Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 29.1s

Alternatives: 31

Speedup: 1.0×

• could not determine a ground truth

  1. NdChar = -1.6465190338419166e-182
  2. Ec = -1.1132016464273782e-307
  3. Vef = -2.8852177289051463e-227
  4. EDonor = 7.712655437933036e-209
  5. mu = 1.796962148924988e-104
  6. KbT = -3.1998184927307097e-298
  7. NaChar = 3.055475773429438e-183
  8. Ev = -1.2391546879229401e-144
  9. EAccept = 1.431647163839272e+162

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / (1.0 + pow(sqrt(exp(((EDonor + (Vef + (mu - Ec))) / KbT))), 2.0))) + (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 + (sqrt(exp(((edonor + (vef + (mu - ec))) / kbt))) ** 2.0d0))) + (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.pow(Math.sqrt(Math.exp(((EDonor + (Vef + (mu - Ec))) / KbT))), 2.0))) + (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.pow(math.sqrt(math.exp(((EDonor + (Vef + (mu - Ec))) / KbT))), 2.0))) + (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 + (sqrt(exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT))) ^ 2.0))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / (1.0 + (sqrt(exp(((EDonor + (Vef + (mu - Ec))) / KbT))) ^ 2.0))) + (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[Power[N[Sqrt[N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-+l- 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := t\_0 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ t_2 := e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}\\ \mathbf{if}\;mu \leq -2 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;mu \leq -4.2 \cdot 10^{-299}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;mu \leq 2050000000000:\\ \;\;\;\;\frac{NdChar}{1 + t\_2} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;mu \leq 8.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1 (+ t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT))))))
        (t_2 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
   (if (<= mu -2e+100)
     t_1
     (if (<= mu -4.2e-299)
       (+ t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
       (if (<= mu 2050000000000.0)
         (+ (/ NdChar (+ 1.0 t_2)) (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
         (if (<= mu 8.4e+120)
           (- (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar (- -1.0 t_2)))
           t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	double t_2 = exp(((EDonor - ((Ec - Vef) - mu)) / KbT));
	double tmp;
	if (mu <= -2e+100) {
		tmp = t_1;
	} else if (mu <= -4.2e-299) {
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	} else if (mu <= 2050000000000.0) {
		tmp = (NdChar / (1.0 + t_2)) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (mu <= 8.4e+120) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) - (NdChar / (-1.0 - t_2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + exp((mu / kbt))))
    t_2 = exp(((edonor - ((ec - vef) - mu)) / kbt))
    if (mu <= (-2d+100)) then
        tmp = t_1
    else if (mu <= (-4.2d-299)) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((vef / kbt))))
    else if (mu <= 2050000000000.0d0) then
        tmp = (ndchar / (1.0d0 + t_2)) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else if (mu <= 8.4d+120) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) - (ndchar / ((-1.0d0) - t_2))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	double t_2 = Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT));
	double tmp;
	if (mu <= -2e+100) {
		tmp = t_1;
	} else if (mu <= -4.2e-299) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	} else if (mu <= 2050000000000.0) {
		tmp = (NdChar / (1.0 + t_2)) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else if (mu <= 8.4e+120) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) - (NdChar / (-1.0 - t_2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = t_0 + (NdChar / (1.0 + math.exp((mu / KbT))))
	t_2 = math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))
	tmp = 0
	if mu <= -2e+100:
		tmp = t_1
	elif mu <= -4.2e-299:
		tmp = t_0 + (NdChar / (1.0 + math.exp((Vef / KbT))))
	elif mu <= 2050000000000.0:
		tmp = (NdChar / (1.0 + t_2)) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	elif mu <= 8.4e+120:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) - (NdChar / (-1.0 - t_2))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))))
	t_2 = exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT))
	tmp = 0.0
	if (mu <= -2e+100)
		tmp = t_1;
	elseif (mu <= -4.2e-299)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	elseif (mu <= 2050000000000.0)
		tmp = Float64(Float64(NdChar / Float64(1.0 + t_2)) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (mu <= 8.4e+120)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) - Float64(NdChar / Float64(-1.0 - t_2)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	t_2 = exp(((EDonor - ((Ec - Vef) - mu)) / KbT));
	tmp = 0.0;
	if (mu <= -2e+100)
		tmp = t_1;
	elseif (mu <= -4.2e-299)
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	elseif (mu <= 2050000000000.0)
		tmp = (NdChar / (1.0 + t_2)) + (NaChar / (1.0 + exp((EAccept / KbT))));
	elseif (mu <= 8.4e+120)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) - (NdChar / (-1.0 - t_2));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[mu, -2e+100], t$95$1, If[LessEqual[mu, -4.2e-299], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2050000000000.0], N[(N[(NdChar / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 8.4e+120], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -2.00000000000000003e100 or 8.4000000000000002e120 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 87.6%

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

   if -2.00000000000000003e100 < mu < -4.2000000000000002e-299

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 84.8%

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

   if -4.2000000000000002e-299 < mu < 2.05e12

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 70.5%

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

   if 2.05e12 < mu < 8.4000000000000002e120

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 83.2%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -2 \cdot 10^{+100}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;mu \leq -4.2 \cdot 10^{-299}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;mu \leq 2050000000000:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;mu \leq 8.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{if}\;NdChar \leq -1.16 \cdot 10^{+210}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq -8 \cdot 10^{-148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 4 \cdot 10^{-51}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.3 \cdot 10^{+107}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ Vef Ev) mu) KbT)))))))
   (if (<= NdChar -1.16e+210)
     (+
      (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
      (/
       NaChar
       (+
        1.0
        (-
         (+ 1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
         (/ mu KbT)))))
     (if (<= NdChar -8e-148)
       t_1
       (if (<= NdChar 4e-51)
         (+ t_0 (/ NdChar (+ 1.0 (* Ec (+ (/ Vef (* Ec KbT)) (/ -1.0 KbT))))))
         (if (<= NdChar 2.3e+107) (+ NdChar t_0) t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT))));
	double tmp;
	if (NdChar <= -1.16e+210) {
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	} else if (NdChar <= -8e-148) {
		tmp = t_1;
	} else if (NdChar <= 4e-51) {
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= 2.3e+107) {
		tmp = NdChar + t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / (1.0d0 + exp((((vef + ev) - mu) / kbt))))
    if (ndchar <= (-1.16d+210)) then
        tmp = (ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))) + (nachar / (1.0d0 + ((1.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt))))
    else if (ndchar <= (-8d-148)) then
        tmp = t_1
    else if (ndchar <= 4d-51) then
        tmp = t_0 + (ndchar / (1.0d0 + (ec * ((vef / (ec * kbt)) + ((-1.0d0) / kbt)))))
    else if (ndchar <= 2.3d+107) then
        tmp = ndchar + t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / (1.0 + Math.exp((((Vef + Ev) - mu) / KbT))));
	double tmp;
	if (NdChar <= -1.16e+210) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	} else if (NdChar <= -8e-148) {
		tmp = t_1;
	} else if (NdChar <= 4e-51) {
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= 2.3e+107) {
		tmp = NdChar + 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)))
	t_1 = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / (1.0 + math.exp((((Vef + Ev) - mu) / KbT))))
	tmp = 0
	if NdChar <= -1.16e+210:
		tmp = (NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))))
	elif NdChar <= -8e-148:
		tmp = t_1
	elif NdChar <= 4e-51:
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))))
	elif NdChar <= 2.3e+107:
		tmp = NdChar + t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)))))
	tmp = 0.0
	if (NdChar <= -1.16e+210)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT)))));
	elseif (NdChar <= -8e-148)
		tmp = t_1;
	elseif (NdChar <= 4e-51)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Ec * Float64(Float64(Vef / Float64(Ec * KbT)) + Float64(-1.0 / KbT))))));
	elseif (NdChar <= 2.3e+107)
		tmp = Float64(NdChar + t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT))));
	tmp = 0.0;
	if (NdChar <= -1.16e+210)
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	elseif (NdChar <= -8e-148)
		tmp = t_1;
	elseif (NdChar <= 4e-51)
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	elseif (NdChar <= 2.3e+107)
		tmp = NdChar + t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.16e+210], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -8e-148], t$95$1, If[LessEqual[NdChar, 4e-51], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(Ec * N[(N[(Vef / N[(Ec * KbT), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.3e+107], N[(NdChar + t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -1.16e210

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 88.3%

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

   if -1.16e210 < NdChar < -7.99999999999999949e-148 or 2.3e107 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 74.5%

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

   6. Taylor expanded in EAccept around 0 71.2%

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

   if -7.99999999999999949e-148 < NdChar < 4e-51

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 71.1%

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

   6. Taylor expanded in Ec around -inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. *-commutative 74.6%

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

      3. distribute-rgt-neg-in 74.6%

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

      4. +-commutative 74.6%

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

      5. mul-1-neg 74.6%

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

      6. unsub-neg 74.6%

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

      7. associate-+r+ 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in Vef around inf 83.9%

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

       1. *-commutative 83.9%

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

   11. Simplified 83.9%

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

   if 4e-51 < NdChar < 2.3e107

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.3%

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

   6. Taylor expanded in Ec around -inf 46.4%

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

      1. mul-1-neg 46.4%

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

      2. *-commutative 46.4%

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

      3. distribute-rgt-neg-in 46.4%

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

      4. +-commutative 46.4%

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

      5. mul-1-neg 46.4%

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

      6. unsub-neg 46.4%

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

      7. associate-+r+ 46.4%

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

   8. Simplified 46.4%

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

   9. Taylor expanded in Vef around inf 51.0%

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

   10. Taylor expanded in Vef around 0 82.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.16 \cdot 10^{+210}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq -8 \cdot 10^{-148}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 4 \cdot 10^{-51}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.3 \cdot 10^{+107}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := t\_0 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{if}\;mu \leq -1.15 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;mu \leq -2.5 \cdot 10^{-300}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.18 \cdot 10^{+122}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1 (+ t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))))
   (if (<= mu -1.15e+100)
     t_1
     (if (<= mu -2.5e-300)
       (+ t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
       (if (<= mu 1.18e+122)
         (+
          (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
          (/ 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 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	double tmp;
	if (mu <= -1.15e+100) {
		tmp = t_1;
	} else if (mu <= -2.5e-300) {
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	} else if (mu <= 1.18e+122) {
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (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 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + exp((mu / kbt))))
    if (mu <= (-1.15d+100)) then
        tmp = t_1
    else if (mu <= (-2.5d-300)) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((vef / kbt))))
    else if (mu <= 1.18d+122) then
        tmp = (ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))) + (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 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	double tmp;
	if (mu <= -1.15e+100) {
		tmp = t_1;
	} else if (mu <= -2.5e-300) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	} else if (mu <= 1.18e+122) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (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 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = t_0 + (NdChar / (1.0 + math.exp((mu / KbT))))
	tmp = 0
	if mu <= -1.15e+100:
		tmp = t_1
	elif mu <= -2.5e-300:
		tmp = t_0 + (NdChar / (1.0 + math.exp((Vef / KbT))))
	elif mu <= 1.18e+122:
		tmp = (NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (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(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))))
	tmp = 0.0
	if (mu <= -1.15e+100)
		tmp = t_1;
	elseif (mu <= -2.5e-300)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	elseif (mu <= 1.18e+122)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))) + 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 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	tmp = 0.0;
	if (mu <= -1.15e+100)
		tmp = t_1;
	elseif (mu <= -2.5e-300)
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	elseif (mu <= 1.18e+122)
		tmp = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (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[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -1.15e+100], t$95$1, If[LessEqual[mu, -2.5e-300], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 1.18e+122], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if mu < -1.14999999999999995e100 or 1.18000000000000003e122 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 87.6%

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

   if -1.14999999999999995e100 < mu < -2.49999999999999998e-300

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 84.8%

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

   if -2.49999999999999998e-300 < mu < 1.18000000000000003e122

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 72.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.15 \cdot 10^{+100}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;mu \leq -2.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.18 \cdot 10^{+122}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;mu \leq -1.1 \cdot 10^{+100} \lor \neg \left(mu \leq 4.6 \cdot 10^{+79}\right):\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (or (<= mu -1.1e+100) (not (<= mu 4.6e+79)))
     (+ t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
     (+ t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if ((mu <= -1.1e+100) || !(mu <= 4.6e+79)) {
		tmp = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	} else {
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if ((mu <= (-1.1d+100)) .or. (.not. (mu <= 4.6d+79))) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((mu / kbt))))
    else
        tmp = t_0 + (ndchar / (1.0d0 + exp((vef / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if ((mu <= -1.1e+100) || !(mu <= 4.6e+79)) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if (mu <= -1.1e+100) or not (mu <= 4.6e+79):
		tmp = t_0 + (NdChar / (1.0 + math.exp((mu / KbT))))
	else:
		tmp = t_0 + (NdChar / (1.0 + math.exp((Vef / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if ((mu <= -1.1e+100) || !(mu <= 4.6e+79))
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if ((mu <= -1.1e+100) || ~((mu <= 4.6e+79)))
		tmp = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	else
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[mu, -1.1e+100], N[Not[LessEqual[mu, 4.6e+79]], $MachinePrecision]], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if mu < -1.1e100 or 4.6000000000000001e79 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 84.4%

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

   if -1.1e100 < mu < 4.6000000000000001e79

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 78.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.1 \cdot 10^{+100} \lor \neg \left(mu \leq 4.6 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;mu \leq -1.25 \cdot 10^{+176}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;mu \leq 9.5 \cdot 10^{+72}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;mu \leq 6.1 \cdot 10^{+203}:\\ \;\;\;\;t\_0 + \frac{NdChar}{EDonor \cdot \left(\left(\frac{2}{EDonor} + \frac{1}{KbT}\right) + \frac{Vef}{EDonor \cdot KbT}\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;NdChar + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= mu -1.25e+176)
     (+
      (/ NdChar (+ 1.0 (exp (/ mu KbT))))
      (/ NaChar (+ 1.0 (exp (/ mu (- KbT))))))
     (if (<= mu 9.5e+72)
       (+ t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
       (if (<= mu 6.1e+203)
         (+
          t_0
          (/
           NdChar
           (-
            (*
             EDonor
             (+ (+ (/ 2.0 EDonor) (/ 1.0 KbT)) (/ Vef (* EDonor KbT))))
            (/ Ec KbT))))
         (+ NdChar t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (mu <= -1.25e+176) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	} else if (mu <= 9.5e+72) {
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	} else if (mu <= 6.1e+203) {
		tmp = t_0 + (NdChar / ((EDonor * (((2.0 / EDonor) + (1.0 / KbT)) + (Vef / (EDonor * KbT)))) - (Ec / KbT)));
	} else {
		tmp = NdChar + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (mu <= (-1.25d+176)) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((mu / -kbt))))
    else if (mu <= 9.5d+72) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((vef / kbt))))
    else if (mu <= 6.1d+203) then
        tmp = t_0 + (ndchar / ((edonor * (((2.0d0 / edonor) + (1.0d0 / kbt)) + (vef / (edonor * kbt)))) - (ec / kbt)))
    else
        tmp = ndchar + t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (mu <= -1.25e+176) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((mu / -KbT))));
	} else if (mu <= 9.5e+72) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	} else if (mu <= 6.1e+203) {
		tmp = t_0 + (NdChar / ((EDonor * (((2.0 / EDonor) + (1.0 / KbT)) + (Vef / (EDonor * KbT)))) - (Ec / KbT)));
	} else {
		tmp = NdChar + t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if mu <= -1.25e+176:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((mu / -KbT))))
	elif mu <= 9.5e+72:
		tmp = t_0 + (NdChar / (1.0 + math.exp((Vef / KbT))))
	elif mu <= 6.1e+203:
		tmp = t_0 + (NdChar / ((EDonor * (((2.0 / EDonor) + (1.0 / KbT)) + (Vef / (EDonor * KbT)))) - (Ec / KbT)))
	else:
		tmp = NdChar + t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (mu <= -1.25e+176)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(mu / Float64(-KbT))))));
	elseif (mu <= 9.5e+72)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	elseif (mu <= 6.1e+203)
		tmp = Float64(t_0 + Float64(NdChar / Float64(Float64(EDonor * Float64(Float64(Float64(2.0 / EDonor) + Float64(1.0 / KbT)) + Float64(Vef / Float64(EDonor * KbT)))) - Float64(Ec / KbT))));
	else
		tmp = Float64(NdChar + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (mu <= -1.25e+176)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((mu / -KbT))));
	elseif (mu <= 9.5e+72)
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	elseif (mu <= 6.1e+203)
		tmp = t_0 + (NdChar / ((EDonor * (((2.0 / EDonor) + (1.0 / KbT)) + (Vef / (EDonor * KbT)))) - (Ec / KbT)));
	else
		tmp = NdChar + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -1.25e+176], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(mu / (-KbT)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 9.5e+72], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 6.1e+203], N[(t$95$0 + N[(NdChar / N[(N[(EDonor * N[(N[(N[(2.0 / EDonor), $MachinePrecision] + N[(1.0 / KbT), $MachinePrecision]), $MachinePrecision] + N[(Vef / N[(EDonor * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -1.25e176

