Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 30.6s

Alternatives: 25

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Final simplification 100.0%

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

Alternative 2: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}}\\ t_1 := t\_0 + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ t_2 := t\_0 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{if}\;Ec \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Ec \leq -6.6 \cdot 10^{-136}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;Ec \leq 1.2 \cdot 10^{-250}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;Ec \leq 9 \cdot 10^{+29}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;Ec \leq 7.5 \cdot 10^{+149}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (/ (- Ec) KbT))))))
        (t_2 (+ t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))))
   (if (<= Ec -1.65e+24)
     t_1
     (if (<= Ec -6.6e-136)
       (+ t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
       (if (<= Ec 1.2e-250)
         t_2
         (if (<= Ec 9e+29)
           (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
           (if (<= Ec 7.5e+149) 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((-Ec / KbT))));
	double t_2 = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	double tmp;
	if (Ec <= -1.65e+24) {
		tmp = t_1;
	} else if (Ec <= -6.6e-136) {
		tmp = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	} else if (Ec <= 1.2e-250) {
		tmp = t_2;
	} else if (Ec <= 9e+29) {
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (Ec <= 7.5e+149) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + ((ev + eaccept) - mu)) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + exp((-ec / kbt))))
    t_2 = t_0 + (ndchar / (1.0d0 + exp((vef / kbt))))
    if (ec <= (-1.65d+24)) then
        tmp = t_1
    else if (ec <= (-6.6d-136)) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((mu / kbt))))
    else if (ec <= 1.2d-250) then
        tmp = t_2
    else if (ec <= 9d+29) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (ec <= 7.5d+149) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + Math.exp((-Ec / KbT))));
	double t_2 = t_0 + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	double tmp;
	if (Ec <= -1.65e+24) {
		tmp = t_1;
	} else if (Ec <= -6.6e-136) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else if (Ec <= 1.2e-250) {
		tmp = t_2;
	} else if (Ec <= 9e+29) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (Ec <= 7.5e+149) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))
	t_1 = t_0 + (NdChar / (1.0 + math.exp((-Ec / KbT))))
	t_2 = t_0 + (NdChar / (1.0 + math.exp((Vef / KbT))))
	tmp = 0
	if Ec <= -1.65e+24:
		tmp = t_1
	elif Ec <= -6.6e-136:
		tmp = t_0 + (NdChar / (1.0 + math.exp((mu / KbT))))
	elif Ec <= 1.2e-250:
		tmp = t_2
	elif Ec <= 9e+29:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif Ec <= 7.5e+149:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT))))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Ec) / KbT)))))
	t_2 = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))))
	tmp = 0.0
	if (Ec <= -1.65e+24)
		tmp = t_1;
	elseif (Ec <= -6.6e-136)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	elseif (Ec <= 1.2e-250)
		tmp = t_2;
	elseif (Ec <= 9e+29)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (Ec <= 7.5e+149)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)));
	t_1 = t_0 + (NdChar / (1.0 + exp((-Ec / KbT))));
	t_2 = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	tmp = 0.0;
	if (Ec <= -1.65e+24)
		tmp = t_1;
	elseif (Ec <= -6.6e-136)
		tmp = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	elseif (Ec <= 1.2e-250)
		tmp = t_2;
	elseif (Ec <= 9e+29)
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (Ec <= 7.5e+149)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Ec, -1.65e+24], t$95$1, If[LessEqual[Ec, -6.6e-136], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ec, 1.2e-250], t$95$2, If[LessEqual[Ec, 9e+29], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ec, 7.5e+149], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if Ec < -1.6499999999999999e24 or 7.50000000000000031e149 < Ec

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 91.6%

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

      1. mul-1-neg 91.6%

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

      2. distribute-neg-frac2 91.6%

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

   7. Simplified 91.6%

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

   if -1.6499999999999999e24 < Ec < -6.60000000000000035e-136

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 82.2%

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

   if -6.60000000000000035e-136 < Ec < 1.1999999999999999e-250 or 9.0000000000000005e29 < Ec < 7.50000000000000031e149

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 87.1%

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

   if 1.1999999999999999e-250 < Ec < 9.0000000000000005e29

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 78.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ec \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \mathbf{elif}\;Ec \leq -6.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;Ec \leq 1.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Ec \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;Ec \leq 7.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ t_2 := \frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \mathbf{if}\;mu \leq -1.4 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;mu \leq -3.3 \cdot 10^{-241}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;mu \leq 1.9 \cdot 10^{-173}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + t\_0\\ \mathbf{elif}\;mu \leq 2.05 \cdot 10^{-60}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \mathbf{elif}\;mu \leq 1.7 \cdot 10^{+118}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- mu) KbT))))))
        (t_2 (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT))))))
   (if (<= mu -1.4e+141)
     t_1
     (if (<= mu -3.3e-241)
       t_2
       (if (<= mu 1.9e-173)
         (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_0)
         (if (<= mu 2.05e-60)
           (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) t_0)
           (if (<= mu 1.7e+118) 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((Ev / KbT)));
	double t_1 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-mu / KbT))));
	double t_2 = NaChar / (1.0 + exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	double tmp;
	if (mu <= -1.4e+141) {
		tmp = t_1;
	} else if (mu <= -3.3e-241) {
		tmp = t_2;
	} else if (mu <= 1.9e-173) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	} else if (mu <= 2.05e-60) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	} else if (mu <= 1.7e+118) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((ev / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((-mu / kbt))))
    t_2 = nachar / (1.0d0 + exp((((eaccept + (vef + ev)) - mu) / kbt)))
    if (mu <= (-1.4d+141)) then
        tmp = t_1
    else if (mu <= (-3.3d-241)) then
        tmp = t_2
    else if (mu <= 1.9d-173) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + t_0
    else if (mu <= 2.05d-60) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + t_0
    else if (mu <= 1.7d+118) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((Ev / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((-mu / KbT))));
	double t_2 = NaChar / (1.0 + Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	double tmp;
	if (mu <= -1.4e+141) {
		tmp = t_1;
	} else if (mu <= -3.3e-241) {
		tmp = t_2;
	} else if (mu <= 1.9e-173) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + t_0;
	} else if (mu <= 2.05e-60) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	} else if (mu <= 1.7e+118) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((Ev / KbT)))
	t_1 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((-mu / KbT))))
	t_2 = NaChar / (1.0 + math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)))
	tmp = 0
	if mu <= -1.4e+141:
		tmp = t_1
	elif mu <= -3.3e-241:
		tmp = t_2
	elif mu <= 1.9e-173:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + t_0
	elif mu <= 2.05e-60:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	elif mu <= 1.7e+118:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))))
	t_2 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT))))
	tmp = 0.0
	if (mu <= -1.4e+141)
		tmp = t_1;
	elseif (mu <= -3.3e-241)
		tmp = t_2;
	elseif (mu <= 1.9e-173)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + t_0);
	elseif (mu <= 2.05e-60)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0);
	elseif (mu <= 1.7e+118)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((Ev / KbT)));
	t_1 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-mu / KbT))));
	t_2 = NaChar / (1.0 + exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	tmp = 0.0;
	if (mu <= -1.4e+141)
		tmp = t_1;
	elseif (mu <= -3.3e-241)
		tmp = t_2;
	elseif (mu <= 1.9e-173)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	elseif (mu <= 2.05e-60)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	elseif (mu <= 1.7e+118)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -1.4e+141], t$95$1, If[LessEqual[mu, -3.3e-241], t$95$2, If[LessEqual[mu, 1.9e-173], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[mu, 2.05e-60], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[mu, 1.7e+118], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -1.39999999999999996e141 or 1.69999999999999993e118 < 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%

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

   5. Taylor expanded in mu around inf 89.1%

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

   6. Taylor expanded in mu around inf 81.6%

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

      1. associate-*r/ 41.8%

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

      2. mul-1-neg 41.8%

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

   8. Simplified 81.6%

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

   if -1.39999999999999996e141 < mu < -3.2999999999999999e-241 or 2.05000000000000006e-60 < mu < 1.69999999999999993e118

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 65.9%

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

      1. mul-1-neg 65.9%

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

      2. associate-+r+ 65.9%

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

      3. +-commutative 65.9%

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

      4. associate-+r+ 65.9%

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

      5. unsub-neg 65.9%

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

      6. associate-+r+ 65.9%

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

      7. +-commutative 65.9%

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

      8. associate-+r+ 65.9%

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

      9. associate--r+ 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in Ec around inf 37.6%

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

      1. mul-1-neg 37.6%

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

      2. associate-/l* 36.3%

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

      3. distribute-lft-neg-in 36.3%

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

   10. Simplified 36.3%

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

   11. Taylor expanded in KbT around 0 70.0%

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

   if -3.2999999999999999e-241 < mu < 1.90000000000000015e-173

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 83.2%

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

   6. Taylor expanded in Ev around inf 64.6%

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

   if 1.90000000000000015e-173 < mu < 2.05000000000000006e-60

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 85.9%

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

   6. Taylor expanded in Ev around inf 76.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;mu \leq -3.3 \cdot 10^{-241}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.9 \cdot 10^{-173}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.05 \cdot 10^{-60}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;mu \leq 1.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + t\_0\\ \mathbf{if}\;EDonor \leq -5.3 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;EDonor \leq -1.6 \cdot 10^{-241}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}\\ \mathbf{elif}\;EDonor \leq 7.4 \cdot 10^{-236}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \mathbf{elif}\;EDonor \leq 2.85 \cdot 10^{-130}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef EAccept) mu) KbT)))))
        (t_1 (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) t_0)))
   (if (<= EDonor -5.3e+165)
     t_1
     (if (<= EDonor -1.6e-241)
       (+
        (/ NaChar (+ 1.0 (exp (/ (+ Vef (- (+ Ev EAccept) mu)) KbT))))
        (/ NdChar (+ 2.0 (/ (- (+ Vef mu) (- Ec EDonor)) KbT))))
       (if (<= EDonor 7.4e-236)
         (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) t_0)
         (if (<= EDonor 2.85e-130)
           (+
            (/ NdChar (+ 1.0 (exp (/ mu KbT))))
            (/ NaChar (+ 1.0 (exp (/ Ev 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 + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	double tmp;
	if (EDonor <= -5.3e+165) {
		tmp = t_1;
	} else if (EDonor <= -1.6e-241) {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)));
	} else if (EDonor <= 7.4e-236) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	} else if (EDonor <= 2.85e-130) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((Ev / 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 + eaccept) - mu) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((edonor / kbt)))) + t_0
    if (edonor <= (-5.3d+165)) then
        tmp = t_1
    else if (edonor <= (-1.6d-241)) then
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev + eaccept) - mu)) / kbt)))) + (ndchar / (2.0d0 + (((vef + mu) - (ec - edonor)) / kbt)))
    else if (edonor <= 7.4d-236) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + t_0
    else if (edonor <= 2.85d-130) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((ev / 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 + EAccept) - mu) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + t_0;
	double tmp;
	if (EDonor <= -5.3e+165) {
		tmp = t_1;
	} else if (EDonor <= -1.6e-241) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)));
	} else if (EDonor <= 7.4e-236) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	} else if (EDonor <= 2.85e-130) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((Ev / 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 + EAccept) - mu) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + t_0
	tmp = 0
	if EDonor <= -5.3e+165:
		tmp = t_1
	elif EDonor <= -1.6e-241:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)))
	elif EDonor <= 7.4e-236:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	elif EDonor <= 2.85e-130:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + EAccept) - mu) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + t_0)
	tmp = 0.0
	if (EDonor <= -5.3e+165)
		tmp = t_1;
	elseif (EDonor <= -1.6e-241)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT)))) + Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(Vef + mu) - Float64(Ec - EDonor)) / KbT))));
	elseif (EDonor <= 7.4e-236)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0);
	elseif (EDonor <= 2.85e-130)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / 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 + EAccept) - mu) / KbT)));
	t_1 = (NdChar / (1.0 + exp((EDonor / KbT)))) + t_0;
	tmp = 0.0;
	if (EDonor <= -5.3e+165)
		tmp = t_1;
	elseif (EDonor <= -1.6e-241)
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)));
	elseif (EDonor <= 7.4e-236)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	elseif (EDonor <= 2.85e-130)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[EDonor, -5.3e+165], t$95$1, If[LessEqual[EDonor, -1.6e-241], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(N[(N[(Vef + mu), $MachinePrecision] - N[(Ec - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, 7.4e-236], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[EDonor, 2.85e-130], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if EDonor < -5.3000000000000001e165 or 2.8499999999999999e-130 < 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%

