

   if -2.1999999999999999e40 < NaChar < 1.2999999999999999e25

   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 EAccept around inf 70.6%

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

   5. Taylor expanded in EAccept around 0 67.8%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -2.2 \cdot 10^{+40} \lor \neg \left(NaChar \leq 1.3 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \end{array} \]

Alternative 17: 50.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}}\\ \mathbf{if}\;Ev \leq -9.5 \cdot 10^{-126}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)}\\ \mathbf{elif}\;Ev \leq 3.5 \cdot 10^{-224}:\\ \;\;\;\;t_0 + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))))
   (if (<= Ev -9.5e-126)
     (+ t_0 (/ NaChar (+ 1.0 (+ 1.0 (/ Ev KbT)))))
     (if (<= Ev 3.5e-224)
       (+ t_0 (/ NaChar (+ (/ EAccept KbT) 2.0)))
       (+
        (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
        (/ NdChar 2.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double tmp;
	if (Ev <= -9.5e-126) {
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (Ev / KbT))));
	} else if (Ev <= 3.5e-224) {
		tmp = t_0 + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))
    if (ev <= (-9.5d-126)) then
        tmp = t_0 + (nachar / (1.0d0 + (1.0d0 + (ev / kbt))))
    else if (ev <= 3.5d-224) then
        tmp = t_0 + (nachar / ((eaccept / kbt) + 2.0d0))
    else
        tmp = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	double tmp;
	if (Ev <= -9.5e-126) {
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (Ev / KbT))));
	} else if (Ev <= 3.5e-224) {
		tmp = t_0 + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))
	tmp = 0
	if Ev <= -9.5e-126:
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (Ev / KbT))))
	elif Ev <= 3.5e-224:
		tmp = t_0 + (NaChar / ((EAccept / KbT) + 2.0))
	else:
		tmp = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT))))
	tmp = 0.0
	if (Ev <= -9.5e-126)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev / KbT)))));
	elseif (Ev <= 3.5e-224)
		tmp = Float64(t_0 + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)));
	tmp = 0.0;
	if (Ev <= -9.5e-126)
		tmp = t_0 + (NaChar / (1.0 + (1.0 + (Ev / KbT))));
	elseif (Ev <= 3.5e-224)
		tmp = t_0 + (NaChar / ((EAccept / KbT) + 2.0));
	else
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(mu + N[(N[(Vef + EDonor), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Ev, -9.5e-126], N[(t$95$0 + N[(NaChar / N[(1.0 + N[(1.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, 3.5e-224], N[(t$95$0 + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $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 / 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if Ev < -9.5000000000000003e-126

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 Ev around inf 75.4%

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

   5. Taylor expanded in Ev around 0 61.6%

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

   if -9.5000000000000003e-126 < Ev < 3.50000000000000019e-224

   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 EAccept around inf 72.5%

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

   5. Taylor expanded in EAccept around 0 56.3%

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

   if 3.50000000000000019e-224 < Ev

   1. Initial program 100.0%

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

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -9.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)}\\ \mathbf{elif}\;Ev \leq 3.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 18: 56.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -5.7 \cdot 10^{-31} \lor \neg \left(NaChar \leq 10^{+113}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -5.7e-31) (not (<= NaChar 1e+113)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (- (+ Vef (+ Ev EAccept)) mu) KbT))))
    (/ NdChar 2.0))
   (+
    (/ 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 <= -5.7e-31) || !(NaChar <= 1e+113)) {
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0);
	} 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 <= (-5.7d-31)) .or. (.not. (nachar <= 1d+113))) then
        tmp = (nachar / (1.0d0 + exp((((vef + (ev + eaccept)) - mu) / kbt)))) + (ndchar / 2.0d0)
    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 <= -5.7e-31) || !(NaChar <= 1e+113)) {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0);
	} 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 <= -5.7e-31) or not (NaChar <= 1e+113):
		tmp = (NaChar / (1.0 + math.exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0)
	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 <= -5.7e-31) || !(NaChar <= 1e+113))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(Ev + EAccept)) - mu) / KbT)))) + Float64(NdChar / 2.0));
	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 <= -5.7e-31) || ~((NaChar <= 1e+113)))
		tmp = (NaChar / (1.0 + exp((((Vef + (Ev + EAccept)) - mu) / KbT)))) + (NdChar / 2.0);
	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, -5.7e-31], N[Not[LessEqual[NaChar, 1e+113]], $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 / 2.0), $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 < -5.69999999999999995e-31 or 1e113 < 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 55.9%