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 100.0%

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

   6. Taylor expanded in mu around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. mul-1-neg 91.3%

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

   8. Simplified 91.3%

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

   if -1.25e176 < mu < 9.50000000000000054e72

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 77.5%

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

   if 9.50000000000000054e72 < mu < 6.10000000000000014e203

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 70.0%

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

   6. Taylor expanded in mu around 0 70.6%

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

      1. associate-+r+ 70.6%

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

   8. Simplified 70.6%

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

   9. Taylor expanded in EDonor around inf 78.0%

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

       1. associate-+r+ 78.0%

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

       2. associate-*r/ 78.0%

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

       3. metadata-eval 78.0%

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

       4. *-commutative 78.0%

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

   11. Simplified 78.0%

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

   if 6.10000000000000014e203 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 39.2%

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

   6. Taylor expanded in Ec around -inf 39.5%

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

      1. mul-1-neg 39.5%

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

      2. *-commutative 39.5%

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

      3. distribute-rgt-neg-in 39.5%

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

      4. +-commutative 39.5%

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

      5. mul-1-neg 39.5%

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

      6. unsub-neg 39.5%

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

      7. associate-+r+ 39.5%

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

   8. Simplified 39.5%

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

   9. Taylor expanded in Vef around inf 51.4%

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

   10. Taylor expanded in Vef around 0 81.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.25 \cdot 10^{+176}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{mu}{-KbT}}}\\ \mathbf{elif}\;mu \leq 9.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;mu \leq 6.1 \cdot 10^{+203}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{EDonor \cdot \left(\left(\frac{2}{EDonor} + \frac{1}{KbT}\right) + \frac{Vef}{EDonor \cdot KbT}\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 8: 62.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_2 := t\_1 + \frac{NdChar}{1 + \left(\left(1 - EDonor \cdot \left(\left(\frac{-1}{KbT} - \frac{\frac{Vef}{EDonor}}{KbT}\right) - \frac{mu}{EDonor \cdot KbT}\right)\right) - \frac{Ec}{KbT}\right)}\\ t_3 := \frac{NdChar}{1 + t\_0} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{if}\;NdChar \leq -9.5 \cdot 10^{+114}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;NdChar \leq -8.2 \cdot 10^{-148}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NdChar \leq 3.3 \cdot 10^{-51}:\\ \;\;\;\;t\_1 + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 3.6 \cdot 10^{+141}:\\ \;\;\;\;NdChar + t\_1\\ \mathbf{elif}\;NdChar \leq 7.2 \cdot 10^{+198}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;NdChar \leq 4.15 \cdot 10^{+282}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_2
         (+
          t_1
          (/
           NdChar
           (+
            1.0
            (-
             (-
              1.0
              (*
               EDonor
               (-
                (- (/ -1.0 KbT) (/ (/ Vef EDonor) KbT))
                (/ mu (* EDonor KbT)))))
             (/ Ec KbT))))))
        (t_3
         (+
          (/ NdChar (+ 1.0 t_0))
          (/
           NaChar
           (+
            1.0
            (-
             (+ 1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
             (/ mu KbT)))))))
   (if (<= NdChar -9.5e+114)
     t_3
     (if (<= NdChar -8.2e-148)
       t_2
       (if (<= NdChar 3.3e-51)
         (+ t_1 (/ NdChar (+ 1.0 (* Ec (+ (/ Vef (* Ec KbT)) (/ -1.0 KbT))))))
         (if (<= NdChar 3.6e+141)
           (+ NdChar t_1)
           (if (<= NdChar 7.2e+198)
             t_3
             (if (<= NdChar 4.15e+282)
               t_2
               (-
                (/ NaChar (+ 1.0 (+ 1.0 (/ Ev KbT))))
                (/ NdChar (- -1.0 t_0)))))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((EDonor - ((Ec - Vef) - mu)) / KbT));
	double t_1 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_2 = t_1 + (NdChar / (1.0 + ((1.0 - (EDonor * (((-1.0 / KbT) - ((Vef / EDonor) / KbT)) - (mu / (EDonor * KbT))))) - (Ec / KbT))));
	double t_3 = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	double tmp;
	if (NdChar <= -9.5e+114) {
		tmp = t_3;
	} else if (NdChar <= -8.2e-148) {
		tmp = t_2;
	} else if (NdChar <= 3.3e-51) {
		tmp = t_1 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= 3.6e+141) {
		tmp = NdChar + t_1;
	} else if (NdChar <= 7.2e+198) {
		tmp = t_3;
	} else if (NdChar <= 4.15e+282) {
		tmp = t_2;
	} else {
		tmp = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = exp(((edonor - ((ec - vef) - mu)) / kbt))
    t_1 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_2 = t_1 + (ndchar / (1.0d0 + ((1.0d0 - (edonor * ((((-1.0d0) / kbt) - ((vef / edonor) / kbt)) - (mu / (edonor * kbt))))) - (ec / kbt))))
    t_3 = (ndchar / (1.0d0 + t_0)) + (nachar / (1.0d0 + ((1.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt))))
    if (ndchar <= (-9.5d+114)) then
        tmp = t_3
    else if (ndchar <= (-8.2d-148)) then
        tmp = t_2
    else if (ndchar <= 3.3d-51) then
        tmp = t_1 + (ndchar / (1.0d0 + (ec * ((vef / (ec * kbt)) + ((-1.0d0) / kbt)))))
    else if (ndchar <= 3.6d+141) then
        tmp = ndchar + t_1
    else if (ndchar <= 7.2d+198) then
        tmp = t_3
    else if (ndchar <= 4.15d+282) then
        tmp = t_2
    else
        tmp = (nachar / (1.0d0 + (1.0d0 + (ev / kbt)))) - (ndchar / ((-1.0d0) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT));
	double t_1 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_2 = t_1 + (NdChar / (1.0 + ((1.0 - (EDonor * (((-1.0 / KbT) - ((Vef / EDonor) / KbT)) - (mu / (EDonor * KbT))))) - (Ec / KbT))));
	double t_3 = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	double tmp;
	if (NdChar <= -9.5e+114) {
		tmp = t_3;
	} else if (NdChar <= -8.2e-148) {
		tmp = t_2;
	} else if (NdChar <= 3.3e-51) {
		tmp = t_1 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= 3.6e+141) {
		tmp = NdChar + t_1;
	} else if (NdChar <= 7.2e+198) {
		tmp = t_3;
	} else if (NdChar <= 4.15e+282) {
		tmp = t_2;
	} else {
		tmp = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - t_0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))
	t_1 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_2 = t_1 + (NdChar / (1.0 + ((1.0 - (EDonor * (((-1.0 / KbT) - ((Vef / EDonor) / KbT)) - (mu / (EDonor * KbT))))) - (Ec / KbT))))
	t_3 = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))))
	tmp = 0
	if NdChar <= -9.5e+114:
		tmp = t_3
	elif NdChar <= -8.2e-148:
		tmp = t_2
	elif NdChar <= 3.3e-51:
		tmp = t_1 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))))
	elif NdChar <= 3.6e+141:
		tmp = NdChar + t_1
	elif NdChar <= 7.2e+198:
		tmp = t_3
	elif NdChar <= 4.15e+282:
		tmp = t_2
	else:
		tmp = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - t_0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_2 = Float64(t_1 + Float64(NdChar / Float64(1.0 + Float64(Float64(1.0 - Float64(EDonor * Float64(Float64(Float64(-1.0 / KbT) - Float64(Float64(Vef / EDonor) / KbT)) - Float64(mu / Float64(EDonor * KbT))))) - Float64(Ec / KbT)))))
	t_3 = Float64(Float64(NdChar / Float64(1.0 + t_0)) + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT)))))
	tmp = 0.0
	if (NdChar <= -9.5e+114)
		tmp = t_3;
	elseif (NdChar <= -8.2e-148)
		tmp = t_2;
	elseif (NdChar <= 3.3e-51)
		tmp = Float64(t_1 + Float64(NdChar / Float64(1.0 + Float64(Ec * Float64(Float64(Vef / Float64(Ec * KbT)) + Float64(-1.0 / KbT))))));
	elseif (NdChar <= 3.6e+141)
		tmp = Float64(NdChar + t_1);
	elseif (NdChar <= 7.2e+198)
		tmp = t_3;
	elseif (NdChar <= 4.15e+282)
		tmp = t_2;
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev / KbT)))) - Float64(NdChar / Float64(-1.0 - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((EDonor - ((Ec - Vef) - mu)) / KbT));
	t_1 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_2 = t_1 + (NdChar / (1.0 + ((1.0 - (EDonor * (((-1.0 / KbT) - ((Vef / EDonor) / KbT)) - (mu / (EDonor * KbT))))) - (Ec / KbT))));
	t_3 = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	tmp = 0.0;
	if (NdChar <= -9.5e+114)
		tmp = t_3;
	elseif (NdChar <= -8.2e-148)
		tmp = t_2;
	elseif (NdChar <= 3.3e-51)
		tmp = t_1 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	elseif (NdChar <= 3.6e+141)
		tmp = NdChar + t_1;
	elseif (NdChar <= 7.2e+198)
		tmp = t_3;
	elseif (NdChar <= 4.15e+282)
		tmp = t_2;
	else
		tmp = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(NdChar / N[(1.0 + N[(N[(1.0 - N[(EDonor * N[(N[(N[(-1.0 / KbT), $MachinePrecision] - N[(N[(Vef / EDonor), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] - N[(mu / N[(EDonor * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -9.5e+114], t$95$3, If[LessEqual[NdChar, -8.2e-148], t$95$2, If[LessEqual[NdChar, 3.3e-51], N[(t$95$1 + N[(NdChar / N[(1.0 + N[(Ec * N[(N[(Vef / N[(Ec * KbT), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 3.6e+141], N[(NdChar + t$95$1), $MachinePrecision], If[LessEqual[NdChar, 7.2e+198], t$95$3, If[LessEqual[NdChar, 4.15e+282], t$95$2, N[(N[(NaChar / N[(1.0 + N[(1.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -9.5000000000000001e114 or 3.6000000000000001e141 < NdChar < 7.2000000000000004e198

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 75.5%

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

   if -9.5000000000000001e114 < NdChar < -8.2000000000000005e-148 or 7.2000000000000004e198 < NdChar < 4.14999999999999985e282

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 55.3%

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

   6. Taylor expanded in EDonor around inf 63.5%

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

      1. associate-+r+ 63.5%

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

      2. associate-/r* 69.7%

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

      3. *-commutative 69.7%

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

   8. Simplified 69.7%

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

   if -8.2000000000000005e-148 < NdChar < 3.29999999999999973e-51

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 71.1%

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

   6. Taylor expanded in Ec around -inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. *-commutative 74.6%

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

      3. distribute-rgt-neg-in 74.6%

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

      4. +-commutative 74.6%

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

      5. mul-1-neg 74.6%

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

      6. unsub-neg 74.6%

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

      7. associate-+r+ 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in Vef around inf 83.9%

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

       1. *-commutative 83.9%

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

   11. Simplified 83.9%

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

   if 3.29999999999999973e-51 < NdChar < 3.6000000000000001e141

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.0%

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

   6. Taylor expanded in Ec around -inf 46.1%

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

      1. mul-1-neg 46.1%

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

      2. *-commutative 46.1%

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

      3. distribute-rgt-neg-in 46.1%

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

      4. +-commutative 46.1%

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

      5. mul-1-neg 46.1%

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

      6. unsub-neg 46.1%

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

      7. associate-+r+ 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in Vef around inf 46.5%

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

   10. Taylor expanded in Vef around 0 79.3%

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

   if 4.14999999999999985e282 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 100.0%

