Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 29.7s

Alternatives: 23

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ t_1 := t_0 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{if}\;EAccept \leq 3.2 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;EAccept \leq 1.62 \cdot 10^{-189}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;EAccept \leq 3.3 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;EAccept \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT)))))
        (t_1 (+ t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))))
   (if (<= EAccept 3.2e-292)
     t_1
     (if (<= EAccept 1.62e-189)
       (+ t_0 (/ NaChar (+ 1.0 (exp (/ (- mu) KbT)))))
       (if (<= EAccept 3.3e-75)
         t_1
         (if (<= EAccept 2.6e+77)
           (+
            (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
            (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
           (+ t_0 (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	double tmp;
	if (EAccept <= 3.2e-292) {
		tmp = t_1;
	} else if (EAccept <= 1.62e-189) {
		tmp = t_0 + (NaChar / (1.0 + exp((-mu / KbT))));
	} else if (EAccept <= 3.3e-75) {
		tmp = t_1;
	} else if (EAccept <= 2.6e+77) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))
    t_1 = t_0 + (nachar / (1.0d0 + exp((ev / kbt))))
    if (eaccept <= 3.2d-292) then
        tmp = t_1
    else if (eaccept <= 1.62d-189) then
        tmp = t_0 + (nachar / (1.0d0 + exp((-mu / kbt))))
    else if (eaccept <= 3.3d-75) then
        tmp = t_1
    else if (eaccept <= 2.6d+77) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + exp((mu / kbt))))
    else
        tmp = t_0 + (nachar / (1.0d0 + exp((eaccept / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double t_1 = t_0 + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	double tmp;
	if (EAccept <= 3.2e-292) {
		tmp = t_1;
	} else if (EAccept <= 1.62e-189) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((-mu / KbT))));
	} else if (EAccept <= 3.3e-75) {
		tmp = t_1;
	} else if (EAccept <= 2.6e+77) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))
	t_1 = t_0 + (NaChar / (1.0 + math.exp((Ev / KbT))))
	tmp = 0
	if EAccept <= 3.2e-292:
		tmp = t_1
	elif EAccept <= 1.62e-189:
		tmp = t_0 + (NaChar / (1.0 + math.exp((-mu / KbT))))
	elif EAccept <= 3.3e-75:
		tmp = t_1
	elif EAccept <= 2.6e+77:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + math.exp((mu / KbT))))
	else:
		tmp = t_0 + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	t_1 = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))))
	tmp = 0.0
	if (EAccept <= 3.2e-292)
		tmp = t_1;
	elseif (EAccept <= 1.62e-189)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(-mu) / KbT)))));
	elseif (EAccept <= 3.3e-75)
		tmp = t_1;
	elseif (EAccept <= 2.6e+77)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	t_1 = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	tmp = 0.0;
	if (EAccept <= 3.2e-292)
		tmp = t_1;
	elseif (EAccept <= 1.62e-189)
		tmp = t_0 + (NaChar / (1.0 + exp((-mu / KbT))));
	elseif (EAccept <= 3.3e-75)
		tmp = t_1;
	elseif (EAccept <= 2.6e+77)
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	else
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EAccept, 3.2e-292], t$95$1, If[LessEqual[EAccept, 1.62e-189], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[((-mu) / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EAccept, 3.3e-75], t$95$1, If[LessEqual[EAccept, 2.6e+77], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if EAccept < 3.2000000000000002e-292 or 1.62e-189 < EAccept < 3.3e-75

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in Ev around inf 70.5%

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

   if 3.2000000000000002e-292 < EAccept < 1.62e-189

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in mu around inf 91.1%

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

      1. associate-*r/ 91.1%

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

      2. mul-1-neg 91.1%

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

   6. Simplified 91.1%

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

   if 3.3e-75 < EAccept < 2.6000000000000002e77

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in mu around inf 72.8%

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

   if 2.6000000000000002e77 < EAccept

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EAccept around inf 91.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq 3.2 \cdot 10^{-292}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;EAccept \leq 1.62 \cdot 10^{-189}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{-mu}{KbT}}}\\ \mathbf{elif}\;EAccept \leq 3.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;EAccept \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \]

Alternative 3: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + Ev}{KbT}}}\\ \mathbf{if}\;Vef \leq -2.75 \cdot 10^{+205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Vef \leq -4.2 \cdot 10^{+161}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{elif}\;Vef \leq 8.5 \cdot 10^{+205}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
          (/ NaChar (+ 1.0 (exp (/ (+ Vef Ev) KbT)))))))
   (if (<= Vef -2.75e+205)
     t_1
     (if (<= Vef -4.2e+161)
       (+ t_0 (/ NdChar (- 1.0 (/ Ec KbT))))
       (if (<= Vef 8.5e+205)
         (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
         t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp(((Vef + Ev) / KbT))));
	double tmp;
	if (Vef <= -2.75e+205) {
		tmp = t_1;
	} else if (Vef <= -4.2e+161) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	} else if (Vef <= 8.5e+205) {
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / (1.0d0 + exp(((vef + ev) / kbt))))
    if (vef <= (-2.75d+205)) then
        tmp = t_1
    else if (vef <= (-4.2d+161)) then
        tmp = t_0 + (ndchar / (1.0d0 - (ec / kbt)))
    else if (vef <= 8.5d+205) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / (1.0 + Math.exp(((Vef + Ev) / KbT))));
	double tmp;
	if (Vef <= -2.75e+205) {
		tmp = t_1;
	} else if (Vef <= -4.2e+161) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	} else if (Vef <= 8.5e+205) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / (1.0 + math.exp(((Vef + Ev) / KbT))))
	tmp = 0
	if Vef <= -2.75e+205:
		tmp = t_1
	elif Vef <= -4.2e+161:
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)))
	elif Vef <= 8.5e+205:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Ev) / KbT)))))
	tmp = 0.0
	if (Vef <= -2.75e+205)
		tmp = t_1;
	elseif (Vef <= -4.2e+161)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 - Float64(Ec / KbT))));
	elseif (Vef <= 8.5e+205)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / (1.0 + exp(((Vef + Ev) / KbT))));
	tmp = 0.0;
	if (Vef <= -2.75e+205)
		tmp = t_1;
	elseif (Vef <= -4.2e+161)
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	elseif (Vef <= 8.5e+205)
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -2.75e+205], t$95$1, If[LessEqual[Vef, -4.2e+161], N[(t$95$0 + N[(NdChar / N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 8.5e+205], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -2.75000000000000002e205 or 8.49999999999999997e205 < 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%

