Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 25.6s

Alternatives: 28

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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + EAccept\right)}{KbT}}}\\ \mathbf{if}\;Vef \leq -1.65 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq -1 \cdot 10^{-115}:\\ \;\;\;\;\frac{NaChar}{1 + t\_0} + \frac{NdChar}{\left(2 + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{elif}\;Vef \leq -4.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;Vef \leq 100000000000:\\ \;\;\;\;\frac{NdChar}{1 + mu \cdot \left(\frac{1}{KbT} - \frac{\frac{Ec}{KbT} + \left(-1 - \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right)}{mu}\right)} - \frac{NaChar}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))
        (t_1
         (-
          (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
          (/ NaChar (- -1.0 (exp (/ (+ Vef (+ Ev EAccept)) KbT)))))))
   (if (<= Vef -1.65e+96)
     t_1
     (if (<= Vef -1e-115)
       (+
        (/ NaChar (+ 1.0 t_0))
        (/ NdChar (- (+ 2.0 (+ (/ Vef KbT) (/ mu KbT))) (/ Ec KbT))))
       (if (<= Vef -4.2e-188)
         (+
          (/ NdChar (+ 1.0 (exp (/ EDonor KbT))))
          (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
         (if (<= Vef 100000000000.0)
           (-
            (/
             NdChar
             (+
              1.0
              (*
               mu
               (-
                (/ 1.0 KbT)
                (/
                 (+ (/ Ec KbT) (- -1.0 (+ (/ Vef KbT) (/ EDonor KbT))))
                 mu)))))
            (/ NaChar (- -1.0 t_0)))
           t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double t_1 = (NdChar / (1.0 + exp((Vef / KbT)))) - (NaChar / (-1.0 - exp(((Vef + (Ev + EAccept)) / KbT))));
	double tmp;
	if (Vef <= -1.65e+96) {
		tmp = t_1;
	} else if (Vef <= -1e-115) {
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / ((2.0 + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)));
	} else if (Vef <= -4.2e-188) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else if (Vef <= 100000000000.0) {
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((vef + (ev + (eaccept - mu))) / kbt))
    t_1 = (ndchar / (1.0d0 + exp((vef / kbt)))) - (nachar / ((-1.0d0) - exp(((vef + (ev + eaccept)) / kbt))))
    if (vef <= (-1.65d+96)) then
        tmp = t_1
    else if (vef <= (-1d-115)) then
        tmp = (nachar / (1.0d0 + t_0)) + (ndchar / ((2.0d0 + ((vef / kbt) + (mu / kbt))) - (ec / kbt)))
    else if (vef <= (-4.2d-188)) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else if (vef <= 100000000000.0d0) then
        tmp = (ndchar / (1.0d0 + (mu * ((1.0d0 / kbt) - (((ec / kbt) + ((-1.0d0) - ((vef / kbt) + (edonor / kbt)))) / mu))))) - (nachar / ((-1.0d0) - t_0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double t_1 = (NdChar / (1.0 + Math.exp((Vef / KbT)))) - (NaChar / (-1.0 - Math.exp(((Vef + (Ev + EAccept)) / KbT))));
	double tmp;
	if (Vef <= -1.65e+96) {
		tmp = t_1;
	} else if (Vef <= -1e-115) {
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / ((2.0 + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)));
	} else if (Vef <= -4.2e-188) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else if (Vef <= 100000000000.0) {
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))
	t_1 = (NdChar / (1.0 + math.exp((Vef / KbT)))) - (NaChar / (-1.0 - math.exp(((Vef + (Ev + EAccept)) / KbT))))
	tmp = 0
	if Vef <= -1.65e+96:
		tmp = t_1
	elif Vef <= -1e-115:
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / ((2.0 + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)))
	elif Vef <= -4.2e-188:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	elif Vef <= 100000000000.0:
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_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 + Float64(Ev + EAccept)) / KbT)))))
	tmp = 0.0
	if (Vef <= -1.65e+96)
		tmp = t_1;
	elseif (Vef <= -1e-115)
		tmp = Float64(Float64(NaChar / Float64(1.0 + t_0)) + Float64(NdChar / Float64(Float64(2.0 + Float64(Float64(Vef / KbT) + Float64(mu / KbT))) - Float64(Ec / KbT))));
	elseif (Vef <= -4.2e-188)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) + Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))));
	elseif (Vef <= 100000000000.0)
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(mu * Float64(Float64(1.0 / KbT) - Float64(Float64(Float64(Ec / KbT) + Float64(-1.0 - Float64(Float64(Vef / KbT) + Float64(EDonor / KbT)))) / mu))))) - Float64(NaChar / Float64(-1.0 - t_0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	t_1 = (NdChar / (1.0 + exp((Vef / KbT)))) - (NaChar / (-1.0 - exp(((Vef + (Ev + EAccept)) / KbT))));
	tmp = 0.0;
	if (Vef <= -1.65e+96)
		tmp = t_1;
	elseif (Vef <= -1e-115)
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / ((2.0 + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)));
	elseif (Vef <= -4.2e-188)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	elseif (Vef <= 100000000000.0)
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $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 + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -1.65e+96], t$95$1, If[LessEqual[Vef, -1e-115], N[(N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(N[(2.0 + N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, -4.2e-188], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 100000000000.0], N[(N[(NdChar / N[(1.0 + N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] - N[(N[(N[(Ec / KbT), $MachinePrecision] + N[(-1.0 - N[(N[(Vef / KbT), $MachinePrecision] + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if Vef < -1.64999999999999992e96 or 1e11 < 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. Add Preprocessing

   5. Taylor expanded in Vef around inf 90.8%

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

   6. Taylor expanded in mu around 0 85.6%

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

      1. +-commutative 46.0%

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

      2. associate-+r+ 46.0%

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

   8. Simplified 85.6%

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

   if -1.64999999999999992e96 < Vef < -1.0000000000000001e-115

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 65.3%

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

      1. associate-+r+ 65.3%

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

   7. Simplified 65.3%

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

   8. Taylor expanded in EDonor around 0 68.9%

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

   if -1.0000000000000001e-115 < Vef < -4.1999999999999998e-188

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 75.1%

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

   6. Taylor expanded in EDonor around inf 66.6%

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

   if -4.1999999999999998e-188 < Vef < 1e11

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 63.2%

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

      1. associate-+r+ 63.2%

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

   7. Simplified 63.2%

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

   8. Taylor expanded in mu around -inf 69.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.65 \cdot 10^{+96}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + EAccept\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq -1 \cdot 10^{-115}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{\left(2 + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{elif}\;Vef \leq -4.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{elif}\;Vef \leq 100000000000:\\ \;\;\;\;\frac{NdChar}{1 + mu \cdot \left(\frac{1}{KbT} - \frac{\frac{Ec}{KbT} + \left(-1 - \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right)}{mu}\right)} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + EAccept\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\\ t_1 := \frac{NaChar}{1 + t\_0} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{if}\;Vef \leq -9 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Vef \leq -8.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - t\_0}\\ \mathbf{elif}\;Vef \leq 1.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))
        (t_1 (+ (/ NaChar (+ 1.0 t_0)) (/ NdChar (+ 1.0 (exp (/ Vef KbT)))))))
   (if (<= Vef -9e+95)
     t_1
     (if (<= Vef -8.5e-110)
       (- (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (/ NaChar (- -1.0 t_0)))
       (if (<= Vef 1.8e+43)
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
          (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))))
         t_1)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double t_1 = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + exp((Vef / KbT))));
	double tmp;
	if (Vef <= -9e+95) {
		tmp = t_1;
	} else if (Vef <= -8.5e-110) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_0));
	} else if (Vef <= 1.8e+43) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((vef + (ev + (eaccept - mu))) / kbt))
    t_1 = (nachar / (1.0d0 + t_0)) + (ndchar / (1.0d0 + exp((vef / kbt))))
    if (vef <= (-9d+95)) then
        tmp = t_1
    else if (vef <= (-8.5d-110)) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) - (nachar / ((-1.0d0) - t_0))
    else if (vef <= 1.8d+43) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar / (1.0d0 + exp((eaccept / kbt))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double t_1 = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	double tmp;
	if (Vef <= -9e+95) {
		tmp = t_1;
	} else if (Vef <= -8.5e-110) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_0));
	} else if (Vef <= 1.8e+43) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + Math.exp((EAccept / KbT))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))
	t_1 = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + math.exp((Vef / KbT))))
	tmp = 0
	if Vef <= -9e+95:
		tmp = t_1
	elif Vef <= -8.5e-110:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_0))
	elif Vef <= 1.8e+43:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + math.exp((EAccept / KbT))))
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))
	t_1 = Float64(Float64(NaChar / Float64(1.0 + t_0)) + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))))
	tmp = 0.0
	if (Vef <= -9e+95)
		tmp = t_1;
	elseif (Vef <= -8.5e-110)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) - Float64(NaChar / Float64(-1.0 - t_0)));
	elseif (Vef <= 1.8e+43)
		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)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	t_1 = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + exp((Vef / KbT))));
	tmp = 0.0;
	if (Vef <= -9e+95)
		tmp = t_1;
	elseif (Vef <= -8.5e-110)
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_0));
	elseif (Vef <= 1.8e+43)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar / (1.0 + exp((EAccept / KbT))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -9e+95], t$95$1, If[LessEqual[Vef, -8.5e-110], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Vef, 1.8e+43], 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], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if Vef < -9.00000000000000033e95 or 1.80000000000000005e43 < 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. Add Preprocessing

   5. Taylor expanded in Vef around inf 91.2%

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

   if -9.00000000000000033e95 < Vef < -8.50000000000000029e-110

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 78.3%

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

   if -8.50000000000000029e-110 < Vef < 1.80000000000000005e43

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 74.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -9 \cdot 10^{+95}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \mathbf{elif}\;Vef \leq -8.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;Vef \leq 1.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Vef \leq -1.35 \cdot 10^{+96} \lor \neg \left(Vef \leq 1.2 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + EAccept\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= Vef -1.35e+96) (not (<= Vef 1.2e+44)))
   (-
    (/ NdChar (+ 1.0 (exp (/ Vef KbT))))
    (/ NaChar (- -1.0 (exp (/ (+ Vef (+ Ev EAccept)) KbT)))))
   (-
    (/ NdChar (+ 1.0 (exp (/ EDonor 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) {
	double tmp;
	if ((Vef <= -1.35e+96) || !(Vef <= 1.2e+44)) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) - (NaChar / (-1.0 - exp(((Vef + (Ev + EAccept)) / KbT))));
	} else {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) - (NaChar / (-1.0 - exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((vef <= (-1.35d+96)) .or. (.not. (vef <= 1.2d+44))) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) - (nachar / ((-1.0d0) - exp(((vef + (ev + eaccept)) / kbt))))
    else
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) - (nachar / ((-1.0d0) - exp(((vef + (ev + (eaccept - mu))) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((Vef <= -1.35e+96) || !(Vef <= 1.2e+44)) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) - (NaChar / (-1.0 - Math.exp(((Vef + (Ev + EAccept)) / KbT))));
	} else {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) - (NaChar / (-1.0 - Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (Vef <= -1.35e+96) or not (Vef <= 1.2e+44):
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) - (NaChar / (-1.0 - math.exp(((Vef + (Ev + EAccept)) / KbT))))
	else:
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) - (NaChar / (-1.0 - math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((Vef <= -1.35e+96) || !(Vef <= 1.2e+44))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef + Float64(Ev + EAccept)) / KbT)))));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) - Float64(NaChar / Float64(-1.0 - exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((Vef <= -1.35e+96) || ~((Vef <= 1.2e+44)))
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) - (NaChar / (-1.0 - exp(((Vef + (Ev + EAccept)) / KbT))));
	else
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) - (NaChar / (-1.0 - exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[Vef, -1.35e+96], N[Not[LessEqual[Vef, 1.2e+44]], $MachinePrecision]], 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 + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / 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. Split input into 2 regimes

2. if Vef < -1.35000000000000011e96 or 1.20000000000000007e44 < 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. Add Preprocessing