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

   6. Taylor expanded in Ev around 0 100.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -9.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq -8.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(\left(1 - EDonor \cdot \left(\left(\frac{-1}{KbT} - \frac{\frac{Vef}{EDonor}}{KbT}\right) - \frac{mu}{EDonor \cdot KbT}\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 3.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 3.6 \cdot 10^{+141}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 7.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 4.15 \cdot 10^{+282}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(\left(1 - EDonor \cdot \left(\left(\frac{-1}{KbT} - \frac{\frac{Vef}{EDonor}}{KbT}\right) - \frac{mu}{EDonor \cdot KbT}\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_2 := \frac{NdChar}{1 + t\_0} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{if}\;NdChar \leq -1 \cdot 10^{+115}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NdChar \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;t\_1 + \frac{NdChar}{EDonor \cdot \left(\left(\frac{2}{EDonor} + \frac{1}{KbT}\right) + \frac{Vef}{EDonor \cdot KbT}\right) - \frac{Ec}{KbT}}\\ \mathbf{elif}\;NdChar \leq -3.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - t\_0}\\ \mathbf{elif}\;NdChar \leq -1.1 \cdot 10^{-218}:\\ \;\;\;\;t\_1 + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;NdChar \leq 2.1 \cdot 10^{-51}:\\ \;\;\;\;t\_1 + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 9.5 \cdot 10^{+142}:\\ \;\;\;\;NdChar + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))
        (t_1 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_2
         (+
          (/ NdChar (+ 1.0 t_0))
          (/
           NaChar
           (+
            1.0
            (-
             (+ 1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
             (/ mu KbT)))))))
   (if (<= NdChar -1e+115)
     t_2
     (if (<= NdChar -3.5e+22)
       (+
        t_1
        (/
         NdChar
         (-
          (* EDonor (+ (+ (/ 2.0 EDonor) (/ 1.0 KbT)) (/ Vef (* EDonor KbT))))
          (/ Ec KbT))))
       (if (<= NdChar -3.2e-131)
         (- (/ NaChar (+ 1.0 (+ 1.0 (/ Ev KbT)))) (/ NdChar (- -1.0 t_0)))
         (if (<= NdChar -1.1e-218)
           (+ t_1 (/ NdChar (+ 1.0 (/ mu KbT))))
           (if (<= NdChar 2.1e-51)
             (+
              t_1
              (/ NdChar (+ 1.0 (* Ec (+ (/ Vef (* Ec KbT)) (/ -1.0 KbT))))))
             (if (<= NdChar 9.5e+142) (+ NdChar 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 = exp(((EDonor - ((Ec - Vef) - mu)) / KbT));
	double t_1 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_2 = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	double tmp;
	if (NdChar <= -1e+115) {
		tmp = t_2;
	} else if (NdChar <= -3.5e+22) {
		tmp = t_1 + (NdChar / ((EDonor * (((2.0 / EDonor) + (1.0 / KbT)) + (Vef / (EDonor * KbT)))) - (Ec / KbT)));
	} else if (NdChar <= -3.2e-131) {
		tmp = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - t_0));
	} else if (NdChar <= -1.1e-218) {
		tmp = t_1 + (NdChar / (1.0 + (mu / KbT)));
	} else if (NdChar <= 2.1e-51) {
		tmp = t_1 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= 9.5e+142) {
		tmp = NdChar + t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(((edonor - ((ec - vef) - mu)) / kbt))
    t_1 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_2 = (ndchar / (1.0d0 + t_0)) + (nachar / (1.0d0 + ((1.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt))))
    if (ndchar <= (-1d+115)) then
        tmp = t_2
    else if (ndchar <= (-3.5d+22)) then
        tmp = t_1 + (ndchar / ((edonor * (((2.0d0 / edonor) + (1.0d0 / kbt)) + (vef / (edonor * kbt)))) - (ec / kbt)))
    else if (ndchar <= (-3.2d-131)) then
        tmp = (nachar / (1.0d0 + (1.0d0 + (ev / kbt)))) - (ndchar / ((-1.0d0) - t_0))
    else if (ndchar <= (-1.1d-218)) then
        tmp = t_1 + (ndchar / (1.0d0 + (mu / kbt)))
    else if (ndchar <= 2.1d-51) then
        tmp = t_1 + (ndchar / (1.0d0 + (ec * ((vef / (ec * kbt)) + ((-1.0d0) / kbt)))))
    else if (ndchar <= 9.5d+142) then
        tmp = ndchar + 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 = Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT));
	double t_1 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_2 = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	double tmp;
	if (NdChar <= -1e+115) {
		tmp = t_2;
	} else if (NdChar <= -3.5e+22) {
		tmp = t_1 + (NdChar / ((EDonor * (((2.0 / EDonor) + (1.0 / KbT)) + (Vef / (EDonor * KbT)))) - (Ec / KbT)));
	} else if (NdChar <= -3.2e-131) {
		tmp = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - t_0));
	} else if (NdChar <= -1.1e-218) {
		tmp = t_1 + (NdChar / (1.0 + (mu / KbT)));
	} else if (NdChar <= 2.1e-51) {
		tmp = t_1 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= 9.5e+142) {
		tmp = NdChar + t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))
	t_1 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_2 = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))))
	tmp = 0
	if NdChar <= -1e+115:
		tmp = t_2
	elif NdChar <= -3.5e+22:
		tmp = t_1 + (NdChar / ((EDonor * (((2.0 / EDonor) + (1.0 / KbT)) + (Vef / (EDonor * KbT)))) - (Ec / KbT)))
	elif NdChar <= -3.2e-131:
		tmp = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - t_0))
	elif NdChar <= -1.1e-218:
		tmp = t_1 + (NdChar / (1.0 + (mu / KbT)))
	elif NdChar <= 2.1e-51:
		tmp = t_1 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))))
	elif NdChar <= 9.5e+142:
		tmp = NdChar + t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + t_0)) + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT)))))
	tmp = 0.0
	if (NdChar <= -1e+115)
		tmp = t_2;
	elseif (NdChar <= -3.5e+22)
		tmp = Float64(t_1 + Float64(NdChar / Float64(Float64(EDonor * Float64(Float64(Float64(2.0 / EDonor) + Float64(1.0 / KbT)) + Float64(Vef / Float64(EDonor * KbT)))) - Float64(Ec / KbT))));
	elseif (NdChar <= -3.2e-131)
		tmp = Float64(Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev / KbT)))) - Float64(NdChar / Float64(-1.0 - t_0)));
	elseif (NdChar <= -1.1e-218)
		tmp = Float64(t_1 + Float64(NdChar / Float64(1.0 + Float64(mu / KbT))));
	elseif (NdChar <= 2.1e-51)
		tmp = Float64(t_1 + Float64(NdChar / Float64(1.0 + Float64(Ec * Float64(Float64(Vef / Float64(Ec * KbT)) + Float64(-1.0 / KbT))))));
	elseif (NdChar <= 9.5e+142)
		tmp = Float64(NdChar + 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 = exp(((EDonor - ((Ec - Vef) - mu)) / KbT));
	t_1 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_2 = (NdChar / (1.0 + t_0)) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	tmp = 0.0;
	if (NdChar <= -1e+115)
		tmp = t_2;
	elseif (NdChar <= -3.5e+22)
		tmp = t_1 + (NdChar / ((EDonor * (((2.0 / EDonor) + (1.0 / KbT)) + (Vef / (EDonor * KbT)))) - (Ec / KbT)));
	elseif (NdChar <= -3.2e-131)
		tmp = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - t_0));
	elseif (NdChar <= -1.1e-218)
		tmp = t_1 + (NdChar / (1.0 + (mu / KbT)));
	elseif (NdChar <= 2.1e-51)
		tmp = t_1 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	elseif (NdChar <= 9.5e+142)
		tmp = NdChar + 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[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1e+115], t$95$2, If[LessEqual[NdChar, -3.5e+22], N[(t$95$1 + N[(NdChar / N[(N[(EDonor * N[(N[(N[(2.0 / EDonor), $MachinePrecision] + N[(1.0 / KbT), $MachinePrecision]), $MachinePrecision] + N[(Vef / N[(EDonor * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -3.2e-131], N[(N[(NaChar / N[(1.0 + N[(1.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -1.1e-218], N[(t$95$1 + N[(NdChar / N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.1e-51], N[(t$95$1 + N[(NdChar / N[(1.0 + N[(Ec * N[(N[(Vef / N[(Ec * KbT), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 9.5e+142], N[(NdChar + t$95$1), $MachinePrecision], t$95$2]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if NdChar < -1e115 or 9.50000000000000001e142 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 68.7%

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

   if -1e115 < NdChar < -3.5e22

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 56.7%

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

   6. Taylor expanded in mu around 0 56.7%

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

      1. associate-+r+ 56.7%

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

   8. Simplified 56.7%

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

   9. Taylor expanded in EDonor around inf 82.9%

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

       1. associate-+r+ 82.9%

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

       2. associate-*r/ 82.9%

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

       3. metadata-eval 82.9%

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

       4. *-commutative 82.9%

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

   11. Simplified 82.9%

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

   if -3.5e22 < NdChar < -3.2e-131

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 87.2%

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

   6. Taylor expanded in Ev around 0 73.6%

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

   if -3.2e-131 < NdChar < -1.10000000000000003e-218

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 58.1%

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

   6. Taylor expanded in Ec around -inf 58.1%

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

      1. mul-1-neg 58.1%

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

      2. *-commutative 58.1%

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

      3. distribute-rgt-neg-in 58.1%

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

      4. +-commutative 58.1%

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

      5. mul-1-neg 58.1%

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

      6. unsub-neg 58.1%

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

      7. associate-+r+ 58.1%

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

   8. Simplified 58.1%

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

   9. Taylor expanded in mu around inf 82.3%

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

   if -1.10000000000000003e-218 < NdChar < 2.10000000000000002e-51

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 74.8%

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

   6. Taylor expanded in Ec around -inf 78.9%

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

      1. mul-1-neg 78.9%

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

      2. *-commutative 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

      4. +-commutative 78.9%

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

      5. mul-1-neg 78.9%

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

      6. unsub-neg 78.9%

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

      7. associate-+r+ 78.9%

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

   8. Simplified 78.9%

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

   9. Taylor expanded in Vef around inf 86.0%

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

       1. *-commutative 86.0%

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

   11. Simplified 86.0%

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

   if 2.10000000000000002e-51 < NdChar < 9.50000000000000001e142

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.0%

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

   6. Taylor expanded in Ec around -inf 46.1%

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

      1. mul-1-neg 46.1%

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

      2. *-commutative 46.1%

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

      3. distribute-rgt-neg-in 46.1%

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

      4. +-commutative 46.1%

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

      5. mul-1-neg 46.1%

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

      6. unsub-neg 46.1%

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

      7. associate-+r+ 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in Vef around inf 46.5%

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

   10. Taylor expanded in Vef around 0 79.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1 \cdot 10^{+115}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{EDonor \cdot \left(\left(\frac{2}{EDonor} + \frac{1}{KbT}\right) + \frac{Vef}{EDonor \cdot KbT}\right) - \frac{Ec}{KbT}}\\ \mathbf{elif}\;NdChar \leq -3.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -1.1 \cdot 10^{-218}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;NdChar \leq 2.1 \cdot 10^{-51}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 9.5 \cdot 10^{+142}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := NdChar + t\_0\\ t_2 := \frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{if}\;NdChar \leq -2.45 \cdot 10^{+104}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NdChar \leq -2.1 \cdot 10^{-160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.7 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1 (+ NdChar t_0))
        (t_2
         (+
          (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
          (/
           NaChar
           (+
            1.0
            (-
             (+ 1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
             (/ mu KbT)))))))
   (if (<= NdChar -2.45e+104)
     t_2
     (if (<= NdChar -2.1e-160)
       t_1
       (if (<= NdChar 1.25e-52)
         (+ t_0 (/ NdChar (+ 1.0 (* Ec (+ (/ Vef (* Ec KbT)) (/ -1.0 KbT))))))
         (if (<= NdChar 2.7e+141) t_1 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar + t_0;
	double t_2 = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	double tmp;
	if (NdChar <= -2.45e+104) {
		tmp = t_2;
	} else if (NdChar <= -2.1e-160) {
		tmp = t_1;
	} else if (NdChar <= 1.25e-52) {
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= 2.7e+141) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = ndchar + t_0
    t_2 = (ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))) + (nachar / (1.0d0 + ((1.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt))))
    if (ndchar <= (-2.45d+104)) then
        tmp = t_2
    else if (ndchar <= (-2.1d-160)) then
        tmp = t_1
    else if (ndchar <= 1.25d-52) then
        tmp = t_0 + (ndchar / (1.0d0 + (ec * ((vef / (ec * kbt)) + ((-1.0d0) / kbt)))))
    else if (ndchar <= 2.7d+141) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar + t_0;
	double t_2 = (NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	double tmp;
	if (NdChar <= -2.45e+104) {
		tmp = t_2;
	} else if (NdChar <= -2.1e-160) {
		tmp = t_1;
	} else if (NdChar <= 1.25e-52) {
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= 2.7e+141) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = NdChar + t_0
	t_2 = (NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))))
	tmp = 0
	if NdChar <= -2.45e+104:
		tmp = t_2
	elif NdChar <= -2.1e-160:
		tmp = t_1
	elif NdChar <= 1.25e-52:
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))))
	elif NdChar <= 2.7e+141:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(NdChar + t_0)
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT)))))
	tmp = 0.0
	if (NdChar <= -2.45e+104)
		tmp = t_2;
	elseif (NdChar <= -2.1e-160)
		tmp = t_1;
	elseif (NdChar <= 1.25e-52)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Ec * Float64(Float64(Vef / Float64(Ec * KbT)) + Float64(-1.0 / KbT))))));
	elseif (NdChar <= 2.7e+141)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = NdChar + t_0;
	t_2 = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	tmp = 0.0;
	if (NdChar <= -2.45e+104)
		tmp = t_2;
	elseif (NdChar <= -2.1e-160)
		tmp = t_1;
	elseif (NdChar <= 1.25e-52)
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	elseif (NdChar <= 2.7e+141)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -2.45e+104], t$95$2, If[LessEqual[NdChar, -2.1e-160], t$95$1, If[LessEqual[NdChar, 1.25e-52], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(Ec * N[(N[(Vef / N[(Ec * KbT), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.7e+141], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -2.44999999999999993e104 or 2.7000000000000001e141 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 68.3%

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

   if -2.44999999999999993e104 < NdChar < -2.1e-160 or 1.25e-52 < NdChar < 2.7000000000000001e141