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

   5. Taylor expanded in EDonor around inf 86.3%

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

   6. Taylor expanded in Ev around 0 74.6%

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

      1. +-commutative 74.6%

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

   8. Simplified 74.6%

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

   if -5.3000000000000001e165 < EDonor < -1.6e-241

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 65.2%

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

      1. mul-1-neg 65.2%

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

      2. associate-+r+ 65.2%

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

      3. +-commutative 65.2%

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

      4. associate-+r+ 65.2%

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

      5. unsub-neg 65.2%

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

      6. associate-+r+ 65.2%

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

      7. +-commutative 65.2%

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

      8. associate-+r+ 65.2%

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

      9. associate--r+ 65.2%

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

   7. Simplified 65.2%

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

   if -1.6e-241 < EDonor < 7.40000000000000042e-236

   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%

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

   5. Taylor expanded in Vef around inf 77.3%

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

   6. Taylor expanded in Ev around 0 70.9%

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

      1. +-commutative 46.2%

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

   8. Simplified 70.9%

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

   if 7.40000000000000042e-236 < EDonor < 2.8499999999999999e-130

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 95.9%

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

   6. Taylor expanded in Ev around inf 83.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -5.3 \cdot 10^{+165}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;EDonor \leq -1.6 \cdot 10^{-241}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}\\ \mathbf{elif}\;EDonor \leq 7.4 \cdot 10^{-236}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;EDonor \leq 2.85 \cdot 10^{-130}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{if}\;mu \leq -1.4 \cdot 10^{+172}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;mu \leq 7.2 \cdot 10^{-168}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 3 \cdot 10^{-60}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.5 \cdot 10^{+117}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- mu) KbT)))))))
   (if (<= mu -1.4e+172)
     t_0
     (if (<= mu 7.2e-168)
       (+
        (/ NaChar (+ 1.0 (exp (/ (+ Vef (- (+ Ev EAccept) mu)) KbT))))
        (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
       (if (<= mu 3e-60)
         (+
          (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ Vef EAccept) mu) KbT)))))
         (if (<= mu 2.5e+117)
           (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT))))
           t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-mu / KbT))));
	double tmp;
	if (mu <= -1.4e+172) {
		tmp = t_0;
	} else if (mu <= 7.2e-168) {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (mu <= 3e-60) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp((((Vef + EAccept) - mu) / KbT))));
	} else if (mu <= 2.5e+117) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((-mu / kbt))))
    if (mu <= (-1.4d+172)) then
        tmp = t_0
    else if (mu <= 7.2d-168) then
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev + eaccept) - mu)) / kbt)))) + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (mu <= 3d-60) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / (1.0d0 + exp((((vef + eaccept) - mu) / kbt))))
    else if (mu <= 2.5d+117) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (vef + ev)) - mu) / kbt)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((-mu / KbT))));
	double tmp;
	if (mu <= -1.4e+172) {
		tmp = t_0;
	} else if (mu <= 7.2e-168) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (mu <= 3e-60) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / (1.0 + Math.exp((((Vef + EAccept) - mu) / KbT))));
	} else if (mu <= 2.5e+117) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((-mu / KbT))))
	tmp = 0
	if mu <= -1.4e+172:
		tmp = t_0
	elif mu <= 7.2e-168:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif mu <= 3e-60:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / (1.0 + math.exp((((Vef + EAccept) - mu) / KbT))))
	elif mu <= 2.5e+117:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))))
	tmp = 0.0
	if (mu <= -1.4e+172)
		tmp = t_0;
	elseif (mu <= 7.2e-168)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (mu <= 3e-60)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + EAccept) - mu) / KbT)))));
	elseif (mu <= 2.5e+117)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((-mu / KbT))));
	tmp = 0.0;
	if (mu <= -1.4e+172)
		tmp = t_0;
	elseif (mu <= 7.2e-168)
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (mu <= 3e-60)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp((((Vef + EAccept) - mu) / KbT))));
	elseif (mu <= 2.5e+117)
		tmp = NaChar / (1.0 + exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -1.4e+172], t$95$0, If[LessEqual[mu, 7.2e-168], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 3e-60], 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 + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 2.5e+117], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -1.4e172 or 2.49999999999999992e117 < 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%

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

   5. Taylor expanded in mu around inf 89.5%

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

   6. Taylor expanded in mu around inf 82.9%

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

      1. associate-*r/ 42.5%

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

      2. mul-1-neg 42.5%

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

   8. Simplified 82.9%

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

   if -1.4e172 < mu < 7.1999999999999998e-168

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 78.5%

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

   if 7.1999999999999998e-168 < mu < 3.00000000000000019e-60

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 84.9%

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

   6. Taylor expanded in Ev around 0 77.9%

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

      1. +-commutative 51.8%

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

   8. Simplified 77.9%

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

   if 3.00000000000000019e-60 < mu < 2.49999999999999992e117

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 66.6%

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

      1. mul-1-neg 66.6%

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

      2. associate-+r+ 66.6%

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

      3. +-commutative 66.6%

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

      4. associate-+r+ 66.6%

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

      5. unsub-neg 66.6%

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

      6. associate-+r+ 66.6%

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

      7. +-commutative 66.6%

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

      8. associate-+r+ 66.6%

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

      9. associate--r+ 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in Ec around inf 42.5%

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

      1. mul-1-neg 42.5%

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

      2. associate-/l* 42.5%

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

      3. distribute-lft-neg-in 42.5%

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

   10. Simplified 42.5%

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

   11. Taylor expanded in KbT around 0 75.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.4 \cdot 10^{+172}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 7.2 \cdot 10^{-168}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;mu \leq 3 \cdot 10^{-60}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;mu \leq 2.5 \cdot 10^{+117}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{if}\;Vef \leq -3.65 \cdot 10^{+171}:\\ \;\;\;\;t\_1 + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq -5.2 \cdot 10^{+51}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.15 \cdot 10^{-46}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + t\_1\\ \end{array} \end{array} \]

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))))))
   (if (<= Vef -3.65e+171)
     (+ t_1 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef EAccept) mu) KbT)))))
     (if (<= Vef -5.2e+51)
       (+ t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
       (if (<= Vef 1.15e-46)
         (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
         (+ 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)));
	double tmp;
	if (Vef <= -3.65e+171) {
		tmp = t_1 + (NaChar / (1.0 + exp((((Vef + EAccept) - mu) / KbT))));
	} else if (Vef <= -5.2e+51) {
		tmp = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	} else if (Vef <= 1.15e-46) {
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else {
		tmp = t_0 + 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)))
    if (vef <= (-3.65d+171)) then
        tmp = t_1 + (nachar / (1.0d0 + exp((((vef + eaccept) - mu) / kbt))))
    else if (vef <= (-5.2d+51)) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((mu / kbt))))
    else if (vef <= 1.15d-46) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else
        tmp = t_0 + 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)));
	double tmp;
	if (Vef <= -3.65e+171) {
		tmp = t_1 + (NaChar / (1.0 + Math.exp((((Vef + EAccept) - mu) / KbT))));
	} else if (Vef <= -5.2e+51) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else if (Vef <= 1.15e-46) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else {
		tmp = t_0 + 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)))
	tmp = 0
	if Vef <= -3.65e+171:
		tmp = t_1 + (NaChar / (1.0 + math.exp((((Vef + EAccept) - mu) / KbT))))
	elif Vef <= -5.2e+51:
		tmp = t_0 + (NdChar / (1.0 + math.exp((mu / KbT))))
	elif Vef <= 1.15e-46:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	else:
		tmp = t_0 + t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT))))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT))))
	tmp = 0.0
	if (Vef <= -3.65e+171)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + EAccept) - mu) / KbT)))));
	elseif (Vef <= -5.2e+51)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	elseif (Vef <= 1.15e-46)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	else
		tmp = Float64(t_0 + 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)));
	tmp = 0.0;
	if (Vef <= -3.65e+171)
		tmp = t_1 + (NaChar / (1.0 + exp((((Vef + EAccept) - mu) / KbT))));
	elseif (Vef <= -5.2e+51)
		tmp = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	elseif (Vef <= 1.15e-46)
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	else
		tmp = t_0 + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -3.65e+171], N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, -5.2e+51], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 1.15e-46], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -3.6500000000000001e171

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 82.2%

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

   6. Taylor expanded in Ev around 0 71.8%

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

      1. +-commutative 49.9%

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

   8. Simplified 71.8%

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

   if -3.6500000000000001e171 < Vef < -5.2000000000000002e51

   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%

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

   5. Taylor expanded in mu around inf 91.0%

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

   if -5.2000000000000002e51 < Vef < 1.15e-46

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 77.3%

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

   if 1.15e-46 < Vef

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 87.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -3.65 \cdot 10^{+171}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq -5.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.15 \cdot 10^{-46}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \mathbf{if}\;EDonor \leq -1.25 \cdot 10^{+166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;EDonor \leq 2.75 \cdot 10^{-254}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}\\ \mathbf{elif}\;EDonor \leq 1.65 \cdot 10^{-131}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0
         (+
          (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
          (/ NaChar (+ 1.0 (exp (/ (- (+ Vef EAccept) mu) KbT)))))))
   (if (<= EDonor -1.25e+166)
     t_0
     (if (<= EDonor 2.75e-254)
       (+
        (/ NaChar (+ 1.0 (exp (/ (+ Vef (- (+ Ev EAccept) mu)) KbT))))
        (/ NdChar (+ 2.0 (/ (- (+ Vef mu) (- Ec EDonor)) KbT))))
       (if (<= EDonor 1.65e-131)
         (+
          (/ NdChar (+ 1.0 (exp (/ mu KbT))))
          (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
         t_0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((((Vef + EAccept) - mu) / KbT))));
	double tmp;
	if (EDonor <= -1.25e+166) {
		tmp = t_0;
	} else if (EDonor <= 2.75e-254) {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)));
	} else if (EDonor <= 1.65e-131) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar / (1.0d0 + exp((((vef + eaccept) - mu) / kbt))))
    if (edonor <= (-1.25d+166)) then
        tmp = t_0
    else if (edonor <= 2.75d-254) then
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev + eaccept) - mu)) / kbt)))) + (ndchar / (2.0d0 + (((vef + mu) - (ec - edonor)) / kbt)))
    else if (edonor <= 1.65d-131) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / (1.0d0 + exp((ev / kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / (1.0 + Math.exp((((Vef + EAccept) - mu) / KbT))));
	double tmp;
	if (EDonor <= -1.25e+166) {
		tmp = t_0;
	} else if (EDonor <= 2.75e-254) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)));
	} else if (EDonor <= 1.65e-131) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / (1.0 + math.exp((((Vef + EAccept) - mu) / KbT))))
	tmp = 0
	if EDonor <= -1.25e+166:
		tmp = t_0
	elif EDonor <= 2.75e-254:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)))
	elif EDonor <= 1.65e-131:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + EAccept) - mu) / KbT)))))
	tmp = 0.0
	if (EDonor <= -1.25e+166)
		tmp = t_0;
	elseif (EDonor <= 2.75e-254)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT)))) + Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(Vef + mu) - Float64(Ec - EDonor)) / KbT))));
	elseif (EDonor <= 1.65e-131)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((((Vef + EAccept) - mu) / KbT))));
	tmp = 0.0;
	if (EDonor <= -1.25e+166)
		tmp = t_0;
	elseif (EDonor <= 2.75e-254)
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)));
	elseif (EDonor <= 1.65e-131)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EDonor, -1.25e+166], t$95$0, If[LessEqual[EDonor, 2.75e-254], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(N[(N[(Vef + mu), $MachinePrecision] - N[(Ec - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, 1.65e-131], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if EDonor < -1.25e166 or 1.6500000000000001e-131 < 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%

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

   5. Taylor expanded in EDonor around inf 86.3%

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

   6. Taylor expanded in Ev around 0 74.6%

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

      1. +-commutative 74.6%

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

   8. Simplified 74.6%

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

   if -1.25e166 < EDonor < 2.74999999999999995e-254

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 63.8%

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

      1. mul-1-neg 63.8%

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

      2. associate-+r+ 63.8%

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

      3. +-commutative 63.8%

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

      4. associate-+r+ 63.8%

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

      5. unsub-neg 63.8%

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

      6. associate-+r+ 63.8%

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

      7. +-commutative 63.8%

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

      8. associate-+r+ 63.8%

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

      9. associate--r+ 63.8%

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

   7. Simplified 63.8%

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

   if 2.74999999999999995e-254 < EDonor < 1.6500000000000001e-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%

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

   5. Taylor expanded in mu around inf 90.5%

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

   6. Taylor expanded in Ev around inf 76.5%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -1.25 \cdot 10^{+166}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;EDonor \leq 2.75 \cdot 10^{-254}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}\\ \mathbf{elif}\;EDonor \leq 1.65 \cdot 10^{-131}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{if}\;NdChar \leq -7.5 \cdot 10^{+211}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + t\_0\\ \mathbf{elif}\;NdChar \leq -1.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -1.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}} + t\_0\\ \mathbf{elif}\;NdChar \leq 1.95 \cdot 10^{+129}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{1}{\frac{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))))
   (if (<= NdChar -7.5e+211)
     (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) t_0)
     (if (<= NdChar -1.3e+21)
       (+
        (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
        (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
       (if (<= NdChar -1.1e-35)
         (+ (/ NdChar (+ 1.0 (exp (/ (- Ec) KbT)))) t_0)
         (if (<= NdChar 1.95e+129)
           (+
            (/ NaChar (+ 1.0 (exp (/ (+ Vef (- (+ Ev EAccept) mu)) KbT))))
            (/ 1.0 (/ (+ 2.0 (/ (- (+ Vef mu) (- Ec EDonor)) KbT)) NdChar)))
           (+
            (/ NdChar (+ 1.0 (exp (/ (- (+ Vef (+ EDonor mu)) Ec) KbT))))
            (* NaChar 0.5))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((EAccept / KbT)));
	double tmp;
	if (NdChar <= -7.5e+211) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	} else if (NdChar <= -1.3e+21) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (NdChar <= -1.1e-35) {
		tmp = (NdChar / (1.0 + exp((-Ec / KbT)))) + t_0;
	} else if (NdChar <= 1.95e+129) {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar));
	} else {
		tmp = (NdChar / (1.0 + exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((eaccept / kbt)))
    if (ndchar <= (-7.5d+211)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + t_0
    else if (ndchar <= (-1.3d+21)) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (ndchar <= (-1.1d-35)) then
        tmp = (ndchar / (1.0d0 + exp((-ec / kbt)))) + t_0
    else if (ndchar <= 1.95d+129) then
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev + eaccept) - mu)) / kbt)))) + (1.0d0 / ((2.0d0 + (((vef + mu) - (ec - edonor)) / kbt)) / ndchar))
    else
        tmp = (ndchar / (1.0d0 + exp((((vef + (edonor + mu)) - ec) / kbt)))) + (nachar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((EAccept / KbT)));
	double tmp;
	if (NdChar <= -7.5e+211) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + t_0;
	} else if (NdChar <= -1.3e+21) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (NdChar <= -1.1e-35) {
		tmp = (NdChar / (1.0 + Math.exp((-Ec / KbT)))) + t_0;
	} else if (NdChar <= 1.95e+129) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar));
	} else {
		tmp = (NdChar / (1.0 + Math.exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((EAccept / KbT)))
	tmp = 0
	if NdChar <= -7.5e+211:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + t_0
	elif NdChar <= -1.3e+21:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif NdChar <= -1.1e-35:
		tmp = (NdChar / (1.0 + math.exp((-Ec / KbT)))) + t_0
	elif NdChar <= 1.95e+129:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar))
	else:
		tmp = (NdChar / (1.0 + math.exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))))
	tmp = 0.0
	if (NdChar <= -7.5e+211)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + t_0);
	elseif (NdChar <= -1.3e+21)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	elseif (NdChar <= -1.1e-35)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Ec) / KbT)))) + t_0);
	elseif (NdChar <= 1.95e+129)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT)))) + Float64(1.0 / Float64(Float64(2.0 + Float64(Float64(Float64(Vef + mu) - Float64(Ec - EDonor)) / KbT)) / NdChar)));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(EDonor + mu)) - Ec) / KbT)))) + Float64(NaChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((EAccept / KbT)));
	tmp = 0.0;
	if (NdChar <= -7.5e+211)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + t_0;
	elseif (NdChar <= -1.3e+21)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (NdChar <= -1.1e-35)
		tmp = (NdChar / (1.0 + exp((-Ec / KbT)))) + t_0;
	elseif (NdChar <= 1.95e+129)
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar));
	else
		tmp = (NdChar / (1.0 + exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -7.5e+211], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[NdChar, -1.3e+21], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -1.1e-35], N[(N[(NdChar / N[(1.0 + N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[NdChar, 1.95e+129], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[(2.0 + N[(N[(N[(Vef + mu), $MachinePrecision] - N[(Ec - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(EDonor + mu), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -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%