   5. Taylor expanded in Vef around inf 91.2%

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

   6. Taylor expanded in mu around 0 86.6%

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

      1. +-commutative 45.4%

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

      2. associate-+r+ 45.4%

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

   8. Simplified 86.6%

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

   if -1.35000000000000011e96 < Vef < 1.20000000000000007e44

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 76.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -1.35 \cdot 10^{+96} \lor \neg \left(Vef \leq 1.2 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + EAccept\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\\ \mathbf{if}\;EDonor \leq -90000 \lor \neg \left(EDonor \leq 14600000000\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + t\_0} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
   (if (or (<= EDonor -90000.0) (not (<= EDonor 14600000000.0)))
     (- (/ NdChar (+ 1.0 (exp (/ EDonor KbT)))) (/ NaChar (- -1.0 t_0)))
     (+ (/ NaChar (+ 1.0 t_0)) (/ NdChar (+ 1.0 (exp (/ Vef KbT))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double tmp;
	if ((EDonor <= -90000.0) || !(EDonor <= 14600000000.0)) {
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + exp((Vef / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((vef + (ev + (eaccept - mu))) / kbt))
    if ((edonor <= (-90000.0d0)) .or. (.not. (edonor <= 14600000000.0d0))) then
        tmp = (ndchar / (1.0d0 + exp((edonor / kbt)))) - (nachar / ((-1.0d0) - t_0))
    else
        tmp = (nachar / (1.0d0 + t_0)) + (ndchar / (1.0d0 + exp((vef / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double tmp;
	if ((EDonor <= -90000.0) || !(EDonor <= 14600000000.0)) {
		tmp = (NdChar / (1.0 + Math.exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + Math.exp((Vef / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))
	tmp = 0
	if (EDonor <= -90000.0) or not (EDonor <= 14600000000.0):
		tmp = (NdChar / (1.0 + math.exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_0))
	else:
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + math.exp((Vef / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))
	tmp = 0.0
	if ((EDonor <= -90000.0) || !(EDonor <= 14600000000.0))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(EDonor / KbT)))) - Float64(NaChar / Float64(-1.0 - t_0)));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + t_0)) + Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	tmp = 0.0;
	if ((EDonor <= -90000.0) || ~((EDonor <= 14600000000.0)))
		tmp = (NdChar / (1.0 + exp((EDonor / KbT)))) - (NaChar / (-1.0 - t_0));
	else
		tmp = (NaChar / (1.0 + t_0)) + (NdChar / (1.0 + exp((Vef / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[EDonor, -90000.0], N[Not[LessEqual[EDonor, 14600000000.0]], $MachinePrecision]], N[(N[(NdChar / N[(1.0 + N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + t$95$0), $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 EDonor < -9e4 or 1.46e10 < EDonor

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EDonor around inf 82.9%

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

   if -9e4 < EDonor < 1.46e10

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 80.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;EDonor \leq -90000 \lor \neg \left(EDonor \leq 14600000000\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor}{KbT}}} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + e^{\frac{Vef}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\\ t_1 := \frac{NaChar}{1 + t\_0}\\ \mathbf{if}\;NaChar \leq -8.8 \cdot 10^{-38}:\\ \;\;\;\;NdChar + t\_1\\ \mathbf{elif}\;NaChar \leq -1.02 \cdot 10^{-234}:\\ \;\;\;\;t\_1 - \frac{NdChar}{-1 - mu \cdot \left(\frac{1}{KbT} + Ec \cdot \frac{\left(\left(\frac{1}{Ec} + \frac{EDonor}{Ec \cdot KbT}\right) + \frac{Vef}{Ec \cdot KbT}\right) + \frac{-1}{KbT}}{mu}\right)}\\ \mathbf{elif}\;NaChar \leq 8 \cdot 10^{-170}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + mu \cdot \left(\frac{1}{KbT} - \frac{\frac{Ec}{KbT} + \left(-1 - \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right)}{mu}\right)} - \frac{NaChar}{-1 - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))
        (t_1 (/ NaChar (+ 1.0 t_0))))
   (if (<= NaChar -8.8e-38)
     (+ NdChar t_1)
     (if (<= NaChar -1.02e-234)
       (-
        t_1
        (/
         NdChar
         (-
          -1.0
          (*
           mu
           (+
            (/ 1.0 KbT)
            (*
             Ec
             (/
              (+
               (+ (+ (/ 1.0 Ec) (/ EDonor (* Ec KbT))) (/ Vef (* Ec KbT)))
               (/ -1.0 KbT))
              mu)))))))
       (if (<= NaChar 8e-170)
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
          (* NaChar 0.5))
         (-
          (/
           NdChar
           (+
            1.0
            (*
             mu
             (-
              (/ 1.0 KbT)
              (/ (+ (/ Ec KbT) (- -1.0 (+ (/ Vef KbT) (/ EDonor KbT)))) mu)))))
          (/ NaChar (- -1.0 t_0))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double t_1 = NaChar / (1.0 + t_0);
	double tmp;
	if (NaChar <= -8.8e-38) {
		tmp = NdChar + t_1;
	} else if (NaChar <= -1.02e-234) {
		tmp = t_1 - (NdChar / (-1.0 - (mu * ((1.0 / KbT) + (Ec * (((((1.0 / Ec) + (EDonor / (Ec * KbT))) + (Vef / (Ec * KbT))) + (-1.0 / KbT)) / mu))))));
	} else if (NaChar <= 8e-170) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((vef + (ev + (eaccept - mu))) / kbt))
    t_1 = nachar / (1.0d0 + t_0)
    if (nachar <= (-8.8d-38)) then
        tmp = ndchar + t_1
    else if (nachar <= (-1.02d-234)) then
        tmp = t_1 - (ndchar / ((-1.0d0) - (mu * ((1.0d0 / kbt) + (ec * (((((1.0d0 / ec) + (edonor / (ec * kbt))) + (vef / (ec * kbt))) + ((-1.0d0) / kbt)) / mu))))))
    else if (nachar <= 8d-170) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (ndchar / (1.0d0 + (mu * ((1.0d0 / kbt) - (((ec / kbt) + ((-1.0d0) - ((vef / kbt) + (edonor / kbt)))) / mu))))) - (nachar / ((-1.0d0) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double t_1 = NaChar / (1.0 + t_0);
	double tmp;
	if (NaChar <= -8.8e-38) {
		tmp = NdChar + t_1;
	} else if (NaChar <= -1.02e-234) {
		tmp = t_1 - (NdChar / (-1.0 - (mu * ((1.0 / KbT) + (Ec * (((((1.0 / Ec) + (EDonor / (Ec * KbT))) + (Vef / (Ec * KbT))) + (-1.0 / KbT)) / mu))))));
	} else if (NaChar <= 8e-170) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))
	t_1 = NaChar / (1.0 + t_0)
	tmp = 0
	if NaChar <= -8.8e-38:
		tmp = NdChar + t_1
	elif NaChar <= -1.02e-234:
		tmp = t_1 - (NdChar / (-1.0 - (mu * ((1.0 / KbT) + (Ec * (((((1.0 / Ec) + (EDonor / (Ec * KbT))) + (Vef / (Ec * KbT))) + (-1.0 / KbT)) / mu))))))
	elif NaChar <= 8e-170:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))
	t_1 = Float64(NaChar / Float64(1.0 + t_0))
	tmp = 0.0
	if (NaChar <= -8.8e-38)
		tmp = Float64(NdChar + t_1);
	elseif (NaChar <= -1.02e-234)
		tmp = Float64(t_1 - Float64(NdChar / Float64(-1.0 - Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Ec * Float64(Float64(Float64(Float64(Float64(1.0 / Ec) + Float64(EDonor / Float64(Ec * KbT))) + Float64(Vef / Float64(Ec * KbT))) + Float64(-1.0 / KbT)) / mu)))))));
	elseif (NaChar <= 8e-170)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(mu * Float64(Float64(1.0 / KbT) - Float64(Float64(Float64(Ec / KbT) + Float64(-1.0 - Float64(Float64(Vef / KbT) + Float64(EDonor / KbT)))) / mu))))) - Float64(NaChar / Float64(-1.0 - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	t_1 = NaChar / (1.0 + t_0);
	tmp = 0.0;
	if (NaChar <= -8.8e-38)
		tmp = NdChar + t_1;
	elseif (NaChar <= -1.02e-234)
		tmp = t_1 - (NdChar / (-1.0 - (mu * ((1.0 / KbT) + (Ec * (((((1.0 / Ec) + (EDonor / (Ec * KbT))) + (Vef / (Ec * KbT))) + (-1.0 / KbT)) / mu))))));
	elseif (NaChar <= 8e-170)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NaChar, -8.8e-38], N[(NdChar + t$95$1), $MachinePrecision], If[LessEqual[NaChar, -1.02e-234], N[(t$95$1 - N[(NdChar / N[(-1.0 - N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(Ec * N[(N[(N[(N[(N[(1.0 / Ec), $MachinePrecision] + N[(EDonor / N[(Ec * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Vef / N[(Ec * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 8e-170], 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 * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] - N[(N[(N[(Ec / KbT), $MachinePrecision] + N[(-1.0 - N[(N[(Vef / KbT), $MachinePrecision] + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if NaChar < -8.80000000000000029e-38

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 56.1%

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

      1. associate-+r+ 56.1%

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

   7. Simplified 56.1%

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

   8. Taylor expanded in Ec around -inf 57.8%

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

      1. associate-*r* 57.8%

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

      2. mul-1-neg 57.8%

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

      3. +-commutative 57.8%

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

      4. mul-1-neg 57.8%

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

      5. unsub-neg 57.8%

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

   10. Simplified 57.8%

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

   11. Taylor expanded in EDonor around inf 60.0%

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

       1. *-commutative 60.0%

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

   13. Simplified 60.0%

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

   14. Taylor expanded in KbT around inf 76.9%

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

   if -8.80000000000000029e-38 < NaChar < -1.01999999999999999e-234

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 61.6%

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

      1. associate-+r+ 61.6%

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

   7. Simplified 61.6%

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

   8. Taylor expanded in Ec around -inf 65.7%

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

      1. associate-*r* 65.7%

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

      2. mul-1-neg 65.7%

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

      3. +-commutative 65.7%

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

      4. mul-1-neg 65.7%

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

      5. unsub-neg 65.7%

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

   10. Simplified 65.7%

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

   11. Taylor expanded in mu around inf 77.9%

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

       1. +-commutative 77.9%

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

       2. mul-1-neg 77.9%

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

       3. unsub-neg 77.9%

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

       4. associate-/l* 80.3%

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

       5. associate-+r+ 80.3%

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

       6. *-commutative 80.3%

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

       7. *-commutative 80.3%

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

   13. Simplified 80.3%

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

   if -1.01999999999999999e-234 < NaChar < 7.99999999999999987e-170

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 90.2%

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

   6. Taylor expanded in EAccept around 0 75.6%

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

   if 7.99999999999999987e-170 < 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. Add Preprocessing

   5. Taylor expanded in KbT around inf 67.4%

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

      1. associate-+r+ 67.4%

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

   7. Simplified 67.4%

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

   8. Taylor expanded in mu around -inf 69.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -8.8 \cdot 10^{-38}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq -1.02 \cdot 10^{-234}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} - \frac{NdChar}{-1 - mu \cdot \left(\frac{1}{KbT} + Ec \cdot \frac{\left(\left(\frac{1}{Ec} + \frac{EDonor}{Ec \cdot KbT}\right) + \frac{Vef}{Ec \cdot KbT}\right) + \frac{-1}{KbT}}{mu}\right)}\\ \mathbf{elif}\;NaChar \leq 8 \cdot 10^{-170}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + mu \cdot \left(\frac{1}{KbT} - \frac{\frac{Ec}{KbT} + \left(-1 - \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right)}{mu}\right)} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;mu \leq 4.3 \cdot 10^{-185}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 - \left(\frac{Ec}{KbT} + \left(Vef \cdot \left(\frac{-1}{KbT} - \frac{mu}{Vef \cdot KbT}\right) + \left(-1 - \frac{EDonor}{KbT}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;NdChar + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= mu 4.3e-185)
     (+
      t_0
      (/
       NdChar
       (-
        1.0
        (+
         (/ Ec KbT)
         (+
          (* Vef (- (/ -1.0 KbT) (/ mu (* Vef KbT))))
          (- -1.0 (/ EDonor KbT)))))))
     (+ NdChar t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (mu <= 4.3e-185) {
		tmp = t_0 + (NdChar / (1.0 - ((Ec / KbT) + ((Vef * ((-1.0 / KbT) - (mu / (Vef * KbT)))) + (-1.0 - (EDonor / KbT))))));
	} else {
		tmp = NdChar + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (mu <= 4.3d-185) then
        tmp = t_0 + (ndchar / (1.0d0 - ((ec / kbt) + ((vef * (((-1.0d0) / kbt) - (mu / (vef * kbt)))) + ((-1.0d0) - (edonor / kbt))))))
    else
        tmp = ndchar + t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (mu <= 4.3e-185) {
		tmp = t_0 + (NdChar / (1.0 - ((Ec / KbT) + ((Vef * ((-1.0 / KbT) - (mu / (Vef * KbT)))) + (-1.0 - (EDonor / KbT))))));
	} else {
		tmp = NdChar + t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if mu <= 4.3e-185:
		tmp = t_0 + (NdChar / (1.0 - ((Ec / KbT) + ((Vef * ((-1.0 / KbT) - (mu / (Vef * KbT)))) + (-1.0 - (EDonor / KbT))))))
	else:
		tmp = NdChar + t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (mu <= 4.3e-185)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 - Float64(Float64(Ec / KbT) + Float64(Float64(Vef * Float64(Float64(-1.0 / KbT) - Float64(mu / Float64(Vef * KbT)))) + Float64(-1.0 - Float64(EDonor / KbT)))))));
	else
		tmp = Float64(NdChar + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (mu <= 4.3e-185)
		tmp = t_0 + (NdChar / (1.0 - ((Ec / KbT) + ((Vef * ((-1.0 / KbT) - (mu / (Vef * KbT)))) + (-1.0 - (EDonor / KbT))))));
	else
		tmp = NdChar + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, 4.3e-185], N[(t$95$0 + N[(NdChar / N[(1.0 - N[(N[(Ec / KbT), $MachinePrecision] + N[(N[(Vef * N[(N[(-1.0 / KbT), $MachinePrecision] - N[(mu / N[(Vef * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if mu < 4.3000000000000001e-185