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 55.2%

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

   6. Taylor expanded in Ec around -inf 58.3%

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

      1. mul-1-neg 58.3%

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

      2. *-commutative 58.3%

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

      3. distribute-rgt-neg-in 58.3%

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

      4. +-commutative 58.3%

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

      5. mul-1-neg 58.3%

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

      6. unsub-neg 58.3%

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

      7. associate-+r+ 58.3%

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

   8. Simplified 58.3%

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

   9. Taylor expanded in Vef around inf 51.6%

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

   10. Taylor expanded in Vef around 0 74.5%

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

   if -2.1e-160 < NdChar < 1.25e-52

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 71.3%

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

   6. Taylor expanded in Ec around -inf 75.0%

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

      1. mul-1-neg 75.0%

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

      2. *-commutative 75.0%

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

      3. distribute-rgt-neg-in 75.0%

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

      4. +-commutative 75.0%

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

      5. mul-1-neg 75.0%

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

      6. unsub-neg 75.0%

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

      7. associate-+r+ 75.0%

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

   8. Simplified 75.0%

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

   9. Taylor expanded in Vef around inf 85.1%

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

       1. *-commutative 85.1%

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

   11. Simplified 85.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.45 \cdot 10^{+104}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq -2.1 \cdot 10^{-160}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.7 \cdot 10^{+141}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{if}\;NdChar \leq -1.05 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq -8 \cdot 10^{-148}:\\ \;\;\;\;t\_0 - \frac{NdChar}{-1 + Ec \cdot \left(\frac{1}{KbT} + \frac{\left(-1 - \frac{EDonor}{KbT}\right) - \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)}{Ec}\right)}\\ \mathbf{elif}\;NdChar \leq 1.35 \cdot 10^{-52}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.2 \cdot 10^{+143}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT))))
          (/
           NaChar
           (+
            1.0
            (-
             (+ 1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
             (/ mu KbT)))))))
   (if (<= NdChar -1.05e+104)
     t_1
     (if (<= NdChar -8e-148)
       (-
        t_0
        (/
         NdChar
         (+
          -1.0
          (*
           Ec
           (+
            (/ 1.0 KbT)
            (/ (- (- -1.0 (/ EDonor KbT)) (+ (/ mu KbT) (/ Vef KbT))) Ec))))))
       (if (<= NdChar 1.35e-52)
         (+ t_0 (/ NdChar (+ 1.0 (* Ec (+ (/ Vef (* Ec KbT)) (/ -1.0 KbT))))))
         (if (<= NdChar 2.2e+143) (+ NdChar t_0) t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	double tmp;
	if (NdChar <= -1.05e+104) {
		tmp = t_1;
	} else if (NdChar <= -8e-148) {
		tmp = t_0 - (NdChar / (-1.0 + (Ec * ((1.0 / KbT) + (((-1.0 - (EDonor / KbT)) - ((mu / KbT) + (Vef / KbT))) / Ec)))));
	} else if (NdChar <= 1.35e-52) {
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= 2.2e+143) {
		tmp = NdChar + t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp(((edonor - ((ec - vef) - mu)) / kbt)))) + (nachar / (1.0d0 + ((1.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt))))
    if (ndchar <= (-1.05d+104)) then
        tmp = t_1
    else if (ndchar <= (-8d-148)) then
        tmp = t_0 - (ndchar / ((-1.0d0) + (ec * ((1.0d0 / kbt) + ((((-1.0d0) - (edonor / kbt)) - ((mu / kbt) + (vef / kbt))) / ec)))))
    else if (ndchar <= 1.35d-52) then
        tmp = t_0 + (ndchar / (1.0d0 + (ec * ((vef / (ec * kbt)) + ((-1.0d0) / kbt)))))
    else if (ndchar <= 2.2d+143) then
        tmp = ndchar + t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	double tmp;
	if (NdChar <= -1.05e+104) {
		tmp = t_1;
	} else if (NdChar <= -8e-148) {
		tmp = t_0 - (NdChar / (-1.0 + (Ec * ((1.0 / KbT) + (((-1.0 - (EDonor / KbT)) - ((mu / KbT) + (Vef / KbT))) / Ec)))));
	} else if (NdChar <= 1.35e-52) {
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= 2.2e+143) {
		tmp = NdChar + 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)))
	t_1 = (NdChar / (1.0 + math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))))
	tmp = 0
	if NdChar <= -1.05e+104:
		tmp = t_1
	elif NdChar <= -8e-148:
		tmp = t_0 - (NdChar / (-1.0 + (Ec * ((1.0 / KbT) + (((-1.0 - (EDonor / KbT)) - ((mu / KbT) + (Vef / KbT))) / Ec)))))
	elif NdChar <= 1.35e-52:
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))))
	elif NdChar <= 2.2e+143:
		tmp = NdChar + t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))) + Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT)))))
	tmp = 0.0
	if (NdChar <= -1.05e+104)
		tmp = t_1;
	elseif (NdChar <= -8e-148)
		tmp = Float64(t_0 - Float64(NdChar / Float64(-1.0 + Float64(Ec * Float64(Float64(1.0 / KbT) + Float64(Float64(Float64(-1.0 - Float64(EDonor / KbT)) - Float64(Float64(mu / KbT) + Float64(Vef / KbT))) / Ec))))));
	elseif (NdChar <= 1.35e-52)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Ec * Float64(Float64(Vef / Float64(Ec * KbT)) + Float64(-1.0 / KbT))))));
	elseif (NdChar <= 2.2e+143)
		tmp = Float64(NdChar + t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = (NdChar / (1.0 + exp(((EDonor - ((Ec - Vef) - mu)) / KbT)))) + (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))));
	tmp = 0.0;
	if (NdChar <= -1.05e+104)
		tmp = t_1;
	elseif (NdChar <= -8e-148)
		tmp = t_0 - (NdChar / (-1.0 + (Ec * ((1.0 / KbT) + (((-1.0 - (EDonor / KbT)) - ((mu / KbT) + (Vef / KbT))) / Ec)))));
	elseif (NdChar <= 1.35e-52)
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	elseif (NdChar <= 2.2e+143)
		tmp = NdChar + t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.05e+104], t$95$1, If[LessEqual[NdChar, -8e-148], N[(t$95$0 - N[(NdChar / N[(-1.0 + N[(Ec * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(N[(-1.0 - N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] - N[(N[(mu / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.35e-52], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(Ec * N[(N[(Vef / N[(Ec * KbT), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.2e+143], N[(NdChar + t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -1.0499999999999999e104 or 2.20000000000000014e143 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 68.3%

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

   if -1.0499999999999999e104 < NdChar < -7.99999999999999949e-148

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 63.9%

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

   6. Taylor expanded in Ec around -inf 71.1%

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

      1. mul-1-neg 71.1%

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

      2. *-commutative 71.1%

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

      3. distribute-rgt-neg-in 71.1%

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

      4. +-commutative 71.1%

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

      5. mul-1-neg 71.1%

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

      6. unsub-neg 71.1%

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

      7. associate-+r+ 71.1%

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

   8. Simplified 71.1%

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

   if -7.99999999999999949e-148 < NdChar < 1.35000000000000005e-52

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 71.1%

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

   6. Taylor expanded in Ec around -inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. *-commutative 74.6%

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

      3. distribute-rgt-neg-in 74.6%

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

      4. +-commutative 74.6%

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

      5. mul-1-neg 74.6%

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

      6. unsub-neg 74.6%

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

      7. associate-+r+ 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in Vef around inf 83.9%

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

       1. *-commutative 83.9%

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

   11. Simplified 83.9%

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

   if 1.35000000000000005e-52 < NdChar < 2.20000000000000014e143

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.0%

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

   6. Taylor expanded in Ec around -inf 46.1%

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

      1. mul-1-neg 46.1%

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

      2. *-commutative 46.1%

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

      3. distribute-rgt-neg-in 46.1%

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

      4. +-commutative 46.1%

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

      5. mul-1-neg 46.1%

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

      6. unsub-neg 46.1%

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

      7. associate-+r+ 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in Vef around inf 46.5%

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

   10. Taylor expanded in Vef around 0 79.3%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.05 \cdot 10^{+104}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq -8 \cdot 10^{-148}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} - \frac{NdChar}{-1 + Ec \cdot \left(\frac{1}{KbT} + \frac{\left(-1 - \frac{EDonor}{KbT}\right) - \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)}{Ec}\right)}\\ \mathbf{elif}\;NdChar \leq 1.35 \cdot 10^{-52}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 2.2 \cdot 10^{+143}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -1.1 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq -8.5 \cdot 10^{-184}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{elif}\;NdChar \leq -9 \cdot 10^{-307}:\\ \;\;\;\;t\_0 + \frac{NdChar}{2 + \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{elif}\;NdChar \leq 1.12 \cdot 10^{-52}:\\ \;\;\;\;t\_0 + KbT \cdot \frac{NdChar}{\left(mu + \left(EDonor + Vef\right)\right) - Ec}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1
         (-
          (/ NaChar (+ 1.0 (+ 1.0 (/ Ev KbT))))
          (/ NdChar (- -1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))))
   (if (<= NdChar -1.1e+104)
     t_1
     (if (<= NdChar -8.5e-184)
       (+ NdChar t_0)
       (if (<= NdChar -9e-307)
         (+ t_0 (/ NdChar (+ 2.0 (+ (/ Vef KbT) (/ EDonor KbT)))))
         (if (<= NdChar 3.9e-159)
           (+ t_0 (/ NdChar (- 1.0 (/ Ec KbT))))
           (if (<= NdChar 1.12e-52)
             (+ t_0 (* KbT (/ NdChar (- (+ mu (+ EDonor Vef)) Ec))))
             t_1)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -1.1e+104) {
		tmp = t_1;
	} else if (NdChar <= -8.5e-184) {
		tmp = NdChar + t_0;
	} else if (NdChar <= -9e-307) {
		tmp = t_0 + (NdChar / (2.0 + ((Vef / KbT) + (EDonor / KbT))));
	} else if (NdChar <= 3.9e-159) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	} else if (NdChar <= 1.12e-52) {
		tmp = t_0 + (KbT * (NdChar / ((mu + (EDonor + Vef)) - Ec)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = (nachar / (1.0d0 + (1.0d0 + (ev / kbt)))) - (ndchar / ((-1.0d0) - exp(((edonor - ((ec - vef) - mu)) / kbt))))
    if (ndchar <= (-1.1d+104)) then
        tmp = t_1
    else if (ndchar <= (-8.5d-184)) then
        tmp = ndchar + t_0
    else if (ndchar <= (-9d-307)) then
        tmp = t_0 + (ndchar / (2.0d0 + ((vef / kbt) + (edonor / kbt))))
    else if (ndchar <= 3.9d-159) then
        tmp = t_0 + (ndchar / (1.0d0 - (ec / kbt)))
    else if (ndchar <= 1.12d-52) then
        tmp = t_0 + (kbt * (ndchar / ((mu + (edonor + vef)) - ec)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -1.1e+104) {
		tmp = t_1;
	} else if (NdChar <= -8.5e-184) {
		tmp = NdChar + t_0;
	} else if (NdChar <= -9e-307) {
		tmp = t_0 + (NdChar / (2.0 + ((Vef / KbT) + (EDonor / KbT))));
	} else if (NdChar <= 3.9e-159) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	} else if (NdChar <= 1.12e-52) {
		tmp = t_0 + (KbT * (NdChar / ((mu + (EDonor + Vef)) - Ec)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))))
	tmp = 0
	if NdChar <= -1.1e+104:
		tmp = t_1
	elif NdChar <= -8.5e-184:
		tmp = NdChar + t_0
	elif NdChar <= -9e-307:
		tmp = t_0 + (NdChar / (2.0 + ((Vef / KbT) + (EDonor / KbT))))
	elif NdChar <= 3.9e-159:
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)))
	elif NdChar <= 1.12e-52:
		tmp = t_0 + (KbT * (NdChar / ((mu + (EDonor + Vef)) - Ec)))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -1.1e+104)
		tmp = t_1;
	elseif (NdChar <= -8.5e-184)
		tmp = Float64(NdChar + t_0);
	elseif (NdChar <= -9e-307)
		tmp = Float64(t_0 + Float64(NdChar / Float64(2.0 + Float64(Float64(Vef / KbT) + Float64(EDonor / KbT)))));
	elseif (NdChar <= 3.9e-159)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 - Float64(Ec / KbT))));
	elseif (NdChar <= 1.12e-52)
		tmp = Float64(t_0 + Float64(KbT * Float64(NdChar / Float64(Float64(mu + Float64(EDonor + Vef)) - Ec))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -1.1e+104)
		tmp = t_1;
	elseif (NdChar <= -8.5e-184)
		tmp = NdChar + t_0;
	elseif (NdChar <= -9e-307)
		tmp = t_0 + (NdChar / (2.0 + ((Vef / KbT) + (EDonor / KbT))));
	elseif (NdChar <= 3.9e-159)
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	elseif (NdChar <= 1.12e-52)
		tmp = t_0 + (KbT * (NdChar / ((mu + (EDonor + Vef)) - Ec)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[(1.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.1e+104], t$95$1, If[LessEqual[NdChar, -8.5e-184], N[(NdChar + t$95$0), $MachinePrecision], If[LessEqual[NdChar, -9e-307], N[(t$95$0 + N[(NdChar / N[(2.0 + N[(N[(Vef / KbT), $MachinePrecision] + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 3.9e-159], N[(t$95$0 + N[(NdChar / N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.12e-52], N[(t$95$0 + N[(KbT * N[(NdChar / N[(N[(mu + N[(EDonor + Vef), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -1.1e104 or 1.11999999999999994e-52 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 80.9%