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

   5. Taylor expanded in Vef around inf 73.3%

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

   6. Taylor expanded in EAccept around inf 58.4%

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

   if -7.5e211 < NdChar < -1.3e21

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 75.0%

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

   6. Taylor expanded in Ev around inf 59.4%

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

   if -1.3e21 < NdChar < -1.09999999999999997e-35

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 65.6%

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

      1. mul-1-neg 65.6%

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

      2. distribute-neg-frac2 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in EAccept around inf 52.2%

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

   if -1.09999999999999997e-35 < NdChar < 1.9499999999999999e129

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 75.0%

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

      1. mul-1-neg 75.0%

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

      2. associate-+r+ 75.0%

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

      3. +-commutative 75.0%

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

      4. associate-+r+ 75.0%

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

      5. unsub-neg 75.0%

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

      6. associate-+r+ 75.0%

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

      7. +-commutative 75.0%

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

      8. associate-+r+ 75.0%

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

      9. associate--r+ 75.0%

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

   7. Simplified 75.0%

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

      1. clear-num 75.5%

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

      2. inv-pow 75.5%

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

   9. Applied egg-rr 75.5%

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

       1. unpow-1 75.5%

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

       2. +-commutative 75.5%

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

   11. Simplified 75.5%

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

   if 1.9499999999999999e129 < 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%

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

   5. Taylor expanded in KbT around inf 69.8%

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

      1. *-commutative 27.5%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   7. Simplified 69.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -7.5 \cdot 10^{+211}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -1.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -1.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;NdChar \leq 1.95 \cdot 10^{+129}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{1}{\frac{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{if}\;Vef \leq -6.5 \cdot 10^{+164}:\\ \;\;\;\;t\_1 + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 4.1 \cdot 10^{-30}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + t\_1\\ \end{array} \end{array} \]

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))))))
   (if (<= Vef -6.5e+164)
     (+ t_1 (/ NaChar (+ 1.0 (exp (/ (- (+ Vef EAccept) mu) KbT)))))
     (if (<= Vef 4.1e-30)
       (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
       (+ 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)));
	double tmp;
	if (Vef <= -6.5e+164) {
		tmp = t_1 + (NaChar / (1.0 + exp((((Vef + EAccept) - mu) / KbT))));
	} else if (Vef <= 4.1e-30) {
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else {
		tmp = t_0 + 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)))
    if (vef <= (-6.5d+164)) then
        tmp = t_1 + (nachar / (1.0d0 + exp((((vef + eaccept) - mu) / kbt))))
    else if (vef <= 4.1d-30) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else
        tmp = t_0 + 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)));
	double tmp;
	if (Vef <= -6.5e+164) {
		tmp = t_1 + (NaChar / (1.0 + Math.exp((((Vef + EAccept) - mu) / KbT))));
	} else if (Vef <= 4.1e-30) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else {
		tmp = t_0 + 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)))
	tmp = 0
	if Vef <= -6.5e+164:
		tmp = t_1 + (NaChar / (1.0 + math.exp((((Vef + EAccept) - mu) / KbT))))
	elif Vef <= 4.1e-30:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	else:
		tmp = t_0 + t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT))))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT))))
	tmp = 0.0
	if (Vef <= -6.5e+164)
		tmp = Float64(t_1 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + EAccept) - mu) / KbT)))));
	elseif (Vef <= 4.1e-30)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	else
		tmp = Float64(t_0 + 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)));
	tmp = 0.0;
	if (Vef <= -6.5e+164)
		tmp = t_1 + (NaChar / (1.0 + exp((((Vef + EAccept) - mu) / KbT))));
	elseif (Vef <= 4.1e-30)
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	else
		tmp = t_0 + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -6.5e+164], N[(t$95$1 + N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + EAccept), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 4.1e-30], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -6.5000000000000003e164

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 82.8%

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

   6. Taylor expanded in Ev around 0 72.7%

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

      1. +-commutative 48.8%

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

   8. Simplified 72.7%

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

   if -6.5000000000000003e164 < Vef < 4.1000000000000003e-30

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 76.0%

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

   if 4.1000000000000003e-30 < Vef

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 87.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -6.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + EAccept\right) - mu}{KbT}}}\\ \mathbf{elif}\;Vef \leq 4.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -5.5 \cdot 10^{+211}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -3.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{1}{\frac{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -5.5e+211)
   (+
    (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
    (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
   (if (<= NdChar -3.5e+20)
     (+
      (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
      (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
     (if (<= NdChar -9.5e+19)
       (+ (/ NdChar (+ 1.0 (exp (/ (- Ec) KbT)))) (* NaChar 0.5))
       (if (<= NdChar 5e+132)
         (+
          (/ NaChar (+ 1.0 (exp (/ (+ Vef (- (+ Ev EAccept) mu)) KbT))))
          (/ 1.0 (/ (+ 2.0 (/ (- (+ Vef mu) (- Ec EDonor)) KbT)) NdChar)))
         (+
          (/ NdChar (+ 1.0 (exp (/ (- (+ Vef (+ EDonor mu)) Ec) KbT))))
          (* NaChar 0.5)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -5.5e+211) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (NdChar <= -3.5e+20) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (NdChar <= -9.5e+19) {
		tmp = (NdChar / (1.0 + exp((-Ec / KbT)))) + (NaChar * 0.5);
	} else if (NdChar <= 5e+132) {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar));
	} else {
		tmp = (NdChar / (1.0 + exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-5.5d+211)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else if (ndchar <= (-3.5d+20)) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (ndchar <= (-9.5d+19)) then
        tmp = (ndchar / (1.0d0 + exp((-ec / kbt)))) + (nachar * 0.5d0)
    else if (ndchar <= 5d+132) then
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev + eaccept) - mu)) / kbt)))) + (1.0d0 / ((2.0d0 + (((vef + mu) - (ec - edonor)) / kbt)) / ndchar))
    else
        tmp = (ndchar / (1.0d0 + exp((((vef + (edonor + mu)) - ec) / kbt)))) + (nachar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -5.5e+211) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else if (NdChar <= -3.5e+20) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (NdChar <= -9.5e+19) {
		tmp = (NdChar / (1.0 + Math.exp((-Ec / KbT)))) + (NaChar * 0.5);
	} else if (NdChar <= 5e+132) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar));
	} else {
		tmp = (NdChar / (1.0 + Math.exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -5.5e+211:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	elif NdChar <= -3.5e+20:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif NdChar <= -9.5e+19:
		tmp = (NdChar / (1.0 + math.exp((-Ec / KbT)))) + (NaChar * 0.5)
	elif NdChar <= 5e+132:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar))
	else:
		tmp = (NdChar / (1.0 + math.exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -5.5e+211)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (NdChar <= -3.5e+20)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	elseif (NdChar <= -9.5e+19)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(-Ec) / KbT)))) + Float64(NaChar * 0.5));
	elseif (NdChar <= 5e+132)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT)))) + Float64(1.0 / Float64(Float64(2.0 + Float64(Float64(Float64(Vef + mu) - Float64(Ec - EDonor)) / KbT)) / NdChar)));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(EDonor + mu)) - Ec) / KbT)))) + Float64(NaChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -5.5e+211)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	elseif (NdChar <= -3.5e+20)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (NdChar <= -9.5e+19)
		tmp = (NdChar / (1.0 + exp((-Ec / KbT)))) + (NaChar * 0.5);
	elseif (NdChar <= 5e+132)
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar));
	else
		tmp = (NdChar / (1.0 + exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -5.5e+211], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -3.5e+20], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, -9.5e+19], N[(N[(NdChar / N[(1.0 + N[Exp[N[((-Ec) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 5e+132], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[(2.0 + N[(N[(N[(Vef + mu), $MachinePrecision] - N[(Ec - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(EDonor + mu), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if NdChar < -5.49999999999999988e211

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 73.3%

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

   6. Taylor expanded in EAccept around inf 58.4%

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

   if -5.49999999999999988e211 < NdChar < -3.5e20

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 75.0%

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

   6. Taylor expanded in Ev around inf 59.4%

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

   if -3.5e20 < NdChar < -9.5e19

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ec around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in KbT around inf 100.0%

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

      1. *-commutative 18.8%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   10. Simplified 100.0%

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

   if -9.5e19 < NdChar < 5.0000000000000001e132

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. associate-+r+ 73.4%

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

      3. +-commutative 73.4%

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

      4. associate-+r+ 73.4%

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

      5. unsub-neg 73.4%

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

      6. associate-+r+ 73.4%

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

      7. +-commutative 73.4%

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

      8. associate-+r+ 73.4%

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

      9. associate--r+ 73.4%

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

   7. Simplified 73.4%

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

      1. clear-num 73.9%

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

      2. inv-pow 73.9%

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

   9. Applied egg-rr 73.9%

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

       1. unpow-1 73.9%

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

       2. +-commutative 73.9%

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

   11. Simplified 73.9%

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

   if 5.0000000000000001e132 < 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%

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

   5. Taylor expanded in KbT around inf 69.8%

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

      1. *-commutative 27.5%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   7. Simplified 69.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -5.5 \cdot 10^{+211}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -3.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;NdChar \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{-Ec}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{1}{\frac{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}{NdChar}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ t_1 := t\_0 + \frac{NdChar}{2}\\ t_2 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{if}\;Ev \leq -2 \cdot 10^{+232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Ev \leq -4 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Ev \leq -5.6 \cdot 10^{+127}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;Ev \leq -4.2 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Ev \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;Ev \leq 6.8 \cdot 10^{-87}:\\ \;\;\;\;t\_2 + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
        (t_1 (+ t_0 (/ NdChar 2.0)))
        (t_2 (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))))
   (if (<= Ev -2e+232)
     t_0
     (if (<= Ev -4e+135)
       t_1
       (if (<= Ev -5.6e+127)
         t_2
         (if (<= Ev -4.2e-47)
           t_1
           (if (<= Ev -1.35e-298)
             (/ NaChar (+ 1.0 (exp (/ (- mu) KbT))))
             (if (<= Ev 6.8e-87) (+ t_2 (/ NdChar 2.0)) t_2))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((Ev / KbT)));
	double t_1 = t_0 + (NdChar / 2.0);
	double t_2 = NaChar / (1.0 + exp((EAccept / KbT)));
	double tmp;
	if (Ev <= -2e+232) {
		tmp = t_0;
	} else if (Ev <= -4e+135) {
		tmp = t_1;
	} else if (Ev <= -5.6e+127) {
		tmp = t_2;
	} else if (Ev <= -4.2e-47) {
		tmp = t_1;
	} else if (Ev <= -1.35e-298) {
		tmp = NaChar / (1.0 + exp((-mu / KbT)));
	} else if (Ev <= 6.8e-87) {
		tmp = t_2 + (NdChar / 2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((ev / kbt)))
    t_1 = t_0 + (ndchar / 2.0d0)
    t_2 = nachar / (1.0d0 + exp((eaccept / kbt)))
    if (ev <= (-2d+232)) then
        tmp = t_0
    else if (ev <= (-4d+135)) then
        tmp = t_1
    else if (ev <= (-5.6d+127)) then
        tmp = t_2
    else if (ev <= (-4.2d-47)) then
        tmp = t_1
    else if (ev <= (-1.35d-298)) then
        tmp = nachar / (1.0d0 + exp((-mu / kbt)))
    else if (ev <= 6.8d-87) then
        tmp = t_2 + (ndchar / 2.0d0)
    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((Ev / KbT)));
	double t_1 = t_0 + (NdChar / 2.0);
	double t_2 = NaChar / (1.0 + Math.exp((EAccept / KbT)));
	double tmp;
	if (Ev <= -2e+232) {
		tmp = t_0;
	} else if (Ev <= -4e+135) {
		tmp = t_1;
	} else if (Ev <= -5.6e+127) {
		tmp = t_2;
	} else if (Ev <= -4.2e-47) {
		tmp = t_1;
	} else if (Ev <= -1.35e-298) {
		tmp = NaChar / (1.0 + Math.exp((-mu / KbT)));
	} else if (Ev <= 6.8e-87) {
		tmp = t_2 + (NdChar / 2.0);
	} 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((Ev / KbT)))
	t_1 = t_0 + (NdChar / 2.0)
	t_2 = NaChar / (1.0 + math.exp((EAccept / KbT)))
	tmp = 0
	if Ev <= -2e+232:
		tmp = t_0
	elif Ev <= -4e+135:
		tmp = t_1
	elif Ev <= -5.6e+127:
		tmp = t_2
	elif Ev <= -4.2e-47:
		tmp = t_1
	elif Ev <= -1.35e-298:
		tmp = NaChar / (1.0 + math.exp((-mu / KbT)))
	elif Ev <= 6.8e-87:
		tmp = t_2 + (NdChar / 2.0)
	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(Ev / KbT))))
	t_1 = Float64(t_0 + Float64(NdChar / 2.0))
	t_2 = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))))
	tmp = 0.0
	if (Ev <= -2e+232)
		tmp = t_0;
	elseif (Ev <= -4e+135)
		tmp = t_1;
	elseif (Ev <= -5.6e+127)
		tmp = t_2;
	elseif (Ev <= -4.2e-47)
		tmp = t_1;
	elseif (Ev <= -1.35e-298)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT))));
	elseif (Ev <= 6.8e-87)
		tmp = Float64(t_2 + Float64(NdChar / 2.0));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((Ev / KbT)));
	t_1 = t_0 + (NdChar / 2.0);
	t_2 = NaChar / (1.0 + exp((EAccept / KbT)));
	tmp = 0.0;
	if (Ev <= -2e+232)
		tmp = t_0;
	elseif (Ev <= -4e+135)
		tmp = t_1;
	elseif (Ev <= -5.6e+127)
		tmp = t_2;
	elseif (Ev <= -4.2e-47)
		tmp = t_1;
	elseif (Ev <= -1.35e-298)
		tmp = NaChar / (1.0 + exp((-mu / KbT)));
	elseif (Ev <= 6.8e-87)
		tmp = t_2 + (NdChar / 2.0);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Ev, -2e+232], t$95$0, If[LessEqual[Ev, -4e+135], t$95$1, If[LessEqual[Ev, -5.6e+127], t$95$2, If[LessEqual[Ev, -4.2e-47], t$95$1, If[LessEqual[Ev, -1.35e-298], N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, 6.8e-87], N[(t$95$2 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if Ev < -2.00000000000000011e232

