Bulmash initializePoisson

Percentage Accurate: 100.0% → 100.0%

Time: 17.0s

Alternatives: 20

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Final simplification 100.0%

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

Alternative 2: 42.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ t_1 := \frac{NaChar}{e^{0 - \frac{mu}{KbT}} + 1}\\ \mathbf{if}\;mu \leq -1.06 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;mu \leq 5.3 \cdot 10^{-250}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{elif}\;mu \leq 5.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{NaChar}{t\_0}\\ \mathbf{elif}\;mu \leq 3.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0))
        (t_1 (/ NaChar (+ (exp (- 0.0 (/ mu KbT))) 1.0))))
   (if (<= mu -1.06e+40)
     t_1
     (if (<= mu 5.3e-250)
       (/ NdChar t_0)
       (if (<= mu 5.6e-36)
         (/ NaChar t_0)
         (if (<= mu 3.2e+79) (/ NdChar (+ (exp (/ EDonor KbT)) 1.0)) 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 / KbT)) + 1.0;
	double t_1 = NaChar / (exp((0.0 - (mu / KbT))) + 1.0);
	double tmp;
	if (mu <= -1.06e+40) {
		tmp = t_1;
	} else if (mu <= 5.3e-250) {
		tmp = NdChar / t_0;
	} else if (mu <= 5.6e-36) {
		tmp = NaChar / t_0;
	} else if (mu <= 3.2e+79) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} 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 / kbt)) + 1.0d0
    t_1 = nachar / (exp((0.0d0 - (mu / kbt))) + 1.0d0)
    if (mu <= (-1.06d+40)) then
        tmp = t_1
    else if (mu <= 5.3d-250) then
        tmp = ndchar / t_0
    else if (mu <= 5.6d-36) then
        tmp = nachar / t_0
    else if (mu <= 3.2d+79) then
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    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 / KbT)) + 1.0;
	double t_1 = NaChar / (Math.exp((0.0 - (mu / KbT))) + 1.0);
	double tmp;
	if (mu <= -1.06e+40) {
		tmp = t_1;
	} else if (mu <= 5.3e-250) {
		tmp = NdChar / t_0;
	} else if (mu <= 5.6e-36) {
		tmp = NaChar / t_0;
	} else if (mu <= 3.2e+79) {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = math.exp((Vef / KbT)) + 1.0
	t_1 = NaChar / (math.exp((0.0 - (mu / KbT))) + 1.0)
	tmp = 0
	if mu <= -1.06e+40:
		tmp = t_1
	elif mu <= 5.3e-250:
		tmp = NdChar / t_0
	elif mu <= 5.6e-36:
		tmp = NaChar / t_0
	elif mu <= 3.2e+79:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	t_1 = Float64(NaChar / Float64(exp(Float64(0.0 - Float64(mu / KbT))) + 1.0))
	tmp = 0.0
	if (mu <= -1.06e+40)
		tmp = t_1;
	elseif (mu <= 5.3e-250)
		tmp = Float64(NdChar / t_0);
	elseif (mu <= 5.6e-36)
		tmp = Float64(NaChar / t_0);
	elseif (mu <= 3.2e+79)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	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 / KbT)) + 1.0;
	t_1 = NaChar / (exp((0.0 - (mu / KbT))) + 1.0);
	tmp = 0.0;
	if (mu <= -1.06e+40)
		tmp = t_1;
	elseif (mu <= 5.3e-250)
		tmp = NdChar / t_0;
	elseif (mu <= 5.6e-36)
		tmp = NaChar / t_0;
	elseif (mu <= 3.2e+79)
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / N[(N[Exp[N[(0.0 - N[(mu / KbT), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[mu, -1.06e+40], t$95$1, If[LessEqual[mu, 5.3e-250], N[(NdChar / t$95$0), $MachinePrecision], If[LessEqual[mu, 5.6e-36], N[(NaChar / t$95$0), $MachinePrecision], If[LessEqual[mu, 3.2e+79], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if mu < -1.05999999999999996e40 or 3.20000000000000003e79 < 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%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 65.1%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 65.1%

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

   8. Taylor expanded in mu around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \frac{mu}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{-1 \cdot mu}{KbT}\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\mathsf{neg}\left(mu\right)}{KbT}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(mu\right)\right), KbT\right)\right)\right)\right) \]

      4. neg-lowering-neg.f64 56.8%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(mu\right), KbT\right)\right)\right)\right) \]

   10. Simplified 56.8%

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

   if -1.05999999999999996e40 < mu < 5.3000000000000001e-250

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 72.6%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 57.2%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 57.2%

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

   if 5.3000000000000001e-250 < mu < 5.6000000000000002e-36

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 70.4%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 70.4%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 56.4%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 56.4%