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

   6. Taylor expanded in Ev around 0 65.9%

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

   if -1.1e104 < NdChar < -8.50000000000000036e-184

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 61.6%

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

   6. Taylor expanded in Ec around -inf 67.3%

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

      1. mul-1-neg 67.3%

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

      2. *-commutative 67.3%

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

      3. distribute-rgt-neg-in 67.3%

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

      4. +-commutative 67.3%

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

      5. mul-1-neg 67.3%

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

      6. unsub-neg 67.3%

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

      7. associate-+r+ 67.3%

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

   8. Simplified 67.3%

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

   9. Taylor expanded in Vef around inf 54.7%

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

   10. Taylor expanded in Vef around 0 70.1%

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

   if -8.50000000000000036e-184 < NdChar < -8.99999999999999978e-307

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 89.1%

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

   6. Taylor expanded in mu around 0 92.8%

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

      1. associate-+r+ 92.8%

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

   8. Simplified 92.8%

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

   9. Taylor expanded in Ec around 0 92.8%

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

       1. +-commutative 92.8%

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

   11. Simplified 92.8%

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

   if -8.99999999999999978e-307 < NdChar < 3.89999999999999977e-159

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 72.4%

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

   6. Taylor expanded in Ec around -inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. *-commutative 75.3%

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

      3. distribute-rgt-neg-in 75.3%

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

      4. +-commutative 75.3%

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

      5. mul-1-neg 75.3%

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

      6. unsub-neg 75.3%

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

      7. associate-+r+ 75.3%

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

   8. Simplified 75.3%

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

   9. Taylor expanded in Ec around inf 89.4%

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

       1. mul-1-neg 89.4%

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

       2. distribute-frac-neg 89.4%

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

   11. Simplified 89.4%

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

   if 3.89999999999999977e-159 < NdChar < 1.11999999999999994e-52

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 48.5%

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

   6. Taylor expanded in KbT around 0 62.4%

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

      1. associate-/l* 93.6%

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

      2. associate-+r+ 93.6%

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

      3. +-commutative 93.6%

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

   8. Simplified 93.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.1 \cdot 10^{+104}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -8.5 \cdot 10^{-184}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -9 \cdot 10^{-307}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{2 + \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{elif}\;NdChar \leq 1.12 \cdot 10^{-52}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + KbT \cdot \frac{NdChar}{\left(mu + \left(EDonor + Vef\right)\right) - Ec}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -9 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq -1.5 \cdot 10^{+61}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + Ec \cdot \left(\frac{EDonor}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq -1.1 \cdot 10^{-166}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{elif}\;NdChar \leq 1.05 \cdot 10^{-58}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1
         (-
          (/ NaChar (+ 1.0 (+ 1.0 (/ Ev KbT))))
          (/ NdChar (- -1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))))
   (if (<= NdChar -9e+114)
     t_1
     (if (<= NdChar -1.5e+61)
       (+ t_0 (/ NdChar (+ 1.0 (* Ec (+ (/ EDonor (* Ec KbT)) (/ -1.0 KbT))))))
       (if (<= NdChar -1.1e-166)
         (+ NdChar t_0)
         (if (<= NdChar 1.05e-58)
           (+
            t_0
            (/ NdChar (+ 1.0 (* Ec (+ (/ Vef (* Ec KbT)) (/ -1.0 KbT))))))
           t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -9e+114) {
		tmp = t_1;
	} else if (NdChar <= -1.5e+61) {
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((EDonor / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= -1.1e-166) {
		tmp = NdChar + t_0;
	} else if (NdChar <= 1.05e-58) {
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = (nachar / (1.0d0 + (1.0d0 + (ev / kbt)))) - (ndchar / ((-1.0d0) - exp(((edonor - ((ec - vef) - mu)) / kbt))))
    if (ndchar <= (-9d+114)) then
        tmp = t_1
    else if (ndchar <= (-1.5d+61)) then
        tmp = t_0 + (ndchar / (1.0d0 + (ec * ((edonor / (ec * kbt)) + ((-1.0d0) / kbt)))))
    else if (ndchar <= (-1.1d-166)) then
        tmp = ndchar + t_0
    else if (ndchar <= 1.05d-58) then
        tmp = t_0 + (ndchar / (1.0d0 + (ec * ((vef / (ec * kbt)) + ((-1.0d0) / kbt)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -9e+114) {
		tmp = t_1;
	} else if (NdChar <= -1.5e+61) {
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((EDonor / (Ec * KbT)) + (-1.0 / KbT)))));
	} else if (NdChar <= -1.1e-166) {
		tmp = NdChar + t_0;
	} else if (NdChar <= 1.05e-58) {
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))))
	tmp = 0
	if NdChar <= -9e+114:
		tmp = t_1
	elif NdChar <= -1.5e+61:
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((EDonor / (Ec * KbT)) + (-1.0 / KbT)))))
	elif NdChar <= -1.1e-166:
		tmp = NdChar + t_0
	elif NdChar <= 1.05e-58:
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -9e+114)
		tmp = t_1;
	elseif (NdChar <= -1.5e+61)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Ec * Float64(Float64(EDonor / Float64(Ec * KbT)) + Float64(-1.0 / KbT))))));
	elseif (NdChar <= -1.1e-166)
		tmp = Float64(NdChar + t_0);
	elseif (NdChar <= 1.05e-58)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Ec * Float64(Float64(Vef / Float64(Ec * KbT)) + Float64(-1.0 / KbT))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -9e+114)
		tmp = t_1;
	elseif (NdChar <= -1.5e+61)
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((EDonor / (Ec * KbT)) + (-1.0 / KbT)))));
	elseif (NdChar <= -1.1e-166)
		tmp = NdChar + t_0;
	elseif (NdChar <= 1.05e-58)
		tmp = t_0 + (NdChar / (1.0 + (Ec * ((Vef / (Ec * KbT)) + (-1.0 / KbT)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[(1.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -9e+114], t$95$1, If[LessEqual[NdChar, -1.5e+61], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(Ec * N[(N[(EDonor / N[(Ec * KbT), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -1.1e-166], N[(NdChar + t$95$0), $MachinePrecision], If[LessEqual[NdChar, 1.05e-58], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(Ec * N[(N[(Vef / N[(Ec * KbT), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if NdChar < -9.0000000000000001e114 or 1.04999999999999994e-58 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 80.8%

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

   6. Taylor expanded in Ev around 0 66.3%

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

   if -9.0000000000000001e114 < NdChar < -1.5e61

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 45.9%

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

   6. Taylor expanded in Ec around -inf 59.0%

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

      1. mul-1-neg 59.0%

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

      2. *-commutative 59.0%

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

      3. distribute-rgt-neg-in 59.0%

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

      4. +-commutative 59.0%

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

      5. mul-1-neg 59.0%

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

      6. unsub-neg 59.0%

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

      7. associate-+r+ 59.0%

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

   8. Simplified 59.0%

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

   9. Taylor expanded in EDonor around inf 72.6%

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

       1. *-commutative 72.6%

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

   11. Simplified 72.6%

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

   if -1.5e61 < NdChar < -1.1000000000000001e-166

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 64.7%

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

   6. Taylor expanded in Ec around -inf 69.4%

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

      1. mul-1-neg 69.4%

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

      2. *-commutative 69.4%

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

      3. distribute-rgt-neg-in 69.4%

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

      4. +-commutative 69.4%

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

      5. mul-1-neg 69.4%

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

      6. unsub-neg 69.4%

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

      7. associate-+r+ 69.4%

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

   8. Simplified 69.4%

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

   9. Taylor expanded in Vef around inf 56.3%

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

   10. Taylor expanded in Vef around 0 70.6%

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

   if -1.1000000000000001e-166 < NdChar < 1.04999999999999994e-58

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 72.2%

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

   6. Taylor expanded in Ec around -inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. +-commutative 75.9%

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

      5. mul-1-neg 75.9%

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

      6. unsub-neg 75.9%

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

      7. associate-+r+ 75.9%

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

   8. Simplified 75.9%

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

   9. Taylor expanded in Vef around inf 86.1%

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

       1. *-commutative 86.1%

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

   11. Simplified 86.1%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -9 \cdot 10^{+114}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -1.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + Ec \cdot \left(\frac{EDonor}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{elif}\;NdChar \leq -1.1 \cdot 10^{-166}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.05 \cdot 10^{-58}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + Ec \cdot \left(\frac{Vef}{Ec \cdot KbT} + \frac{-1}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -2.3 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq -7 \cdot 10^{-231}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{elif}\;NdChar \leq 2.5 \cdot 10^{-52}:\\ \;\;\;\;t\_0 + KbT \cdot \frac{NdChar}{\left(mu + \left(EDonor + Vef\right)\right) - Ec}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1
         (-
          (/ NaChar (+ 1.0 (+ 1.0 (/ Ev KbT))))
          (/ NdChar (- -1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))))
   (if (<= NdChar -2.3e+104)
     t_1
     (if (<= NdChar -7e-231)
       (+ NdChar t_0)
       (if (<= NdChar 2.5e-52)
         (+ t_0 (* KbT (/ NdChar (- (+ mu (+ EDonor Vef)) Ec))))
         t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -2.3e+104) {
		tmp = t_1;
	} else if (NdChar <= -7e-231) {
		tmp = NdChar + t_0;
	} else if (NdChar <= 2.5e-52) {
		tmp = t_0 + (KbT * (NdChar / ((mu + (EDonor + Vef)) - Ec)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = (nachar / (1.0d0 + (1.0d0 + (ev / kbt)))) - (ndchar / ((-1.0d0) - exp(((edonor - ((ec - vef) - mu)) / kbt))))
    if (ndchar <= (-2.3d+104)) then
        tmp = t_1
    else if (ndchar <= (-7d-231)) then
        tmp = ndchar + t_0
    else if (ndchar <= 2.5d-52) then
        tmp = t_0 + (kbt * (ndchar / ((mu + (edonor + vef)) - ec)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -2.3e+104) {
		tmp = t_1;
	} else if (NdChar <= -7e-231) {
		tmp = NdChar + t_0;
	} else if (NdChar <= 2.5e-52) {
		tmp = t_0 + (KbT * (NdChar / ((mu + (EDonor + Vef)) - Ec)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))))
	tmp = 0
	if NdChar <= -2.3e+104:
		tmp = t_1
	elif NdChar <= -7e-231:
		tmp = NdChar + t_0
	elif NdChar <= 2.5e-52:
		tmp = t_0 + (KbT * (NdChar / ((mu + (EDonor + Vef)) - Ec)))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -2.3e+104)
		tmp = t_1;
	elseif (NdChar <= -7e-231)
		tmp = Float64(NdChar + t_0);
	elseif (NdChar <= 2.5e-52)
		tmp = Float64(t_0 + Float64(KbT * Float64(NdChar / Float64(Float64(mu + Float64(EDonor + Vef)) - Ec))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -2.3e+104)
		tmp = t_1;
	elseif (NdChar <= -7e-231)
		tmp = NdChar + t_0;
	elseif (NdChar <= 2.5e-52)
		tmp = t_0 + (KbT * (NdChar / ((mu + (EDonor + Vef)) - Ec)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[(1.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -2.3e+104], t$95$1, If[LessEqual[NdChar, -7e-231], N[(NdChar + t$95$0), $MachinePrecision], If[LessEqual[NdChar, 2.5e-52], N[(t$95$0 + N[(KbT * N[(NdChar / N[(N[(mu + N[(EDonor + Vef), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -2.29999999999999985e104 or 2.5e-52 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 80.9%

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

   6. Taylor expanded in Ev around 0 65.9%

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

   if -2.29999999999999985e104 < NdChar < -7.0000000000000002e-231

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 63.6%

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

   6. Taylor expanded in Ec around -inf 68.5%

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

      1. mul-1-neg 68.5%

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

      2. *-commutative 68.5%

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

      3. distribute-rgt-neg-in 68.5%

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

      4. +-commutative 68.5%

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

      5. mul-1-neg 68.5%

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

      6. unsub-neg 68.5%

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

      7. associate-+r+ 68.5%

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

   8. Simplified 68.5%

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

   9. Taylor expanded in Vef around inf 56.3%

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

   10. Taylor expanded in Vef around 0 69.6%

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

   if -7.0000000000000002e-231 < NdChar < 2.5e-52

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 73.3%

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

   6. Taylor expanded in KbT around 0 79.4%

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

      1. associate-/l* 86.3%

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

      2. associate-+r+ 86.3%

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

      3. +-commutative 86.3%

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

   8. Simplified 86.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.3 \cdot 10^{+104}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -7 \cdot 10^{-231}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 2.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + KbT \cdot \frac{NdChar}{\left(mu + \left(EDonor + Vef\right)\right) - Ec}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -3.7 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq -1.6 \cdot 10^{-235}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;NdChar \leq 6 \cdot 10^{-68}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + \frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1
         (-
          (/ NaChar (+ 1.0 (+ 1.0 (/ Ev KbT))))
          (/ NdChar (- -1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))))
   (if (<= NdChar -3.7e-131)
     t_1
     (if (<= NdChar -1.6e-235)
       (+ t_0 (/ NdChar (+ 1.0 (/ mu KbT))))
       (if (<= NdChar 6e-68) (+ t_0 (/ NdChar (+ 1.0 (/ Vef KbT)))) t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -3.7e-131) {
		tmp = t_1;
	} else if (NdChar <= -1.6e-235) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if (NdChar <= 6e-68) {
		tmp = t_0 + (NdChar / (1.0 + (Vef / KbT)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = (nachar / (1.0d0 + (1.0d0 + (ev / kbt)))) - (ndchar / ((-1.0d0) - exp(((edonor - ((ec - vef) - mu)) / kbt))))
    if (ndchar <= (-3.7d-131)) then
        tmp = t_1
    else if (ndchar <= (-1.6d-235)) then
        tmp = t_0 + (ndchar / (1.0d0 + (mu / kbt)))
    else if (ndchar <= 6d-68) then
        tmp = t_0 + (ndchar / (1.0d0 + (vef / kbt)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -3.7e-131) {
		tmp = t_1;
	} else if (NdChar <= -1.6e-235) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if (NdChar <= 6e-68) {
		tmp = t_0 + (NdChar / (1.0 + (Vef / KbT)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))))
	tmp = 0
	if NdChar <= -3.7e-131:
		tmp = t_1
	elif NdChar <= -1.6e-235:
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)))
	elif NdChar <= 6e-68:
		tmp = t_0 + (NdChar / (1.0 + (Vef / KbT)))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + Float64(1.0 + Float64(Ev / KbT)))) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -3.7e-131)
		tmp = t_1;
	elseif (NdChar <= -1.6e-235)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(mu / KbT))));
	elseif (NdChar <= 6e-68)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Vef / KbT))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = (NaChar / (1.0 + (1.0 + (Ev / KbT)))) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -3.7e-131)
		tmp = t_1;
	elseif (NdChar <= -1.6e-235)
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	elseif (NdChar <= 6e-68)
		tmp = t_0 + (NdChar / (1.0 + (Vef / KbT)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + N[(1.0 + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -3.7e-131], t$95$1, If[LessEqual[NdChar, -1.6e-235], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 6e-68], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -3.7000000000000002e-131 or 6e-68 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 80.7%

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

   6. Taylor expanded in Ev around 0 64.8%

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

   if -3.7000000000000002e-131 < NdChar < -1.6000000000000001e-235

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 69.5%

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

   6. Taylor expanded in Ec around -inf 69.5%

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

      1. mul-1-neg 69.5%

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

      2. *-commutative 69.5%

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

      3. distribute-rgt-neg-in 69.5%

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

      4. +-commutative 69.5%

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

      5. mul-1-neg 69.5%

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

      6. unsub-neg 69.5%

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

      7. associate-+r+ 69.5%

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

   8. Simplified 69.5%

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

   9. Taylor expanded in mu around inf 80.4%

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

   if -1.6000000000000001e-235 < NdChar < 6e-68

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 74.3%

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

   6. Taylor expanded in Ec around -inf 77.4%

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

      1. mul-1-neg 77.4%

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

      2. *-commutative 77.4%

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

      3. distribute-rgt-neg-in 77.4%

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

      4. +-commutative 77.4%

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

      5. mul-1-neg 77.4%

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

      6. unsub-neg 77.4%

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

      7. associate-+r+ 77.4%

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

   8. Simplified 77.4%

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

   9. Taylor expanded in Vef around inf 83.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -3.7 \cdot 10^{-131}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -1.6 \cdot 10^{-235}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;NdChar \leq 6 \cdot 10^{-68}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + \left(1 + \frac{Ev}{KbT}\right)} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := NdChar + t\_0\\ t_2 := \frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -8.6 \cdot 10^{+156}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NdChar \leq -1.1 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq -1 \cdot 10^{-305}:\\ \;\;\;\;t\_0 + KbT \cdot \frac{NdChar}{Vef}\\ \mathbf{elif}\;NdChar \leq 4.6 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1 (+ NdChar t_0))
        (t_2
         (-
          (/ NaChar 2.0)
          (/ NdChar (- -1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))))
   (if (<= NdChar -8.6e+156)
     t_2
     (if (<= NdChar -1.1e-242)
       t_1
       (if (<= NdChar -1e-305)
         (+ t_0 (* KbT (/ NdChar Vef)))
         (if (<= NdChar 4.6e+96) t_1 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar + t_0;
	double t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -8.6e+156) {
		tmp = t_2;
	} else if (NdChar <= -1.1e-242) {
		tmp = t_1;
	} else if (NdChar <= -1e-305) {
		tmp = t_0 + (KbT * (NdChar / Vef));
	} else if (NdChar <= 4.6e+96) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = ndchar + t_0
    t_2 = (nachar / 2.0d0) - (ndchar / ((-1.0d0) - exp(((edonor - ((ec - vef) - mu)) / kbt))))
    if (ndchar <= (-8.6d+156)) then
        tmp = t_2
    else if (ndchar <= (-1.1d-242)) then
        tmp = t_1
    else if (ndchar <= (-1d-305)) then
        tmp = t_0 + (kbt * (ndchar / vef))
    else if (ndchar <= 4.6d+96) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar + t_0;
	double t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -8.6e+156) {
		tmp = t_2;
	} else if (NdChar <= -1.1e-242) {
		tmp = t_1;
	} else if (NdChar <= -1e-305) {
		tmp = t_0 + (KbT * (NdChar / Vef));
	} else if (NdChar <= 4.6e+96) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = NdChar + t_0
	t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))))
	tmp = 0
	if NdChar <= -8.6e+156:
		tmp = t_2
	elif NdChar <= -1.1e-242:
		tmp = t_1
	elif NdChar <= -1e-305:
		tmp = t_0 + (KbT * (NdChar / Vef))
	elif NdChar <= 4.6e+96:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(NdChar + t_0)
	t_2 = Float64(Float64(NaChar / 2.0) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -8.6e+156)
		tmp = t_2;
	elseif (NdChar <= -1.1e-242)
		tmp = t_1;
	elseif (NdChar <= -1e-305)
		tmp = Float64(t_0 + Float64(KbT * Float64(NdChar / Vef)));
	elseif (NdChar <= 4.6e+96)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = NdChar + t_0;
	t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -8.6e+156)
		tmp = t_2;
	elseif (NdChar <= -1.1e-242)
		tmp = t_1;
	elseif (NdChar <= -1e-305)
		tmp = t_0 + (KbT * (NdChar / Vef));
	elseif (NdChar <= 4.6e+96)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / 2.0), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -8.6e+156], t$95$2, If[LessEqual[NdChar, -1.1e-242], t$95$1, If[LessEqual[NdChar, -1e-305], N[(t$95$0 + N[(KbT * N[(NdChar / Vef), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 4.6e+96], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -8.5999999999999997e156 or 4.6000000000000003e96 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 61.7%