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 67.3%

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

      1. associate-+r+ 67.3%

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

   7. Simplified 67.3%

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

   8. Taylor expanded in Vef around inf 67.9%

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

      1. *-commutative 67.9%

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

   10. Simplified 67.9%

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

   if 4.3000000000000001e-185 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.3%

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

      1. associate-+r+ 46.3%

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

   7. Simplified 46.3%

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

   8. Taylor expanded in Ec around -inf 47.4%

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

      1. associate-*r* 47.4%

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

      2. mul-1-neg 47.4%

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

      3. +-commutative 47.4%

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

      4. mul-1-neg 47.4%

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

      5. unsub-neg 47.4%

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

   10. Simplified 47.4%

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

   11. Taylor expanded in EDonor around inf 48.8%

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

       1. *-commutative 48.8%

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

   13. Simplified 48.8%

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

   14. Taylor expanded in KbT around inf 70.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq 4.3 \cdot 10^{-185}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 - \left(\frac{Ec}{KbT} + \left(Vef \cdot \left(\frac{-1}{KbT} - \frac{mu}{Vef \cdot KbT}\right) + \left(-1 - \frac{EDonor}{KbT}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;mu \leq 8.5 \cdot 10^{-185}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + \left(\left(\left(1 + \frac{EDonor}{KbT}\right) + mu \cdot \left(\frac{1}{KbT} + \frac{Vef}{mu \cdot KbT}\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;NdChar + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= mu 8.5e-185)
     (+
      t_0
      (/
       NdChar
       (+
        1.0
        (-
         (+ (+ 1.0 (/ EDonor KbT)) (* mu (+ (/ 1.0 KbT) (/ Vef (* mu KbT)))))
         (/ Ec KbT)))))
     (+ NdChar t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (mu <= 8.5e-185) {
		tmp = t_0 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + (mu * ((1.0 / KbT) + (Vef / (mu * KbT))))) - (Ec / KbT))));
	} else {
		tmp = NdChar + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (mu <= 8.5d-185) then
        tmp = t_0 + (ndchar / (1.0d0 + (((1.0d0 + (edonor / kbt)) + (mu * ((1.0d0 / kbt) + (vef / (mu * kbt))))) - (ec / kbt))))
    else
        tmp = ndchar + t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (mu <= 8.5e-185) {
		tmp = t_0 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + (mu * ((1.0 / KbT) + (Vef / (mu * KbT))))) - (Ec / KbT))));
	} else {
		tmp = NdChar + t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if mu <= 8.5e-185:
		tmp = t_0 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + (mu * ((1.0 / KbT) + (Vef / (mu * KbT))))) - (Ec / KbT))))
	else:
		tmp = NdChar + t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (mu <= 8.5e-185)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(EDonor / KbT)) + Float64(mu * Float64(Float64(1.0 / KbT) + Float64(Vef / Float64(mu * KbT))))) - Float64(Ec / KbT)))));
	else
		tmp = Float64(NdChar + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (mu <= 8.5e-185)
		tmp = t_0 + (NdChar / (1.0 + (((1.0 + (EDonor / KbT)) + (mu * ((1.0 / KbT) + (Vef / (mu * KbT))))) - (Ec / KbT))));
	else
		tmp = NdChar + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, 8.5e-185], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(N[(N[(1.0 + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] + N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] + N[(Vef / N[(mu * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if mu < 8.5000000000000001e-185

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 67.3%

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

      1. associate-+r+ 67.3%

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

   7. Simplified 67.3%

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

   8. Taylor expanded in mu around inf 70.3%

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

   if 8.5000000000000001e-185 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.3%

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

      1. associate-+r+ 46.3%

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

   7. Simplified 46.3%

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

   8. Taylor expanded in Ec around -inf 47.4%

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

      1. associate-*r* 47.4%

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

      2. mul-1-neg 47.4%

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

      3. +-commutative 47.4%

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

      4. mul-1-neg 47.4%

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

      5. unsub-neg 47.4%

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

   10. Simplified 47.4%

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

   11. Taylor expanded in EDonor around inf 48.8%

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

       1. *-commutative 48.8%

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

   13. Simplified 48.8%

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

   14. Taylor expanded in KbT around inf 70.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq 8.5 \cdot 10^{-185}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(\left(\left(1 + \frac{EDonor}{KbT}\right) + mu \cdot \left(\frac{1}{KbT} + \frac{Vef}{mu \cdot KbT}\right)\right) - \frac{Ec}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\\ \mathbf{if}\;mu \leq 4.1 \cdot 10^{-185}:\\ \;\;\;\;\frac{NdChar}{1 + mu \cdot \left(\frac{1}{KbT} - \frac{\frac{Ec}{KbT} + \left(-1 - \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right)}{mu}\right)} - \frac{NaChar}{-1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
   (if (<= mu 4.1e-185)
     (-
      (/
       NdChar
       (+
        1.0
        (*
         mu
         (-
          (/ 1.0 KbT)
          (/ (+ (/ Ec KbT) (- -1.0 (+ (/ Vef KbT) (/ EDonor KbT)))) mu)))))
      (/ NaChar (- -1.0 t_0)))
     (+ NdChar (/ NaChar (+ 1.0 t_0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double tmp;
	if (mu <= 4.1e-185) {
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = NdChar + (NaChar / (1.0 + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((vef + (ev + (eaccept - mu))) / kbt))
    if (mu <= 4.1d-185) then
        tmp = (ndchar / (1.0d0 + (mu * ((1.0d0 / kbt) - (((ec / kbt) + ((-1.0d0) - ((vef / kbt) + (edonor / kbt)))) / mu))))) - (nachar / ((-1.0d0) - t_0))
    else
        tmp = ndchar + (nachar / (1.0d0 + t_0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double tmp;
	if (mu <= 4.1e-185) {
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0));
	} else {
		tmp = NdChar + (NaChar / (1.0 + t_0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))
	tmp = 0
	if mu <= 4.1e-185:
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0))
	else:
		tmp = NdChar + (NaChar / (1.0 + t_0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))
	tmp = 0.0
	if (mu <= 4.1e-185)
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(mu * Float64(Float64(1.0 / KbT) - Float64(Float64(Float64(Ec / KbT) + Float64(-1.0 - Float64(Float64(Vef / KbT) + Float64(EDonor / KbT)))) / mu))))) - Float64(NaChar / Float64(-1.0 - t_0)));
	else
		tmp = Float64(NdChar + Float64(NaChar / Float64(1.0 + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	tmp = 0.0;
	if (mu <= 4.1e-185)
		tmp = (NdChar / (1.0 + (mu * ((1.0 / KbT) - (((Ec / KbT) + (-1.0 - ((Vef / KbT) + (EDonor / KbT)))) / mu))))) - (NaChar / (-1.0 - t_0));
	else
		tmp = NdChar + (NaChar / (1.0 + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[mu, 4.1e-185], N[(N[(NdChar / N[(1.0 + N[(mu * N[(N[(1.0 / KbT), $MachinePrecision] - N[(N[(N[(Ec / KbT), $MachinePrecision] + N[(-1.0 - N[(N[(Vef / KbT), $MachinePrecision] + N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar + N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if mu < 4.1e-185

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 67.3%

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

      1. associate-+r+ 67.3%

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

   7. Simplified 67.3%

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

   8. Taylor expanded in mu around -inf 74.0%

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

   if 4.1e-185 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.3%

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

      1. associate-+r+ 46.3%

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

   7. Simplified 46.3%

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

   8. Taylor expanded in Ec around -inf 47.4%

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

      1. associate-*r* 47.4%

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

      2. mul-1-neg 47.4%

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

      3. +-commutative 47.4%

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

      4. mul-1-neg 47.4%

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

      5. unsub-neg 47.4%

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

   10. Simplified 47.4%

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

   11. Taylor expanded in EDonor around inf 48.8%

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

       1. *-commutative 48.8%

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

   13. Simplified 48.8%

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

   14. Taylor expanded in KbT around inf 70.8%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq 4.1 \cdot 10^{-185}:\\ \;\;\;\;\frac{NdChar}{1 + mu \cdot \left(\frac{1}{KbT} - \frac{\frac{Ec}{KbT} + \left(-1 - \left(\frac{Vef}{KbT} + \frac{EDonor}{KbT}\right)\right)}{mu}\right)} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;mu \leq 2 \cdot 10^{-185}:\\ \;\;\;\;\frac{NdChar}{1 - \left(\frac{Ec}{KbT} + \left(\left(-1 - \frac{EDonor}{KbT}\right) - \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)} - NaChar \cdot \frac{1}{-1 - e^{\frac{\left(EAccept - mu\right) + \left(Vef + Ev\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= mu 2e-185)
   (-
    (/
     NdChar
     (-
      1.0
      (+ (/ Ec KbT) (- (- -1.0 (/ EDonor KbT)) (+ (/ Vef KbT) (/ mu KbT))))))
    (* NaChar (/ 1.0 (- -1.0 (exp (/ (+ (- EAccept mu) (+ Vef Ev)) KbT))))))
   (+ NdChar (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (mu <= 2e-185) {
		tmp = (NdChar / (1.0 - ((Ec / KbT) + ((-1.0 - (EDonor / KbT)) - ((Vef / KbT) + (mu / KbT)))))) - (NaChar * (1.0 / (-1.0 - exp((((EAccept - mu) + (Vef + Ev)) / KbT)))));
	} else {
		tmp = NdChar + (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (mu <= 2d-185) then
        tmp = (ndchar / (1.0d0 - ((ec / kbt) + (((-1.0d0) - (edonor / kbt)) - ((vef / kbt) + (mu / kbt)))))) - (nachar * (1.0d0 / ((-1.0d0) - exp((((eaccept - mu) + (vef + ev)) / kbt)))))
    else
        tmp = ndchar + (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (mu <= 2e-185) {
		tmp = (NdChar / (1.0 - ((Ec / KbT) + ((-1.0 - (EDonor / KbT)) - ((Vef / KbT) + (mu / KbT)))))) - (NaChar * (1.0 / (-1.0 - Math.exp((((EAccept - mu) + (Vef + Ev)) / KbT)))));
	} else {
		tmp = NdChar + (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if mu <= 2e-185:
		tmp = (NdChar / (1.0 - ((Ec / KbT) + ((-1.0 - (EDonor / KbT)) - ((Vef / KbT) + (mu / KbT)))))) - (NaChar * (1.0 / (-1.0 - math.exp((((EAccept - mu) + (Vef + Ev)) / KbT)))))
	else:
		tmp = NdChar + (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (mu <= 2e-185)
		tmp = Float64(Float64(NdChar / Float64(1.0 - Float64(Float64(Ec / KbT) + Float64(Float64(-1.0 - Float64(EDonor / KbT)) - Float64(Float64(Vef / KbT) + Float64(mu / KbT)))))) - Float64(NaChar * Float64(1.0 / Float64(-1.0 - exp(Float64(Float64(Float64(EAccept - mu) + Float64(Vef + Ev)) / KbT))))));
	else
		tmp = Float64(NdChar + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (mu <= 2e-185)
		tmp = (NdChar / (1.0 - ((Ec / KbT) + ((-1.0 - (EDonor / KbT)) - ((Vef / KbT) + (mu / KbT)))))) - (NaChar * (1.0 / (-1.0 - exp((((EAccept - mu) + (Vef + Ev)) / KbT)))));
	else
		tmp = NdChar + (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[mu, 2e-185], N[(N[(NdChar / N[(1.0 - N[(N[(Ec / KbT), $MachinePrecision] + N[(N[(-1.0 - N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] - N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar * N[(1.0 / N[(-1.0 - N[Exp[N[(N[(N[(EAccept - mu), $MachinePrecision] + N[(Vef + Ev), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if mu < 2e-185