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

   if -5.69999999999999995e-31 < NaChar < 1e113

   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 55.6%

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

      1. *-commutative 55.6%

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

   6. Simplified 55.6%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -5.7 \cdot 10^{-31} \lor \neg \left(NaChar \leq 10^{+113}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + \left(Ev + EAccept\right)\right) - mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 19: 48.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -5.4 \cdot 10^{+211}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NaChar \leq 3.4 \cdot 10^{+171}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -5.4e+211)
   (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (/ NdChar 2.0))
   (if (<= NaChar 3.4e+171)
     (+
      (/ NdChar (+ 1.0 (exp (/ (+ mu (- (+ Vef EDonor) Ec)) KbT))))
      (* NaChar 0.5))
     (+ (/ NaChar (+ 1.0 (exp (/ Vef KbT)))) (* NdChar 0.5)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -5.4e+211) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / 2.0);
	} else if (NaChar <= 3.4e+171) {
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-5.4d+211)) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar / 2.0d0)
    else if (nachar <= 3.4d+171) then
        tmp = (ndchar / (1.0d0 + exp(((mu + ((vef + edonor) - ec)) / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (nachar / (1.0d0 + exp((vef / kbt)))) + (ndchar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -5.4e+211) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar / 2.0);
	} else if (NaChar <= 3.4e+171) {
		tmp = (NdChar / (1.0 + Math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Vef / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -5.4e+211:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar / 2.0)
	elif NaChar <= 3.4e+171:
		tmp = (NdChar / (1.0 + math.exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NaChar / (1.0 + math.exp((Vef / KbT)))) + (NdChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -5.4e+211)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar / 2.0));
	elseif (NaChar <= 3.4e+171)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(mu + Float64(Float64(Vef + EDonor) - Ec)) / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NdChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -5.4e+211)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / 2.0);
	elseif (NaChar <= 3.4e+171)
		tmp = (NdChar / (1.0 + exp(((mu + ((Vef + EDonor) - Ec)) / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + (NdChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -5.4e+211], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 3.4e+171], 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], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -5.3999999999999998e211

   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 EAccept around inf 57.8%

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

   5. Taylor expanded in KbT around inf 52.4%

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

   if -5.3999999999999998e211 < NaChar < 3.4000000000000001e171

   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 50.6%

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

      1. *-commutative 50.6%

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

   6. Simplified 50.6%

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

   if 3.4000000000000001e171 < 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 Vef around inf 67.7%

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

   5. Taylor expanded in KbT around inf 38.2%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -5.4 \cdot 10^{+211}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NaChar \leq 3.4 \cdot 10^{+171}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu + \left(\left(Vef + EDonor\right) - Ec\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \]

Alternative 20: 38.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -9 \cdot 10^{+44}:\\ \;\;\;\;NaChar \cdot 0.5 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 6 \cdot 10^{-138}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq 2.3 \cdot 10^{+39}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -9e+44)
   (+ (* NaChar 0.5) (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
   (if (<= NdChar 6e-138)
     (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (/ NdChar 2.0))
     (if (<= NdChar 2.3e+39)
       (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (* NaChar 0.5))
       (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (* NaChar 0.5))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -9e+44) {
		tmp = (NaChar * 0.5) + (NdChar / (1.0 + exp((Vef / KbT))));
	} else if (NdChar <= 6e-138) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / 2.0);
	} else if (NdChar <= 2.3e+39) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-9d+44)) then
        tmp = (nachar * 0.5d0) + (ndchar / (1.0d0 + exp((vef / kbt))))
    else if (ndchar <= 6d-138) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar / 2.0d0)
    else if (ndchar <= 2.3d+39) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -9e+44) {
		tmp = (NaChar * 0.5) + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	} else if (NdChar <= 6e-138) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar / 2.0);
	} else if (NdChar <= 2.3e+39) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -9e+44:
		tmp = (NaChar * 0.5) + (NdChar / (1.0 + math.exp((Vef / KbT))))
	elif NdChar <= 6e-138:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar / 2.0)
	elif NdChar <= 2.3e+39:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -9e+44)
		tmp = Float64(Float64(NaChar * 0.5) + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	elseif (NdChar <= 6e-138)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar / 2.0));
	elseif (NdChar <= 2.3e+39)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -9e+44)
		tmp = (NaChar * 0.5) + (NdChar / (1.0 + exp((Vef / KbT))));
	elseif (NdChar <= 6e-138)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / 2.0);
	elseif (NdChar <= 2.3e+39)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -9e+44], N[(N[(NaChar * 0.5), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 6e-138], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.3e+39], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -9e44