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

   if -8.5999999999999997e156 < NdChar < -1.10000000000000001e-242 or -9.99999999999999996e-306 < NdChar < 4.6000000000000003e96

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 58.9%

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

   6. Taylor expanded in Ec around -inf 62.7%

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

      1. mul-1-neg 62.7%

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

      2. *-commutative 62.7%

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

      3. distribute-rgt-neg-in 62.7%

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

      4. +-commutative 62.7%

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

      5. mul-1-neg 62.7%

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

      6. unsub-neg 62.7%

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

      7. associate-+r+ 62.7%

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

   8. Simplified 62.7%

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

   9. Taylor expanded in Vef around inf 59.1%

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

   10. Taylor expanded in Vef around 0 71.2%

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

   if -1.10000000000000001e-242 < NdChar < -9.99999999999999996e-306

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 93.3%

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

   6. Taylor expanded in Vef around inf 87.1%

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

      1. associate-/l* 87.1%

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

   8. Simplified 87.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -8.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -1.1 \cdot 10^{-242}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -1 \cdot 10^{-305}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + KbT \cdot \frac{NdChar}{Vef}\\ \mathbf{elif}\;NdChar \leq 4.6 \cdot 10^{+96}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := NdChar + t\_0\\ t_2 := \frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -1.3 \cdot 10^{+152}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NdChar \leq -8 \cdot 10^{-234}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq -9.6 \cdot 10^{-306}:\\ \;\;\;\;t\_0 + \frac{NdChar \cdot KbT}{mu}\\ \mathbf{elif}\;NdChar \leq 1.55 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1 (+ NdChar t_0))
        (t_2
         (-
          (/ NaChar 2.0)
          (/ NdChar (- -1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))))
   (if (<= NdChar -1.3e+152)
     t_2
     (if (<= NdChar -8e-234)
       t_1
       (if (<= NdChar -9.6e-306)
         (+ t_0 (/ (* NdChar KbT) mu))
         (if (<= NdChar 1.55e+96) t_1 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar + t_0;
	double t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -1.3e+152) {
		tmp = t_2;
	} else if (NdChar <= -8e-234) {
		tmp = t_1;
	} else if (NdChar <= -9.6e-306) {
		tmp = t_0 + ((NdChar * KbT) / mu);
	} else if (NdChar <= 1.55e+96) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = ndchar + t_0
    t_2 = (nachar / 2.0d0) - (ndchar / ((-1.0d0) - exp(((edonor - ((ec - vef) - mu)) / kbt))))
    if (ndchar <= (-1.3d+152)) then
        tmp = t_2
    else if (ndchar <= (-8d-234)) then
        tmp = t_1
    else if (ndchar <= (-9.6d-306)) then
        tmp = t_0 + ((ndchar * kbt) / mu)
    else if (ndchar <= 1.55d+96) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar + t_0;
	double t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -1.3e+152) {
		tmp = t_2;
	} else if (NdChar <= -8e-234) {
		tmp = t_1;
	} else if (NdChar <= -9.6e-306) {
		tmp = t_0 + ((NdChar * KbT) / mu);
	} else if (NdChar <= 1.55e+96) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = NdChar + t_0
	t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))))
	tmp = 0
	if NdChar <= -1.3e+152:
		tmp = t_2
	elif NdChar <= -8e-234:
		tmp = t_1
	elif NdChar <= -9.6e-306:
		tmp = t_0 + ((NdChar * KbT) / mu)
	elif NdChar <= 1.55e+96:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(NdChar + t_0)
	t_2 = Float64(Float64(NaChar / 2.0) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -1.3e+152)
		tmp = t_2;
	elseif (NdChar <= -8e-234)
		tmp = t_1;
	elseif (NdChar <= -9.6e-306)
		tmp = Float64(t_0 + Float64(Float64(NdChar * KbT) / mu));
	elseif (NdChar <= 1.55e+96)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = NdChar + t_0;
	t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -1.3e+152)
		tmp = t_2;
	elseif (NdChar <= -8e-234)
		tmp = t_1;
	elseif (NdChar <= -9.6e-306)
		tmp = t_0 + ((NdChar * KbT) / mu);
	elseif (NdChar <= 1.55e+96)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / 2.0), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.3e+152], t$95$2, If[LessEqual[NdChar, -8e-234], t$95$1, If[LessEqual[NdChar, -9.6e-306], N[(t$95$0 + N[(N[(NdChar * KbT), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.55e+96], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.3e152 or 1.5499999999999999e96 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 61.7%

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

   if -1.3e152 < NdChar < -7.9999999999999997e-234 or -9.5999999999999998e-306 < NdChar < 1.5499999999999999e96

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 58.4%

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

   6. Taylor expanded in Ec around -inf 62.2%

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

      1. mul-1-neg 62.2%

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

      2. *-commutative 62.2%

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

      3. distribute-rgt-neg-in 62.2%

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

      4. +-commutative 62.2%

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

      5. mul-1-neg 62.2%

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

      6. unsub-neg 62.2%

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

      7. associate-+r+ 62.2%

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

   8. Simplified 62.2%

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

   9. Taylor expanded in Vef around inf 59.2%

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

   10. Taylor expanded in Vef around 0 71.4%

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

   if -7.9999999999999997e-234 < NdChar < -9.5999999999999998e-306

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 94.1%

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

   6. Taylor expanded in KbT around 0 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-+r+ 100.0%

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

      3. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in mu around inf 83.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -8 \cdot 10^{-234}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -9.6 \cdot 10^{-306}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar \cdot KbT}{mu}\\ \mathbf{elif}\;NdChar \leq 1.55 \cdot 10^{+96}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := NdChar + t\_0\\ t_2 := \frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -1.12 \cdot 10^{+152}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NdChar \leq -4.5 \cdot 10^{-110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 8.5 \cdot 10^{-226}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;NdChar \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1 (+ NdChar t_0))
        (t_2
         (-
          (/ NaChar 2.0)
          (/ NdChar (- -1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))))
   (if (<= NdChar -1.12e+152)
     t_2
     (if (<= NdChar -4.5e-110)
       t_1
       (if (<= NdChar 8.5e-226)
         (+ t_0 (/ NdChar (+ 1.0 (/ EDonor KbT))))
         (if (<= NdChar 2.2e+97) t_1 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar + t_0;
	double t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -1.12e+152) {
		tmp = t_2;
	} else if (NdChar <= -4.5e-110) {
		tmp = t_1;
	} else if (NdChar <= 8.5e-226) {
		tmp = t_0 + (NdChar / (1.0 + (EDonor / KbT)));
	} else if (NdChar <= 2.2e+97) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = ndchar + t_0
    t_2 = (nachar / 2.0d0) - (ndchar / ((-1.0d0) - exp(((edonor - ((ec - vef) - mu)) / kbt))))
    if (ndchar <= (-1.12d+152)) then
        tmp = t_2
    else if (ndchar <= (-4.5d-110)) then
        tmp = t_1
    else if (ndchar <= 8.5d-226) then
        tmp = t_0 + (ndchar / (1.0d0 + (edonor / kbt)))
    else if (ndchar <= 2.2d+97) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar + t_0;
	double t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -1.12e+152) {
		tmp = t_2;
	} else if (NdChar <= -4.5e-110) {
		tmp = t_1;
	} else if (NdChar <= 8.5e-226) {
		tmp = t_0 + (NdChar / (1.0 + (EDonor / KbT)));
	} else if (NdChar <= 2.2e+97) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = NdChar + t_0
	t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))))
	tmp = 0
	if NdChar <= -1.12e+152:
		tmp = t_2
	elif NdChar <= -4.5e-110:
		tmp = t_1
	elif NdChar <= 8.5e-226:
		tmp = t_0 + (NdChar / (1.0 + (EDonor / KbT)))
	elif NdChar <= 2.2e+97:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(NdChar + t_0)
	t_2 = Float64(Float64(NaChar / 2.0) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -1.12e+152)
		tmp = t_2;
	elseif (NdChar <= -4.5e-110)
		tmp = t_1;
	elseif (NdChar <= 8.5e-226)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(EDonor / KbT))));
	elseif (NdChar <= 2.2e+97)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = NdChar + t_0;
	t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -1.12e+152)
		tmp = t_2;
	elseif (NdChar <= -4.5e-110)
		tmp = t_1;
	elseif (NdChar <= 8.5e-226)
		tmp = t_0 + (NdChar / (1.0 + (EDonor / KbT)));
	elseif (NdChar <= 2.2e+97)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / 2.0), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.12e+152], t$95$2, If[LessEqual[NdChar, -4.5e-110], t$95$1, If[LessEqual[NdChar, 8.5e-226], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.2e+97], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.11999999999999995e152 or 2.2000000000000001e97 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 61.7%

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

   if -1.11999999999999995e152 < NdChar < -4.5000000000000001e-110 or 8.4999999999999998e-226 < NdChar < 2.2000000000000001e97

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.9%

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

   6. Taylor expanded in Ec around -inf 58.3%

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

      1. mul-1-neg 58.3%

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

      2. *-commutative 58.3%

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

      3. distribute-rgt-neg-in 58.3%

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

      4. +-commutative 58.3%

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

      5. mul-1-neg 58.3%

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

      6. unsub-neg 58.3%

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

      7. associate-+r+ 58.3%

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

   8. Simplified 58.3%

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

   9. Taylor expanded in Vef around inf 54.8%

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

   10. Taylor expanded in Vef around 0 70.4%

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

   if -4.5000000000000001e-110 < NdChar < 8.4999999999999998e-226

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 77.5%

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

   6. Taylor expanded in Ec around -inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. *-commutative 77.5%

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

      3. distribute-rgt-neg-in 77.5%

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

      4. +-commutative 77.5%

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

      5. mul-1-neg 77.5%

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

      6. unsub-neg 77.5%

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

      7. associate-+r+ 77.5%

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

   8. Simplified 77.5%

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

   9. Taylor expanded in EDonor around inf 77.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.12 \cdot 10^{+152}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -4.5 \cdot 10^{-110}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 8.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;NdChar \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 65.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := NdChar + t\_0\\ t_2 := \frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{if}\;NdChar \leq -1.15 \cdot 10^{+152}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;NdChar \leq -2.8 \cdot 10^{-183}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NdChar \leq 1.15 \cdot 10^{-52}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + \frac{Vef}{KbT}}\\ \mathbf{elif}\;NdChar \leq 1.76 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1 (+ NdChar t_0))
        (t_2
         (-
          (/ NaChar 2.0)
          (/ NdChar (- -1.0 (exp (/ (- EDonor (- (- Ec Vef) mu)) KbT)))))))
   (if (<= NdChar -1.15e+152)
     t_2
     (if (<= NdChar -2.8e-183)
       t_1
       (if (<= NdChar 1.15e-52)
         (+ t_0 (/ NdChar (+ 1.0 (/ Vef KbT))))
         (if (<= NdChar 1.76e+96) t_1 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar + t_0;
	double t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -1.15e+152) {
		tmp = t_2;
	} else if (NdChar <= -2.8e-183) {
		tmp = t_1;
	} else if (NdChar <= 1.15e-52) {
		tmp = t_0 + (NdChar / (1.0 + (Vef / KbT)));
	} else if (NdChar <= 1.76e+96) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = ndchar + t_0
    t_2 = (nachar / 2.0d0) - (ndchar / ((-1.0d0) - exp(((edonor - ((ec - vef) - mu)) / kbt))))
    if (ndchar <= (-1.15d+152)) then
        tmp = t_2
    else if (ndchar <= (-2.8d-183)) then
        tmp = t_1
    else if (ndchar <= 1.15d-52) then
        tmp = t_0 + (ndchar / (1.0d0 + (vef / kbt)))
    else if (ndchar <= 1.76d+96) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar + t_0;
	double t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - Math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	double tmp;
	if (NdChar <= -1.15e+152) {
		tmp = t_2;
	} else if (NdChar <= -2.8e-183) {
		tmp = t_1;
	} else if (NdChar <= 1.15e-52) {
		tmp = t_0 + (NdChar / (1.0 + (Vef / KbT)));
	} else if (NdChar <= 1.76e+96) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = NdChar + t_0
	t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - math.exp(((EDonor - ((Ec - Vef) - mu)) / KbT))))
	tmp = 0
	if NdChar <= -1.15e+152:
		tmp = t_2
	elif NdChar <= -2.8e-183:
		tmp = t_1
	elif NdChar <= 1.15e-52:
		tmp = t_0 + (NdChar / (1.0 + (Vef / KbT)))
	elif NdChar <= 1.76e+96:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(NdChar + t_0)
	t_2 = Float64(Float64(NaChar / 2.0) - Float64(NdChar / Float64(-1.0 - exp(Float64(Float64(EDonor - Float64(Float64(Ec - Vef) - mu)) / KbT)))))
	tmp = 0.0
	if (NdChar <= -1.15e+152)
		tmp = t_2;
	elseif (NdChar <= -2.8e-183)
		tmp = t_1;
	elseif (NdChar <= 1.15e-52)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Vef / KbT))));
	elseif (NdChar <= 1.76e+96)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = NdChar + t_0;
	t_2 = (NaChar / 2.0) - (NdChar / (-1.0 - exp(((EDonor - ((Ec - Vef) - mu)) / KbT))));
	tmp = 0.0;
	if (NdChar <= -1.15e+152)
		tmp = t_2;
	elseif (NdChar <= -2.8e-183)
		tmp = t_1;
	elseif (NdChar <= 1.15e-52)
		tmp = t_0 + (NdChar / (1.0 + (Vef / KbT)));
	elseif (NdChar <= 1.76e+96)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(NaChar / 2.0), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(N[(EDonor - N[(N[(Ec - Vef), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.15e+152], t$95$2, If[LessEqual[NdChar, -2.8e-183], t$95$1, If[LessEqual[NdChar, 1.15e-52], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.76e+96], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -1.14999999999999993e152 or 1.7599999999999999e96 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 61.7%

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

   if -1.14999999999999993e152 < NdChar < -2.79999999999999985e-183 or 1.14999999999999997e-52 < NdChar < 1.7599999999999999e96