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 67.3%

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

      1. associate-+r+ 67.3%

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

   7. Simplified 67.3%

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

      1. div-inv 67.3%

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

      2. associate-+r+ 67.3%

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

   9. Applied egg-rr 67.3%

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

   if 2e-185 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.3%

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

      1. associate-+r+ 46.3%

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

   7. Simplified 46.3%

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

   8. Taylor expanded in Ec around -inf 47.4%

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

      1. associate-*r* 47.4%

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

      2. mul-1-neg 47.4%

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

      3. +-commutative 47.4%

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

      4. mul-1-neg 47.4%

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

      5. unsub-neg 47.4%

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

   10. Simplified 47.4%

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

   11. Taylor expanded in EDonor around inf 48.8%

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

       1. *-commutative 48.8%

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

   13. Simplified 48.8%

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

   14. Taylor expanded in KbT around inf 70.8%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq 2 \cdot 10^{-185}:\\ \;\;\;\;\frac{NdChar}{1 - \left(\frac{Ec}{KbT} + \left(\left(-1 - \frac{EDonor}{KbT}\right) - \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)} - NaChar \cdot \frac{1}{-1 - e^{\frac{\left(EAccept - mu\right) + \left(Vef + Ev\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{if}\;KbT \leq -7.8 \cdot 10^{+198}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq 3.2 \cdot 10^{-198}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{elif}\;KbT \leq 3.9 \cdot 10^{-73}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 2 \cdot 10^{+109}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
          (* NaChar 0.5))))
   (if (<= KbT -7.8e+198)
     t_1
     (if (<= KbT 3.2e-198)
       (+ NdChar t_0)
       (if (<= KbT 3.9e-73)
         (+ t_0 (/ NdChar (+ 1.0 (/ mu KbT))))
         (if (<= KbT 2e+109) (+ t_0 (/ NdChar (- 1.0 (/ Ec 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(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (KbT <= -7.8e+198) {
		tmp = t_1;
	} else if (KbT <= 3.2e-198) {
		tmp = NdChar + t_0;
	} else if (KbT <= 3.9e-73) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if (KbT <= 2e+109) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / 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(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar * 0.5d0)
    if (kbt <= (-7.8d+198)) then
        tmp = t_1
    else if (kbt <= 3.2d-198) then
        tmp = ndchar + t_0
    else if (kbt <= 3.9d-73) then
        tmp = t_0 + (ndchar / (1.0d0 + (mu / kbt)))
    else if (kbt <= 2d+109) then
        tmp = t_0 + (ndchar / (1.0d0 - (ec / 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(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (KbT <= -7.8e+198) {
		tmp = t_1;
	} else if (KbT <= 3.2e-198) {
		tmp = NdChar + t_0;
	} else if (KbT <= 3.9e-73) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if (KbT <= 2e+109) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / 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(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5)
	tmp = 0
	if KbT <= -7.8e+198:
		tmp = t_1
	elif KbT <= 3.2e-198:
		tmp = NdChar + t_0
	elif KbT <= 3.9e-73:
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)))
	elif KbT <= 2e+109:
		tmp = t_0 + (NdChar / (1.0 - (Ec / 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(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar * 0.5))
	tmp = 0.0
	if (KbT <= -7.8e+198)
		tmp = t_1;
	elseif (KbT <= 3.2e-198)
		tmp = Float64(NdChar + t_0);
	elseif (KbT <= 3.9e-73)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(mu / KbT))));
	elseif (KbT <= 2e+109)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 - Float64(Ec / 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(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	tmp = 0.0;
	if (KbT <= -7.8e+198)
		tmp = t_1;
	elseif (KbT <= 3.2e-198)
		tmp = NdChar + t_0;
	elseif (KbT <= 3.9e-73)
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	elseif (KbT <= 2e+109)
		tmp = t_0 + (NdChar / (1.0 - (Ec / 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[(N[(EDonor + N[(mu + N[(Vef - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -7.8e+198], t$95$1, If[LessEqual[KbT, 3.2e-198], N[(NdChar + t$95$0), $MachinePrecision], If[LessEqual[KbT, 3.9e-73], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2e+109], N[(t$95$0 + N[(NdChar / N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -7.8e198 or 1.99999999999999996e109 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 87.1%

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

   6. Taylor expanded in EAccept around 0 77.3%

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

   if -7.8e198 < KbT < 3.19999999999999994e-198

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 50.5%

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

      1. associate-+r+ 50.5%

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

   7. Simplified 50.5%

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

   8. Taylor expanded in Ec around -inf 52.7%

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

      1. associate-*r* 52.7%

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

      2. mul-1-neg 52.7%

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

      3. +-commutative 52.7%

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

      4. mul-1-neg 52.7%

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

      5. unsub-neg 52.7%

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

   10. Simplified 52.7%

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

   11. Taylor expanded in EDonor around inf 57.5%

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

       1. *-commutative 57.5%

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

   13. Simplified 57.5%

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

   14. Taylor expanded in KbT around inf 68.1%

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

   if 3.19999999999999994e-198 < KbT < 3.89999999999999982e-73

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 53.4%

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

      1. associate-+r+ 53.4%

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

   7. Simplified 53.4%

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

   8. Taylor expanded in Ec around -inf 53.4%

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

      1. associate-*r* 53.4%

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

      2. mul-1-neg 53.4%

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

      3. +-commutative 53.4%

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

      4. mul-1-neg 53.4%

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

      5. unsub-neg 53.4%

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

   10. Simplified 53.4%

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

   11. Taylor expanded in mu around inf 72.8%

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

   if 3.89999999999999982e-73 < KbT < 1.99999999999999996e109

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around inf 60.1%

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

      1. associate-+r+ 60.1%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in Ec around -inf 63.6%

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

      1. associate-*r* 63.6%

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

      2. mul-1-neg 63.6%

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

      3. +-commutative 63.6%

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

      4. mul-1-neg 63.6%

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

      5. unsub-neg 63.6%

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

   10. Simplified 63.6%

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

   11. Taylor expanded in EDonor around inf 58.4%

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

       1. *-commutative 58.4%

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

   13. Simplified 58.4%

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

   14. Taylor expanded in EDonor around 0 58.2%

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

       1. mul-1-neg 58.2%

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

       2. unsub-neg 58.2%

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

   16. Simplified 58.2%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -7.8 \cdot 10^{+198}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 3.2 \cdot 10^{-198}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;KbT \leq 3.9 \cdot 10^{-73}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 2 \cdot 10^{+109}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}}\\ \mathbf{if}\;KbT \leq -2.75 \cdot 10^{+205}:\\ \;\;\;\;t\_1 + \left(NaChar \cdot 0.5 + -0.25 \cdot \frac{NaChar \cdot Ev}{KbT}\right)\\ \mathbf{elif}\;KbT \leq 1.26 \cdot 10^{-198}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{elif}\;KbT \leq 3.4 \cdot 10^{-69}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.8 \cdot 10^{+108}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + NaChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1 (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))))
   (if (<= KbT -2.75e+205)
     (+ t_1 (+ (* NaChar 0.5) (* -0.25 (/ (* NaChar Ev) KbT))))
     (if (<= KbT 1.26e-198)
       (+ NdChar t_0)
       (if (<= KbT 3.4e-69)
         (+ t_0 (/ NdChar (+ 1.0 (/ mu KbT))))
         (if (<= KbT 1.8e+108)
           (+ t_0 (/ NdChar (- 1.0 (/ Ec KbT))))
           (+ t_1 (* NaChar 0.5))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double tmp;
	if (KbT <= -2.75e+205) {
		tmp = t_1 + ((NaChar * 0.5) + (-0.25 * ((NaChar * Ev) / KbT)));
	} else if (KbT <= 1.26e-198) {
		tmp = NdChar + t_0;
	} else if (KbT <= 3.4e-69) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if (KbT <= 1.8e+108) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	} else {
		tmp = t_1 + (NaChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    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)))
    if (kbt <= (-2.75d+205)) then
        tmp = t_1 + ((nachar * 0.5d0) + ((-0.25d0) * ((nachar * ev) / kbt)))
    else if (kbt <= 1.26d-198) then
        tmp = ndchar + t_0
    else if (kbt <= 3.4d-69) then
        tmp = t_0 + (ndchar / (1.0d0 + (mu / kbt)))
    else if (kbt <= 1.8d+108) then
        tmp = t_0 + (ndchar / (1.0d0 - (ec / kbt)))
    else
        tmp = t_1 + (nachar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	double tmp;
	if (KbT <= -2.75e+205) {
		tmp = t_1 + ((NaChar * 0.5) + (-0.25 * ((NaChar * Ev) / KbT)));
	} else if (KbT <= 1.26e-198) {
		tmp = NdChar + t_0;
	} else if (KbT <= 3.4e-69) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if (KbT <= 1.8e+108) {
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	} else {
		tmp = t_1 + (NaChar * 0.5);
	}
	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)))
	tmp = 0
	if KbT <= -2.75e+205:
		tmp = t_1 + ((NaChar * 0.5) + (-0.25 * ((NaChar * Ev) / KbT)))
	elif KbT <= 1.26e-198:
		tmp = NdChar + t_0
	elif KbT <= 3.4e-69:
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)))
	elif KbT <= 1.8e+108:
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)))
	else:
		tmp = t_1 + (NaChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	t_1 = Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT))))
	tmp = 0.0
	if (KbT <= -2.75e+205)
		tmp = Float64(t_1 + Float64(Float64(NaChar * 0.5) + Float64(-0.25 * Float64(Float64(NaChar * Ev) / KbT))));
	elseif (KbT <= 1.26e-198)
		tmp = Float64(NdChar + t_0);
	elseif (KbT <= 3.4e-69)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(mu / KbT))));
	elseif (KbT <= 1.8e+108)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 - Float64(Ec / KbT))));
	else
		tmp = Float64(t_1 + Float64(NaChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)));
	tmp = 0.0;
	if (KbT <= -2.75e+205)
		tmp = t_1 + ((NaChar * 0.5) + (-0.25 * ((NaChar * Ev) / KbT)));
	elseif (KbT <= 1.26e-198)
		tmp = NdChar + t_0;
	elseif (KbT <= 3.4e-69)
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	elseif (KbT <= 1.8e+108)
		tmp = t_0 + (NdChar / (1.0 - (Ec / KbT)));
	else
		tmp = t_1 + (NaChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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[KbT, -2.75e+205], N[(t$95$1 + N[(N[(NaChar * 0.5), $MachinePrecision] + N[(-0.25 * N[(N[(NaChar * Ev), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.26e-198], N[(NdChar + t$95$0), $MachinePrecision], If[LessEqual[KbT, 3.4e-69], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.8e+108], N[(t$95$0 + N[(NdChar / N[(1.0 - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(NaChar * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if KbT < -2.75000000000000002e205

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Ev around inf 88.4%

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

   6. Taylor expanded in Ev around 0 85.6%

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

   if -2.75000000000000002e205 < KbT < 1.25999999999999992e-198

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 50.5%

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

      1. associate-+r+ 50.5%

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

   7. Simplified 50.5%

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

   8. Taylor expanded in Ec around -inf 52.7%

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

      1. associate-*r* 52.7%

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

      2. mul-1-neg 52.7%

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

      3. +-commutative 52.7%

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

      4. mul-1-neg 52.7%

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

      5. unsub-neg 52.7%

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

   10. Simplified 52.7%

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

   11. Taylor expanded in EDonor around inf 57.5%

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

       1. *-commutative 57.5%

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

   13. Simplified 57.5%

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

   14. Taylor expanded in KbT around inf 68.1%

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

   if 1.25999999999999992e-198 < KbT < 3.40000000000000008e-69

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 53.4%

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

      1. associate-+r+ 53.4%

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

   7. Simplified 53.4%

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

   8. Taylor expanded in Ec around -inf 53.4%

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

      1. associate-*r* 53.4%

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

      2. mul-1-neg 53.4%

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

      3. +-commutative 53.4%

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

      4. mul-1-neg 53.4%

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

      5. unsub-neg 53.4%

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

   10. Simplified 53.4%

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

   11. Taylor expanded in mu around inf 72.8%

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

   if 3.40000000000000008e-69 < KbT < 1.8e108

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in KbT around inf 60.1%

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

      1. associate-+r+ 60.1%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in Ec around -inf 63.6%