   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 47.7%

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

      1. *-commutative 47.7%

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

   6. Simplified 47.7%

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

   7. Taylor expanded in Vef around inf 36.7%

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

   if -9e44 < NdChar < 6.0000000000000001e-138

   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 EAccept around inf 62.3%

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

   5. Taylor expanded in KbT around inf 39.5%

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

   if 6.0000000000000001e-138 < NdChar < 2.30000000000000012e39

   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 51.4%

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

      1. *-commutative 51.4%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in EDonor around inf 46.1%

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

   if 2.30000000000000012e39 < 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 59.5%

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

      1. *-commutative 59.5%

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

   6. Simplified 59.5%

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

   7. Taylor expanded in mu around inf 50.5%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -9 \cdot 10^{+44}:\\ \;\;\;\;NaChar \cdot 0.5 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 6 \cdot 10^{-138}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq 2.3 \cdot 10^{+39}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 21: 38.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq 1.72 \cdot 10^{+134}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -1.7e+96)
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (* NdChar 0.5))
   (if (<= NaChar 1.72e+134)
     (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (* NaChar 0.5))
     (+ (/ NaChar (+ 1.0 (exp (/ Vef KbT)))) (* NdChar 0.5)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -1.7e+96) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar * 0.5);
	} else if (NaChar <= 1.72e+134) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-1.7d+96)) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar * 0.5d0)
    else if (nachar <= 1.72d+134) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (nachar / (1.0d0 + exp((vef / kbt)))) + (ndchar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -1.7e+96) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar * 0.5);
	} else if (NaChar <= 1.72e+134) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Vef / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -1.7e+96:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar * 0.5)
	elif NaChar <= 1.72e+134:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NaChar / (1.0 + math.exp((Vef / KbT)))) + (NdChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -1.7e+96)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar * 0.5));
	elseif (NaChar <= 1.72e+134)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NdChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NaChar <= -1.7e+96)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar * 0.5);
	elseif (NaChar <= 1.72e+134)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + (NdChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -1.7e+96], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.72e+134], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -1.7e96

   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 Ev around inf 57.5%

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

   5. Taylor expanded in KbT around inf 36.7%

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

   if -1.7e96 < NaChar < 1.71999999999999998e134

   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 52.4%

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

      1. *-commutative 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in EDonor around inf 40.1%

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

   if 1.71999999999999998e134 < 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 Vef around inf 67.7%

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

   5. Taylor expanded in KbT around inf 38.5%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;NaChar \leq 1.72 \cdot 10^{+134}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \]

Alternative 22: 36.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -7.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -7.6e+123)
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (* NdChar 0.5))
   (+ (/ NaChar (+ 1.0 (exp (/ Vef KbT)))) (* NdChar 0.5))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -7.6e+123) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ev <= (-7.6d+123)) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = (nachar / (1.0d0 + exp((vef / kbt)))) + (ndchar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -7.6e+123) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Vef / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -7.6e+123:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar * 0.5)
	else:
		tmp = (NaChar / (1.0 + math.exp((Vef / KbT)))) + (NdChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -7.6e+123)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NdChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -7.6e+123)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar * 0.5);
	else
		tmp = (NaChar / (1.0 + exp((Vef / KbT)))) + (NdChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -7.6e+123], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Ev < -7.59999999999999989e123

   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 Ev around inf 92.3%

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

   5. Taylor expanded in KbT around inf 33.4%

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

   if -7.59999999999999989e123 < Ev

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
   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 71.2%

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

   5. Taylor expanded in KbT around inf 34.4%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -7.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \]

Alternative 23: 35.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + NdChar \cdot 0.5 \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (* NdChar 0.5)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar * 0.5);
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar * 0.5d0)
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar * 0.5);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar * 0.5)

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar * 0.5))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar * 0.5);
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]
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 Ev around inf 64.5%