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

   4. Taylor expanded in Vef around inf 97.5%

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

   5. Taylor expanded in Ev around inf 97.5%

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

   if -2.75000000000000002e205 < Vef < -4.2e161

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 36.8%

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

      1. associate-+r+ 36.8%

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

      2. associate-+r+ 36.8%

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

   6. Simplified 36.8%

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

      1. flip-- 3.2%

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

      2. +-commutative 3.2%

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

      3. associate-+r+ 3.2%

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

      4. +-commutative 3.2%

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

      5. associate-+r+ 3.2%

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

      6. +-commutative 3.2%

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

      7. associate-+r+ 3.2%

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

   8. Applied egg-rr 3.2%

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

      1. +-commutative 3.2%

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

      2. +-commutative 3.2%

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

      3. +-commutative 3.2%

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

      4. +-commutative 3.2%

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

      5. +-commutative 3.2%

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

      6. +-commutative 3.2%

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

      7. +-commutative 3.2%

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

   10. Simplified 3.2%

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

   11. Taylor expanded in Ec around inf 83.0%

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

       1. neg-mul-1 83.0%

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

       2. distribute-neg-frac 83.0%

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

   13. Simplified 83.0%

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

   if -4.2e161 < Vef < 8.49999999999999997e205

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EDonor around inf 74.8%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -2.75 \cdot 10^{+205}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + Ev}{KbT}}}\\ \mathbf{elif}\;Vef \leq -4.2 \cdot 10^{+161}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{elif}\;Vef \leq 8.5 \cdot 10^{+205}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Vef + Ev}{KbT}}}\\ \end{array} \]

Alternative 4: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := t_0 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{if}\;Vef \leq -1.3 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;Vef \leq 3 \cdot 10^{-189}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;Vef \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1 (+ t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))))
   (if (<= Vef -1.3e+153)
     t_1
     (if (<= Vef 3e-189)
       (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))))
       (if (<= Vef 5.5e+153)
         (+ t_0 (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
         t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	double tmp;
	if (Vef <= -1.3e+153) {
		tmp = t_1;
	} else if (Vef <= 3e-189) {
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	} else if (Vef <= 5.5e+153) {
		tmp = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = t_0 + (ndchar / (1.0d0 + exp((vef / kbt))))
    if (vef <= (-1.3d+153)) then
        tmp = t_1
    else if (vef <= 3d-189) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    else if (vef <= 5.5d+153) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((mu / kbt))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = t_0 + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	double tmp;
	if (Vef <= -1.3e+153) {
		tmp = t_1;
	} else if (Vef <= 3e-189) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	} else if (Vef <= 5.5e+153) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = t_0 + (NdChar / (1.0 + math.exp((Vef / KbT))))
	tmp = 0
	if Vef <= -1.3e+153:
		tmp = t_1
	elif Vef <= 3e-189:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	elif Vef <= 5.5e+153:
		tmp = t_0 + (NdChar / (1.0 + math.exp((mu / KbT))))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))))
	tmp = 0.0
	if (Vef <= -1.3e+153)
		tmp = t_1;
	elseif (Vef <= 3e-189)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	elseif (Vef <= 5.5e+153)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	tmp = 0.0;
	if (Vef <= -1.3e+153)
		tmp = t_1;
	elseif (Vef <= 3e-189)
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	elseif (Vef <= 5.5e+153)
		tmp = t_0 + (NdChar / (1.0 + exp((mu / KbT))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -1.3e+153], t$95$1, If[LessEqual[Vef, 3e-189], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 5.5e+153], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -1.2999999999999999e153 or 5.5000000000000003e153 < 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%

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

   4. Taylor expanded in Vef around inf 95.3%

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

   if -1.2999999999999999e153 < Vef < 3e-189

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EDonor around inf 79.7%

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

   if 3e-189 < Vef < 5.5000000000000003e153

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in mu around inf 76.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq 3 \cdot 10^{-189}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \mathbf{elif}\;Vef \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]

Alternative 5: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;Vef \leq -6.6 \cdot 10^{+147} \lor \neg \left(Vef \leq 6.6 \cdot 10^{+47}\right):\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (or (<= Vef -6.6e+147) (not (<= Vef 6.6e+47)))
     (+ t_0 (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))
     (+ t_0 (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if ((Vef <= -6.6e+147) || !(Vef <= 6.6e+47)) {
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	} else {
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if ((vef <= (-6.6d+147)) .or. (.not. (vef <= 6.6d+47))) then
        tmp = t_0 + (ndchar / (1.0d0 + exp((vef / kbt))))
    else
        tmp = t_0 + (ndchar / (1.0d0 + exp((edonor / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if ((Vef <= -6.6e+147) || !(Vef <= 6.6e+47)) {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	} else {
		tmp = t_0 + (NdChar / (1.0 + Math.exp((EDonor / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if (Vef <= -6.6e+147) or not (Vef <= 6.6e+47):
		tmp = t_0 + (NdChar / (1.0 + math.exp((Vef / KbT))))
	else:
		tmp = t_0 + (NdChar / (1.0 + math.exp((EDonor / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if ((Vef <= -6.6e+147) || !(Vef <= 6.6e+47))
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if ((Vef <= -6.6e+147) || ~((Vef <= 6.6e+47)))
		tmp = t_0 + (NdChar / (1.0 + exp((Vef / KbT))));
	else
		tmp = t_0 + (NdChar / (1.0 + exp((EDonor / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[Vef, -6.6e+147], N[Not[LessEqual[Vef, 6.6e+47]], $MachinePrecision]], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -6.60000000000000049e147 or 6.5999999999999998e47 < 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%

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

   4. Taylor expanded in Vef around inf 87.4%

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

   if -6.60000000000000049e147 < Vef < 6.5999999999999998e47

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in EDonor around inf 75.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -6.6 \cdot 10^{+147} \lor \neg \left(Vef \leq 6.6 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}}\\ \end{array} \]

Alternative 6: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{if}\;EAccept \leq 3.1 \cdot 10^{-75}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;EAccept \leq 2.4 \cdot 10^{+75}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))))
   (if (<= EAccept 3.1e-75)
     (+ t_0 (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
     (if (<= EAccept 2.4e+75)
       (+
        (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
        (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
       (+ t_0 (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double tmp;
	if (EAccept <= 3.1e-75) {
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	} else if (EAccept <= 2.4e+75) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))
    if (eaccept <= 3.1d-75) then
        tmp = t_0 + (nachar / (1.0d0 + exp((ev / kbt))))
    else if (eaccept <= 2.4d+75) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + exp((mu / kbt))))
    else
        tmp = t_0 + (nachar / (1.0d0 + exp((eaccept / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double tmp;
	if (EAccept <= 3.1e-75) {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else if (EAccept <= 2.4e+75) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else {
		tmp = t_0 + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))
	tmp = 0
	if EAccept <= 3.1e-75:
		tmp = t_0 + (NaChar / (1.0 + math.exp((Ev / KbT))))
	elif EAccept <= 2.4e+75:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + math.exp((mu / KbT))))
	else:
		tmp = t_0 + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	tmp = 0.0
	if (EAccept <= 3.1e-75)
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	elseif (EAccept <= 2.4e+75)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	else
		tmp = Float64(t_0 + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	tmp = 0.0;
	if (EAccept <= 3.1e-75)
		tmp = t_0 + (NaChar / (1.0 + exp((Ev / KbT))));
	elseif (EAccept <= 2.4e+75)
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	else
		tmp = t_0 + (NaChar / (1.0 + exp((EAccept / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EAccept, 3.1e-75], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EAccept, 2.4e+75], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if EAccept < 3.10000000000000007e-75