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. associate-+r+ 60.9%

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

      3. +-commutative 60.9%

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

      4. associate-+r+ 60.9%

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

      5. unsub-neg 60.9%

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

      6. associate-+r+ 60.9%

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

      7. +-commutative 60.9%

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

      8. associate-+r+ 60.9%

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

      9. associate--r+ 60.9%

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

   7. Simplified 60.9%

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

   8. Taylor expanded in Ec around inf 33.4%

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

      1. mul-1-neg 33.4%

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

      2. associate-/l* 41.9%

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

      3. distribute-lft-neg-in 41.9%

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

   10. Simplified 41.9%

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

   11. Taylor expanded in KbT around 0 80.7%

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

   12. Taylor expanded in Ev around inf 74.4%

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

   if -2.00000000000000011e232 < Ev < -3.99999999999999985e135 or -5.6000000000000004e127 < Ev < -4.2000000000000001e-47

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.8%

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

   6. Taylor expanded in Ev around inf 46.6%

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

   if -3.99999999999999985e135 < Ev < -5.6000000000000004e127 or 6.7999999999999997e-87 < Ev

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 61.0%

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

      1. mul-1-neg 61.0%

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

      2. associate-+r+ 61.0%

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

      3. +-commutative 61.0%

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

      4. associate-+r+ 61.0%

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

      5. unsub-neg 61.0%

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

      6. associate-+r+ 61.0%

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

      7. +-commutative 61.0%

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

      8. associate-+r+ 61.0%

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

      9. associate--r+ 61.0%

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

   7. Simplified 61.0%

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

   8. Taylor expanded in Ec around inf 34.8%

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

      1. mul-1-neg 34.8%

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

      2. associate-/l* 33.8%

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

      3. distribute-lft-neg-in 33.8%

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

   10. Simplified 33.8%

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

   11. Taylor expanded in KbT around 0 61.4%

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

   12. Taylor expanded in EAccept around inf 36.0%

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

   if -4.2000000000000001e-47 < Ev < -1.3500000000000001e-298

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 62.3%

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

      1. mul-1-neg 62.3%

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

      2. associate-+r+ 62.3%

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

      3. +-commutative 62.3%

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

      4. associate-+r+ 62.3%

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

      5. unsub-neg 62.3%

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

      6. associate-+r+ 62.3%

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

      7. +-commutative 62.3%

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

      8. associate-+r+ 62.3%

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

      9. associate--r+ 62.3%

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

   7. Simplified 62.3%

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

   8. Taylor expanded in Ec around inf 27.0%

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

      1. mul-1-neg 27.0%

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

      2. associate-/l* 25.0%

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

      3. distribute-lft-neg-in 25.0%

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

   10. Simplified 25.0%

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

   11. Taylor expanded in KbT around 0 62.2%

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

   12. Taylor expanded in mu around inf 43.0%

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

       1. associate-*r/ 43.0%

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

       2. mul-1-neg 43.0%

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

   14. Simplified 43.0%

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

   if -1.3500000000000001e-298 < Ev < 6.7999999999999997e-87

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 51.7%

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

   6. Taylor expanded in EAccept around inf 44.4%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -2 \cdot 10^{+232}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;Ev \leq -4 \cdot 10^{+135}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;Ev \leq -5.6 \cdot 10^{+127}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;Ev \leq -4.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;Ev \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;Ev \leq 6.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ t_1 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{if}\;Ev \leq -2 \cdot 10^{+232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Ev \leq -1.3 \cdot 10^{+134}:\\ \;\;\;\;t\_0 + \frac{NdChar}{2}\\ \mathbf{elif}\;Ev \leq -6.5 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Ev \leq -2.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;Ev \leq 5.5 \cdot 10^{-254}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;Ev \leq 3.4 \cdot 10^{-86}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
        (t_1 (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))))
   (if (<= Ev -2e+232)
     t_0
     (if (<= Ev -1.3e+134)
       (+ t_0 (/ NdChar 2.0))
       (if (<= Ev -6.5e+92)
         t_1
         (if (<= Ev -2.6e-20)
           (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) (* NaChar 0.5))
           (if (<= Ev 5.5e-254)
             (/ NaChar (+ 1.0 (exp (/ (- mu) KbT))))
             (if (<= Ev 3.4e-86)
               (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (* NaChar 0.5))
               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((Ev / KbT)));
	double t_1 = NaChar / (1.0 + exp((EAccept / KbT)));
	double tmp;
	if (Ev <= -2e+232) {
		tmp = t_0;
	} else if (Ev <= -1.3e+134) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (Ev <= -6.5e+92) {
		tmp = t_1;
	} else if (Ev <= -2.6e-20) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar * 0.5);
	} else if (Ev <= 5.5e-254) {
		tmp = NaChar / (1.0 + exp((-mu / KbT)));
	} else if (Ev <= 3.4e-86) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	} 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((ev / kbt)))
    t_1 = nachar / (1.0d0 + exp((eaccept / kbt)))
    if (ev <= (-2d+232)) then
        tmp = t_0
    else if (ev <= (-1.3d+134)) then
        tmp = t_0 + (ndchar / 2.0d0)
    else if (ev <= (-6.5d+92)) then
        tmp = t_1
    else if (ev <= (-2.6d-20)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar * 0.5d0)
    else if (ev <= 5.5d-254) then
        tmp = nachar / (1.0d0 + exp((-mu / kbt)))
    else if (ev <= 3.4d-86) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar * 0.5d0)
    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((Ev / KbT)));
	double t_1 = NaChar / (1.0 + Math.exp((EAccept / KbT)));
	double tmp;
	if (Ev <= -2e+232) {
		tmp = t_0;
	} else if (Ev <= -1.3e+134) {
		tmp = t_0 + (NdChar / 2.0);
	} else if (Ev <= -6.5e+92) {
		tmp = t_1;
	} else if (Ev <= -2.6e-20) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar * 0.5);
	} else if (Ev <= 5.5e-254) {
		tmp = NaChar / (1.0 + Math.exp((-mu / KbT)));
	} else if (Ev <= 3.4e-86) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	} 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((Ev / KbT)))
	t_1 = NaChar / (1.0 + math.exp((EAccept / KbT)))
	tmp = 0
	if Ev <= -2e+232:
		tmp = t_0
	elif Ev <= -1.3e+134:
		tmp = t_0 + (NdChar / 2.0)
	elif Ev <= -6.5e+92:
		tmp = t_1
	elif Ev <= -2.6e-20:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar * 0.5)
	elif Ev <= 5.5e-254:
		tmp = NaChar / (1.0 + math.exp((-mu / KbT)))
	elif Ev <= 3.4e-86:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	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(Ev / KbT))))
	t_1 = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))))
	tmp = 0.0
	if (Ev <= -2e+232)
		tmp = t_0;
	elseif (Ev <= -1.3e+134)
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	elseif (Ev <= -6.5e+92)
		tmp = t_1;
	elseif (Ev <= -2.6e-20)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar * 0.5));
	elseif (Ev <= 5.5e-254)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT))));
	elseif (Ev <= 3.4e-86)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar * 0.5));
	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((Ev / KbT)));
	t_1 = NaChar / (1.0 + exp((EAccept / KbT)));
	tmp = 0.0;
	if (Ev <= -2e+232)
		tmp = t_0;
	elseif (Ev <= -1.3e+134)
		tmp = t_0 + (NdChar / 2.0);
	elseif (Ev <= -6.5e+92)
		tmp = t_1;
	elseif (Ev <= -2.6e-20)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar * 0.5);
	elseif (Ev <= 5.5e-254)
		tmp = NaChar / (1.0 + exp((-mu / KbT)));
	elseif (Ev <= 3.4e-86)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	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[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Ev, -2e+232], t$95$0, If[LessEqual[Ev, -1.3e+134], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, -6.5e+92], t$95$1, If[LessEqual[Ev, -2.6e-20], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, 5.5e-254], N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Ev, 3.4e-86], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if Ev < -2.00000000000000011e232

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. associate-+r+ 60.9%

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

      3. +-commutative 60.9%

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

      4. associate-+r+ 60.9%

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

      5. unsub-neg 60.9%

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

      6. associate-+r+ 60.9%

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

      7. +-commutative 60.9%

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

      8. associate-+r+ 60.9%

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

      9. associate--r+ 60.9%

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

   7. Simplified 60.9%

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

   8. Taylor expanded in Ec around inf 33.4%

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

      1. mul-1-neg 33.4%

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

      2. associate-/l* 41.9%

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

      3. distribute-lft-neg-in 41.9%

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

   10. Simplified 41.9%

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

   11. Taylor expanded in KbT around 0 80.7%

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

   12. Taylor expanded in Ev around inf 74.4%

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

   if -2.00000000000000011e232 < Ev < -1.3000000000000001e134

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.8%

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

   6. Taylor expanded in Ev around inf 43.0%

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

   if -1.3000000000000001e134 < Ev < -6.49999999999999999e92 or 3.4e-86 < Ev

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 60.4%

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

      1. mul-1-neg 60.4%

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

      2. associate-+r+ 60.4%

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

      3. +-commutative 60.4%

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

      4. associate-+r+ 60.4%

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

      5. unsub-neg 60.4%

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

      6. associate-+r+ 60.4%

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

      7. +-commutative 60.4%

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

      8. associate-+r+ 60.4%

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

      9. associate--r+ 60.4%

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

   7. Simplified 60.4%

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

   8. Taylor expanded in Ec around inf 36.4%

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

      1. mul-1-neg 36.4%

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

      2. associate-/l* 35.5%

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

      3. distribute-lft-neg-in 35.5%

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

   10. Simplified 35.5%

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

   11. Taylor expanded in KbT around 0 60.9%

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

   12. Taylor expanded in EAccept around inf 34.4%

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

   if -6.49999999999999999e92 < Ev < -2.59999999999999995e-20

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 90.2%

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

   6. Taylor expanded in KbT around inf 64.8%

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

      1. *-commutative 45.9%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   8. Simplified 64.8%

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

   if -2.59999999999999995e-20 < Ev < 5.4999999999999999e-254

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 64.2%

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

      1. mul-1-neg 64.2%

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

      2. associate-+r+ 64.2%

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

      3. +-commutative 64.2%

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

      4. associate-+r+ 64.2%

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

      5. unsub-neg 64.2%

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

      6. associate-+r+ 64.2%

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

      7. +-commutative 64.2%

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

      8. associate-+r+ 64.2%

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

      9. associate--r+ 64.2%

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

   7. Simplified 64.2%

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

   8. Taylor expanded in Ec around inf 27.6%

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

      1. mul-1-neg 27.6%

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

      2. associate-/l* 26.0%

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

      3. distribute-lft-neg-in 26.0%

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

   10. Simplified 26.0%

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

   11. Taylor expanded in KbT around 0 62.8%

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

   12. Taylor expanded in mu around inf 43.5%

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

       1. associate-*r/ 43.5%

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

       2. mul-1-neg 43.5%

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

   14. Simplified 43.5%

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

   if 5.4999999999999999e-254 < Ev < 3.4e-86

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 89.5%

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

   6. Taylor expanded in KbT around inf 52.9%

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

      1. *-commutative 38.0%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   8. Simplified 52.9%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -2 \cdot 10^{+232}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;Ev \leq -1.3 \cdot 10^{+134}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;Ev \leq -6.5 \cdot 10^{+92}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;Ev \leq -2.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;Ev \leq 5.5 \cdot 10^{-254}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;Ev \leq 3.4 \cdot 10^{-86}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.4 \cdot 10^{+61} \lor \neg \left(NdChar \leq 10^{+128}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{1}{\frac{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}{NdChar}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -2.4e+61) (not (<= NdChar 1e+128)))
   (+
    (/ NdChar (+ 1.0 (exp (/ (- (+ Vef (+ EDonor mu)) Ec) KbT))))
    (* NaChar 0.5))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (- (+ Ev EAccept) mu)) KbT))))
    (/ 1.0 (/ (+ 2.0 (/ (- (+ Vef mu) (- Ec EDonor)) KbT)) NdChar)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.4e+61) || !(NdChar <= 1e+128)) {
		tmp = (NdChar / (1.0 + exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-2.4d+61)) .or. (.not. (ndchar <= 1d+128))) then
        tmp = (ndchar / (1.0d0 + exp((((vef + (edonor + mu)) - ec) / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev + eaccept) - mu)) / kbt)))) + (1.0d0 / ((2.0d0 + (((vef + mu) - (ec - edonor)) / kbt)) / ndchar))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.4e+61) || !(NdChar <= 1e+128)) {
		tmp = (NdChar / (1.0 + Math.exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -2.4e+61) or not (NdChar <= 1e+128):
		tmp = (NdChar / (1.0 + math.exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -2.4e+61) || !(NdChar <= 1e+128))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(EDonor + mu)) - Ec) / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT)))) + Float64(1.0 / Float64(Float64(2.0 + Float64(Float64(Float64(Vef + mu) - Float64(Ec - EDonor)) / KbT)) / NdChar)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -2.4e+61) || ~((NdChar <= 1e+128)))
		tmp = (NdChar / (1.0 + exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (1.0 / ((2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)) / NdChar));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -2.4e+61], N[Not[LessEqual[NdChar, 1e+128]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(EDonor + mu), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[(2.0 + N[(N[(N[(Vef + mu), $MachinePrecision] - N[(Ec - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / NdChar), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.3999999999999999e61 or 1.0000000000000001e128 < 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%

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

   5. Taylor expanded in KbT around inf 63.4%

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

      1. *-commutative 25.9%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   7. Simplified 63.4%