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

   if 5.6000000000000002e-36 < mu < 3.20000000000000003e79

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 67.8%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 67.8%

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

   8. Taylor expanded in EDonor around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EDonor}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(EDonor, KbT\right)\right)\right)\right) \]

   10. Simplified 65.1%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;mu \leq -1.06 \cdot 10^{+40}:\\ \;\;\;\;\frac{NaChar}{e^{0 - \frac{mu}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 5.3 \cdot 10^{-250}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 5.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;mu \leq 3.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{0 - \frac{mu}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{if}\;NdChar \leq -1.25 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;NdChar \leq 7.2 \cdot 10^{-168}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NdChar (+ (exp (/ (+ EDonor (+ Vef (- mu Ec))) KbT)) 1.0))))
   (if (<= NdChar -1.25e+122)
     t_0
     (if (<= NdChar 7.2e-168)
       (/ NaChar (+ (exp (/ (+ (+ EAccept Ev) (- Vef mu)) KbT)) 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 = NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	double tmp;
	if (NdChar <= -1.25e+122) {
		tmp = t_0;
	} else if (NdChar <= 7.2e-168) {
		tmp = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ndchar / (exp(((edonor + (vef + (mu - ec))) / kbt)) + 1.0d0)
    if (ndchar <= (-1.25d+122)) then
        tmp = t_0
    else if (ndchar <= 7.2d-168) then
        tmp = nachar / (exp((((eaccept + ev) + (vef - mu)) / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NdChar / (Math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	double tmp;
	if (NdChar <= -1.25e+122) {
		tmp = t_0;
	} else if (NdChar <= 7.2e-168) {
		tmp = NaChar / (Math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NdChar / (math.exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0)
	tmp = 0
	if NdChar <= -1.25e+122:
		tmp = t_0
	elif NdChar <= 7.2e-168:
		tmp = NaChar / (math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NdChar / Float64(exp(Float64(Float64(EDonor + Float64(Vef + Float64(mu - Ec))) / KbT)) + 1.0))
	tmp = 0.0
	if (NdChar <= -1.25e+122)
		tmp = t_0;
	elseif (NdChar <= 7.2e-168)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Ev) + Float64(Vef - mu)) / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NdChar / (exp(((EDonor + (Vef + (mu - Ec))) / KbT)) + 1.0);
	tmp = 0.0;
	if (NdChar <= -1.25e+122)
		tmp = t_0;
	elseif (NdChar <= 7.2e-168)
		tmp = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NdChar / N[(N[Exp[N[(N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[NdChar, -1.25e+122], t$95$0, If[LessEqual[NdChar, 7.2e-168], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -1.24999999999999997e122 or 7.1999999999999998e-168 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 73.5%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 73.5%

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

   if -1.24999999999999997e122 < NdChar < 7.1999999999999998e-168

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 76.7%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 76.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NdChar \leq -1.25 \cdot 10^{+122}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \mathbf{elif}\;NdChar \leq 7.2 \cdot 10^{-168}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor + \left(Vef + \left(mu - Ec\right)\right)}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;KbT \leq -9.8 \cdot 10^{+204}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= KbT -9.8e+204)
   (* 0.5 (+ NdChar NaChar))
   (if (<= KbT 1.4e+189)
     (/ NaChar (+ (exp (/ (+ (+ EAccept Ev) (- Vef mu)) KbT)) 1.0))
     (+ (/ NdChar 2.0) (/ NaChar (+ (exp (/ Ev KbT)) 1.0))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -9.8e+204) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (KbT <= 1.4e+189) {
		tmp = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / (exp((Ev / KbT)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (kbt <= (-9.8d+204)) then
        tmp = 0.5d0 * (ndchar + nachar)
    else if (kbt <= 1.4d+189) then
        tmp = nachar / (exp((((eaccept + ev) + (vef - mu)) / kbt)) + 1.0d0)
    else
        tmp = (ndchar / 2.0d0) + (nachar / (exp((ev / kbt)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (KbT <= -9.8e+204) {
		tmp = 0.5 * (NdChar + NaChar);
	} else if (KbT <= 1.4e+189) {
		tmp = NaChar / (Math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	} else {
		tmp = (NdChar / 2.0) + (NaChar / (Math.exp((Ev / KbT)) + 1.0));
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if KbT <= -9.8e+204:
		tmp = 0.5 * (NdChar + NaChar)
	elif KbT <= 1.4e+189:
		tmp = NaChar / (math.exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0)
	else:
		tmp = (NdChar / 2.0) + (NaChar / (math.exp((Ev / KbT)) + 1.0))
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (KbT <= -9.8e+204)
		tmp = Float64(0.5 * Float64(NdChar + NaChar));
	elseif (KbT <= 1.4e+189)
		tmp = Float64(NaChar / Float64(exp(Float64(Float64(Float64(EAccept + Ev) + Float64(Vef - mu)) / KbT)) + 1.0));
	else
		tmp = Float64(Float64(NdChar / 2.0) + Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (KbT <= -9.8e+204)
		tmp = 0.5 * (NdChar + NaChar);
	elseif (KbT <= 1.4e+189)
		tmp = NaChar / (exp((((EAccept + Ev) + (Vef - mu)) / KbT)) + 1.0);
	else
		tmp = (NdChar / 2.0) + (NaChar / (exp((Ev / KbT)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[KbT, -9.8e+204], N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 1.4e+189], N[(NaChar / N[(N[Exp[N[(N[(N[(EAccept + Ev), $MachinePrecision] + N[(Vef - mu), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(NdChar / 2.0), $MachinePrecision] + N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -9.7999999999999995e204