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in Ev around inf 70.7%

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

   if 3.10000000000000007e-75 < EAccept < 2.4e75

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in mu around inf 72.8%

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

   if 2.4e75 < EAccept

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EAccept around inf 91.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq 3.1 \cdot 10^{-75}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{elif}\;EAccept \leq 2.4 \cdot 10^{+75}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \]

Alternative 7: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EAccept \leq 9.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EAccept 9.2e+75)
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
    (/ NdChar (+ 1.0 (exp (/ mu KbT)))))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
    (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 9.2e+75) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	} else {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (eaccept <= 9.2d+75) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + exp((mu / kbt))))
    else
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 9.2e+75) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + Math.exp((mu / KbT))));
	} else {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EAccept <= 9.2e+75:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + math.exp((mu / KbT))))
	else:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EAccept <= 9.2e+75)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (EAccept <= 9.2e+75)
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + exp((mu / KbT))));
	else
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EAccept, 9.2e+75], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if EAccept < 9.1999999999999994e75

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in mu around inf 70.8%

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

   if 9.1999999999999994e75 < EAccept

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EAccept around inf 91.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq 9.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \end{array} \]

Alternative 8: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EAccept \leq 3.7 \cdot 10^{+125}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EAccept 3.7e+125)
   (+
    (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
    (/ NaChar (+ 1.0 (exp (/ (- (+ Vef Ev) mu) KbT)))))
   (+
    (/ NaChar (+ 1.0 (exp (/ EAccept KbT))))
    (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 3.7e+125) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT))));
	} else {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / (1.0 + exp((Vef / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (eaccept <= 3.7d+125) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar / (1.0d0 + exp((((vef + ev) - mu) / kbt))))
    else
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar / (1.0d0 + exp((vef / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (EAccept <= 3.7e+125) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / (1.0 + Math.exp((((Vef + Ev) - mu) / KbT))));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if EAccept <= 3.7e+125:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / (1.0 + math.exp((((Vef + Ev) - mu) / KbT))))
	else:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar / (1.0 + math.exp((Vef / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (EAccept <= 3.7e+125)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (EAccept <= 3.7e+125)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT))));
	else
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / (1.0 + exp((Vef / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[EAccept, 3.7e+125], 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 + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if EAccept < 3.6999999999999998e125

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EDonor around inf 72.8%

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

   5. Taylor expanded in EAccept around 0 68.4%

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

      1. +-commutative 68.4%

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

   7. Simplified 68.4%

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

   if 3.6999999999999998e125 < EAccept

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in Vef around inf 72.6%

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

   5. Taylor expanded in EAccept around inf 70.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq 3.7 \cdot 10^{+125}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]

Alternative 9: 54.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ t_2 := t_0 + \frac{NdChar}{\left(2 + \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{if}\;EDonor \leq -2.4 \cdot 10^{+98}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;EDonor \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;EDonor \leq 6.9 \cdot 10^{-217}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;EDonor \leq 3.9 \cdot 10^{-119}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\left(\frac{EDonor}{KbT} + \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{Ec}{KbT}}} + \frac{NaChar}{1 + \frac{EAccept}{KbT}}\\ \mathbf{elif}\;EDonor \leq 4.2 \cdot 10^{+44}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
          (/ NaChar (+ (/ EAccept KbT) 2.0))))
        (t_2
         (+
          t_0
          (/ NdChar (- (+ 2.0 (+ (/ Vef KbT) (/ EDonor KbT))) (/ Ec KbT))))))
   (if (<= EDonor -2.4e+98)
     (+ t_0 (/ NdChar (+ 1.0 (/ mu KbT))))
     (if (<= EDonor -3.2e-102)
       t_1
       (if (<= EDonor 6.9e-217)
         t_2
         (if (<= EDonor 3.9e-119)
           (+
            (/
             NdChar
             (+
              1.0
              (exp
               (- (+ (/ EDonor KbT) (+ (/ mu KbT) (/ Vef KbT))) (/ Ec KbT)))))
            (/ NaChar (+ 1.0 (/ EAccept KbT))))
           (if (<= EDonor 4.2e+44) 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 = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	double t_2 = t_0 + (NdChar / ((2.0 + ((Vef / KbT) + (EDonor / KbT))) - (Ec / KbT)));
	double tmp;
	if (EDonor <= -2.4e+98) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if (EDonor <= -3.2e-102) {
		tmp = t_1;
	} else if (EDonor <= 6.9e-217) {
		tmp = t_2;
	} else if (EDonor <= 3.9e-119) {
		tmp = (NdChar / (1.0 + exp((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) - (Ec / KbT))))) + (NaChar / (1.0 + (EAccept / KbT)));
	} else if (EDonor <= 4.2e+44) {
		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 = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / ((eaccept / kbt) + 2.0d0))
    t_2 = t_0 + (ndchar / ((2.0d0 + ((vef / kbt) + (edonor / kbt))) - (ec / kbt)))
    if (edonor <= (-2.4d+98)) then
        tmp = t_0 + (ndchar / (1.0d0 + (mu / kbt)))
    else if (edonor <= (-3.2d-102)) then
        tmp = t_1
    else if (edonor <= 6.9d-217) then
        tmp = t_2
    else if (edonor <= 3.9d-119) then
        tmp = (ndchar / (1.0d0 + exp((((edonor / kbt) + ((mu / kbt) + (vef / kbt))) - (ec / kbt))))) + (nachar / (1.0d0 + (eaccept / kbt)))
    else if (edonor <= 4.2d+44) 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 = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	double t_2 = t_0 + (NdChar / ((2.0 + ((Vef / KbT) + (EDonor / KbT))) - (Ec / KbT)));
	double tmp;
	if (EDonor <= -2.4e+98) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if (EDonor <= -3.2e-102) {
		tmp = t_1;
	} else if (EDonor <= 6.9e-217) {
		tmp = t_2;
	} else if (EDonor <= 3.9e-119) {
		tmp = (NdChar / (1.0 + Math.exp((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) - (Ec / KbT))))) + (NaChar / (1.0 + (EAccept / KbT)));
	} else if (EDonor <= 4.2e+44) {
		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 = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0))
	t_2 = t_0 + (NdChar / ((2.0 + ((Vef / KbT) + (EDonor / KbT))) - (Ec / KbT)))
	tmp = 0
	if EDonor <= -2.4e+98:
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)))
	elif EDonor <= -3.2e-102:
		tmp = t_1
	elif EDonor <= 6.9e-217:
		tmp = t_2
	elif EDonor <= 3.9e-119:
		tmp = (NdChar / (1.0 + math.exp((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) - (Ec / KbT))))) + (NaChar / (1.0 + (EAccept / KbT)))
	elif EDonor <= 4.2e+44:
		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(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)))
	t_2 = Float64(t_0 + Float64(NdChar / Float64(Float64(2.0 + Float64(Float64(Vef / KbT) + Float64(EDonor / KbT))) - Float64(Ec / KbT))))
	tmp = 0.0
	if (EDonor <= -2.4e+98)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(mu / KbT))));
	elseif (EDonor <= -3.2e-102)
		tmp = t_1;
	elseif (EDonor <= 6.9e-217)
		tmp = t_2;
	elseif (EDonor <= 3.9e-119)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(Float64(EDonor / KbT) + Float64(Float64(mu / KbT) + Float64(Vef / KbT))) - Float64(Ec / KbT))))) + Float64(NaChar / Float64(1.0 + Float64(EAccept / KbT))));
	elseif (EDonor <= 4.2e+44)
		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 = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	t_2 = t_0 + (NdChar / ((2.0 + ((Vef / KbT) + (EDonor / KbT))) - (Ec / KbT)));
	tmp = 0.0;
	if (EDonor <= -2.4e+98)
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	elseif (EDonor <= -3.2e-102)
		tmp = t_1;
	elseif (EDonor <= 6.9e-217)
		tmp = t_2;
	elseif (EDonor <= 3.9e-119)
		tmp = (NdChar / (1.0 + exp((((EDonor / KbT) + ((mu / KbT) + (Vef / KbT))) - (Ec / KbT))))) + (NaChar / (1.0 + (EAccept / KbT)));
	elseif (EDonor <= 4.2e+44)
		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[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(NdChar / N[(N[(2.0 + N[(N[(Vef / KbT), $MachinePrecision] + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EDonor, -2.4e+98], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, -3.2e-102], t$95$1, If[LessEqual[EDonor, 6.9e-217], t$95$2, If[LessEqual[EDonor, 3.9e-119], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(N[(EDonor / KbT), $MachinePrecision] + N[(N[(mu / KbT), $MachinePrecision] + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[(EAccept / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[EDonor, 4.2e+44], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if EDonor < -2.3999999999999999e98