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

   if -2.3999999999999999e61 < NdChar < 1.0000000000000001e128

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 71.9%

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

      1. mul-1-neg 71.9%

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

      2. associate-+r+ 71.9%

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

      3. +-commutative 71.9%

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

      4. associate-+r+ 71.9%

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

      5. unsub-neg 71.9%

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

      6. associate-+r+ 71.9%

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

      7. +-commutative 71.9%

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

      8. associate-+r+ 71.9%

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

      9. associate--r+ 71.9%

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

   7. Simplified 71.9%

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

      1. clear-num 72.4%

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

      2. inv-pow 72.4%

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

   9. Applied egg-rr 72.4%

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

       1. unpow-1 72.4%

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

       2. +-commutative 72.4%

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

   11. Simplified 72.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.4 \cdot 10^{+61} \lor \neg \left(NdChar \leq 10^{+128}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{1}{\frac{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}{NdChar}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -6.8 \cdot 10^{+161} \lor \neg \left(KbT \leq -3.4 \cdot 10^{+119} \lor \neg \left(KbT \leq -6 \cdot 10^{+14}\right) \land KbT \leq 7.4 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= KbT -6.8e+161)
         (not
          (or (<= KbT -3.4e+119)
              (and (not (<= KbT -6e+14)) (<= KbT 7.4e+142)))))
   (+
    (/ NdChar (+ 1.0 (exp (/ (- (+ Vef (+ EDonor mu)) Ec) KbT))))
    (* NaChar 0.5))
   (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((KbT <= -6.8e+161) || !((KbT <= -3.4e+119) || (!(KbT <= -6e+14) && (KbT <= 7.4e+142)))) {
		tmp = (NdChar / (1.0 + exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = NaChar / (1.0 + exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((kbt <= (-6.8d+161)) .or. (.not. (kbt <= (-3.4d+119)) .or. (.not. (kbt <= (-6d+14))) .and. (kbt <= 7.4d+142))) then
        tmp = (ndchar / (1.0d0 + exp((((vef + (edonor + mu)) - ec) / kbt)))) + (nachar * 0.5d0)
    else
        tmp = nachar / (1.0d0 + exp((((eaccept + (vef + ev)) - mu) / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((KbT <= -6.8e+161) || !((KbT <= -3.4e+119) || (!(KbT <= -6e+14) && (KbT <= 7.4e+142)))) {
		tmp = (NdChar / (1.0 + Math.exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (KbT <= -6.8e+161) or not ((KbT <= -3.4e+119) or (not (KbT <= -6e+14) and (KbT <= 7.4e+142))):
		tmp = (NdChar / (1.0 + math.exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5)
	else:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((KbT <= -6.8e+161) || !((KbT <= -3.4e+119) || (!(KbT <= -6e+14) && (KbT <= 7.4e+142))))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(EDonor + mu)) - Ec) / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((KbT <= -6.8e+161) || ~(((KbT <= -3.4e+119) || (~((KbT <= -6e+14)) && (KbT <= 7.4e+142)))))
		tmp = (NdChar / (1.0 + exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	else
		tmp = NaChar / (1.0 + exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[KbT, -6.8e+161], N[Not[Or[LessEqual[KbT, -3.4e+119], And[N[Not[LessEqual[KbT, -6e+14]], $MachinePrecision], LessEqual[KbT, 7.4e+142]]]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(EDonor + mu), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -6.79999999999999986e161 or -3.40000000000000013e119 < KbT < -6e14 or 7.3999999999999995e142 < 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%

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

   5. Taylor expanded in KbT around inf 79.4%

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

      1. *-commutative 60.3%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   7. Simplified 79.4%

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

   if -6.79999999999999986e161 < KbT < -3.40000000000000013e119 or -6e14 < KbT < 7.3999999999999995e142

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 56.9%

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

      1. mul-1-neg 56.9%

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

      2. associate-+r+ 56.9%

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

      3. +-commutative 56.9%

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

      4. associate-+r+ 56.9%

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

      5. unsub-neg 56.9%

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

      6. associate-+r+ 56.9%

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

      7. +-commutative 56.9%

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

      8. associate-+r+ 56.9%

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

      9. associate--r+ 56.9%

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

   7. Simplified 56.9%

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

   8. Taylor expanded in Ec around inf 40.8%

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

      1. mul-1-neg 40.8%

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

      2. associate-/l* 38.0%

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

      3. distribute-lft-neg-in 38.0%

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

   10. Simplified 38.0%

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

   11. Taylor expanded in KbT around 0 65.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6.8 \cdot 10^{+161} \lor \neg \left(KbT \leq -3.4 \cdot 10^{+119} \lor \neg \left(KbT \leq -6 \cdot 10^{+14}\right) \land KbT \leq 7.4 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -1.05 \cdot 10^{+61} \lor \neg \left(NdChar \leq 2.6 \cdot 10^{+132}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -1.05e+61) (not (<= NdChar 2.6e+132)))
   (+
    (/ NdChar (+ 1.0 (exp (/ (- (+ Vef (+ EDonor mu)) Ec) KbT))))
    (* NaChar 0.5))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (- (+ Ev EAccept) mu)) KbT))))
    (/ NdChar (+ 2.0 (/ (- (+ Vef mu) (- Ec EDonor)) KbT))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.05e+61) || !(NdChar <= 2.6e+132)) {
		tmp = (NdChar / (1.0 + exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-1.05d+61)) .or. (.not. (ndchar <= 2.6d+132))) then
        tmp = (ndchar / (1.0d0 + exp((((vef + (edonor + mu)) - ec) / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev + eaccept) - mu)) / kbt)))) + (ndchar / (2.0d0 + (((vef + mu) - (ec - edonor)) / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.05e+61) || !(NdChar <= 2.6e+132)) {
		tmp = (NdChar / (1.0 + Math.exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -1.05e+61) or not (NdChar <= 2.6e+132):
		tmp = (NdChar / (1.0 + math.exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -1.05e+61) || !(NdChar <= 2.6e+132))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Float64(EDonor + mu)) - Ec) / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT)))) + Float64(NdChar / Float64(2.0 + Float64(Float64(Float64(Vef + mu) - Float64(Ec - EDonor)) / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -1.05e+61) || ~((NdChar <= 2.6e+132)))
		tmp = (NdChar / (1.0 + exp((((Vef + (EDonor + mu)) - Ec) / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / (2.0 + (((Vef + mu) - (Ec - EDonor)) / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -1.05e+61], N[Not[LessEqual[NdChar, 2.6e+132]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + N[(EDonor + mu), $MachinePrecision]), $MachinePrecision] - Ec), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(N[(N[(Vef + mu), $MachinePrecision] - N[(Ec - EDonor), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.0500000000000001e61 or 2.6e132 < 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%

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

   5. Taylor expanded in KbT around inf 63.4%

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

      1. *-commutative 25.9%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   7. Simplified 63.4%

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

   if -1.0500000000000001e61 < NdChar < 2.6e132

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 71.9%

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

      1. mul-1-neg 71.9%

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

      2. associate-+r+ 71.9%

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

      3. +-commutative 71.9%

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

      4. associate-+r+ 71.9%

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

      5. unsub-neg 71.9%

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

      6. associate-+r+ 71.9%

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

      7. +-commutative 71.9%

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

      8. associate-+r+ 71.9%

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

      9. associate--r+ 71.9%

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

   7. Simplified 71.9%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.05 \cdot 10^{+61} \lor \neg \left(NdChar \leq 2.6 \cdot 10^{+132}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{\left(Vef + \left(EDonor + mu\right)\right) - Ec}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{2 + \frac{\left(Vef + mu\right) - \left(Ec - EDonor\right)}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{if}\;KbT \leq -5 \cdot 10^{+161}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;KbT \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;KbT \leq -6.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq -2.8 \cdot 10^{-259}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.7 \cdot 10^{+127}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))))
   (if (<= KbT -5e+161)
     (+ (/ NaChar (+ 1.0 (exp (/ (- mu) KbT)))) (/ NdChar 2.0))
     (if (<= KbT -2.6e+118)
       (/ NaChar (+ 1.0 (exp (/ Vef KbT))))
       (if (<= KbT -6.6e+22)
         (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (* NaChar 0.5))
         (if (<= KbT -2.8e-259)
           t_0
           (if (<= KbT 1.7e+127)
             (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
             (+ t_0 (/ NdChar 2.0)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((EAccept / KbT)));
	double tmp;
	if (KbT <= -5e+161) {
		tmp = (NaChar / (1.0 + exp((-mu / KbT)))) + (NdChar / 2.0);
	} else if (KbT <= -2.6e+118) {
		tmp = NaChar / (1.0 + exp((Vef / KbT)));
	} else if (KbT <= -6.6e+22) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else if (KbT <= -2.8e-259) {
		tmp = t_0;
	} else if (KbT <= 1.7e+127) {
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((eaccept / kbt)))
    if (kbt <= (-5d+161)) then
        tmp = (nachar / (1.0d0 + exp((-mu / kbt)))) + (ndchar / 2.0d0)
    else if (kbt <= (-2.6d+118)) then
        tmp = nachar / (1.0d0 + exp((vef / kbt)))
    else if (kbt <= (-6.6d+22)) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar * 0.5d0)
    else if (kbt <= (-2.8d-259)) then
        tmp = t_0
    else if (kbt <= 1.7d+127) then
        tmp = nachar / (1.0d0 + exp((ev / kbt)))
    else
        tmp = t_0 + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((EAccept / KbT)));
	double tmp;
	if (KbT <= -5e+161) {
		tmp = (NaChar / (1.0 + Math.exp((-mu / KbT)))) + (NdChar / 2.0);
	} else if (KbT <= -2.6e+118) {
		tmp = NaChar / (1.0 + Math.exp((Vef / KbT)));
	} else if (KbT <= -6.6e+22) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else if (KbT <= -2.8e-259) {
		tmp = t_0;
	} else if (KbT <= 1.7e+127) {
		tmp = NaChar / (1.0 + Math.exp((Ev / KbT)));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((EAccept / KbT)))
	tmp = 0
	if KbT <= -5e+161:
		tmp = (NaChar / (1.0 + math.exp((-mu / KbT)))) + (NdChar / 2.0)
	elif KbT <= -2.6e+118:
		tmp = NaChar / (1.0 + math.exp((Vef / KbT)))
	elif KbT <= -6.6e+22:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	elif KbT <= -2.8e-259:
		tmp = t_0
	elif KbT <= 1.7e+127:
		tmp = NaChar / (1.0 + math.exp((Ev / KbT)))
	else:
		tmp = t_0 + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))))
	tmp = 0.0
	if (KbT <= -5e+161)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))) + Float64(NdChar / 2.0));
	elseif (KbT <= -2.6e+118)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT))));
	elseif (KbT <= -6.6e+22)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar * 0.5));
	elseif (KbT <= -2.8e-259)
		tmp = t_0;
	elseif (KbT <= 1.7e+127)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))));
	else
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((EAccept / KbT)));
	tmp = 0.0;
	if (KbT <= -5e+161)
		tmp = (NaChar / (1.0 + exp((-mu / KbT)))) + (NdChar / 2.0);
	elseif (KbT <= -2.6e+118)
		tmp = NaChar / (1.0 + exp((Vef / KbT)));
	elseif (KbT <= -6.6e+22)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	elseif (KbT <= -2.8e-259)
		tmp = t_0;
	elseif (KbT <= 1.7e+127)
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	else
		tmp = t_0 + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -5e+161], N[(N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -2.6e+118], N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -6.6e+22], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -2.8e-259], t$95$0, If[LessEqual[KbT, 1.7e+127], N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if KbT < -4.9999999999999997e161

   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%

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

   5. Taylor expanded in KbT around inf 81.0%

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

   6. Taylor expanded in mu around inf 81.0%

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

      1. associate-*r/ 81.0%

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

      2. mul-1-neg 81.0%

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

   8. Simplified 81.0%

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

   if -4.9999999999999997e161 < KbT < -2.60000000000000016e118

   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%

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

   5. Taylor expanded in KbT around inf 64.7%

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

      1. mul-1-neg 64.7%

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

      2. associate-+r+ 64.7%

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

      3. +-commutative 64.7%

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

      4. associate-+r+ 64.7%

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

      5. unsub-neg 64.7%

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

      6. associate-+r+ 64.7%

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

      7. +-commutative 64.7%

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

      8. associate-+r+ 64.7%

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

      9. associate--r+ 64.7%

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

   7. Simplified 64.7%

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

   8. Taylor expanded in Ec around inf 37.5%

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

      1. mul-1-neg 37.5%

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

      2. associate-/l* 37.5%

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

      3. distribute-lft-neg-in 37.5%

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

   10. Simplified 37.5%

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

   11. Taylor expanded in KbT around 0 79.9%

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

   12. Taylor expanded in Vef around inf 65.3%

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

   if -2.60000000000000016e118 < KbT < -6.5999999999999996e22

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 72.8%

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

   6. Taylor expanded in KbT around inf 52.5%

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

      1. *-commutative 27.9%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   8. Simplified 52.5%

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

   if -6.5999999999999996e22 < KbT < -2.8e-259

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 53.2%

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

      1. mul-1-neg 53.2%

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

      2. associate-+r+ 53.2%

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

      3. +-commutative 53.2%

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

      4. associate-+r+ 53.2%

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

      5. unsub-neg 53.2%

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

      6. associate-+r+ 53.2%

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

      7. +-commutative 53.2%

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

      8. associate-+r+ 53.2%

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

      9. associate--r+ 53.2%

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

   7. Simplified 53.2%

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

   8. Taylor expanded in Ec around inf 40.6%

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

      1. mul-1-neg 40.6%

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

      2. associate-/l* 39.1%

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

      3. distribute-lft-neg-in 39.1%

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

   10. Simplified 39.1%

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

   11. Taylor expanded in KbT around 0 57.7%

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

   12. Taylor expanded in EAccept around inf 30.5%

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

   if -2.8e-259 < KbT < 1.69999999999999989e127

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 58.0%

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

      1. mul-1-neg 58.0%

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

      2. associate-+r+ 58.0%

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

      3. +-commutative 58.0%

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

      4. associate-+r+ 58.0%

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

      5. unsub-neg 58.0%

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

      6. associate-+r+ 58.0%

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

      7. +-commutative 58.0%

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

      8. associate-+r+ 58.0%

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

      9. associate--r+ 58.0%

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

   7. Simplified 58.0%

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

   8. Taylor expanded in Ec around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. associate-/l* 37.1%

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

      3. distribute-lft-neg-in 37.1%

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

   10. Simplified 37.1%

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

   11. Taylor expanded in KbT around 0 67.6%

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

   12. Taylor expanded in Ev around inf 43.5%

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

   if 1.69999999999999989e127 < 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%