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

5. Taylor expanded in KbT around inf 32.1%

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

6. Final simplification 32.1%

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

Alternative 24: 26.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -5.8 \cdot 10^{-220}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(\frac{EDonor}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -5.8e-220)
   (+
    (/
     NdChar
     (+
      1.0
      (- (+ (/ mu KbT) (+ (/ EDonor KbT) (+ 1.0 (/ Vef KbT)))) (/ Ec KbT))))
    (/
     NaChar
     (+ 1.0 (+ (/ Ev KbT) (+ 1.0 (* 0.5 (/ (* Ev Ev) (* KbT KbT))))))))
   (* 0.5 (+ NdChar NaChar))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -5.8e-220) {
		tmp = (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-5.8d-220)) then
        tmp = (ndchar / (1.0d0 + (((mu / kbt) + ((edonor / kbt) + (1.0d0 + (vef / kbt)))) - (ec / kbt)))) + (nachar / (1.0d0 + ((ev / kbt) + (1.0d0 + (0.5d0 * ((ev * ev) / (kbt * kbt)))))))
    else
        tmp = 0.5d0 * (ndchar + nachar)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -5.8e-220) {
		tmp = (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))));
	} else {
		tmp = 0.5 * (NdChar + NaChar);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -5.8e-220:
		tmp = (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))))
	else:
		tmp = 0.5 * (NdChar + NaChar)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -5.8e-220)
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(mu / KbT) + Float64(Float64(EDonor / KbT) + Float64(1.0 + Float64(Vef / KbT)))) - Float64(Ec / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(Ev / KbT) + Float64(1.0 + Float64(0.5 * Float64(Float64(Ev * Ev) / Float64(KbT * KbT))))))));
	else
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -5.8e-220)
		tmp = (NdChar / (1.0 + (((mu / KbT) + ((EDonor / KbT) + (1.0 + (Vef / KbT)))) - (Ec / KbT)))) + (NaChar / (1.0 + ((Ev / KbT) + (1.0 + (0.5 * ((Ev * Ev) / (KbT * KbT)))))));
	else
		tmp = 0.5 * (NdChar + NaChar);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -5.8e-220], N[(N[(NdChar / N[(1.0 + N[(N[(N[(mu / KbT), $MachinePrecision] + N[(N[(EDonor / KbT), $MachinePrecision] + N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(Ev / KbT), $MachinePrecision] + N[(1.0 + N[(0.5 * N[(N[(Ev * Ev), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -5.7999999999999997e-220

   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 Ev around inf 64.2%

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

   5. Taylor expanded in Ev around 0 58.2%

      \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \color{blue}{\left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{{Ev}^{2}}{{KbT}^{2}}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 58.2%

         \[\leadsto \frac{NdChar}{1 + e^{\frac{mu - \left(Ec - \left(Vef + EDonor\right)\right)}{KbT}}} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{\color{blue}{Ev \cdot Ev}}{{KbT}^{2}}\right)\right)} \]

      2. unpow2 58.2%

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

   7. Simplified 58.2%

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

   8. Taylor expanded in KbT around inf 26.5%

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

      1. associate-+r+ 26.5%

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

   10. Simplified 26.5%

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

   if -5.7999999999999997e-220 < 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 47.3%

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

      1. *-commutative 47.3%

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

   6. Simplified 47.3%

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

   7. Taylor expanded in KbT around inf 18.8%

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

      1. fma-def 18.8%

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

      2. associate-/l* 23.4%

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

      3. associate--l+ 23.4%

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

   9. Simplified 23.4%

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

   10. Taylor expanded in KbT around inf 28.7%

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

       1. distribute-lft-out 28.7%

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

   12. Simplified 28.7%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -5.8 \cdot 10^{-220}:\\ \;\;\;\;\frac{NdChar}{1 + \left(\left(\frac{mu}{KbT} + \left(\frac{EDonor}{KbT} + \left(1 + \frac{Vef}{KbT}\right)\right)\right) - \frac{Ec}{KbT}\right)} + \frac{NaChar}{1 + \left(\frac{Ev}{KbT} + \left(1 + 0.5 \cdot \frac{Ev \cdot Ev}{KbT \cdot KbT}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]

Alternative 25: 27.6% accurate, 45.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = 0.5d0 * (ndchar + nachar)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 45.3%

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

   1. *-commutative 45.3%

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

6. Simplified 45.3%

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

7. Taylor expanded in KbT around inf 16.8%

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

   1. fma-def 16.8%

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

   2. associate-/l* 21.5%

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

   3. associate--l+ 21.5%

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

9. Simplified 21.5%

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

10. Taylor expanded in KbT around inf 26.6%

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

    1. distribute-lft-out 26.6%

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

12. Simplified 26.6%

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

13. Final simplification 26.6%

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

Reproduce

herbie shell --seed 2023257 
(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))))))