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 45.4%

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

      1. associate-+r+ 45.4%

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

      2. associate-+r+ 45.4%

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

   6. Simplified 45.4%

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

      1. flip-- 39.3%

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

      2. +-commutative 39.3%

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

      3. associate-+r+ 39.3%

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

      4. +-commutative 39.3%

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

      5. associate-+r+ 39.3%

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

      6. +-commutative 39.3%

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

      7. associate-+r+ 39.3%

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

   8. Applied egg-rr 39.3%

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

      1. +-commutative 39.3%

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

      2. +-commutative 39.3%

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

      3. +-commutative 39.3%

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

      4. +-commutative 39.3%

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

      5. +-commutative 39.3%

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

      6. +-commutative 39.3%

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

      7. +-commutative 39.3%

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

   10. Simplified 39.3%

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

   11. Taylor expanded in mu around inf 64.5%

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

   if -2.3999999999999999e98 < EDonor < -3.19999999999999986e-102 or 4.19999999999999974e44 < EDonor

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EAccept around inf 74.3%

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

   5. Taylor expanded in EAccept around 0 62.8%

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

   if -3.19999999999999986e-102 < EDonor < 6.89999999999999974e-217 or 3.8999999999999999e-119 < EDonor < 4.19999999999999974e44

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 75.2%

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

      1. associate-+r+ 75.2%

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

      2. associate-+r+ 75.2%

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

   6. Simplified 75.2%

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

   7. Taylor expanded in mu around 0 76.6%

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

   if 6.89999999999999974e-217 < EDonor < 3.8999999999999999e-119

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in EDonor around 0 99.8%

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

   5. Taylor expanded in KbT around inf 48.9%

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

      1. +-commutative 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in EAccept around inf 55.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -2.4 \cdot 10^{+98}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;EDonor \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{elif}\;EDonor \leq 6.9 \cdot 10^{-217}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{\left(2 + \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{elif}\;EDonor \leq 3.9 \cdot 10^{-119}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\left(\frac{EDonor}{KbT} + \left(\frac{mu}{KbT} + \frac{Vef}{KbT}\right)\right) - \frac{Ec}{KbT}}} + \frac{NaChar}{1 + \frac{EAccept}{KbT}}\\ \mathbf{elif}\;EDonor \leq 4.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{\left(2 + \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \end{array} \]

Alternative 10: 54.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;EDonor \leq -2.6 \cdot 10^{+98}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;EDonor \leq -2.9 \cdot 10^{-102} \lor \neg \left(EDonor \leq 1.4 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{\left(2 + \left(\frac{mu}{KbT} + \frac{EDonor}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= EDonor -2.6e+98)
     (+ t_0 (/ NdChar (+ 1.0 (/ mu KbT))))
     (if (or (<= EDonor -2.9e-102) (not (<= EDonor 1.4e+46)))
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
        (/ NaChar (+ (/ EAccept KbT) 2.0)))
       (+
        t_0
        (/ NdChar (- (+ 2.0 (+ (/ mu KbT) (/ EDonor KbT))) (/ Ec KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (EDonor <= -2.6e+98) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if ((EDonor <= -2.9e-102) || !(EDonor <= 1.4e+46)) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = t_0 + (NdChar / ((2.0 + ((mu / KbT) + (EDonor / KbT))) - (Ec / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (edonor <= (-2.6d+98)) then
        tmp = t_0 + (ndchar / (1.0d0 + (mu / kbt)))
    else if ((edonor <= (-2.9d-102)) .or. (.not. (edonor <= 1.4d+46))) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / ((eaccept / kbt) + 2.0d0))
    else
        tmp = t_0 + (ndchar / ((2.0d0 + ((mu / kbt) + (edonor / kbt))) - (ec / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (EDonor <= -2.6e+98) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if ((EDonor <= -2.9e-102) || !(EDonor <= 1.4e+46)) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = t_0 + (NdChar / ((2.0 + ((mu / KbT) + (EDonor / KbT))) - (Ec / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if EDonor <= -2.6e+98:
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)))
	elif (EDonor <= -2.9e-102) or not (EDonor <= 1.4e+46):
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0))
	else:
		tmp = t_0 + (NdChar / ((2.0 + ((mu / KbT) + (EDonor / KbT))) - (Ec / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (EDonor <= -2.6e+98)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(mu / KbT))));
	elseif ((EDonor <= -2.9e-102) || !(EDonor <= 1.4e+46))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	else
		tmp = Float64(t_0 + Float64(NdChar / Float64(Float64(2.0 + Float64(Float64(mu / KbT) + Float64(EDonor / KbT))) - Float64(Ec / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (EDonor <= -2.6e+98)
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	elseif ((EDonor <= -2.9e-102) || ~((EDonor <= 1.4e+46)))
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	else
		tmp = t_0 + (NdChar / ((2.0 + ((mu / KbT) + (EDonor / KbT))) - (Ec / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[EDonor, -2.6e+98], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[EDonor, -2.9e-102], N[Not[LessEqual[EDonor, 1.4e+46]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / N[(N[(2.0 + N[(N[(mu / KbT), $MachinePrecision] + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if EDonor < -2.6e98