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

   5. Taylor expanded in KbT around inf 76.2%

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

   6. Taylor expanded in EAccept around inf 64.9%

      \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -5 \cdot 10^{+161}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;KbT \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;KbT \leq -6.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq -2.8 \cdot 10^{-259}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 1.7 \cdot 10^{+127}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{if}\;KbT \leq -5.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq -3.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;KbT \leq -1.65 \cdot 10^{-258}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 3.4 \cdot 10^{+137}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))))
   (if (<= KbT -5.5e+161)
     (+ (/ NdChar 2.0) (/ NaChar (- 2.0 (/ mu KbT))))
     (if (<= KbT -3.3e-7)
       (/ NaChar (+ 1.0 (exp (/ Vef KbT))))
       (if (<= KbT -1.65e-258)
         t_0
         (if (<= KbT 3.4e+137)
           (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
           (+ t_0 (/ NdChar 2.0))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((EAccept / KbT)));
	double tmp;
	if (KbT <= -5.5e+161) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	} else if (KbT <= -3.3e-7) {
		tmp = NaChar / (1.0 + exp((Vef / KbT)));
	} else if (KbT <= -1.65e-258) {
		tmp = t_0;
	} else if (KbT <= 3.4e+137) {
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((eaccept / kbt)))
    if (kbt <= (-5.5d+161)) then
        tmp = (ndchar / 2.0d0) + (nachar / (2.0d0 - (mu / kbt)))
    else if (kbt <= (-3.3d-7)) then
        tmp = nachar / (1.0d0 + exp((vef / kbt)))
    else if (kbt <= (-1.65d-258)) then
        tmp = t_0
    else if (kbt <= 3.4d+137) then
        tmp = nachar / (1.0d0 + exp((ev / kbt)))
    else
        tmp = t_0 + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((EAccept / KbT)));
	double tmp;
	if (KbT <= -5.5e+161) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	} else if (KbT <= -3.3e-7) {
		tmp = NaChar / (1.0 + Math.exp((Vef / KbT)));
	} else if (KbT <= -1.65e-258) {
		tmp = t_0;
	} else if (KbT <= 3.4e+137) {
		tmp = NaChar / (1.0 + Math.exp((Ev / KbT)));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((EAccept / KbT)))
	tmp = 0
	if KbT <= -5.5e+161:
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)))
	elif KbT <= -3.3e-7:
		tmp = NaChar / (1.0 + math.exp((Vef / KbT)))
	elif KbT <= -1.65e-258:
		tmp = t_0
	elif KbT <= 3.4e+137:
		tmp = NaChar / (1.0 + math.exp((Ev / KbT)))
	else:
		tmp = t_0 + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))))
	tmp = 0.0
	if (KbT <= -5.5e+161)
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 - Float64(mu / KbT))));
	elseif (KbT <= -3.3e-7)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT))));
	elseif (KbT <= -1.65e-258)
		tmp = t_0;
	elseif (KbT <= 3.4e+137)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))));
	else
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((EAccept / KbT)));
	tmp = 0.0;
	if (KbT <= -5.5e+161)
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	elseif (KbT <= -3.3e-7)
		tmp = NaChar / (1.0 + exp((Vef / KbT)));
	elseif (KbT <= -1.65e-258)
		tmp = t_0;
	elseif (KbT <= 3.4e+137)
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	else
		tmp = t_0 + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -5.5e+161], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -3.3e-7], N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -1.65e-258], t$95$0, If[LessEqual[KbT, 3.4e+137], N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if KbT < -5.5000000000000005e161

   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%

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

   5. Taylor expanded in KbT around inf 81.0%

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

   6. Taylor expanded in mu around inf 81.0%

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

      1. associate-*r/ 81.0%

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

      2. mul-1-neg 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in mu around 0 77.3%

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

       1. mul-1-neg 77.3%

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

       2. unsub-neg 77.3%

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

   11. Simplified 77.3%

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

   if -5.5000000000000005e161 < KbT < -3.3000000000000002e-7

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.0%

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

      1. mul-1-neg 46.0%

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

      2. associate-+r+ 46.0%

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

      3. +-commutative 46.0%

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

      4. associate-+r+ 46.0%

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

      5. unsub-neg 46.0%

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

      6. associate-+r+ 46.0%

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

      7. +-commutative 46.0%

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

      8. associate-+r+ 46.0%

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

      9. associate--r+ 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in Ec around inf 30.4%

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

      1. mul-1-neg 30.4%

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

      2. associate-/l* 30.2%

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

      3. distribute-lft-neg-in 30.2%

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

   10. Simplified 30.2%

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

   11. Taylor expanded in KbT around 0 56.9%

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

   12. Taylor expanded in Vef around inf 43.5%

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

   if -3.3000000000000002e-7 < KbT < -1.65e-258

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 54.1%

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

      1. mul-1-neg 54.1%

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

      2. associate-+r+ 54.1%

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

      3. +-commutative 54.1%

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

      4. associate-+r+ 54.1%

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

      5. unsub-neg 54.1%

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

      6. associate-+r+ 54.1%

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

      7. +-commutative 54.1%

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

      8. associate-+r+ 54.1%

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

      9. associate--r+ 54.1%

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

   7. Simplified 54.1%

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

   8. Taylor expanded in Ec around inf 40.3%

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

      1. mul-1-neg 40.3%

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

      2. associate-/l* 38.8%

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

      3. distribute-lft-neg-in 38.8%

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

   10. Simplified 38.8%

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

   11. Taylor expanded in KbT around 0 59.0%

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

   12. Taylor expanded in EAccept around inf 32.7%

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

   if -1.65e-258 < KbT < 3.39999999999999986e137

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 58.0%

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

      1. mul-1-neg 58.0%

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

      2. associate-+r+ 58.0%

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

      3. +-commutative 58.0%

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

      4. associate-+r+ 58.0%

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

      5. unsub-neg 58.0%

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

      6. associate-+r+ 58.0%

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

      7. +-commutative 58.0%

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

      8. associate-+r+ 58.0%

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

      9. associate--r+ 58.0%

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

   7. Simplified 58.0%

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

   8. Taylor expanded in Ec around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. associate-/l* 37.1%

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

      3. distribute-lft-neg-in 37.1%

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

   10. Simplified 37.1%

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

   11. Taylor expanded in KbT around 0 67.6%

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

   12. Taylor expanded in Ev around inf 43.5%

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

   if 3.39999999999999986e137 < 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%

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

   5. Taylor expanded in KbT around inf 76.2%

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

   6. Taylor expanded in EAccept around inf 64.9%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -5.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq -3.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;KbT \leq -1.65 \cdot 10^{-258}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 3.4 \cdot 10^{+137}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 41.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -6.2 \cdot 10^{+161}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;KbT \leq -2 \cdot 10^{-258}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 4 \cdot 10^{+126}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -6.2e+161)
   (+ (/ NdChar 2.0) (/ NaChar (- 2.0 (/ mu KbT))))
   (if (<= KbT -1.7e-7)
     (/ NaChar (+ 1.0 (exp (/ Vef KbT))))
     (if (<= KbT -2e-258)
       (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))
       (if (<= KbT 4e+126)
         (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
         (+ (* NaChar 0.5) (/ 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 <= -6.2e+161) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	} else if (KbT <= -1.7e-7) {
		tmp = NaChar / (1.0 + exp((Vef / KbT)));
	} else if (KbT <= -2e-258) {
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	} else if (KbT <= 4e+126) {
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	} else {
		tmp = (NaChar * 0.5) + (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 <= (-6.2d+161)) then
        tmp = (ndchar / 2.0d0) + (nachar / (2.0d0 - (mu / kbt)))
    else if (kbt <= (-1.7d-7)) then
        tmp = nachar / (1.0d0 + exp((vef / kbt)))
    else if (kbt <= (-2d-258)) then
        tmp = nachar / (1.0d0 + exp((eaccept / kbt)))
    else if (kbt <= 4d+126) then
        tmp = nachar / (1.0d0 + exp((ev / kbt)))
    else
        tmp = (nachar * 0.5d0) + (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 <= -6.2e+161) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	} else if (KbT <= -1.7e-7) {
		tmp = NaChar / (1.0 + Math.exp((Vef / KbT)));
	} else if (KbT <= -2e-258) {
		tmp = NaChar / (1.0 + Math.exp((EAccept / KbT)));
	} else if (KbT <= 4e+126) {
		tmp = NaChar / (1.0 + Math.exp((Ev / KbT)));
	} else {
		tmp = (NaChar * 0.5) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -6.2e+161:
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)))
	elif KbT <= -1.7e-7:
		tmp = NaChar / (1.0 + math.exp((Vef / KbT)))
	elif KbT <= -2e-258:
		tmp = NaChar / (1.0 + math.exp((EAccept / KbT)))
	elif KbT <= 4e+126:
		tmp = NaChar / (1.0 + math.exp((Ev / KbT)))
	else:
		tmp = (NaChar * 0.5) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -6.2e+161)
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 - Float64(mu / KbT))));
	elseif (KbT <= -1.7e-7)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Vef / KbT))));
	elseif (KbT <= -2e-258)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))));
	elseif (KbT <= 4e+126)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))));
	else
		tmp = Float64(Float64(NaChar * 0.5) + 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 <= -6.2e+161)
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	elseif (KbT <= -1.7e-7)
		tmp = NaChar / (1.0 + exp((Vef / KbT)));
	elseif (KbT <= -2e-258)
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	elseif (KbT <= 4e+126)
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	else
		tmp = (NaChar * 0.5) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -6.2e+161], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -1.7e-7], N[(NaChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -2e-258], N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 4e+126], N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar * 0.5), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if KbT < -6.20000000000000013e161

   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%

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

   5. Taylor expanded in KbT around inf 81.0%

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

   6. Taylor expanded in mu around inf 81.0%

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

      1. associate-*r/ 81.0%

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

      2. mul-1-neg 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in mu around 0 77.3%

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

       1. mul-1-neg 77.3%

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

       2. unsub-neg 77.3%

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

   11. Simplified 77.3%

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

   if -6.20000000000000013e161 < KbT < -1.69999999999999987e-7

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.0%

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

      1. mul-1-neg 46.0%

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

      2. associate-+r+ 46.0%

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

      3. +-commutative 46.0%

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

      4. associate-+r+ 46.0%

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

      5. unsub-neg 46.0%

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

      6. associate-+r+ 46.0%

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

      7. +-commutative 46.0%

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

      8. associate-+r+ 46.0%

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

      9. associate--r+ 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in Ec around inf 30.4%

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

      1. mul-1-neg 30.4%

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

      2. associate-/l* 30.2%

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

      3. distribute-lft-neg-in 30.2%

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

   10. Simplified 30.2%

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

   11. Taylor expanded in KbT around 0 56.9%

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

   12. Taylor expanded in Vef around inf 43.5%

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

   if -1.69999999999999987e-7 < KbT < -1.99999999999999991e-258

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 54.1%

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

      1. mul-1-neg 54.1%

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

      2. associate-+r+ 54.1%

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

      3. +-commutative 54.1%

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

      4. associate-+r+ 54.1%

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

      5. unsub-neg 54.1%

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

      6. associate-+r+ 54.1%

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

      7. +-commutative 54.1%

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

      8. associate-+r+ 54.1%

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

      9. associate--r+ 54.1%

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

   7. Simplified 54.1%

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

   8. Taylor expanded in Ec around inf 40.3%

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

      1. mul-1-neg 40.3%

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

      2. associate-/l* 38.8%

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

      3. distribute-lft-neg-in 38.8%

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

   10. Simplified 38.8%

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

   11. Taylor expanded in KbT around 0 59.0%

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

   12. Taylor expanded in EAccept around inf 32.7%

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

   if -1.99999999999999991e-258 < KbT < 3.9999999999999997e126

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 58.0%

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

      1. mul-1-neg 58.0%

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

      2. associate-+r+ 58.0%

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

      3. +-commutative 58.0%

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

      4. associate-+r+ 58.0%

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

      5. unsub-neg 58.0%

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

      6. associate-+r+ 58.0%

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

      7. +-commutative 58.0%

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

      8. associate-+r+ 58.0%

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

      9. associate--r+ 58.0%

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

   7. Simplified 58.0%

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

   8. Taylor expanded in Ec around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. associate-/l* 37.1%

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

      3. distribute-lft-neg-in 37.1%

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

   10. Simplified 37.1%

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

   11. Taylor expanded in KbT around 0 67.6%

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

   12. Taylor expanded in Ev around inf 43.5%

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

   if 3.9999999999999997e126 < 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%

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

   5. Taylor expanded in KbT around inf 76.2%

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

   6. Taylor expanded in KbT around inf 62.3%

      \[\leadsto \frac{NdChar}{2} + \color{blue}{0.5 \cdot NaChar} \]
   7. Step-by-step derivation

      1. *-commutative 62.3%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   8. Simplified 62.3%

      \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]
3. Recombined 5 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6.2 \cdot 10^{+161}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;KbT \leq -2 \cdot 10^{-258}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 4 \cdot 10^{+126}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -5 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;KbT \leq 1.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -5e+187)
   (+ (/ NaChar (+ 1.0 (exp (/ (- mu) KbT)))) (/ NdChar 2.0))
   (if (<= KbT 1.4e+88)
     (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT))))
     (+
      (/ NaChar (+ 1.0 (exp (/ (+ Vef (- (+ Ev EAccept) mu)) KbT))))
      (/ NdChar 2.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -5e+187) {
		tmp = (NaChar / (1.0 + exp((-mu / KbT)))) + (NdChar / 2.0);
	} else if (KbT <= 1.4e+88) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-5d+187)) then
        tmp = (nachar / (1.0d0 + exp((-mu / kbt)))) + (ndchar / 2.0d0)
    else if (kbt <= 1.4d+88) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (vef + ev)) - mu) / kbt)))
    else
        tmp = (nachar / (1.0d0 + exp(((vef + ((ev + eaccept) - mu)) / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -5e+187) {
		tmp = (NaChar / (1.0 + Math.exp((-mu / KbT)))) + (NdChar / 2.0);
	} else if (KbT <= 1.4e+88) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -5e+187:
		tmp = (NaChar / (1.0 + math.exp((-mu / KbT)))) + (NdChar / 2.0)
	elif KbT <= 1.4e+88:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)))
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -5e+187)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))) + Float64(NdChar / 2.0));
	elseif (KbT <= 1.4e+88)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Float64(Ev + EAccept) - mu)) / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -5e+187)
		tmp = (NaChar / (1.0 + exp((-mu / KbT)))) + (NdChar / 2.0);
	elseif (KbT <= 1.4e+88)
		tmp = NaChar / (1.0 + exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	else
		tmp = (NaChar / (1.0 + exp(((Vef + ((Ev + EAccept) - mu)) / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -5e+187], N[(N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.4e+88], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(N[(Ev + EAccept), $MachinePrecision] - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -5.0000000000000001e187