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

      1. associate-*r* 63.6%

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

      2. mul-1-neg 63.6%

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

      3. +-commutative 63.6%

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

      4. mul-1-neg 63.6%

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

      5. unsub-neg 63.6%

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

   10. Simplified 63.6%

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

   11. Taylor expanded in EDonor around inf 58.4%

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

       1. *-commutative 58.4%

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

   13. Simplified 58.4%

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

   14. Taylor expanded in EDonor around 0 58.2%

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

       1. mul-1-neg 58.2%

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

       2. unsub-neg 58.2%

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

   16. Simplified 58.2%

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

   if 1.8e108 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 80.3%

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

   6. Taylor expanded in EAccept around 0 69.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.75 \cdot 10^{+205}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + \left(NaChar \cdot 0.5 + -0.25 \cdot \frac{NaChar \cdot Ev}{KbT}\right)\\ \mathbf{elif}\;KbT \leq 1.26 \cdot 10^{-198}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;KbT \leq 3.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;mu \leq 2.05 \cdot 10^{-185}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 - \left(\frac{Ec}{KbT} + \left(\left(-1 - \frac{EDonor}{KbT}\right) - \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;NdChar + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= mu 2.05e-185)
     (+
      t_0
      (/
       NdChar
       (-
        1.0
        (+
         (/ Ec KbT)
         (- (- -1.0 (/ EDonor KbT)) (+ (/ Vef KbT) (/ mu KbT)))))))
     (+ NdChar t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (mu <= 2.05e-185) {
		tmp = t_0 + (NdChar / (1.0 - ((Ec / KbT) + ((-1.0 - (EDonor / KbT)) - ((Vef / KbT) + (mu / KbT))))));
	} else {
		tmp = NdChar + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (mu <= 2.05d-185) then
        tmp = t_0 + (ndchar / (1.0d0 - ((ec / kbt) + (((-1.0d0) - (edonor / kbt)) - ((vef / kbt) + (mu / kbt))))))
    else
        tmp = ndchar + t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (mu <= 2.05e-185) {
		tmp = t_0 + (NdChar / (1.0 - ((Ec / KbT) + ((-1.0 - (EDonor / KbT)) - ((Vef / KbT) + (mu / KbT))))));
	} else {
		tmp = NdChar + t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if mu <= 2.05e-185:
		tmp = t_0 + (NdChar / (1.0 - ((Ec / KbT) + ((-1.0 - (EDonor / KbT)) - ((Vef / KbT) + (mu / KbT))))))
	else:
		tmp = NdChar + t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (mu <= 2.05e-185)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 - Float64(Float64(Ec / KbT) + Float64(Float64(-1.0 - Float64(EDonor / KbT)) - Float64(Float64(Vef / KbT) + Float64(mu / KbT)))))));
	else
		tmp = Float64(NdChar + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (mu <= 2.05e-185)
		tmp = t_0 + (NdChar / (1.0 - ((Ec / KbT) + ((-1.0 - (EDonor / KbT)) - ((Vef / KbT) + (mu / KbT))))));
	else
		tmp = NdChar + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, 2.05e-185], N[(t$95$0 + N[(NdChar / N[(1.0 - N[(N[(Ec / KbT), $MachinePrecision] + N[(N[(-1.0 - N[(EDonor / KbT), $MachinePrecision]), $MachinePrecision] - N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if mu < 2.05e-185

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 67.3%

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

      1. associate-+r+ 67.3%

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

   7. Simplified 67.3%

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

   if 2.05e-185 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.3%

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

      1. associate-+r+ 46.3%

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

   7. Simplified 46.3%

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

   8. Taylor expanded in Ec around -inf 47.4%

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

      1. associate-*r* 47.4%

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

      2. mul-1-neg 47.4%

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

      3. +-commutative 47.4%

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

      4. mul-1-neg 47.4%

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

      5. unsub-neg 47.4%

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

   10. Simplified 47.4%

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

   11. Taylor expanded in EDonor around inf 48.8%

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

       1. *-commutative 48.8%

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

   13. Simplified 48.8%

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

   14. Taylor expanded in KbT around inf 70.8%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq 2.05 \cdot 10^{-185}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 - \left(\frac{Ec}{KbT} + \left(\left(-1 - \frac{EDonor}{KbT}\right) - \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\\ \mathbf{if}\;NaChar \leq -1.05 \cdot 10^{-107}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + t\_0}\\ \mathbf{elif}\;NaChar \leq 1.45 \cdot 10^{-166}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 - EDonor \cdot \left(\frac{Ec}{EDonor \cdot KbT} + \frac{-1}{KbT}\right)} - \frac{NaChar}{-1 - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))
   (if (<= NaChar -1.05e-107)
     (+ NdChar (/ NaChar (+ 1.0 t_0)))
     (if (<= NaChar 1.45e-166)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
        (* NaChar 0.5))
       (-
        (/ NdChar (- 1.0 (* EDonor (+ (/ Ec (* EDonor KbT)) (/ -1.0 KbT)))))
        (/ NaChar (- -1.0 t_0)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double tmp;
	if (NaChar <= -1.05e-107) {
		tmp = NdChar + (NaChar / (1.0 + t_0));
	} else if (NaChar <= 1.45e-166) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 - (EDonor * ((Ec / (EDonor * KbT)) + (-1.0 / KbT))))) - (NaChar / (-1.0 - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((vef + (ev + (eaccept - mu))) / kbt))
    if (nachar <= (-1.05d-107)) then
        tmp = ndchar + (nachar / (1.0d0 + t_0))
    else if (nachar <= 1.45d-166) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar * 0.5d0)
    else
        tmp = (ndchar / (1.0d0 - (edonor * ((ec / (edonor * kbt)) + ((-1.0d0) / kbt))))) - (nachar / ((-1.0d0) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double tmp;
	if (NaChar <= -1.05e-107) {
		tmp = NdChar + (NaChar / (1.0 + t_0));
	} else if (NaChar <= 1.45e-166) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 - (EDonor * ((Ec / (EDonor * KbT)) + (-1.0 / KbT))))) - (NaChar / (-1.0 - t_0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))
	tmp = 0
	if NaChar <= -1.05e-107:
		tmp = NdChar + (NaChar / (1.0 + t_0))
	elif NaChar <= 1.45e-166:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5)
	else:
		tmp = (NdChar / (1.0 - (EDonor * ((Ec / (EDonor * KbT)) + (-1.0 / KbT))))) - (NaChar / (-1.0 - t_0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))
	tmp = 0.0
	if (NaChar <= -1.05e-107)
		tmp = Float64(NdChar + Float64(NaChar / Float64(1.0 + t_0)));
	elseif (NaChar <= 1.45e-166)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 - Float64(EDonor * Float64(Float64(Ec / Float64(EDonor * KbT)) + Float64(-1.0 / KbT))))) - Float64(NaChar / Float64(-1.0 - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	tmp = 0.0;
	if (NaChar <= -1.05e-107)
		tmp = NdChar + (NaChar / (1.0 + t_0));
	elseif (NaChar <= 1.45e-166)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	else
		tmp = (NdChar / (1.0 - (EDonor * ((Ec / (EDonor * KbT)) + (-1.0 / KbT))))) - (NaChar / (-1.0 - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[NaChar, -1.05e-107], N[(NdChar + N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[NaChar, 1.45e-166], 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 * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 - N[(EDonor * N[(N[(Ec / N[(EDonor * KbT), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -1.05e-107

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 56.7%

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

      1. associate-+r+ 56.7%

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

   7. Simplified 56.7%

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

   8. Taylor expanded in Ec around -inf 59.4%

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

      1. associate-*r* 59.4%

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

      2. mul-1-neg 59.4%

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

      3. +-commutative 59.4%

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

      4. mul-1-neg 59.4%

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

      5. unsub-neg 59.4%

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

   10. Simplified 59.4%

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

   11. Taylor expanded in EDonor around inf 60.3%

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

       1. *-commutative 60.3%

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

   13. Simplified 60.3%

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

   14. Taylor expanded in KbT around inf 72.3%

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

   if -1.05e-107 < NaChar < 1.45e-166

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 85.8%

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

   6. Taylor expanded in EAccept around 0 68.8%

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

   if 1.45e-166 < 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. Add Preprocessing

   5. Taylor expanded in KbT around inf 66.7%

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

      1. associate-+r+ 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in Ec around -inf 66.7%

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

      1. associate-*r* 66.7%

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

      2. mul-1-neg 66.7%

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

      3. +-commutative 66.7%

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

      4. mul-1-neg 66.7%

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

      5. unsub-neg 66.7%

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

   10. Simplified 66.7%

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

   11. Taylor expanded in EDonor around inf 64.1%

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

       1. *-commutative 64.1%

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

   13. Simplified 64.1%

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

   14. Taylor expanded in EDonor around inf 64.9%

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

       1. +-commutative 64.9%

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

       2. mul-1-neg 64.9%

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

       3. unsub-neg 64.9%

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

       4. *-commutative 64.9%

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

   16. Simplified 64.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -1.05 \cdot 10^{-107}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.45 \cdot 10^{-166}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 - EDonor \cdot \left(\frac{Ec}{EDonor \cdot KbT} + \frac{-1}{KbT}\right)} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}\\ t_1 := NdChar + \frac{NaChar}{1 + t\_0}\\ t_2 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{if}\;KbT \leq -9 \cdot 10^{+198}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;KbT \leq 2.2 \cdot 10^{-193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq 8.2 \cdot 10^{-116}:\\ \;\;\;\;KbT \cdot \frac{NdChar}{mu} - \frac{NaChar}{-1 - t\_0}\\ \mathbf{elif}\;KbT \leq 9.8 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))
        (t_1 (+ NdChar (/ NaChar (+ 1.0 t_0))))
        (t_2
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
          (* NaChar 0.5))))
   (if (<= KbT -9e+198)
     t_2
     (if (<= KbT 2.2e-193)
       t_1
       (if (<= KbT 8.2e-116)
         (- (* KbT (/ NdChar mu)) (/ NaChar (- -1.0 t_0)))
         (if (<= KbT 9.8e+166) t_1 t_2))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double t_1 = NdChar + (NaChar / (1.0 + t_0));
	double t_2 = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (KbT <= -9e+198) {
		tmp = t_2;
	} else if (KbT <= 2.2e-193) {
		tmp = t_1;
	} else if (KbT <= 8.2e-116) {
		tmp = (KbT * (NdChar / mu)) - (NaChar / (-1.0 - t_0));
	} else if (KbT <= 9.8e+166) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(((vef + (ev + (eaccept - mu))) / kbt))
    t_1 = ndchar + (nachar / (1.0d0 + t_0))
    t_2 = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar * 0.5d0)
    if (kbt <= (-9d+198)) then
        tmp = t_2
    else if (kbt <= 2.2d-193) then
        tmp = t_1
    else if (kbt <= 8.2d-116) then
        tmp = (kbt * (ndchar / mu)) - (nachar / ((-1.0d0) - t_0))
    else if (kbt <= 9.8d+166) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	double t_1 = NdChar + (NaChar / (1.0 + t_0));
	double t_2 = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (KbT <= -9e+198) {
		tmp = t_2;
	} else if (KbT <= 2.2e-193) {
		tmp = t_1;
	} else if (KbT <= 8.2e-116) {
		tmp = (KbT * (NdChar / mu)) - (NaChar / (-1.0 - t_0));
	} else if (KbT <= 9.8e+166) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))
	t_1 = NdChar + (NaChar / (1.0 + t_0))
	t_2 = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5)
	tmp = 0
	if KbT <= -9e+198:
		tmp = t_2
	elif KbT <= 2.2e-193:
		tmp = t_1
	elif KbT <= 8.2e-116:
		tmp = (KbT * (NdChar / mu)) - (NaChar / (-1.0 - t_0))
	elif KbT <= 9.8e+166:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))
	t_1 = Float64(NdChar + Float64(NaChar / Float64(1.0 + t_0)))
	t_2 = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar * 0.5))
	tmp = 0.0
	if (KbT <= -9e+198)
		tmp = t_2;
	elseif (KbT <= 2.2e-193)
		tmp = t_1;
	elseif (KbT <= 8.2e-116)
		tmp = Float64(Float64(KbT * Float64(NdChar / mu)) - Float64(NaChar / Float64(-1.0 - t_0)));
	elseif (KbT <= 9.8e+166)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp(((Vef + (Ev + (EAccept - mu))) / KbT));
	t_1 = NdChar + (NaChar / (1.0 + t_0));
	t_2 = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	tmp = 0.0;
	if (KbT <= -9e+198)
		tmp = t_2;
	elseif (KbT <= 2.2e-193)
		tmp = t_1;
	elseif (KbT <= 8.2e-116)
		tmp = (KbT * (NdChar / mu)) - (NaChar / (-1.0 - t_0));
	elseif (KbT <= 9.8e+166)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(NdChar + N[(NaChar / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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 * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -9e+198], t$95$2, If[LessEqual[KbT, 2.2e-193], t$95$1, If[LessEqual[KbT, 8.2e-116], N[(N[(KbT * N[(NdChar / mu), $MachinePrecision]), $MachinePrecision] - N[(NaChar / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 9.8e+166], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -9.00000000000000003e198 or 9.79999999999999938e166 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 89.4%