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 45.4%

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

      1. associate-+r+ 45.4%

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

      2. associate-+r+ 45.4%

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

   6. Simplified 45.4%

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

      1. flip-- 39.3%

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

      2. +-commutative 39.3%

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

      3. associate-+r+ 39.3%

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

      4. +-commutative 39.3%

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

      5. associate-+r+ 39.3%

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

      6. +-commutative 39.3%

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

      7. associate-+r+ 39.3%

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

   8. Applied egg-rr 39.3%

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

      1. +-commutative 39.3%

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

      2. +-commutative 39.3%

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

      3. +-commutative 39.3%

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

      4. +-commutative 39.3%

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

      5. +-commutative 39.3%

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

      6. +-commutative 39.3%

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

      7. +-commutative 39.3%

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

   10. Simplified 39.3%

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

   11. Taylor expanded in mu around inf 64.5%

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

   if -2.6e98 < EDonor < -2.89999999999999986e-102 or 1.40000000000000009e46 < EDonor

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EAccept around inf 74.3%

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

   5. Taylor expanded in EAccept around 0 62.8%

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

   if -2.89999999999999986e-102 < EDonor < 1.40000000000000009e46

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 72.4%

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

      1. associate-+r+ 72.4%

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

      2. associate-+r+ 72.4%

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

   6. Simplified 72.4%

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

   7. Taylor expanded in Vef around 0 72.7%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -2.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;EDonor \leq -2.9 \cdot 10^{-102} \lor \neg \left(EDonor \leq 1.4 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{\left(2 + \left(\frac{mu}{KbT} + \frac{EDonor}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \end{array} \]

Alternative 11: 64.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -27000000000:\\ \;\;\;\;t_0 + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{elif}\;NaChar \leq 7.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \left(1 + \frac{Vef}{KbT}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= NaChar -27000000000.0)
     (+ t_0 (/ NdChar (- 1.0 (/ Ec KbT))))
     (if (<= NaChar 7.2e-123)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
        (/ NaChar (+ (/ EAccept KbT) 2.0)))
       (+ t_0 (/ NdChar (+ 1.0 (+ 1.0 (/ Vef KbT)))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (NaChar <= -27000000000.0) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	} else if (NaChar <= 7.2e-123) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (Vef / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (nachar <= (-27000000000.0d0)) then
        tmp = t_0 + (ndchar / (1.0d0 - (ec / kbt)))
    else if (nachar <= 7.2d-123) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / ((eaccept / kbt) + 2.0d0))
    else
        tmp = t_0 + (ndchar / (1.0d0 + (1.0d0 + (vef / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (NaChar <= -27000000000.0) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	} else if (NaChar <= 7.2e-123) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (Vef / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if NaChar <= -27000000000.0:
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)))
	elif NaChar <= 7.2e-123:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0))
	else:
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (Vef / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (NaChar <= -27000000000.0)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 - Float64(Ec / KbT))));
	elseif (NaChar <= 7.2e-123)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	else
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(1.0 + Float64(Vef / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (NaChar <= -27000000000.0)
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	elseif (NaChar <= 7.2e-123)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	else
		tmp = t_0 + (NdChar / (1.0 + (1.0 + (Vef / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -27000000000.0], N[(t$95$0 + N[(NdChar / N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 7.2e-123], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(1.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -2.7e10

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 62.0%

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

      1. associate-+r+ 62.0%

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

      2. associate-+r+ 62.0%

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

   6. Simplified 62.0%

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

      1. flip-- 47.6%

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

      2. +-commutative 47.6%

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

      3. associate-+r+ 47.6%

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

      4. +-commutative 47.6%

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

      5. associate-+r+ 47.6%

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

      6. +-commutative 47.6%

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

      7. associate-+r+ 47.6%

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

   8. Applied egg-rr 47.6%

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

      1. +-commutative 47.6%

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

      2. +-commutative 47.6%

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

      3. +-commutative 47.6%

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

      4. +-commutative 47.6%

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

      5. +-commutative 47.6%

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

      6. +-commutative 47.6%

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

      7. +-commutative 47.6%

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

   10. Simplified 47.6%

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

   11. Taylor expanded in Ec around inf 66.9%

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

       1. neg-mul-1 66.9%

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

       2. distribute-neg-frac 66.9%

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

   13. Simplified 66.9%

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

   if -2.7e10 < NaChar < 7.1999999999999994e-123

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in EAccept around inf 74.1%

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

   5. Taylor expanded in EAccept around 0 64.7%

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

   if 7.1999999999999994e-123 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in Vef around inf 73.3%

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

   5. Taylor expanded in Vef around 0 66.1%

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

      1. +-commutative 66.1%

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

   7. Simplified 66.1%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -27000000000:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{elif}\;NaChar \leq 7.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(1 + \frac{Vef}{KbT}\right)}\\ \end{array} \]

Alternative 12: 62.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -6000000000000 \lor \neg \left(NaChar \leq 2.7 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NaChar -6000000000000.0) (not (<= NaChar 2.7e-121)))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
    (/ NdChar (+ 1.0 (/ mu KbT))))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
    (/ NaChar (+ (/ EAccept KbT) 2.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -6000000000000.0) || !(NaChar <= 2.7e-121)) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (mu / KbT)));
	} else {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((nachar <= (-6000000000000.0d0)) .or. (.not. (nachar <= 2.7d-121))) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / (1.0d0 + (mu / kbt)))
    else
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / ((eaccept / kbt) + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NaChar <= -6000000000000.0) || !(NaChar <= 2.7e-121)) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (mu / KbT)));
	} else {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NaChar <= -6000000000000.0) or not (NaChar <= 2.7e-121):
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (mu / KbT)))
	else:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NaChar <= -6000000000000.0) || !(NaChar <= 2.7e-121))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / Float64(1.0 + Float64(mu / KbT))));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NaChar <= -6000000000000.0) || ~((NaChar <= 2.7e-121)))
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / (1.0 + (mu / KbT)));
	else
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NaChar, -6000000000000.0], N[Not[LessEqual[NaChar, 2.7e-121]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -6e12 or 2.7000000000000002e-121 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 64.4%