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.6%

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

   6. Taylor expanded in Ec around -inf 55.7%

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

      1. mul-1-neg 55.7%

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

      2. *-commutative 55.7%

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

      3. distribute-rgt-neg-in 55.7%

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

      4. +-commutative 55.7%

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

      5. mul-1-neg 55.7%

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

      6. unsub-neg 55.7%

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

      7. associate-+r+ 55.7%

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

   8. Simplified 55.7%

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

   9. Taylor expanded in Vef around inf 49.3%

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

   10. Taylor expanded in Vef around 0 73.2%

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

   if -2.79999999999999985e-183 < NdChar < 1.14999999999999997e-52

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 73.6%

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

   6. Taylor expanded in Ec around -inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. *-commutative 77.5%

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

      3. distribute-rgt-neg-in 77.5%

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

      4. +-commutative 77.5%

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

      5. mul-1-neg 77.5%

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

      6. unsub-neg 77.5%

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

      7. associate-+r+ 77.5%

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

   8. Simplified 77.5%

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

   9. Taylor expanded in Vef around inf 79.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.15 \cdot 10^{+152}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -2.8 \cdot 10^{-183}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.15 \cdot 10^{-52}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{Vef}{KbT}}\\ \mathbf{elif}\;NdChar \leq 1.76 \cdot 10^{+96}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor - \left(\left(Ec - Vef\right) - mu\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 65.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;KbT \leq -4.6 \cdot 10^{+207}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 4.8 \cdot 10^{+155}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= KbT -4.6e+207)
     (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (/ NaChar 2.0))
     (if (<= KbT 4.8e+155) (+ NdChar t_0) (+ t_0 (/ NdChar 2.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (KbT <= -4.6e+207) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	} else if (KbT <= 4.8e+155) {
		tmp = NdChar + t_0;
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (kbt <= (-4.6d+207)) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / 2.0d0)
    else if (kbt <= 4.8d+155) then
        tmp = ndchar + t_0
    else
        tmp = t_0 + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (KbT <= -4.6e+207) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / 2.0);
	} else if (KbT <= 4.8e+155) {
		tmp = NdChar + t_0;
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if KbT <= -4.6e+207:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / 2.0)
	elif KbT <= 4.8e+155:
		tmp = NdChar + t_0
	else:
		tmp = t_0 + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (KbT <= -4.6e+207)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / 2.0));
	elseif (KbT <= 4.8e+155)
		tmp = Float64(NdChar + t_0);
	else
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (KbT <= -4.6e+207)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	elseif (KbT <= 4.8e+155)
		tmp = NdChar + t_0;
	else
		tmp = t_0 + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -4.6e+207], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4.8e+155], N[(NdChar + t$95$0), $MachinePrecision], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -4.59999999999999989e207

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 95.3%

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

   6. Taylor expanded in KbT around inf 91.5%

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

   if -4.59999999999999989e207 < KbT < 4.80000000000000042e155

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 45.6%

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

   6. Taylor expanded in Ec around -inf 50.7%

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

      1. mul-1-neg 50.7%

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

      2. *-commutative 50.7%

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

      3. distribute-rgt-neg-in 50.7%

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

      4. +-commutative 50.7%

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

      5. mul-1-neg 50.7%

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

      6. unsub-neg 50.7%

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

      7. associate-+r+ 50.7%

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

   8. Simplified 50.7%

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

   9. Taylor expanded in Vef around inf 52.0%

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

   10. Taylor expanded in Vef around 0 61.3%

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

   if 4.80000000000000042e155 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 84.1%

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

   6. Taylor expanded in Vef around 0 80.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.6 \cdot 10^{+207}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 4.8 \cdot 10^{+155}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 64.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -1.55 \cdot 10^{+207}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 7.5 \cdot 10^{+211}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -1.55e+207)
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (/ NaChar 2.0))
   (if (<= KbT 7.5e+211)
     (+ NdChar (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
     (+ (/ NaChar 2.0) (/ NdChar (+ 1.0 (exp (/ (- Ec) KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.55e+207) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	} else if (KbT <= 7.5e+211) {
		tmp = NdChar + (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	} else {
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + exp((-Ec / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-1.55d+207)) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / 2.0d0)
    else if (kbt <= 7.5d+211) then
        tmp = ndchar + (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt))))
    else
        tmp = (nachar / 2.0d0) + (ndchar / (1.0d0 + exp((-ec / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.55e+207) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / 2.0);
	} else if (KbT <= 7.5e+211) {
		tmp = NdChar + (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	} else {
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + Math.exp((-Ec / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -1.55e+207:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / 2.0)
	elif KbT <= 7.5e+211:
		tmp = NdChar + (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))))
	else:
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + math.exp((-Ec / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -1.55e+207)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / 2.0));
	elseif (KbT <= 7.5e+211)
		tmp = Float64(NdChar + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))));
	else
		tmp = Float64(Float64(NaChar / 2.0) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Ec) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -1.55e+207)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	elseif (KbT <= 7.5e+211)
		tmp = NdChar + (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	else
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + exp((-Ec / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -1.55e+207], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 7.5e+211], N[(NdChar + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.5500000000000001e207

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 95.3%

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

   6. Taylor expanded in KbT around inf 91.5%

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

   if -1.5500000000000001e207 < KbT < 7.5e211

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 48.2%

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

   6. Taylor expanded in Ec around -inf 52.8%

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

      1. mul-1-neg 52.8%

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

      2. *-commutative 52.8%

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

      3. distribute-rgt-neg-in 52.8%

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

      4. +-commutative 52.8%

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

      5. mul-1-neg 52.8%

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

      6. unsub-neg 52.8%

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

      7. associate-+r+ 52.8%

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

   8. Simplified 52.8%

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

   9. Taylor expanded in Vef around inf 53.0%

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

   10. Taylor expanded in Vef around 0 61.5%

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

   if 7.5e211 < KbT

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around inf 86.9%

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

   6. Taylor expanded in Ec around inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. mul-1-neg 86.9%

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

   8. Simplified 86.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.55 \cdot 10^{+207}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 7.5 \cdot 10^{+211}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 35.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EDonor \leq -4.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;EDonor \leq -1.85 \cdot 10^{-140}:\\ \;\;\;\;\frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)} - \frac{NdChar}{-1 + Ec \cdot \left(\frac{1}{KbT} + \frac{\left(-1 - \frac{EDonor}{KbT}\right) - \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)}{Ec}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EDonor -4.2e+64)
   (- (/ NaChar 2.0) (/ NdChar (- -1.0 (exp (/ EDonor KbT)))))
   (if (<= EDonor -1.85e-140)
     (-
      (/
       NaChar
       (+
        1.0
        (- (+ 1.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT)))) (/ mu KbT))))
      (/
       NdChar
       (+
        -1.0
        (*
         Ec
         (+
          (/ 1.0 KbT)
          (/ (- (- -1.0 (/ EDonor KbT)) (+ (/ mu KbT) (/ Vef KbT))) Ec))))))
     (+ (/ NaChar 2.0) (/ NdChar (+ 1.0 (exp (/ (- Ec) KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EDonor <= -4.2e+64) {
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - exp((EDonor / KbT))));
	} else if (EDonor <= -1.85e-140) {
		tmp = (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)))) - (NdChar / (-1.0 + (Ec * ((1.0 / KbT) + (((-1.0 - (EDonor / KbT)) - ((mu / KbT) + (Vef / KbT))) / Ec)))));
	} else {
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + exp((-Ec / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (edonor <= (-4.2d+64)) then
        tmp = (nachar / 2.0d0) - (ndchar / ((-1.0d0) - exp((edonor / kbt))))
    else if (edonor <= (-1.85d-140)) then
        tmp = (nachar / (1.0d0 + ((1.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt)))) - (ndchar / ((-1.0d0) + (ec * ((1.0d0 / kbt) + ((((-1.0d0) - (edonor / kbt)) - ((mu / kbt) + (vef / kbt))) / ec)))))
    else
        tmp = (nachar / 2.0d0) + (ndchar / (1.0d0 + exp((-ec / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EDonor <= -4.2e+64) {
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - Math.exp((EDonor / KbT))));
	} else if (EDonor <= -1.85e-140) {
		tmp = (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)))) - (NdChar / (-1.0 + (Ec * ((1.0 / KbT) + (((-1.0 - (EDonor / KbT)) - ((mu / KbT) + (Vef / KbT))) / Ec)))));
	} else {
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + Math.exp((-Ec / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EDonor <= -4.2e+64:
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - math.exp((EDonor / KbT))))
	elif EDonor <= -1.85e-140:
		tmp = (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)))) - (NdChar / (-1.0 + (Ec * ((1.0 / KbT) + (((-1.0 - (EDonor / KbT)) - ((mu / KbT) + (Vef / KbT))) / Ec)))))
	else:
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + math.exp((-Ec / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EDonor <= -4.2e+64)
		tmp = Float64(Float64(NaChar / 2.0) - Float64(NdChar / Float64(-1.0 - exp(Float64(EDonor / KbT)))));
	elseif (EDonor <= -1.85e-140)
		tmp = Float64(Float64(NaChar / Float64(1.0 + Float64(Float64(1.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT)))) - Float64(NdChar / Float64(-1.0 + Float64(Ec * Float64(Float64(1.0 / KbT) + Float64(Float64(Float64(-1.0 - Float64(EDonor / KbT)) - Float64(Float64(mu / KbT) + Float64(Vef / KbT))) / Ec))))));
	else
		tmp = Float64(Float64(NaChar / 2.0) + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Ec) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (EDonor <= -4.2e+64)
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - exp((EDonor / KbT))));
	elseif (EDonor <= -1.85e-140)
		tmp = (NaChar / (1.0 + ((1.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)))) - (NdChar / (-1.0 + (Ec * ((1.0 / KbT) + (((-1.0 - (EDonor / KbT)) - ((mu / KbT) + (Vef / KbT))) / Ec)))));
	else
		tmp = (NaChar / 2.0) + (NdChar / (1.0 + exp((-Ec / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EDonor, -4.2e+64], N[(N[(NaChar / 2.0), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, -1.85e-140], N[(N[(NaChar / N[(1.0 + N[(N[(1.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NdChar / N[(-1.0 + N[(Ec * N[(N[(1.0 / KbT), $MachinePrecision] + N[(N[(N[(-1.0 - N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] - N[(N[(mu / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if EDonor < -4.2000000000000001e64

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 37.4%

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

   6. Taylor expanded in EDonor around inf 35.2%

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

   if -4.2000000000000001e64 < EDonor < -1.84999999999999989e-140

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 61.6%

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

   6. Taylor expanded in Ec around -inf 69.0%

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

      1. mul-1-neg 69.0%

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

      2. *-commutative 69.0%

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

      3. distribute-rgt-neg-in 69.0%

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

      4. +-commutative 69.0%

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

      5. mul-1-neg 69.0%

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

      6. unsub-neg 69.0%

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

      7. associate-+r+ 69.0%

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

   8. Simplified 69.0%

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

   9. Taylor expanded in KbT around inf 45.0%

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

   if -1.84999999999999989e-140 < EDonor

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 53.5%

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

   6. Taylor expanded in Ec around inf 45.0%

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

      1. associate-*r/ 45.0%

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

      2. mul-1-neg 45.0%

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

   8. Simplified 45.0%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -4.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;EDonor \leq -1.85 \cdot 10^{-140}:\\ \;\;\;\;\frac{NaChar}{1 + \left(\left(1 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}\right)} - \frac{NdChar}{-1 + Ec \cdot \left(\frac{1}{KbT} + \frac{\left(-1 - \frac{EDonor}{KbT}\right) - \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)}{Ec}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 38.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.45 \cdot 10^{+30} \lor \neg \left(NdChar \leq 3.1 \cdot 10^{-164}\right):\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -2.45e+30) (not (<= NdChar 3.1e-164)))
   (- (/ NaChar 2.0) (/ NdChar (- -1.0 (exp (/ EDonor KbT)))))
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.45e+30) || !(NdChar <= 3.1e-164)) {
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - exp((EDonor / KbT))));
	} else {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-2.45d+30)) .or. (.not. (ndchar <= 3.1d-164))) then
        tmp = (nachar / 2.0d0) - (ndchar / ((-1.0d0) - exp((edonor / kbt))))
    else
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.45e+30) || !(NdChar <= 3.1e-164)) {
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - Math.exp((EDonor / KbT))));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -2.45e+30) or not (NdChar <= 3.1e-164):
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - math.exp((EDonor / KbT))))
	else:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -2.45e+30) || !(NdChar <= 3.1e-164))
		tmp = Float64(Float64(NaChar / 2.0) - Float64(NdChar / Float64(-1.0 - exp(Float64(EDonor / KbT)))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -2.45e+30) || ~((NdChar <= 3.1e-164)))
		tmp = (NaChar / 2.0) - (NdChar / (-1.0 - exp((EDonor / KbT))));
	else
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -2.45e+30], N[Not[LessEqual[NdChar, 3.1e-164]], $MachinePrecision]], N[(N[(NaChar / 2.0), $MachinePrecision] - N[(NdChar / N[(-1.0 - N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.44999999999999992e30 or 3.1000000000000001e-164 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 55.8%

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

   6. Taylor expanded in EDonor around inf 38.7%

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

   if -2.44999999999999992e30 < NdChar < 3.1000000000000001e-164

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 62.9%

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

   6. Taylor expanded in KbT around inf 41.2%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.45 \cdot 10^{+30} \lor \neg \left(NdChar \leq 3.1 \cdot 10^{-164}\right):\\ \;\;\;\;\frac{NaChar}{2} - \frac{NdChar}{-1 - e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 38.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -1.4 \cdot 10^{-33} \lor \neg \left(NdChar \leq 3.5 \cdot 10^{-164}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -1.4e-33) (not (<= NdChar 3.5e-164)))
   (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) (/ NaChar 2.0))
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.4e-33) || !(NdChar <= 3.5e-164)) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-1.4d-33)) .or. (.not. (ndchar <= 3.5d-164))) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.4e-33) || !(NdChar <= 3.5e-164)) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -1.4e-33) or not (NdChar <= 3.5e-164):
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -1.4e-33) || !(NdChar <= 3.5e-164))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -1.4e-33) || ~((NdChar <= 3.5e-164)))
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -1.4e-33], N[Not[LessEqual[NdChar, 3.5e-164]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.4e-33 or 3.5e-164 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 56.6%

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

   6. Taylor expanded in Vef around inf 41.4%

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

   if -1.4e-33 < NdChar < 3.5e-164

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 61.3%

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

   6. Taylor expanded in KbT around inf 41.1%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.4 \cdot 10^{-33} \lor \neg \left(NdChar \leq 3.5 \cdot 10^{-164}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 38.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -5.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;NdChar \leq 4.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -5.3e-27)
   (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) (/ NaChar 2.0))
   (if (<= NdChar 4.8e-163)
     (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar 2.0))
     (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (/ NaChar 2.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -5.3e-27) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / 2.0);
	} else if (NdChar <= 4.8e-163) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-5.3d-27)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / 2.0d0)
    else if (ndchar <= 4.8d-163) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
    else
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -5.3e-27) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / 2.0);
	} else if (NdChar <= 4.8e-163) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -5.3e-27:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / 2.0)
	elif NdChar <= 4.8e-163:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / 2.0)
	else:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -5.3e-27)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / 2.0));
	elseif (NdChar <= 4.8e-163)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / 2.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -5.3e-27)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / 2.0);
	elseif (NdChar <= 4.8e-163)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	else
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -5.3e-27], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 4.8e-163], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -5.30000000000000006e-27