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

   6. Taylor expanded in EAccept around 0 82.5%

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

   if -9.00000000000000003e198 < KbT < 2.19999999999999977e-193 or 8.1999999999999998e-116 < KbT < 9.79999999999999938e166

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.3%

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

      1. associate-+r+ 52.3%

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

   7. Simplified 52.3%

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

   8. Taylor expanded in Ec around -inf 54.4%

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

      1. associate-*r* 54.4%

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

      2. mul-1-neg 54.4%

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

      3. +-commutative 54.4%

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

      4. mul-1-neg 54.4%

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

      5. unsub-neg 54.4%

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

   10. Simplified 54.4%

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

   11. Taylor expanded in EDonor around inf 54.9%

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

       1. *-commutative 54.9%

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

   13. Simplified 54.9%

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

   14. Taylor expanded in KbT around inf 64.1%

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

   if 2.19999999999999977e-193 < KbT < 8.1999999999999998e-116

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 62.1%

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

      1. associate-+r+ 62.1%

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

   7. Simplified 62.1%

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

   8. Taylor expanded in mu around inf 85.6%

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

      1. associate-/l* 85.6%

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

   10. Simplified 85.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -9 \cdot 10^{+198}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 2.2 \cdot 10^{-193}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;KbT \leq 8.2 \cdot 10^{-116}:\\ \;\;\;\;KbT \cdot \frac{NdChar}{mu} - \frac{NaChar}{-1 - e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;KbT \leq 9.8 \cdot 10^{+166}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ t_1 := \frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{if}\;KbT \leq -3.05 \cdot 10^{+204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq 4.2 \cdot 10^{-198}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{elif}\;KbT \leq 5.4 \cdot 10^{+128}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.2 \cdot 10^{+170}:\\ \;\;\;\;t\_0 + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
        (t_1
         (+
          (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
          (* NaChar 0.5))))
   (if (<= KbT -3.05e+204)
     t_1
     (if (<= KbT 4.2e-198)
       (+ NdChar t_0)
       (if (<= KbT 5.4e+128)
         (+ t_0 (/ NdChar (+ 1.0 (/ mu KbT))))
         (if (<= KbT 1.2e+170) (+ t_0 (* NdChar 0.5)) t_1))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (KbT <= -3.05e+204) {
		tmp = t_1;
	} else if (KbT <= 4.2e-198) {
		tmp = NdChar + t_0;
	} else if (KbT <= 5.4e+128) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if (KbT <= 1.2e+170) {
		tmp = t_0 + (NdChar * 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    t_1 = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar * 0.5d0)
    if (kbt <= (-3.05d+204)) then
        tmp = t_1
    else if (kbt <= 4.2d-198) then
        tmp = ndchar + t_0
    else if (kbt <= 5.4d+128) then
        tmp = t_0 + (ndchar / (1.0d0 + (mu / kbt)))
    else if (kbt <= 1.2d+170) then
        tmp = t_0 + (ndchar * 0.5d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double t_1 = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	double tmp;
	if (KbT <= -3.05e+204) {
		tmp = t_1;
	} else if (KbT <= 4.2e-198) {
		tmp = NdChar + t_0;
	} else if (KbT <= 5.4e+128) {
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	} else if (KbT <= 1.2e+170) {
		tmp = t_0 + (NdChar * 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	t_1 = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5)
	tmp = 0
	if KbT <= -3.05e+204:
		tmp = t_1
	elif KbT <= 4.2e-198:
		tmp = NdChar + t_0
	elif KbT <= 5.4e+128:
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)))
	elif KbT <= 1.2e+170:
		tmp = t_0 + (NdChar * 0.5)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(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 * 0.5))
	tmp = 0.0
	if (KbT <= -3.05e+204)
		tmp = t_1;
	elseif (KbT <= 4.2e-198)
		tmp = Float64(NdChar + t_0);
	elseif (KbT <= 5.4e+128)
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(mu / KbT))));
	elseif (KbT <= 1.2e+170)
		tmp = Float64(t_0 + Float64(NdChar * 0.5));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	t_1 = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	tmp = 0.0;
	if (KbT <= -3.05e+204)
		tmp = t_1;
	elseif (KbT <= 4.2e-198)
		tmp = NdChar + t_0;
	elseif (KbT <= 5.4e+128)
		tmp = t_0 + (NdChar / (1.0 + (mu / KbT)));
	elseif (KbT <= 1.2e+170)
		tmp = t_0 + (NdChar * 0.5);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(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 * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -3.05e+204], t$95$1, If[LessEqual[KbT, 4.2e-198], N[(NdChar + t$95$0), $MachinePrecision], If[LessEqual[KbT, 5.4e+128], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.2e+170], N[(t$95$0 + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if KbT < -3.04999999999999972e204 or 1.2e170 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 89.4%

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

   6. Taylor expanded in EAccept around 0 82.5%

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

   if -3.04999999999999972e204 < KbT < 4.19999999999999986e-198

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 50.5%

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

      1. associate-+r+ 50.5%

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

   7. Simplified 50.5%

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

   8. Taylor expanded in Ec around -inf 52.7%

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

      1. associate-*r* 52.7%

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

      2. mul-1-neg 52.7%

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

      3. +-commutative 52.7%

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

      4. mul-1-neg 52.7%

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

      5. unsub-neg 52.7%

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

   10. Simplified 52.7%

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

   11. Taylor expanded in EDonor around inf 57.5%

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

       1. *-commutative 57.5%

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

   13. Simplified 57.5%

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

   14. Taylor expanded in KbT around inf 68.1%

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

   if 4.19999999999999986e-198 < KbT < 5.40000000000000002e128

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 56.6%

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

      1. associate-+r+ 56.6%

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

   7. Simplified 56.6%

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

   8. Taylor expanded in Ec around -inf 58.4%

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

      1. associate-*r* 58.4%

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

      2. mul-1-neg 58.4%

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

      3. +-commutative 58.4%

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

      4. mul-1-neg 58.4%

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

      5. unsub-neg 58.4%

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

   10. Simplified 58.4%

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

   11. Taylor expanded in mu around inf 59.2%

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

   if 5.40000000000000002e128 < KbT < 1.2e170

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in KbT around inf 63.8%

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

      1. *-commutative 63.8%

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

   7. Simplified 63.8%

      \[\leadsto \color{blue}{NdChar \cdot 0.5} + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.05 \cdot 10^{+204}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 4.2 \cdot 10^{-198}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;KbT \leq 5.4 \cdot 10^{+128}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \frac{mu}{KbT}}\\ \mathbf{elif}\;KbT \leq 1.2 \cdot 10^{+170}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;NaChar \leq -5.8 \cdot 10^{-111}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{elif}\;NaChar \leq 1.1 \cdot 10^{-166}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{NdChar}{1 + \left(\frac{EDonor}{KbT} - \frac{Ec}{KbT}\right)}\\ \end{array} \end{array} \]

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 -5.8e-111)
     (+ NdChar t_0)
     (if (<= NaChar 1.1e-166)
       (+
        (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
        (* NaChar 0.5))
       (+ t_0 (/ NdChar (+ 1.0 (- (/ 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 (NaChar <= -5.8e-111) {
		tmp = NdChar + t_0;
	} else if (NaChar <= 1.1e-166) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = t_0 + (NdChar / (1.0 + ((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 (nachar <= (-5.8d-111)) then
        tmp = ndchar + t_0
    else if (nachar <= 1.1d-166) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar * 0.5d0)
    else
        tmp = t_0 + (ndchar / (1.0d0 + ((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 (NaChar <= -5.8e-111) {
		tmp = NdChar + t_0;
	} else if (NaChar <= 1.1e-166) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = t_0 + (NdChar / (1.0 + ((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 NaChar <= -5.8e-111:
		tmp = NdChar + t_0
	elif NaChar <= 1.1e-166:
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5)
	else:
		tmp = t_0 + (NdChar / (1.0 + ((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 (NaChar <= -5.8e-111)
		tmp = Float64(NdChar + t_0);
	elseif (NaChar <= 1.1e-166)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(t_0 + Float64(NdChar / Float64(1.0 + Float64(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 (NaChar <= -5.8e-111)
		tmp = NdChar + t_0;
	elseif (NaChar <= 1.1e-166)
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	else
		tmp = t_0 + (NdChar / (1.0 + ((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[NaChar, -5.8e-111], N[(NdChar + t$95$0), $MachinePrecision], If[LessEqual[NaChar, 1.1e-166], 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 * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(NdChar / N[(1.0 + N[(N[(EDonor / KbT), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if NaChar < -5.80000000000000003e-111

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 56.7%

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

      1. associate-+r+ 56.7%

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

   7. Simplified 56.7%

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

   8. Taylor expanded in Ec around -inf 59.4%

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

      1. associate-*r* 59.4%

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

      2. mul-1-neg 59.4%

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

      3. +-commutative 59.4%

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

      4. mul-1-neg 59.4%

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

      5. unsub-neg 59.4%

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

   10. Simplified 59.4%

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

   11. Taylor expanded in EDonor around inf 60.3%

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

       1. *-commutative 60.3%

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

   13. Simplified 60.3%

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

   14. Taylor expanded in KbT around inf 72.3%

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

   if -5.80000000000000003e-111 < NaChar < 1.1000000000000001e-166

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 85.8%

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

   6. Taylor expanded in EAccept around 0 68.8%

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

   if 1.1000000000000001e-166 < 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. Add Preprocessing

   5. Taylor expanded in KbT around inf 66.7%

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

      1. associate-+r+ 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in EDonor around inf 64.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -5.8 \cdot 10^{-111}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{elif}\;NaChar \leq 1.1 \cdot 10^{-166}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{1 + \left(\frac{EDonor}{KbT} - \frac{Ec}{KbT}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;mu \leq 8.5 \cdot 10^{-185}:\\ \;\;\;\;t\_0 + \frac{NdChar}{\left(2 + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;NdChar + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= mu 8.5e-185)
     (+ t_0 (/ NdChar (- (+ 2.0 (+ (/ Vef KbT) (/ mu KbT))) (/ Ec KbT))))
     (+ NdChar t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (mu <= 8.5e-185) {
		tmp = t_0 + (NdChar / ((2.0 + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)));
	} else {
		tmp = NdChar + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (mu <= 8.5d-185) then
        tmp = t_0 + (ndchar / ((2.0d0 + ((vef / kbt) + (mu / kbt))) - (ec / kbt)))
    else
        tmp = ndchar + t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (mu <= 8.5e-185) {
		tmp = t_0 + (NdChar / ((2.0 + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)));
	} else {
		tmp = NdChar + t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if mu <= 8.5e-185:
		tmp = t_0 + (NdChar / ((2.0 + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)))
	else:
		tmp = NdChar + t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (mu <= 8.5e-185)
		tmp = Float64(t_0 + Float64(NdChar / Float64(Float64(2.0 + Float64(Float64(Vef / KbT) + Float64(mu / KbT))) - Float64(Ec / KbT))));
	else
		tmp = Float64(NdChar + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (mu <= 8.5e-185)
		tmp = t_0 + (NdChar / ((2.0 + ((Vef / KbT) + (mu / KbT))) - (Ec / KbT)));
	else
		tmp = NdChar + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, 8.5e-185], N[(t$95$0 + N[(NdChar / N[(N[(2.0 + N[(N[(Vef / KbT), $MachinePrecision] + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Ec / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(NdChar + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if mu < 8.5000000000000001e-185