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

      1. associate-+r+ 64.4%

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

      2. associate-+r+ 64.4%

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

   6. Simplified 64.4%

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

      1. flip-- 51.4%

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

      2. +-commutative 51.4%

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

      3. associate-+r+ 51.4%

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

      4. +-commutative 51.4%

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

      5. associate-+r+ 51.4%

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

      6. +-commutative 51.4%

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

      7. associate-+r+ 51.4%

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

   8. Applied egg-rr 51.4%

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

      1. +-commutative 51.4%

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

      2. +-commutative 51.4%

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

      3. +-commutative 51.4%

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

      4. +-commutative 51.4%

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

      5. +-commutative 51.4%

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

      6. +-commutative 51.4%

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

      7. +-commutative 51.4%

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

   10. Simplified 51.4%

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

   11. Taylor expanded in mu around inf 65.5%

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

   if -6e12 < NaChar < 2.7000000000000002e-121

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in EAccept around inf 74.1%

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

   5. Taylor expanded in EAccept around 0 64.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -6000000000000 \lor \neg \left(NaChar \leq 2.7 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \end{array} \]

Alternative 13: 60.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NaChar \leq -3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}} + \frac{NdChar}{\frac{EDonor}{KbT} + 2}\\ \mathbf{elif}\;NaChar \leq 1.82 \cdot 10^{-65}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NaChar -3.2e+14)
   (+
    (/ NaChar (+ 1.0 (exp (/ (- (+ Vef Ev) mu) KbT))))
    (/ NdChar (+ (/ EDonor KbT) 2.0)))
   (if (<= NaChar 1.82e-65)
     (+
      (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
      (/ NaChar (+ (/ EAccept KbT) 2.0)))
     (+
      (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
      (/ NdChar 2.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -3.2e+14) {
		tmp = (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT)))) + (NdChar / ((EDonor / KbT) + 2.0));
	} else if (NaChar <= 1.82e-65) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (nachar <= (-3.2d+14)) then
        tmp = (nachar / (1.0d0 + exp((((vef + ev) - mu) / kbt)))) + (ndchar / ((edonor / kbt) + 2.0d0))
    else if (nachar <= 1.82d-65) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / ((eaccept / kbt) + 2.0d0))
    else
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NaChar <= -3.2e+14) {
		tmp = (NaChar / (1.0 + Math.exp((((Vef + Ev) - mu) / KbT)))) + (NdChar / ((EDonor / KbT) + 2.0));
	} else if (NaChar <= 1.82e-65) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NaChar <= -3.2e+14:
		tmp = (NaChar / (1.0 + math.exp((((Vef + Ev) - mu) / KbT)))) + (NdChar / ((EDonor / KbT) + 2.0))
	elif NaChar <= 1.82e-65:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0))
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NaChar <= -3.2e+14)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Float64(Vef + Ev) - mu) / KbT)))) + Float64(NdChar / Float64(Float64(EDonor / KbT) + 2.0)));
	elseif (NaChar <= 1.82e-65)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(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 (NaChar <= -3.2e+14)
		tmp = (NaChar / (1.0 + exp((((Vef + Ev) - mu) / KbT)))) + (NdChar / ((EDonor / KbT) + 2.0));
	elseif (NaChar <= 1.82e-65)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	else
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NaChar, -3.2e+14], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(N[(Vef + Ev), $MachinePrecision] - mu), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[(EDonor / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.82e-65], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -3.2e14

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EDonor around inf 81.9%

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

   5. Taylor expanded in EAccept around 0 70.8%

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

      1. +-commutative 70.8%

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

   7. Simplified 70.8%

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

   8. Taylor expanded in EDonor around 0 54.9%

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

   if -3.2e14 < NaChar < 1.82e-65

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in EAccept around inf 72.0%

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

   5. Taylor expanded in EAccept around 0 62.9%

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

   if 1.82e-65 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 63.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{\left(Vef + Ev\right) - mu}{KbT}}} + \frac{NdChar}{\frac{EDonor}{KbT} + 2}\\ \mathbf{elif}\;NaChar \leq 1.82 \cdot 10^{-65}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 14: 62.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -82000000000:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;NaChar \leq 1.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= NaChar -82000000000.0)
     (+ t_0 (/ NdChar (+ 1.0 (/ EDonor KbT))))
     (if (<= NaChar 1.8e-65)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
        (/ NaChar (+ (/ EAccept KbT) 2.0)))
       (+ t_0 (/ NdChar 2.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (NaChar <= -82000000000.0) {
		tmp = t_0 + (NdChar / (1.0 + (EDonor / KbT)));
	} else if (NaChar <= 1.8e-65) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (nachar <= (-82000000000.0d0)) then
        tmp = t_0 + (ndchar / (1.0d0 + (edonor / kbt)))
    else if (nachar <= 1.8d-65) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / ((eaccept / kbt) + 2.0d0))
    else
        tmp = t_0 + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (NaChar <= -82000000000.0) {
		tmp = t_0 + (NdChar / (1.0 + (EDonor / KbT)));
	} else if (NaChar <= 1.8e-65) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = t_0 + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if NaChar <= -82000000000.0:
		tmp = t_0 + (NdChar / (1.0 + (EDonor / KbT)))
	elif NaChar <= 1.8e-65:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0))
	else:
		tmp = t_0 + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (NaChar <= -82000000000.0)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(EDonor / KbT))));
	elseif (NaChar <= 1.8e-65)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	else
		tmp = Float64(t_0 + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (NaChar <= -82000000000.0)
		tmp = t_0 + (NdChar / (1.0 + (EDonor / KbT)));
	elseif (NaChar <= 1.8e-65)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	else
		tmp = t_0 + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -82000000000.0], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.8e-65], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -8.2e10