   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%

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

   5. Taylor expanded in KbT around inf 88.1%

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

   6. Taylor expanded in mu around inf 88.1%

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

      1. associate-*r/ 88.1%

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

      2. mul-1-neg 88.1%

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

   8. Simplified 88.1%

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

   if -5.0000000000000001e187 < KbT < 1.39999999999999994e88

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.9%

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

      1. mul-1-neg 52.9%

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

      2. associate-+r+ 52.9%

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

      3. +-commutative 52.9%

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

      4. associate-+r+ 52.9%

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

      5. unsub-neg 52.9%

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

      6. associate-+r+ 52.9%

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

      7. +-commutative 52.9%

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

      8. associate-+r+ 52.9%

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

      9. associate--r+ 52.9%

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

   7. Simplified 52.9%

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

   8. Taylor expanded in Ec around inf 37.7%

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

      1. mul-1-neg 37.7%

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

      2. associate-/l* 35.1%

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

      3. distribute-lft-neg-in 35.1%

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

   10. Simplified 35.1%

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

   11. Taylor expanded in KbT around 0 62.3%

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

   if 1.39999999999999994e88 < 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%

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

   5. Taylor expanded in KbT around inf 78.7%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -5 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;KbT \leq 1.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{if}\;KbT \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq -4 \cdot 10^{-261}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.1 \cdot 10^{+126}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))))
   (if (<= KbT -6.2e+18)
     (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (* NaChar 0.5))
     (if (<= KbT -4e-261)
       t_0
       (if (<= KbT 1.1e+126)
         (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
         (+ t_0 (/ NdChar 2.0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp((EAccept / KbT)));
	double tmp;
	if (KbT <= -6.2e+18) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else if (KbT <= -4e-261) {
		tmp = t_0;
	} else if (KbT <= 1.1e+126) {
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp((eaccept / kbt)))
    if (kbt <= (-6.2d+18)) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar * 0.5d0)
    else if (kbt <= (-4d-261)) then
        tmp = t_0
    else if (kbt <= 1.1d+126) then
        tmp = nachar / (1.0d0 + exp((ev / kbt)))
    else
        tmp = t_0 + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp((EAccept / KbT)));
	double tmp;
	if (KbT <= -6.2e+18) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else if (KbT <= -4e-261) {
		tmp = t_0;
	} else if (KbT <= 1.1e+126) {
		tmp = NaChar / (1.0 + Math.exp((Ev / KbT)));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp((EAccept / KbT)))
	tmp = 0
	if KbT <= -6.2e+18:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	elif KbT <= -4e-261:
		tmp = t_0
	elif KbT <= 1.1e+126:
		tmp = NaChar / (1.0 + math.exp((Ev / KbT)))
	else:
		tmp = t_0 + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))))
	tmp = 0.0
	if (KbT <= -6.2e+18)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar * 0.5));
	elseif (KbT <= -4e-261)
		tmp = t_0;
	elseif (KbT <= 1.1e+126)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))));
	else
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp((EAccept / KbT)));
	tmp = 0.0;
	if (KbT <= -6.2e+18)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	elseif (KbT <= -4e-261)
		tmp = t_0;
	elseif (KbT <= 1.1e+126)
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	else
		tmp = t_0 + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -6.2e+18], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -4e-261], t$95$0, If[LessEqual[KbT, 1.1e+126], N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -6.2e18

   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%

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

   5. Taylor expanded in EDonor around inf 76.1%

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

   6. Taylor expanded in KbT around inf 60.8%

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

      1. *-commutative 52.0%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   8. Simplified 60.8%

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

   if -6.2e18 < KbT < -3.99999999999999994e-261

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 53.2%

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

      1. mul-1-neg 53.2%

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

      2. associate-+r+ 53.2%

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

      3. +-commutative 53.2%

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

      4. associate-+r+ 53.2%

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

      5. unsub-neg 53.2%

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

      6. associate-+r+ 53.2%

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

      7. +-commutative 53.2%

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

      8. associate-+r+ 53.2%

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

      9. associate--r+ 53.2%

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

   7. Simplified 53.2%

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

   8. Taylor expanded in Ec around inf 40.6%

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

      1. mul-1-neg 40.6%

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

      2. associate-/l* 39.1%

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

      3. distribute-lft-neg-in 39.1%

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

   10. Simplified 39.1%

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

   11. Taylor expanded in KbT around 0 57.7%

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

   12. Taylor expanded in EAccept around inf 30.5%

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

   if -3.99999999999999994e-261 < KbT < 1.09999999999999999e126

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 58.0%

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

      1. mul-1-neg 58.0%

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

      2. associate-+r+ 58.0%

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

      3. +-commutative 58.0%

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

      4. associate-+r+ 58.0%

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

      5. unsub-neg 58.0%

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

      6. associate-+r+ 58.0%

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

      7. +-commutative 58.0%

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

      8. associate-+r+ 58.0%

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

      9. associate--r+ 58.0%

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

   7. Simplified 58.0%

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

   8. Taylor expanded in Ec around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. associate-/l* 37.1%

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

      3. distribute-lft-neg-in 37.1%

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

   10. Simplified 37.1%

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

   11. Taylor expanded in KbT around 0 67.6%

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

   12. Taylor expanded in Ev around inf 43.5%

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

   if 1.09999999999999999e126 < 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%

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

   5. Taylor expanded in KbT around inf 76.2%

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

   6. Taylor expanded in EAccept around inf 64.9%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq -4 \cdot 10^{-261}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 1.1 \cdot 10^{+126}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -3.5 \cdot 10^{+184}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;KbT \leq 2.65 \cdot 10^{+152}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -3.5e+184)
   (+ (/ NaChar (+ 1.0 (exp (/ (- mu) KbT)))) (/ NdChar 2.0))
   (if (<= KbT 2.65e+152)
     (/ NaChar (+ 1.0 (exp (/ (- (+ EAccept (+ Vef Ev)) mu) KbT))))
     (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (/ NdChar 2.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -3.5e+184) {
		tmp = (NaChar / (1.0 + exp((-mu / KbT)))) + (NdChar / 2.0);
	} else if (KbT <= 2.65e+152) {
		tmp = NaChar / (1.0 + exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	} else {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-3.5d+184)) then
        tmp = (nachar / (1.0d0 + exp((-mu / kbt)))) + (ndchar / 2.0d0)
    else if (kbt <= 2.65d+152) then
        tmp = nachar / (1.0d0 + exp((((eaccept + (vef + ev)) - mu) / kbt)))
    else
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -3.5e+184) {
		tmp = (NaChar / (1.0 + Math.exp((-mu / KbT)))) + (NdChar / 2.0);
	} else if (KbT <= 2.65e+152) {
		tmp = NaChar / (1.0 + Math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -3.5e+184:
		tmp = (NaChar / (1.0 + math.exp((-mu / KbT)))) + (NdChar / 2.0)
	elif KbT <= 2.65e+152:
		tmp = NaChar / (1.0 + math.exp((((EAccept + (Vef + Ev)) - mu) / KbT)))
	else:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -3.5e+184)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))) + Float64(NdChar / 2.0));
	elseif (KbT <= 2.65e+152)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(EAccept + Float64(Vef + Ev)) - mu) / KbT))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -3.5e+184)
		tmp = (NaChar / (1.0 + exp((-mu / KbT)))) + (NdChar / 2.0);
	elseif (KbT <= 2.65e+152)
		tmp = NaChar / (1.0 + exp((((EAccept + (Vef + Ev)) - mu) / KbT)));
	else
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -3.5e+184], N[(N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.65e+152], N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(EAccept + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -3.49999999999999978e184

   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%

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

   5. Taylor expanded in KbT around inf 88.1%

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

   6. Taylor expanded in mu around inf 88.1%

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

      1. associate-*r/ 88.1%

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

      2. mul-1-neg 88.1%

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

   8. Simplified 88.1%

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

   if -3.49999999999999978e184 < KbT < 2.64999999999999988e152

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 54.0%

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

      1. mul-1-neg 54.0%

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

      2. associate-+r+ 54.0%

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

      3. +-commutative 54.0%

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

      4. associate-+r+ 54.0%

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

      5. unsub-neg 54.0%

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

      6. associate-+r+ 54.0%

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

      7. +-commutative 54.0%

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

      8. associate-+r+ 54.0%

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

      9. associate--r+ 54.0%

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

   7. Simplified 54.0%

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

   8. Taylor expanded in Ec around inf 38.1%

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

      1. mul-1-neg 38.1%

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

      2. associate-/l* 35.6%

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

      3. distribute-lft-neg-in 35.6%

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

   10. Simplified 35.6%

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

   11. Taylor expanded in KbT around 0 63.1%

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

   if 2.64999999999999988e152 < 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%

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

   5. Taylor expanded in KbT around inf 78.5%

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

   6. Taylor expanded in EAccept around inf 66.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.5 \cdot 10^{+184}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{-mu}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{elif}\;KbT \leq 2.65 \cdot 10^{+152}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Vef + Ev\right)\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 41.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -4 \cdot 10^{+182}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq -5 \cdot 10^{-260}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 9.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -4e+182)
   (+ (/ NdChar 2.0) (/ NaChar (- 2.0 (/ mu KbT))))
   (if (<= KbT -5e-260)
     (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))
     (if (<= KbT 9.5e+137)
       (/ NaChar (+ 1.0 (exp (/ Ev KbT))))
       (+ (* NaChar 0.5) (/ 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 <= -4e+182) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	} else if (KbT <= -5e-260) {
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	} else if (KbT <= 9.5e+137) {
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	} else {
		tmp = (NaChar * 0.5) + (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 <= (-4d+182)) then
        tmp = (ndchar / 2.0d0) + (nachar / (2.0d0 - (mu / kbt)))
    else if (kbt <= (-5d-260)) then
        tmp = nachar / (1.0d0 + exp((eaccept / kbt)))
    else if (kbt <= 9.5d+137) then
        tmp = nachar / (1.0d0 + exp((ev / kbt)))
    else
        tmp = (nachar * 0.5d0) + (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 <= -4e+182) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	} else if (KbT <= -5e-260) {
		tmp = NaChar / (1.0 + Math.exp((EAccept / KbT)));
	} else if (KbT <= 9.5e+137) {
		tmp = NaChar / (1.0 + Math.exp((Ev / KbT)));
	} else {
		tmp = (NaChar * 0.5) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -4e+182:
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)))
	elif KbT <= -5e-260:
		tmp = NaChar / (1.0 + math.exp((EAccept / KbT)))
	elif KbT <= 9.5e+137:
		tmp = NaChar / (1.0 + math.exp((Ev / KbT)))
	else:
		tmp = (NaChar * 0.5) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -4e+182)
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 - Float64(mu / KbT))));
	elseif (KbT <= -5e-260)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))));
	elseif (KbT <= 9.5e+137)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT))));
	else
		tmp = Float64(Float64(NaChar * 0.5) + 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 <= -4e+182)
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	elseif (KbT <= -5e-260)
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	elseif (KbT <= 9.5e+137)
		tmp = NaChar / (1.0 + exp((Ev / KbT)));
	else
		tmp = (NaChar * 0.5) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -4e+182], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, -5e-260], N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 9.5e+137], N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar * 0.5), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -4.0000000000000003e182

   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%

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

   5. Taylor expanded in KbT around inf 88.1%

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

   6. Taylor expanded in mu around inf 88.1%

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

      1. associate-*r/ 88.1%

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

      2. mul-1-neg 88.1%

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

   8. Simplified 88.1%

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

   9. Taylor expanded in mu around 0 83.8%

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

       1. mul-1-neg 83.8%

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

       2. unsub-neg 83.8%

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

   11. Simplified 83.8%

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

   if -4.0000000000000003e182 < KbT < -5.0000000000000003e-260

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. associate-+r+ 50.4%

         \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \color{blue}{\left(\left(EDonor + Vef\right) + mu\right)}}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      3. +-commutative 50.4%

         \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \left(\color{blue}{\left(Vef + EDonor\right)} + mu\right)}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      4. associate-+r+ 50.4%

         \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \color{blue}{\left(Vef + \left(EDonor + mu\right)\right)}}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      5. unsub-neg 50.4%

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      6. associate-+r+ 50.4%

         \[\leadsto \frac{NdChar}{2 - \frac{Ec - \color{blue}{\left(\left(Vef + EDonor\right) + mu\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      7. +-commutative 50.4%

         \[\leadsto \frac{NdChar}{2 - \frac{Ec - \left(\color{blue}{\left(EDonor + Vef\right)} + mu\right)}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      8. associate-+r+ 50.4%

         \[\leadsto \frac{NdChar}{2 - \frac{Ec - \color{blue}{\left(EDonor + \left(Vef + mu\right)\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      9. associate--r+ 50.4%

         \[\leadsto \frac{NdChar}{2 - \frac{\color{blue}{\left(Ec - EDonor\right) - \left(Vef + mu\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   7. Simplified 50.4%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\left(Ec - EDonor\right) - \left(Vef + mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   8. Taylor expanded in Ec around inf 35.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{KbT \cdot NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 35.5%

         \[\leadsto \color{blue}{\left(-\frac{KbT \cdot NdChar}{Ec}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      2. associate-/l* 34.5%

         \[\leadsto \left(-\color{blue}{KbT \cdot \frac{NdChar}{Ec}}\right) + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      3. distribute-lft-neg-in 34.5%

         \[\leadsto \color{blue}{\left(-KbT\right) \cdot \frac{NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   10. Simplified 34.5%

       \[\leadsto \color{blue}{\left(-KbT\right) \cdot \frac{NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   11. Taylor expanded in KbT around 0 57.9%

       \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   12. Taylor expanded in EAccept around inf 30.0%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   if -5.0000000000000003e-260 < KbT < 9.50000000000000031e137