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 67.3%

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

      1. associate-+r+ 67.3%

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

   7. Simplified 67.3%

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

   8. Taylor expanded in EDonor around 0 67.0%

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

   if 8.5000000000000001e-185 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.3%

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

      1. associate-+r+ 46.3%

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

   7. Simplified 46.3%

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

   8. Taylor expanded in Ec around -inf 47.4%

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

      1. associate-*r* 47.4%

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

      2. mul-1-neg 47.4%

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

      3. +-commutative 47.4%

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

      4. mul-1-neg 47.4%

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

      5. unsub-neg 47.4%

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

   10. Simplified 47.4%

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

   11. Taylor expanded in EDonor around inf 48.8%

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

       1. *-commutative 48.8%

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

   13. Simplified 48.8%

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

   14. Taylor expanded in KbT around inf 70.8%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq 8.5 \cdot 10^{-185}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + \frac{NdChar}{\left(2 + \left(\frac{Vef}{KbT} + \frac{mu}{KbT}\right)\right) - \frac{Ec}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 65.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -3.25 \cdot 10^{+199} \lor \neg \left(KbT \leq 1.55 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= KbT -3.25e+199) (not (<= KbT 1.55e+168)))
   (+
    (/ NdChar (+ 1.0 (exp (/ (+ EDonor (+ mu (- Vef Ec))) KbT))))
    (* NaChar 0.5))
   (+ NdChar (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((KbT <= -3.25e+199) || !(KbT <= 1.55e+168)) {
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = NdChar + (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((kbt <= (-3.25d+199)) .or. (.not. (kbt <= 1.55d+168))) then
        tmp = (ndchar / (1.0d0 + exp(((edonor + (mu + (vef - ec))) / kbt)))) + (nachar * 0.5d0)
    else
        tmp = ndchar + (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((KbT <= -3.25e+199) || !(KbT <= 1.55e+168)) {
		tmp = (NdChar / (1.0 + Math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	} else {
		tmp = NdChar + (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (KbT <= -3.25e+199) or not (KbT <= 1.55e+168):
		tmp = (NdChar / (1.0 + math.exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5)
	else:
		tmp = NdChar + (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((KbT <= -3.25e+199) || !(KbT <= 1.55e+168))
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Float64(EDonor + Float64(mu + Float64(Vef - Ec))) / KbT)))) + Float64(NaChar * 0.5));
	else
		tmp = Float64(NdChar + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((KbT <= -3.25e+199) || ~((KbT <= 1.55e+168)))
		tmp = (NdChar / (1.0 + exp(((EDonor + (mu + (Vef - Ec))) / KbT)))) + (NaChar * 0.5);
	else
		tmp = NdChar + (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[KbT, -3.25e+199], N[Not[LessEqual[KbT, 1.55e+168]], $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 * 0.5), $MachinePrecision]), $MachinePrecision], N[(NdChar + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -3.2500000000000002e199 or 1.54999999999999998e168 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in EAccept around inf 89.4%

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

   6. Taylor expanded in EAccept around 0 82.5%

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

   if -3.2500000000000002e199 < KbT < 1.54999999999999998e168

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.9%

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

      1. associate-+r+ 52.9%

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

   7. Simplified 52.9%

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

   8. Taylor expanded in Ec around -inf 55.0%

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

      1. associate-*r* 55.0%

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

      2. mul-1-neg 55.0%

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

      3. +-commutative 55.0%

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

      4. mul-1-neg 55.0%

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

      5. unsub-neg 55.0%

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

   10. Simplified 55.0%

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

   11. Taylor expanded in EDonor around inf 56.5%

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

       1. *-commutative 56.5%

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

   13. Simplified 56.5%

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

   14. Taylor expanded in KbT around inf 62.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.25 \cdot 10^{+199} \lor \neg \left(KbT \leq 1.55 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{EDonor + \left(mu + \left(Vef - Ec\right)\right)}{KbT}}} + NaChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 65.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{if}\;KbT \leq -3.15 \cdot 10^{+202}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + EAccept\right)}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 10^{+119}:\\ \;\;\;\;NdChar + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 + NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT))))))
   (if (<= KbT -3.15e+202)
     (+ (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev EAccept)) KbT)))) (* NdChar 0.5))
     (if (<= KbT 1e+119) (+ NdChar t_0) (+ t_0 (* NdChar 0.5))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (KbT <= -3.15e+202) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + EAccept)) / KbT)))) + (NdChar * 0.5);
	} else if (KbT <= 1e+119) {
		tmp = NdChar + t_0;
	} else {
		tmp = t_0 + (NdChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt)))
    if (kbt <= (-3.15d+202)) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + eaccept)) / kbt)))) + (ndchar * 0.5d0)
    else if (kbt <= 1d+119) then
        tmp = ndchar + t_0
    else
        tmp = t_0 + (ndchar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	double tmp;
	if (KbT <= -3.15e+202) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + EAccept)) / KbT)))) + (NdChar * 0.5);
	} else if (KbT <= 1e+119) {
		tmp = NdChar + t_0;
	} else {
		tmp = t_0 + (NdChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT)))
	tmp = 0
	if KbT <= -3.15e+202:
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + EAccept)) / KbT)))) + (NdChar * 0.5)
	elif KbT <= 1e+119:
		tmp = NdChar + t_0
	else:
		tmp = t_0 + (NdChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT))))
	tmp = 0.0
	if (KbT <= -3.15e+202)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + EAccept)) / KbT)))) + Float64(NdChar * 0.5));
	elseif (KbT <= 1e+119)
		tmp = Float64(NdChar + t_0);
	else
		tmp = Float64(t_0 + Float64(NdChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT)));
	tmp = 0.0;
	if (KbT <= -3.15e+202)
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + EAccept)) / KbT)))) + (NdChar * 0.5);
	elseif (KbT <= 1e+119)
		tmp = NdChar + t_0;
	else
		tmp = t_0 + (NdChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -3.15e+202], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1e+119], N[(NdChar + t$95$0), $MachinePrecision], N[(t$95$0 + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -3.15000000000000002e202

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

   8. Taylor expanded in mu around 0 85.7%

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

      1. +-commutative 85.7%

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

      2. associate-+r+ 85.7%

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

   10. Simplified 85.7%

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

   if -3.15000000000000002e202 < KbT < 9.99999999999999944e118

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 51.9%

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

      1. associate-+r+ 51.9%

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

   7. Simplified 51.9%

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

   8. Taylor expanded in Ec around -inf 54.0%

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

      1. associate-*r* 54.0%

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

      2. mul-1-neg 54.0%

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

      3. +-commutative 54.0%

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

      4. mul-1-neg 54.0%

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

      5. unsub-neg 54.0%

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

   10. Simplified 54.0%

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

   11. Taylor expanded in EDonor around inf 57.9%

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

       1. *-commutative 57.9%

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

   13. Simplified 57.9%

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

   14. Taylor expanded in KbT around inf 63.8%

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

   if 9.99999999999999944e118 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 68.7%

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

      1. *-commutative 68.7%

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

   7. Simplified 68.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.15 \cdot 10^{+202}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + EAccept\right)}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;KbT \leq 10^{+119}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 65.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -2.1 \cdot 10^{+201} \lor \neg \left(KbT \leq 1.7 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + EAccept\right)}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (or (<= KbT -2.1e+201) (not (<= KbT 1.7e+135)))
   (+ (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev EAccept)) KbT)))) (* NdChar 0.5))
   (+ NdChar (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((KbT <= -2.1e+201) || !(KbT <= 1.7e+135)) {
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + EAccept)) / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = NdChar + (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if ((kbt <= (-2.1d+201)) .or. (.not. (kbt <= 1.7d+135))) then
        tmp = (nachar / (1.0d0 + exp(((vef + (ev + eaccept)) / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = ndchar + (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt))))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if ((KbT <= -2.1e+201) || !(KbT <= 1.7e+135)) {
		tmp = (NaChar / (1.0 + Math.exp(((Vef + (Ev + EAccept)) / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = NdChar + (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if (KbT <= -2.1e+201) or not (KbT <= 1.7e+135):
		tmp = (NaChar / (1.0 + math.exp(((Vef + (Ev + EAccept)) / KbT)))) + (NdChar * 0.5)
	else:
		tmp = NdChar + (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if ((KbT <= -2.1e+201) || !(KbT <= 1.7e+135))
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + EAccept)) / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(NdChar + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if ((KbT <= -2.1e+201) || ~((KbT <= 1.7e+135)))
		tmp = (NaChar / (1.0 + exp(((Vef + (Ev + EAccept)) / KbT)))) + (NdChar * 0.5);
	else
		tmp = NdChar + (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[Or[LessEqual[KbT, -2.1e+201], N[Not[LessEqual[KbT, 1.7e+135]], $MachinePrecision]], N[(N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + EAccept), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(NdChar + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if KbT < -2.0999999999999999e201 or 1.70000000000000005e135 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 77.5%

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

      1. *-commutative 77.5%

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

   7. Simplified 77.5%

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

   8. Taylor expanded in mu around 0 77.5%

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

      1. +-commutative 77.5%

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

      2. associate-+r+ 77.5%

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

   10. Simplified 77.5%

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

   if -2.0999999999999999e201 < KbT < 1.70000000000000005e135

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.2%

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

      1. associate-+r+ 52.2%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in Ec around -inf 54.3%

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

      1. associate-*r* 54.3%

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

      2. mul-1-neg 54.3%

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

      3. +-commutative 54.3%

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

      4. mul-1-neg 54.3%

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

      5. unsub-neg 54.3%

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

   10. Simplified 54.3%

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

   11. Taylor expanded in EDonor around inf 57.1%

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

       1. *-commutative 57.1%

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

   13. Simplified 57.1%

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

   14. Taylor expanded in KbT around inf 63.4%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.1 \cdot 10^{+201} \lor \neg \left(KbT \leq 1.7 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + EAccept\right)}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 64.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -1.75 \cdot 10^{+201}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 2.6 \cdot 10^{+253}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -1.75e+201)
   (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (/ NaChar 2.0))
   (if (<= KbT 2.6e+253)
     (+ NdChar (/ NaChar (+ 1.0 (exp (/ (+ Vef (+ Ev (- EAccept mu))) KbT)))))
     (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.75e+201) {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	} else if (KbT <= 2.6e+253) {
		tmp = NdChar + (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	} else {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-1.75d+201)) then
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / 2.0d0)
    else if (kbt <= 2.6d+253) then
        tmp = ndchar + (nachar / (1.0d0 + exp(((vef + (ev + (eaccept - mu))) / kbt))))
    else
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -1.75e+201) {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / 2.0);
	} else if (KbT <= 2.6e+253) {
		tmp = NdChar + (NaChar / (1.0 + Math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -1.75e+201:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / 2.0)
	elif KbT <= 2.6e+253:
		tmp = NdChar + (NaChar / (1.0 + math.exp(((Vef + (Ev + (EAccept - mu))) / KbT))))
	else:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -1.75e+201)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / 2.0));
	elseif (KbT <= 2.6e+253)
		tmp = Float64(NdChar + Float64(NaChar / Float64(1.0 + exp(Float64(Float64(Vef + Float64(Ev + Float64(EAccept - mu))) / KbT)))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -1.75e+201)
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	elseif (KbT <= 2.6e+253)
		tmp = NdChar + (NaChar / (1.0 + exp(((Vef + (Ev + (EAccept - mu))) / KbT))));
	else
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -1.75e+201], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 2.6e+253], N[(NdChar + N[(NaChar / N[(1.0 + N[Exp[N[(N[(Vef + N[(Ev + N[(EAccept - mu), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -1.7500000000000001e201