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 62.0%

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

      1. associate-+r+ 62.0%

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

      2. associate-+r+ 62.0%

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

   6. Simplified 62.0%

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

      1. flip-- 47.6%

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

      2. +-commutative 47.6%

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

      3. associate-+r+ 47.6%

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

      4. +-commutative 47.6%

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

      5. associate-+r+ 47.6%

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

      6. +-commutative 47.6%

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

      7. associate-+r+ 47.6%

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

   8. Applied egg-rr 47.6%

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

      1. +-commutative 47.6%

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

      2. +-commutative 47.6%

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

      3. +-commutative 47.6%

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

      4. +-commutative 47.6%

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

      5. +-commutative 47.6%

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

      6. +-commutative 47.6%

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

      7. +-commutative 47.6%

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

   10. Simplified 47.6%

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

   11. Taylor expanded in EDonor around inf 64.1%

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

   if -8.2e10 < NaChar < 1.7999999999999999e-65

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in EAccept around inf 72.0%

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

   5. Taylor expanded in EAccept around 0 62.9%

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

   if 1.7999999999999999e-65 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 63.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -82000000000:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{EDonor}{KbT}}\\ \mathbf{elif}\;NaChar \leq 1.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 15: 63.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -9.8 \cdot 10^{+18}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{elif}\;NaChar \leq 3.75 \cdot 10^{-124}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= NaChar -9.8e+18)
     (+ t_0 (/ NdChar (- 1.0 (/ Ec KbT))))
     (if (<= NaChar 3.75e-124)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
        (/ NaChar (+ (/ EAccept KbT) 2.0)))
       (+ t_0 (/ NdChar (+ 1.0 (/ mu KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (NaChar <= -9.8e+18) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	} else if (NaChar <= 3.75e-124) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (nachar <= (-9.8d+18)) then
        tmp = t_0 + (ndchar / (1.0d0 - (ec / kbt)))
    else if (nachar <= 3.75d-124) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / ((eaccept / kbt) + 2.0d0))
    else
        tmp = t_0 + (ndchar / (1.0d0 + (mu / kbt)))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (NaChar <= -9.8e+18) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	} else if (NaChar <= 3.75e-124) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	} else {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if NaChar <= -9.8e+18:
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)))
	elif NaChar <= 3.75e-124:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0))
	else:
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (NaChar <= -9.8e+18)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 - Float64(Ec / KbT))));
	elseif (NaChar <= 3.75e-124)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / Float64(Float64(EAccept / KbT) + 2.0)));
	else
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(mu / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (NaChar <= -9.8e+18)
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	elseif (NaChar <= 3.75e-124)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / ((EAccept / KbT) + 2.0));
	else
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -9.8e+18], N[(t$95$0 + N[(NdChar / N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 3.75e-124], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -9.8e18

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 62.0%

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

      1. associate-+r+ 62.0%

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

      2. associate-+r+ 62.0%

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

   6. Simplified 62.0%

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

      1. flip-- 47.6%

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

      2. +-commutative 47.6%

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

      3. associate-+r+ 47.6%

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

      4. +-commutative 47.6%

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

      5. associate-+r+ 47.6%

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

      6. +-commutative 47.6%

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

      7. associate-+r+ 47.6%

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

   8. Applied egg-rr 47.6%

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

      1. +-commutative 47.6%

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

      2. +-commutative 47.6%

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

      3. +-commutative 47.6%

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

      4. +-commutative 47.6%

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

      5. +-commutative 47.6%

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

      6. +-commutative 47.6%

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

      7. +-commutative 47.6%

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

   10. Simplified 47.6%

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

   11. Taylor expanded in Ec around inf 66.9%

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

       1. neg-mul-1 66.9%

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

       2. distribute-neg-frac 66.9%

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

   13. Simplified 66.9%

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

   if -9.8e18 < NaChar < 3.7499999999999998e-124

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in EAccept around inf 74.1%

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

   5. Taylor expanded in EAccept around 0 64.7%

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

   if 3.7499999999999998e-124 < NaChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 66.3%

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

      1. associate-+r+ 66.3%

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

      2. associate-+r+ 66.3%

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

   6. Simplified 66.3%

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

      1. flip-- 54.4%

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

      2. +-commutative 54.4%

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

      3. associate-+r+ 54.4%

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

      4. +-commutative 54.4%

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

      5. associate-+r+ 54.4%

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

      6. +-commutative 54.4%

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

      7. associate-+r+ 54.4%

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

   8. Applied egg-rr 54.4%

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

      1. +-commutative 54.4%

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

      2. +-commutative 54.4%

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

      3. +-commutative 54.4%

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

      4. +-commutative 54.4%

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

      5. +-commutative 54.4%

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

      6. +-commutative 54.4%

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

      7. +-commutative 54.4%

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

   10. Simplified 54.4%

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

   11. Taylor expanded in mu around inf 65.6%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -9.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{elif}\;NaChar \leq 3.75 \cdot 10^{-124}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{\frac{EAccept}{KbT} + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \end{array} \]

Alternative 16: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.05 \cdot 10^{-5} \lor \neg \left(NdChar \leq 7.5 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -2.05e-5) (not (<= NdChar 7.5e+45)))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
    (/ NaChar 2.0))
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
    (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.05e-5) || !(NdChar <= 7.5e+45)) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-2.05d-5)) .or. (.not. (ndchar <= 7.5d+45))) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -2.05e-5) || !(NdChar <= 7.5e+45)) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -2.05e-5) or not (NdChar <= 7.5e+45):
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -2.05e-5) || !(NdChar <= 7.5e+45))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(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 ((NdChar <= -2.05e-5) || ~((NdChar <= 7.5e+45)))
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -2.05e-5], N[Not[LessEqual[NdChar, 7.5e+45]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -2.05000000000000002e-5 or 7.50000000000000058e45 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EAccept around inf 75.4%

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

   5. Taylor expanded in EAccept around 0 57.7%

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

   if -2.05000000000000002e-5 < NdChar < 7.50000000000000058e45

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 61.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.05 \cdot 10^{-5} \lor \neg \left(NdChar \leq 7.5 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 17: 48.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq 3.9 \cdot 10^{+52}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar 3.9e+52)
   (+
    (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
    (/ NdChar 2.0))
   (+ (/ NdChar (+ 1.0 (exp (/ EDonor 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 <= 3.9e+52) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + exp((EDonor / 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 <= 3.9d+52) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))) + (ndchar / 2.0d0)
    else
        tmp = (ndchar / (1.0d0 + exp((edonor / 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 <= 3.9e+52) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= 3.9e+52:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0)
	else:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= 3.9e+52)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))) + Float64(NdChar / 2.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / 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 <= 3.9e+52)
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)))) + (NdChar / 2.0);
	else
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, 3.9e+52], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < 3.9e52

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 53.9%

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

   if 3.9e52 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EDonor around inf 73.4%

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

   5. Taylor expanded in KbT around inf 53.2%

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

      1. *-commutative 32.5%

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

   7. Simplified 53.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq 3.9 \cdot 10^{+52}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 18: 43.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -2.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq 2.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -2.5e-30)
   (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) (* NaChar 0.5))
   (if (<= NdChar 2.5e+51)
     (+ (/ NaChar (+ 1.0 (exp (/ (+ Vef Ev) KbT)))) (/ NdChar 2.0))
     (+ (/ NdChar (+ 1.0 (exp (/ EDonor 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 <= -2.5e-30) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar * 0.5);
	} else if (NdChar <= 2.5e+51) {
		tmp = (NaChar / (1.0 + exp(((Vef + Ev) / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + exp((EDonor / 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 <= (-2.5d-30)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar * 0.5d0)
    else if (ndchar <= 2.5d+51) then
        tmp = (nachar / (1.0d0 + exp(((vef + ev) / kbt)))) + (ndchar / 2.0d0)
    else
        tmp = (ndchar / (1.0d0 + exp((edonor / 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 <= -2.5e-30) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar * 0.5);
	} else if (NdChar <= 2.5e+51) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + Ev) / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -2.5e-30:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar * 0.5)
	elif NdChar <= 2.5e+51:
		tmp = (NaChar / (1.0 + math.exp(((Vef + Ev) / KbT)))) + (NdChar / 2.0)
	else:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -2.5e-30)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar * 0.5));
	elseif (NdChar <= 2.5e+51)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Ev) / KbT)))) + Float64(NdChar / 2.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / 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 <= -2.5e-30)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar * 0.5);
	elseif (NdChar <= 2.5e+51)
		tmp = (NaChar / (1.0 + exp(((Vef + Ev) / KbT)))) + (NdChar / 2.0);
	else
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -2.5e-30], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 2.5e+51], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + Ev), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -2.49999999999999986e-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%