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 60.9%

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

   6. Taylor expanded in Vef around inf 44.6%

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

   if -5.30000000000000006e-27 < NdChar < 4.8000000000000001e-163

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 60.7%

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

   6. Taylor expanded in KbT around inf 40.7%

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

   if 4.8000000000000001e-163 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 65.1%

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

   6. Taylor expanded in KbT around inf 40.1%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -5.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;NdChar \leq 4.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 35.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in Ev around inf 71.5%

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

6. Taylor expanded in KbT around inf 37.0%

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

7. Final simplification 37.0%

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

Alternative 27: 29.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \frac{EDonor}{KbT}\\ t_1 := \frac{NaChar}{\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \mathbf{if}\;KbT \leq -0.32:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{t\_0}\\ \mathbf{elif}\;KbT \leq 6 \cdot 10^{+156}:\\ \;\;\;\;\frac{NdChar}{1 + \frac{Vef}{KbT}} + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{NdChar}{\left(\frac{Vef}{KbT} + t\_0\right) - \frac{Ec}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ 2.0 (/ EDonor KbT)))
        (t_1
         (/
          NaChar
          (-
           (+ 2.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT))))
           (/ mu KbT)))))
   (if (<= KbT -0.32)
     (+ (/ NaChar 2.0) (/ NdChar t_0))
     (if (<= KbT 6e+156)
       (+ (/ NdChar (+ 1.0 (/ Vef KbT))) t_1)
       (+ t_1 (/ NdChar (- (+ (/ Vef KbT) t_0) (/ Ec 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 = 2.0 + (EDonor / KbT);
	double t_1 = NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT));
	double tmp;
	if (KbT <= -0.32) {
		tmp = (NaChar / 2.0) + (NdChar / t_0);
	} else if (KbT <= 6e+156) {
		tmp = (NdChar / (1.0 + (Vef / KbT))) + t_1;
	} else {
		tmp = t_1 + (NdChar / (((Vef / KbT) + t_0) - (Ec / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (edonor / kbt)
    t_1 = nachar / ((2.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt))
    if (kbt <= (-0.32d0)) then
        tmp = (nachar / 2.0d0) + (ndchar / t_0)
    else if (kbt <= 6d+156) then
        tmp = (ndchar / (1.0d0 + (vef / kbt))) + t_1
    else
        tmp = t_1 + (ndchar / (((vef / kbt) + t_0) - (ec / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 2.0 + (EDonor / KbT);
	double t_1 = NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT));
	double tmp;
	if (KbT <= -0.32) {
		tmp = (NaChar / 2.0) + (NdChar / t_0);
	} else if (KbT <= 6e+156) {
		tmp = (NdChar / (1.0 + (Vef / KbT))) + t_1;
	} else {
		tmp = t_1 + (NdChar / (((Vef / KbT) + t_0) - (Ec / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 2.0 + (EDonor / KbT)
	t_1 = NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT))
	tmp = 0
	if KbT <= -0.32:
		tmp = (NaChar / 2.0) + (NdChar / t_0)
	elif KbT <= 6e+156:
		tmp = (NdChar / (1.0 + (Vef / KbT))) + t_1
	else:
		tmp = t_1 + (NdChar / (((Vef / KbT) + t_0) - (Ec / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(2.0 + Float64(EDonor / KbT))
	t_1 = Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT)))
	tmp = 0.0
	if (KbT <= -0.32)
		tmp = Float64(Float64(NaChar / 2.0) + Float64(NdChar / t_0));
	elseif (KbT <= 6e+156)
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Vef / KbT))) + t_1);
	else
		tmp = Float64(t_1 + Float64(NdChar / Float64(Float64(Float64(Vef / KbT) + t_0) - Float64(Ec / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 2.0 + (EDonor / KbT);
	t_1 = NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT));
	tmp = 0.0;
	if (KbT <= -0.32)
		tmp = (NaChar / 2.0) + (NdChar / t_0);
	elseif (KbT <= 6e+156)
		tmp = (NdChar / (1.0 + (Vef / KbT))) + t_1;
	else
		tmp = t_1 + (NdChar / (((Vef / KbT) + t_0) - (Ec / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(N[(2.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -0.32], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(NdChar / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 6e+156], N[(N[(NdChar / N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 + N[(NdChar / N[(N[(N[(Vef / KbT), $MachinePrecision] + t$95$0), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -0.320000000000000007

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 57.4%

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

   6. Taylor expanded in EDonor around inf 49.8%

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

   7. Taylor expanded in EDonor around 0 47.4%

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

   if -0.320000000000000007 < KbT < 5.9999999999999999e156

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 41.4%

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

   6. Taylor expanded in Ec around -inf 48.1%

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

      1. mul-1-neg 48.1%

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

      2. *-commutative 48.1%

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

      3. distribute-rgt-neg-in 48.1%

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

      4. +-commutative 48.1%

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

      5. mul-1-neg 48.1%

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

      6. unsub-neg 48.1%

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

      7. associate-+r+ 48.1%

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

   8. Simplified 48.1%

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

   9. Taylor expanded in Vef around inf 51.3%

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

   10. Taylor expanded in KbT around inf 23.7%

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

   if 5.9999999999999999e156 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 81.0%

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

   6. Taylor expanded in mu around 0 81.4%

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

      1. associate-+r+ 81.4%

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

   8. Simplified 81.4%

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

   9. Taylor expanded in KbT around inf 66.8%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -0.32:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;KbT \leq 6 \cdot 10^{+156}:\\ \;\;\;\;\frac{NdChar}{1 + \frac{Vef}{KbT}} + \frac{NaChar}{\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}} + \frac{NdChar}{\left(\frac{Vef}{KbT} + \left(2 + \frac{EDonor}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 30.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -0.94:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;KbT \leq 9.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{NdChar}{1 + \frac{Vef}{KbT}} + \frac{NaChar}{\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -0.94)
   (+ (/ NaChar 2.0) (/ NdChar (+ 2.0 (/ EDonor KbT))))
   (if (<= KbT 9.8e+151)
     (+
      (/ NdChar (+ 1.0 (/ Vef KbT)))
      (/
       NaChar
       (- (+ 2.0 (+ (/ EAccept KbT) (+ (/ Vef KbT) (/ Ev KbT)))) (/ mu KbT))))
     (+ (/ NaChar 2.0) (/ NdChar 2.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -0.94) {
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (EDonor / KbT)));
	} else if (KbT <= 9.8e+151) {
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)));
	} else {
		tmp = (NaChar / 2.0) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-0.94d0)) then
        tmp = (nachar / 2.0d0) + (ndchar / (2.0d0 + (edonor / kbt)))
    else if (kbt <= 9.8d+151) then
        tmp = (ndchar / (1.0d0 + (vef / kbt))) + (nachar / ((2.0d0 + ((eaccept / kbt) + ((vef / kbt) + (ev / kbt)))) - (mu / kbt)))
    else
        tmp = (nachar / 2.0d0) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -0.94) {
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (EDonor / KbT)));
	} else if (KbT <= 9.8e+151) {
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)));
	} else {
		tmp = (NaChar / 2.0) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -0.94:
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (EDonor / KbT)))
	elif KbT <= 9.8e+151:
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)))
	else:
		tmp = (NaChar / 2.0) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -0.94)
		tmp = Float64(Float64(NaChar / 2.0) + Float64(NdChar / Float64(2.0 + Float64(EDonor / KbT))));
	elseif (KbT <= 9.8e+151)
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Vef / KbT))) + Float64(NaChar / Float64(Float64(2.0 + Float64(Float64(EAccept / KbT) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT)))) - Float64(mu / KbT))));
	else
		tmp = Float64(Float64(NaChar / 2.0) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -0.94)
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (EDonor / KbT)));
	elseif (KbT <= 9.8e+151)
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / ((2.0 + ((EAccept / KbT) + ((Vef / KbT) + (Ev / KbT)))) - (mu / KbT)));
	else
		tmp = (NaChar / 2.0) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -0.94], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 9.8e+151], N[(N[(NdChar / N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(2.0 + N[(N[(EAccept / KbT), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -0.93999999999999995

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 57.4%

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

   6. Taylor expanded in EDonor around inf 49.8%

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

   7. Taylor expanded in EDonor around 0 47.4%

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

   if -0.93999999999999995 < KbT < 9.7999999999999998e151

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 41.4%

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

   6. Taylor expanded in Ec around -inf 48.1%

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

      1. mul-1-neg 48.1%

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

      2. *-commutative 48.1%

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

      3. distribute-rgt-neg-in 48.1%

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

      4. +-commutative 48.1%

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

      5. mul-1-neg 48.1%

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

      6. unsub-neg 48.1%

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

      7. associate-+r+ 48.1%

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

   8. Simplified 48.1%

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

   9. Taylor expanded in Vef around inf 51.3%

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

   10. Taylor expanded in KbT around inf 23.7%

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

   if 9.7999999999999998e151 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 74.2%

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

   6. Taylor expanded in KbT around inf 65.4%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -0.94:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;KbT \leq 9.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{NdChar}{1 + \frac{Vef}{KbT}} + \frac{NaChar}{\left(2 + \left(\frac{EAccept}{KbT} + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 28.4% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -9.2 \cdot 10^{+71} \lor \neg \left(KbT \leq 1.36 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + \frac{Vef}{KbT}} + \frac{NaChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= KbT -9.2e+71) (not (<= KbT 1.36e+101)))
   (+ (/ NaChar 2.0) (/ NdChar 2.0))
   (+ (/ NdChar (+ 1.0 (/ Vef KbT))) (/ NaChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((KbT <= -9.2e+71) || !(KbT <= 1.36e+101)) {
		tmp = (NaChar / 2.0) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((kbt <= (-9.2d+71)) .or. (.not. (kbt <= 1.36d+101))) then
        tmp = (nachar / 2.0d0) + (ndchar / 2.0d0)
    else
        tmp = (ndchar / (1.0d0 + (vef / kbt))) + (nachar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((KbT <= -9.2e+71) || !(KbT <= 1.36e+101)) {
		tmp = (NaChar / 2.0) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (KbT <= -9.2e+71) or not (KbT <= 1.36e+101):
		tmp = (NaChar / 2.0) + (NdChar / 2.0)
	else:
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((KbT <= -9.2e+71) || !(KbT <= 1.36e+101))
		tmp = Float64(Float64(NaChar / 2.0) + Float64(NdChar / 2.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Vef / KbT))) + Float64(NaChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((KbT <= -9.2e+71) || ~((KbT <= 1.36e+101)))
		tmp = (NaChar / 2.0) + (NdChar / 2.0);
	else
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[KbT, -9.2e+71], N[Not[LessEqual[KbT, 1.36e+101]], $MachinePrecision]], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -9.200000000000001e71 or 1.35999999999999998e101 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 70.2%

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

   6. Taylor expanded in KbT around inf 59.7%

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

   if -9.200000000000001e71 < KbT < 1.35999999999999998e101

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 43.8%

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

   6. Taylor expanded in Ec around -inf 49.5%

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

      1. mul-1-neg 49.5%

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

      2. *-commutative 49.5%

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

      3. distribute-rgt-neg-in 49.5%

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

      4. +-commutative 49.5%

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

      5. mul-1-neg 49.5%

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

      6. unsub-neg 49.5%

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

      7. associate-+r+ 49.5%

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

   8. Simplified 49.5%

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

   9. Taylor expanded in Vef around inf 51.2%

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

   10. Taylor expanded in KbT around inf 16.1%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -9.2 \cdot 10^{+71} \lor \neg \left(KbT \leq 1.36 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + \frac{Vef}{KbT}} + \frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 28.4% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -1.15 \cdot 10^{+71}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;KbT \leq 7.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{NdChar}{1 + \frac{Vef}{KbT}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -1.15e+71)
   (+ (/ NaChar 2.0) (/ NdChar (+ 2.0 (/ EDonor KbT))))
   (if (<= KbT 7.2e+99)
     (+ (/ NdChar (+ 1.0 (/ Vef KbT))) (/ NaChar 2.0))
     (+ (/ NaChar 2.0) (/ NdChar 2.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.15e+71) {
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (EDonor / KbT)));
	} else if (KbT <= 7.2e+99) {
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / 2.0) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-1.15d+71)) then
        tmp = (nachar / 2.0d0) + (ndchar / (2.0d0 + (edonor / kbt)))
    else if (kbt <= 7.2d+99) then
        tmp = (ndchar / (1.0d0 + (vef / kbt))) + (nachar / 2.0d0)
    else
        tmp = (nachar / 2.0d0) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.15e+71) {
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (EDonor / KbT)));
	} else if (KbT <= 7.2e+99) {
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / 2.0) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -1.15e+71:
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (EDonor / KbT)))
	elif KbT <= 7.2e+99:
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / 2.0) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -1.15e+71)
		tmp = Float64(Float64(NaChar / 2.0) + Float64(NdChar / Float64(2.0 + Float64(EDonor / KbT))));
	elseif (KbT <= 7.2e+99)
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Vef / KbT))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / 2.0) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -1.15e+71)
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (EDonor / KbT)));
	elseif (KbT <= 7.2e+99)
		tmp = (NdChar / (1.0 + (Vef / KbT))) + (NaChar / 2.0);
	else
		tmp = (NaChar / 2.0) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -1.15e+71], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 7.2e+99], N[(N[(NdChar / N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.1500000000000001e71

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 70.0%

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

   6. Taylor expanded in EDonor around inf 59.8%

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

   7. Taylor expanded in EDonor around 0 59.6%

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

   if -1.1500000000000001e71 < KbT < 7.2000000000000003e99

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 43.8%

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

   6. Taylor expanded in Ec around -inf 49.5%

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

      1. mul-1-neg 49.5%

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

      2. *-commutative 49.5%

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

      3. distribute-rgt-neg-in 49.5%

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

      4. +-commutative 49.5%

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

      5. mul-1-neg 49.5%

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

      6. unsub-neg 49.5%

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

      7. associate-+r+ 49.5%

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

   8. Simplified 49.5%

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

   9. Taylor expanded in Vef around inf 51.2%

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

   10. Taylor expanded in KbT around inf 16.1%

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

   if 7.2000000000000003e99 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 70.3%

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

   6. Taylor expanded in KbT around inf 60.0%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.15 \cdot 10^{+71}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;KbT \leq 7.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{NdChar}{1 + \frac{Vef}{KbT}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 27.2% accurate, 32.7× speedup

Math

\[\begin{array}{l} \\ \frac{NaChar}{2} + \frac{NdChar}{2} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (nachar / 2.0d0) + (ndchar / 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in KbT around inf 48.5%

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

6. Taylor expanded in KbT around inf 30.1%

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

7. Final simplification 30.1%

   \[\leadsto \frac{NaChar}{2} + \frac{NdChar}{2} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(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))))))