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 91.4%

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

   6. Taylor expanded in KbT around inf 85.0%

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

   if -1.7500000000000001e201 < KbT < 2.6e253

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.8%

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

      1. associate-+r+ 52.8%

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

   7. Simplified 52.8%

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

   8. Taylor expanded in Ec around -inf 54.7%

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

      1. associate-*r* 54.7%

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

      2. mul-1-neg 54.7%

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

      3. +-commutative 54.7%

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

      4. mul-1-neg 54.7%

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

      5. unsub-neg 54.7%

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

   10. Simplified 54.7%

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

   11. Taylor expanded in EDonor around inf 55.3%

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

       1. *-commutative 55.3%

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

   13. Simplified 55.3%

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

   14. Taylor expanded in KbT around inf 61.9%

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

   if 2.6e253 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 92.2%

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

      1. *-commutative 92.2%

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

   7. Simplified 92.2%

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

   8. Taylor expanded in EAccept around inf 91.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.75 \cdot 10^{+201}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{elif}\;KbT \leq 2.6 \cdot 10^{+253}:\\ \;\;\;\;NdChar + \frac{NaChar}{1 + e^{\frac{Vef + \left(Ev + \left(EAccept - mu\right)\right)}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 35.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;mu \leq 5.9 \cdot 10^{-123}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;mu \leq 5 \cdot 10^{+173}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= mu 5.9e-123)
   (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))
   (if (<= mu 5e+173)
     (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) (/ NaChar 2.0))
     (+ (/ NdChar (+ 1.0 (exp (/ mu KbT)))) (/ NaChar 2.0)))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (mu <= 5.9e-123) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else if (mu <= 5e+173) {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (mu <= 5.9d-123) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    else if (mu <= 5d+173) then
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / 2.0d0)
    else
        tmp = (ndchar / (1.0d0 + exp((mu / kbt)))) + (nachar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (mu <= 5.9e-123) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else if (mu <= 5e+173) {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / 2.0);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((mu / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if mu <= 5.9e-123:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	elif mu <= 5e+173:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / 2.0)
	else:
		tmp = (NdChar / (1.0 + math.exp((mu / KbT)))) + (NaChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (mu <= 5.9e-123)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	elseif (mu <= 5e+173)
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / 2.0));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(mu / KbT)))) + Float64(NaChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (mu <= 5.9e-123)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	elseif (mu <= 5e+173)
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / 2.0);
	else
		tmp = (NdChar / (1.0 + exp((mu / KbT)))) + (NaChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[mu, 5.9e-123], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[mu, 5e+173], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(mu / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if mu < 5.89999999999999988e-123

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.4%

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

      1. *-commutative 52.4%

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

   7. Simplified 52.4%

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

   8. Taylor expanded in EAccept around inf 40.6%

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

   if 5.89999999999999988e-123 < mu < 5.00000000000000034e173

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 78.3%

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

   6. Taylor expanded in KbT around inf 51.5%

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

   if 5.00000000000000034e173 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 95.5%

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

   6. Taylor expanded in KbT around inf 41.1%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq 5.9 \cdot 10^{-123}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{elif}\;mu \leq 5 \cdot 10^{+173}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{mu}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 36.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Ev \leq -1.32 \cdot 10^{+169}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= Ev -1.32e+169)
   (+ (* NdChar 0.5) (/ NaChar (+ 1.0 (exp (/ Ev KbT)))))
   (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -1.32e+169) {
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + exp((Ev / KbT))));
	} else {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ev <= (-1.32d+169)) then
        tmp = (ndchar * 0.5d0) + (nachar / (1.0d0 + exp((ev / kbt))))
    else
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (Ev <= -1.32e+169) {
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + Math.exp((Ev / KbT))));
	} else {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if Ev <= -1.32e+169:
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + math.exp((Ev / KbT))))
	else:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (Ev <= -1.32e+169)
		tmp = Float64(Float64(NdChar * 0.5) + Float64(NaChar / Float64(1.0 + exp(Float64(Ev / KbT)))));
	else
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (Ev <= -1.32e+169)
		tmp = (NdChar * 0.5) + (NaChar / (1.0 + exp((Ev / KbT))));
	else
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[Ev, -1.32e+169], N[(N[(NdChar * 0.5), $MachinePrecision] + N[(NaChar / N[(1.0 + N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Ev < -1.3199999999999999e169

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 46.2%

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

      1. *-commutative 46.2%

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

   7. Simplified 46.2%

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

   8. Taylor expanded in Ev around inf 41.4%

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

   if -1.3199999999999999e169 < Ev

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 49.1%

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

      1. *-commutative 49.1%

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

   7. Simplified 49.1%

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

   8. Taylor expanded in EAccept around inf 38.9%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Ev \leq -1.32 \cdot 10^{+169}:\\ \;\;\;\;NdChar \cdot 0.5 + \frac{NaChar}{1 + e^{\frac{Ev}{KbT}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 34.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;mu \leq 9.8 \cdot 10^{-123}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= mu 9.8e-123)
   (+ (/ NaChar (+ 1.0 (exp (/ EAccept KbT)))) (* NdChar 0.5))
   (+ (/ NdChar (+ 1.0 (exp (/ Vef KbT)))) (/ NaChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (mu <= 9.8e-123) {
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (mu <= 9.8d-123) then
        tmp = (nachar / (1.0d0 + exp((eaccept / kbt)))) + (ndchar * 0.5d0)
    else
        tmp = (ndchar / (1.0d0 + exp((vef / kbt)))) + (nachar / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (mu <= 9.8e-123) {
		tmp = (NaChar / (1.0 + Math.exp((EAccept / KbT)))) + (NdChar * 0.5);
	} else {
		tmp = (NdChar / (1.0 + Math.exp((Vef / KbT)))) + (NaChar / 2.0);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if mu <= 9.8e-123:
		tmp = (NaChar / (1.0 + math.exp((EAccept / KbT)))) + (NdChar * 0.5)
	else:
		tmp = (NdChar / (1.0 + math.exp((Vef / KbT)))) + (NaChar / 2.0)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (mu <= 9.8e-123)
		tmp = Float64(Float64(NaChar / Float64(1.0 + exp(Float64(EAccept / KbT)))) + Float64(NdChar * 0.5));
	else
		tmp = Float64(Float64(NdChar / Float64(1.0 + exp(Float64(Vef / KbT)))) + Float64(NaChar / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (mu <= 9.8e-123)
		tmp = (NaChar / (1.0 + exp((EAccept / KbT)))) + (NdChar * 0.5);
	else
		tmp = (NdChar / (1.0 + exp((Vef / KbT)))) + (NaChar / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[mu, 9.8e-123], N[(N[(NaChar / N[(1.0 + N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NdChar * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / N[(1.0 + N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if mu < 9.7999999999999996e-123

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 52.4%

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

      1. *-commutative 52.4%

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

   7. Simplified 52.4%

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

   8. Taylor expanded in EAccept around inf 40.6%

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

   if 9.7999999999999996e-123 < mu

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Vef around inf 69.2%

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

   6. Taylor expanded in KbT around inf 43.1%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq 9.8 \cdot 10^{-123}:\\ \;\;\;\;\frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{1 + e^{\frac{Vef}{KbT}}} + \frac{NaChar}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 34.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in KbT around inf 48.8%

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

   1. *-commutative 48.8%

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

7. Simplified 48.8%

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

8. Taylor expanded in EAccept around inf 38.0%

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

9. Final simplification 38.0%

   \[\leadsto \frac{NaChar}{1 + e^{\frac{EAccept}{KbT}}} + NdChar \cdot 0.5 \]
10. Add Preprocessing

Alternative 27: 28.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -6 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 9.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{NdChar}{1 + Ec \cdot \left(\frac{EDonor}{Ec \cdot KbT} + \frac{-1}{KbT}\right)} + \frac{NaChar}{\left(\left(\frac{EAccept}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2 + \frac{mu}{KbT}}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -6e-61)
   (* 0.5 (+ NdChar NaChar))
   (if (<= KbT 9.8e-21)
     (+
      (/ NdChar (+ 1.0 (* Ec (+ (/ EDonor (* Ec KbT)) (/ -1.0 KbT)))))
      (/
       NaChar
       (- (+ (+ (/ EAccept KbT) 2.0) (+ (/ Vef KbT) (/ Ev KbT))) (/ mu KbT))))
     (+ (/ NaChar 2.0) (/ NdChar (+ 2.0 (/ mu KbT)))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -6e-61) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (KbT <= 9.8e-21) {
		tmp = (NdChar / (1.0 + (Ec * ((EDonor / (Ec * KbT)) + (-1.0 / KbT))))) + (NaChar / ((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else {
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (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) :: tmp
    if (kbt <= (-6d-61)) then
        tmp = 0.5d0 * (ndchar + nachar)
    else if (kbt <= 9.8d-21) then
        tmp = (ndchar / (1.0d0 + (ec * ((edonor / (ec * kbt)) + ((-1.0d0) / kbt))))) + (nachar / ((((eaccept / kbt) + 2.0d0) + ((vef / kbt) + (ev / kbt))) - (mu / kbt)))
    else
        tmp = (nachar / 2.0d0) + (ndchar / (2.0d0 + (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 tmp;
	if (KbT <= -6e-61) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (KbT <= 9.8e-21) {
		tmp = (NdChar / (1.0 + (Ec * ((EDonor / (Ec * KbT)) + (-1.0 / KbT))))) + (NaChar / ((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	} else {
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (mu / KbT)));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -6e-61:
		tmp = 0.5 * (NdChar + NaChar)
	elif KbT <= 9.8e-21:
		tmp = (NdChar / (1.0 + (Ec * ((EDonor / (Ec * KbT)) + (-1.0 / KbT))))) + (NaChar / ((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)))
	else:
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (mu / KbT)))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -6e-61)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	elseif (KbT <= 9.8e-21)
		tmp = Float64(Float64(NdChar / Float64(1.0 + Float64(Ec * Float64(Float64(EDonor / Float64(Ec * KbT)) + Float64(-1.0 / KbT))))) + Float64(NaChar / Float64(Float64(Float64(Float64(EAccept / KbT) + 2.0) + Float64(Float64(Vef / KbT) + Float64(Ev / KbT))) - Float64(mu / KbT))));
	else
		tmp = Float64(Float64(NaChar / 2.0) + Float64(NdChar / Float64(2.0 + Float64(mu / KbT))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -6e-61)
		tmp = 0.5 * (NdChar + NaChar);
	elseif (KbT <= 9.8e-21)
		tmp = (NdChar / (1.0 + (Ec * ((EDonor / (Ec * KbT)) + (-1.0 / KbT))))) + (NaChar / ((((EAccept / KbT) + 2.0) + ((Vef / KbT) + (Ev / KbT))) - (mu / KbT)));
	else
		tmp = (NaChar / 2.0) + (NdChar / (2.0 + (mu / KbT)));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -6e-61], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 9.8e-21], N[(N[(NdChar / N[(1.0 + N[(Ec * N[(N[(EDonor / N[(Ec * KbT), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(NaChar / N[(N[(N[(N[(EAccept / KbT), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(Vef / KbT), $MachinePrecision] + N[(Ev / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(NaChar / 2.0), $MachinePrecision] + N[(NdChar / N[(2.0 + N[(mu / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -6.00000000000000024e-61

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 76.9%

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

   6. Taylor expanded in KbT around inf 51.2%

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

   7. Taylor expanded in mu around 0 43.4%

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

   8. Taylor expanded in mu around 0 44.7%

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

      1. distribute-lft-out 44.7%

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

   10. Simplified 44.7%

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

   if -6.00000000000000024e-61 < KbT < 9.8000000000000003e-21

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf 50.1%

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

      1. associate-+r+ 50.1%

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

   7. Simplified 50.1%

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

   8. Taylor expanded in Ec around -inf 53.0%

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

      1. associate-*r* 53.0%

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

      2. mul-1-neg 53.0%

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

      3. +-commutative 53.0%

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

      4. mul-1-neg 53.0%

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

      5. unsub-neg 53.0%

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

   10. Simplified 53.0%

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

   11. Taylor expanded in EDonor around inf 62.3%

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

       1. *-commutative 62.3%

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

   13. Simplified 62.3%

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

   14. Taylor expanded in KbT around inf 18.6%

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

       1. associate-+r+ 18.6%

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

   16. Simplified 18.6%

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

   if 9.8000000000000003e-21 < KbT

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in mu around inf 63.6%

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

   6. Taylor expanded in KbT around inf 42.9%

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

   7. Taylor expanded in mu around 0 42.3%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 9.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{NdChar}{1 + Ec \cdot \left(\frac{EDonor}{Ec \cdot KbT} + \frac{-1}{KbT}\right)} + \frac{NaChar}{\left(\left(\frac{EAccept}{KbT} + 2\right) + \left(\frac{Vef}{KbT} + \frac{Ev}{KbT}\right)\right) - \frac{mu}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{2} + \frac{NdChar}{2 + \frac{mu}{KbT}}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 27.3% accurate, 45.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in mu around inf 66.7%

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

6. Taylor expanded in KbT around inf 36.4%

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

7. Taylor expanded in mu around 0 29.8%

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

8. Taylor expanded in mu around 0 30.7%

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

   1. distribute-lft-out 30.7%

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

10. Simplified 30.7%

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

11. Final simplification 30.7%

    \[\leadsto 0.5 \cdot \left(NdChar + NaChar\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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))))))