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

   4. Taylor expanded in Vef around inf 58.9%

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

   5. Taylor expanded in KbT around inf 36.6%

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

      1. *-commutative 23.2%

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

   7. Simplified 36.6%

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

   if -2.49999999999999986e-30 < NdChar < 2.5e51

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 63.2%

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

   5. Taylor expanded in Ev around inf 51.6%

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

   if 2.5e51 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EDonor around inf 73.4%

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

   5. Taylor expanded in KbT around inf 53.2%

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

      1. *-commutative 32.5%

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

   7. Simplified 53.2%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -2.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq 2.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 19: 39.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -1.04 \cdot 10^{-16} \lor \neg \left(NdChar \leq 4.2 \cdot 10^{+52}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= NdChar -1.04e-16) (not (<= NdChar 4.2e+52)))
   (+ (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (* NaChar 0.5))
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.04e-16) || !(NdChar <= 4.2e+52)) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((ndchar <= (-1.04d-16)) .or. (.not. (ndchar <= 4.2d+52))) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((NdChar <= -1.04e-16) || !(NdChar <= 4.2e+52)) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (NdChar <= -1.04e-16) or not (NdChar <= 4.2e+52):
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((NdChar <= -1.04e-16) || !(NdChar <= 4.2e+52))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((NdChar <= -1.04e-16) || ~((NdChar <= 4.2e+52)))
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[NdChar, -1.04e-16], N[Not[LessEqual[NdChar, 4.2e+52]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.04000000000000001e-16 or 4.2e52 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EDonor around inf 70.0%

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

   5. Taylor expanded in KbT around inf 44.2%

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

      1. *-commutative 27.7%

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

   7. Simplified 44.2%

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

   if -1.04000000000000001e-16 < NdChar < 4.2e52

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 61.6%

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

   5. Taylor expanded in Ev around inf 45.1%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.04 \cdot 10^{-16} \lor \neg \left(NdChar \leq 4.2 \cdot 10^{+52}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 20: 39.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq 1.15 \cdot 10^{+52}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -5.5e-30)
   (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) (* NaChar 0.5))
   (if (<= NdChar 1.15e+52)
     (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar 2.0))
     (+ (/ NdChar (+ 1.0 (exp (/ EDonor 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-30) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar * 0.5);
	} else if (NdChar <= 1.15e+52) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + exp((EDonor / 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-30)) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar * 0.5d0)
    else if (ndchar <= 1.15d+52) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
    else
        tmp = (ndchar / (1.0d0 + exp((edonor / 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-30) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar * 0.5);
	} else if (NdChar <= 1.15e+52) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -5.5e-30:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar * 0.5)
	elif NdChar <= 1.15e+52:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / 2.0)
	else:
		tmp = (NdChar / (1.0 + math.exp((EDonor / 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-30)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar * 0.5));
	elseif (NdChar <= 1.15e+52)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / 2.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / 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-30)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar * 0.5);
	elseif (NdChar <= 1.15e+52)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	else
		tmp = (NdChar / (1.0 + exp((EDonor / 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-30], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[NdChar, 1.15e+52], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if NdChar < -5.49999999999999976e-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%

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

   4. Taylor expanded in Vef around inf 58.9%

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

   5. Taylor expanded in KbT around inf 36.6%

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

      1. *-commutative 23.2%

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

   7. Simplified 36.6%

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

   if -5.49999999999999976e-30 < NdChar < 1.15e52

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in KbT around inf 63.2%

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

   5. Taylor expanded in Ev around inf 46.1%

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

   if 1.15e52 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in EDonor around inf 73.4%

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

   5. Taylor expanded in KbT around inf 53.2%

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

      1. *-commutative 32.5%

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

   7. Simplified 53.2%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;NdChar \leq 1.15 \cdot 10^{+52}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]

Alternative 21: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;EAccept \leq 1.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= EAccept 1.8e+89)
   (+ (/ NaChar (+ 1.0 (exp (/ Ev KbT)))) (/ NdChar 2.0))
   (+ (/ 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 (EAccept <= 1.8e+89) {
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	} 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 (eaccept <= 1.8d+89) then
        tmp = (nachar / (1.0d0 + exp((ev / kbt)))) + (ndchar / 2.0d0)
    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 (EAccept <= 1.8e+89) {
		tmp = (NaChar / (1.0 + Math.exp((Ev / KbT)))) + (NdChar / 2.0);
	} 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 EAccept <= 1.8e+89:
		tmp = (NaChar / (1.0 + math.exp((Ev / KbT)))) + (NdChar / 2.0)
	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 (EAccept <= 1.8e+89)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))) + Float64(NdChar / 2.0));
	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 (EAccept <= 1.8e+89)
		tmp = (NaChar / (1.0 + exp((Ev / KbT)))) + (NdChar / 2.0);
	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[EAccept, 1.8e+89], N[(N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $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 2 regimes

2. if EAccept < 1.8e89

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 50.7%

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

   5. Taylor expanded in Ev around inf 40.0%

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

   if 1.8e89 < EAccept

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in KbT around inf 47.8%

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

   5. Taylor expanded in EAccept around inf 43.8%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EAccept \leq 1.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Ev}{KbT}}} + \frac{NdChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2}\\ \end{array} \]

Alternative 22: 36.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+ (/ 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) {
	return (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar / 2.0);
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar / 2.0d0)
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar / 2.0);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar / 2.0)

Julia

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

MATLAB

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

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in KbT around inf 50.1%

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

5. Taylor expanded in EAccept around inf 39.8%

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

6. Final simplification 39.8%

   \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + \frac{NdChar}{2} \]

Alternative 23: 28.5% accurate, 32.7× speedup

Math

\[\begin{array}{l} \\ \frac{NdChar}{2} + NaChar \cdot 0.5 \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (+ (/ NdChar 2.0) (* NaChar 0.5)))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / 2.0) + (NaChar * 0.5);
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    code = (ndchar / 2.0d0) + (nachar * 0.5d0)
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	return (NdChar / 2.0) + (NaChar * 0.5);
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	return (NdChar / 2.0) + (NaChar * 0.5)

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	return Float64(Float64(NdChar / 2.0) + Float64(NaChar * 0.5))
end

MATLAB

function tmp = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = (NdChar / 2.0) + (NaChar * 0.5);
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in KbT around inf 50.1%

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

5. Taylor expanded in KbT around inf 29.9%

   \[\leadsto \frac{NdChar}{2} + \color{blue}{0.5 \cdot NaChar} \]
6. Step-by-step derivation

   1. *-commutative 29.9%

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

7. Simplified 29.9%

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

8. Final simplification 29.9%

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

Reproduce

herbie shell --seed 2023271 
(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))))))