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{-KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf 58.0%

      \[\leadsto \frac{NdChar}{\color{blue}{2 + -1 \cdot \frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 58.0%

         \[\leadsto \frac{NdChar}{2 + \color{blue}{\left(-\frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      2. associate-+r+ 58.0%

         \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \color{blue}{\left(\left(EDonor + Vef\right) + mu\right)}}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      3. +-commutative 58.0%

         \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \left(\color{blue}{\left(Vef + EDonor\right)} + mu\right)}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      4. associate-+r+ 58.0%

         \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \color{blue}{\left(Vef + \left(EDonor + mu\right)\right)}}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      5. unsub-neg 58.0%

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      6. associate-+r+ 58.0%

         \[\leadsto \frac{NdChar}{2 - \frac{Ec - \color{blue}{\left(\left(Vef + EDonor\right) + mu\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      7. +-commutative 58.0%

         \[\leadsto \frac{NdChar}{2 - \frac{Ec - \left(\color{blue}{\left(EDonor + Vef\right)} + mu\right)}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      8. associate-+r+ 58.0%

         \[\leadsto \frac{NdChar}{2 - \frac{Ec - \color{blue}{\left(EDonor + \left(Vef + mu\right)\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      9. associate--r+ 58.0%

         \[\leadsto \frac{NdChar}{2 - \frac{\color{blue}{\left(Ec - EDonor\right) - \left(Vef + mu\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   7. Simplified 58.0%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\left(Ec - EDonor\right) - \left(Vef + mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   8. Taylor expanded in Ec around inf 40.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{KbT \cdot NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 40.9%

         \[\leadsto \color{blue}{\left(-\frac{KbT \cdot NdChar}{Ec}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      2. associate-/l* 37.1%

         \[\leadsto \left(-\color{blue}{KbT \cdot \frac{NdChar}{Ec}}\right) + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      3. distribute-lft-neg-in 37.1%

         \[\leadsto \color{blue}{\left(-KbT\right) \cdot \frac{NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   10. Simplified 37.1%

       \[\leadsto \color{blue}{\left(-KbT\right) \cdot \frac{NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   11. Taylor expanded in KbT around 0 67.6%

       \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   12. Taylor expanded in Ev around inf 43.5%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{Ev}{KbT}}}} \]

   if 9.50000000000000031e137 < 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%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{-KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf 76.2%

      \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   6. Taylor expanded in KbT around inf 62.3%

      \[\leadsto \frac{NdChar}{2} + \color{blue}{0.5 \cdot NaChar} \]
   7. Step-by-step derivation

      1. *-commutative 62.3%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   8. Simplified 62.3%

      \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4 \cdot 10^{+182}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq -5 \cdot 10^{-260}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;KbT \leq 9.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 41.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -2.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 3.7 \cdot 10^{+70}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -2.2e+189)
   (+ (/ NdChar 2.0) (/ NaChar (- 2.0 (/ mu KbT))))
   (if (<= KbT 3.7e+70)
     (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))
     (+ (* NaChar 0.5) (/ 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 <= -2.2e+189) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	} else if (KbT <= 3.7e+70) {
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	} else {
		tmp = (NaChar * 0.5) + (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 <= (-2.2d+189)) then
        tmp = (ndchar / 2.0d0) + (nachar / (2.0d0 - (mu / kbt)))
    else if (kbt <= 3.7d+70) then
        tmp = nachar / (1.0d0 + exp((eaccept / kbt)))
    else
        tmp = (nachar * 0.5d0) + (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 <= -2.2e+189) {
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	} else if (KbT <= 3.7e+70) {
		tmp = NaChar / (1.0 + Math.exp((EAccept / KbT)));
	} else {
		tmp = (NaChar * 0.5) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -2.2e+189:
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)))
	elif KbT <= 3.7e+70:
		tmp = NaChar / (1.0 + math.exp((EAccept / KbT)))
	else:
		tmp = (NaChar * 0.5) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -2.2e+189)
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(2.0 - Float64(mu / KbT))));
	elseif (KbT <= 3.7e+70)
		tmp = Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT))));
	else
		tmp = Float64(Float64(NaChar * 0.5) + 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 <= -2.2e+189)
		tmp = (NdChar / 2.0) + (NaChar / (2.0 - (mu / KbT)));
	elseif (KbT <= 3.7e+70)
		tmp = NaChar / (1.0 + exp((EAccept / KbT)));
	else
		tmp = (NaChar * 0.5) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -2.2e+189], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(2.0 - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 3.7e+70], N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar * 0.5), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -2.20000000000000005e189

   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%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{-KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf 88.1%

      \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   6. Taylor expanded in mu around inf 88.1%

      \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{1 + e^{\color{blue}{-1 \cdot \frac{mu}{KbT}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 88.1%

         \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{1 + e^{\color{blue}{\frac{-1 \cdot mu}{KbT}}}} \]

      2. mul-1-neg 88.1%

         \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{1 + e^{\frac{\color{blue}{-mu}}{KbT}}} \]

   8. Simplified 88.1%

      \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{1 + e^{\color{blue}{\frac{-mu}{KbT}}}} \]

   9. Taylor expanded in mu around 0 83.8%

      \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{\color{blue}{2 + -1 \cdot \frac{mu}{KbT}}} \]
   10. Step-by-step derivation

       1. mul-1-neg 83.8%

          \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{2 + \color{blue}{\left(-\frac{mu}{KbT}\right)}} \]

       2. unsub-neg 83.8%

          \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{\color{blue}{2 - \frac{mu}{KbT}}} \]

   11. Simplified 83.8%

       \[\leadsto \frac{NdChar}{2} + \frac{NaChar}{\color{blue}{2 - \frac{mu}{KbT}}} \]

   if -2.20000000000000005e189 < KbT < 3.69999999999999989e70

   1. Initial program 100.0%

      \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{-KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf 52.7%

      \[\leadsto \frac{NdChar}{\color{blue}{2 + -1 \cdot \frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 52.7%

         \[\leadsto \frac{NdChar}{2 + \color{blue}{\left(-\frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      2. associate-+r+ 52.7%

         \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \color{blue}{\left(\left(EDonor + Vef\right) + mu\right)}}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      3. +-commutative 52.7%

         \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \left(\color{blue}{\left(Vef + EDonor\right)} + mu\right)}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      4. associate-+r+ 52.7%

         \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \color{blue}{\left(Vef + \left(EDonor + mu\right)\right)}}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      5. unsub-neg 52.7%

         \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      6. associate-+r+ 52.7%

         \[\leadsto \frac{NdChar}{2 - \frac{Ec - \color{blue}{\left(\left(Vef + EDonor\right) + mu\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      7. +-commutative 52.7%

         \[\leadsto \frac{NdChar}{2 - \frac{Ec - \left(\color{blue}{\left(EDonor + Vef\right)} + mu\right)}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      8. associate-+r+ 52.7%

         \[\leadsto \frac{NdChar}{2 - \frac{Ec - \color{blue}{\left(EDonor + \left(Vef + mu\right)\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      9. associate--r+ 52.7%

         \[\leadsto \frac{NdChar}{2 - \frac{\color{blue}{\left(Ec - EDonor\right) - \left(Vef + mu\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   7. Simplified 52.7%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\left(Ec - EDonor\right) - \left(Vef + mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   8. Taylor expanded in Ec around inf 37.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{KbT \cdot NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 37.9%

         \[\leadsto \color{blue}{\left(-\frac{KbT \cdot NdChar}{Ec}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      2. associate-/l* 35.2%

         \[\leadsto \left(-\color{blue}{KbT \cdot \frac{NdChar}{Ec}}\right) + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

      3. distribute-lft-neg-in 35.2%

         \[\leadsto \color{blue}{\left(-KbT\right) \cdot \frac{NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   10. Simplified 35.2%

       \[\leadsto \color{blue}{\left(-KbT\right) \cdot \frac{NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   11. Taylor expanded in KbT around 0 61.9%

       \[\leadsto \color{blue}{\frac{NaChar}{1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}}} \]

   12. Taylor expanded in EAccept around inf 33.3%

       \[\leadsto \frac{NaChar}{1 + e^{\color{blue}{\frac{EAccept}{KbT}}}} \]

   if 3.69999999999999989e70 < 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%

      \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{-KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in KbT around inf 76.1%

      \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   6. Taylor expanded in KbT around inf 55.3%

      \[\leadsto \frac{NdChar}{2} + \color{blue}{0.5 \cdot NaChar} \]
   7. Step-by-step derivation

      1. *-commutative 55.3%

         \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

   8. Simplified 55.3%

      \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{2 - \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 3.7 \cdot 10^{+70}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NaChar \cdot 0.5 + \frac{NdChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 27.6% accurate, 32.7× speedup

Math

\[\begin{array}{l} \\ NaChar \cdot 0.5 + \frac{NdChar}{2} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+ (* NaChar 0.5) (/ 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 * 0.5) + (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 * 0.5d0) + (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 * 0.5) + (NdChar / 2.0);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NaChar * 0.5) + (NdChar / 2.0)

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NaChar * 0.5) + Float64(NdChar / 2.0))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NaChar * 0.5) + (NdChar / 2.0);
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NaChar * 0.5), $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%

   \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{-KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}}} \]
4. Add Preprocessing

5. Taylor expanded in KbT around inf 49.8%

   \[\leadsto \frac{NdChar}{\color{blue}{2}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

6. Taylor expanded in KbT around inf 29.1%

   \[\leadsto \frac{NdChar}{2} + \color{blue}{0.5 \cdot NaChar} \]
7. Step-by-step derivation

   1. *-commutative 29.1%

      \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

8. Simplified 29.1%

   \[\leadsto \frac{NdChar}{2} + \color{blue}{NaChar \cdot 0.5} \]

9. Final simplification 29.1%

   \[\leadsto NaChar \cdot 0.5 + \frac{NdChar}{2} \]
10. Add Preprocessing

Alternative 25: 18.1% accurate, 76.3× speedup

Math

\[\begin{array}{l} \\ NaChar \cdot 0.5 \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (* NaChar 0.5))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return NaChar * 0.5;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = nachar * 0.5d0
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return NaChar * 0.5;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return NaChar * 0.5

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(NaChar * 0.5)
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = NaChar * 0.5;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(NaChar * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{NdChar}{1 + e^{\frac{-\left(\left(\left(Ec - Vef\right) - EDonor\right) - mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(\left(Ev + Vef\right) + EAccept\right) + \left(-mu\right)}{KbT}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{NdChar}{1 + e^{\frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{-KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}}} \]
4. Add Preprocessing

5. Taylor expanded in KbT around inf 60.2%

   \[\leadsto \frac{NdChar}{\color{blue}{2 + -1 \cdot \frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]
6. Step-by-step derivation

   1. mul-1-neg 60.2%

      \[\leadsto \frac{NdChar}{2 + \color{blue}{\left(-\frac{Ec - \left(EDonor + \left(Vef + mu\right)\right)}{KbT}\right)}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   2. associate-+r+ 60.2%

      \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \color{blue}{\left(\left(EDonor + Vef\right) + mu\right)}}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   3. +-commutative 60.2%

      \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \left(\color{blue}{\left(Vef + EDonor\right)} + mu\right)}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   4. associate-+r+ 60.2%

      \[\leadsto \frac{NdChar}{2 + \left(-\frac{Ec - \color{blue}{\left(Vef + \left(EDonor + mu\right)\right)}}{KbT}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   5. unsub-neg 60.2%

      \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{Ec - \left(Vef + \left(EDonor + mu\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   6. associate-+r+ 60.2%

      \[\leadsto \frac{NdChar}{2 - \frac{Ec - \color{blue}{\left(\left(Vef + EDonor\right) + mu\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   7. +-commutative 60.2%

      \[\leadsto \frac{NdChar}{2 - \frac{Ec - \left(\color{blue}{\left(EDonor + Vef\right)} + mu\right)}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   8. associate-+r+ 60.2%

      \[\leadsto \frac{NdChar}{2 - \frac{Ec - \color{blue}{\left(EDonor + \left(Vef + mu\right)\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   9. associate--r+ 60.2%

      \[\leadsto \frac{NdChar}{2 - \frac{\color{blue}{\left(Ec - EDonor\right) - \left(Vef + mu\right)}}{KbT}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

7. Simplified 60.2%

   \[\leadsto \frac{NdChar}{\color{blue}{2 - \frac{\left(Ec - EDonor\right) - \left(Vef + mu\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

8. Taylor expanded in Ec around inf 33.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{KbT \cdot NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]
9. Step-by-step derivation

   1. mul-1-neg 33.2%

      \[\leadsto \color{blue}{\left(-\frac{KbT \cdot NdChar}{Ec}\right)} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   2. associate-/l* 32.6%

      \[\leadsto \left(-\color{blue}{KbT \cdot \frac{NdChar}{Ec}}\right) + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

   3. distribute-lft-neg-in 32.6%

      \[\leadsto \color{blue}{\left(-KbT\right) \cdot \frac{NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

10. Simplified 32.6%

    \[\leadsto \color{blue}{\left(-KbT\right) \cdot \frac{NdChar}{Ec}} + \frac{NaChar}{1 + e^{\frac{Vef + \left(\left(Ev + EAccept\right) - mu\right)}{KbT}}} \]

11. Taylor expanded in KbT around inf 11.7%

    \[\leadsto \color{blue}{-1 \cdot \frac{KbT \cdot NdChar}{Ec} + 0.5 \cdot NaChar} \]
12. Step-by-step derivation

    1. +-commutative 11.7%

       \[\leadsto \color{blue}{0.5 \cdot NaChar + -1 \cdot \frac{KbT \cdot NdChar}{Ec}} \]

    2. mul-1-neg 11.7%

       \[\leadsto 0.5 \cdot NaChar + \color{blue}{\left(-\frac{KbT \cdot NdChar}{Ec}\right)} \]

    3. unsub-neg 11.7%

       \[\leadsto \color{blue}{0.5 \cdot NaChar - \frac{KbT \cdot NdChar}{Ec}} \]

    4. associate-*r/ 12.1%

       \[\leadsto 0.5 \cdot NaChar - \color{blue}{KbT \cdot \frac{NdChar}{Ec}} \]

13. Simplified 12.1%

    \[\leadsto \color{blue}{0.5 \cdot NaChar - KbT \cdot \frac{NdChar}{Ec}} \]

14. Taylor expanded in NaChar around inf 18.8%

    \[\leadsto \color{blue}{0.5 \cdot NaChar} \]

15. Final simplification 18.8%

    \[\leadsto NaChar \cdot 0.5 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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))))))