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 78.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 78.1%

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

   if -9.7999999999999995e204 < KbT < 1.40000000000000003e189

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 65.3%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 65.3%

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

   if 1.40000000000000003e189 < 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%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, \color{blue}{2}\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(Vef, \mathsf{+.f64}\left(EAccept, \mathsf{\_.f64}\left(Ev, mu\right)\right)\right), KbT\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 83.0%

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

   8. Taylor expanded in Ev around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 74.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(NdChar, 2\right), \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right)\right)\right)\right)\right) \]

   10. Simplified 74.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -9.8 \cdot 10^{+204}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{\left(EAccept + Ev\right) + \left(Vef - mu\right)}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2} + \frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -1.7 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;KbT \leq -2.15 \cdot 10^{-194}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 9.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 3.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\left(\left(\left(Ec - mu\right) - Vef\right) - EDonor\right) + \frac{-0.5 \cdot \left(t\_0 \cdot t\_0\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \left(Ec \cdot \frac{NdChar}{KbT}\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (+ Vef (- mu Ec)))) (t_1 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -1.7e+117)
     t_1
     (if (<= KbT -2.15e-194)
       (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
       (if (<= KbT 9.2e-117)
         (/ NaChar (+ (exp (/ Ev KbT)) 1.0))
         (if (<= KbT 3.6e+208)
           (/
            NdChar
            (-
             2.0
             (/
              (+ (- (- (- Ec mu) Vef) EDonor) (/ (* -0.5 (* t_0 t_0)) KbT))
              KbT)))
           (- t_1 (* (* Ec (/ NdChar KbT)) -0.25))))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + (Vef + (mu - Ec));
	double t_1 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.7e+117) {
		tmp = t_1;
	} else if (KbT <= -2.15e-194) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else if (KbT <= 9.2e-117) {
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	} else if (KbT <= 3.6e+208) {
		tmp = NdChar / (2.0 - (((((Ec - mu) - Vef) - EDonor) + ((-0.5 * (t_0 * t_0)) / KbT)) / KbT));
	} else {
		tmp = t_1 - ((Ec * (NdChar / KbT)) * -0.25);
	}
	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 = edonor + (vef + (mu - ec))
    t_1 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-1.7d+117)) then
        tmp = t_1
    else if (kbt <= (-2.15d-194)) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else if (kbt <= 9.2d-117) then
        tmp = nachar / (exp((ev / kbt)) + 1.0d0)
    else if (kbt <= 3.6d+208) then
        tmp = ndchar / (2.0d0 - (((((ec - mu) - vef) - edonor) + (((-0.5d0) * (t_0 * t_0)) / kbt)) / kbt))
    else
        tmp = t_1 - ((ec * (ndchar / kbt)) * (-0.25d0))
    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 = EDonor + (Vef + (mu - Ec));
	double t_1 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -1.7e+117) {
		tmp = t_1;
	} else if (KbT <= -2.15e-194) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else if (KbT <= 9.2e-117) {
		tmp = NaChar / (Math.exp((Ev / KbT)) + 1.0);
	} else if (KbT <= 3.6e+208) {
		tmp = NdChar / (2.0 - (((((Ec - mu) - Vef) - EDonor) + ((-0.5 * (t_0 * t_0)) / KbT)) / KbT));
	} else {
		tmp = t_1 - ((Ec * (NdChar / KbT)) * -0.25);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = EDonor + (Vef + (mu - Ec))
	t_1 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -1.7e+117:
		tmp = t_1
	elif KbT <= -2.15e-194:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	elif KbT <= 9.2e-117:
		tmp = NaChar / (math.exp((Ev / KbT)) + 1.0)
	elif KbT <= 3.6e+208:
		tmp = NdChar / (2.0 - (((((Ec - mu) - Vef) - EDonor) + ((-0.5 * (t_0 * t_0)) / KbT)) / KbT))
	else:
		tmp = t_1 - ((Ec * (NdChar / KbT)) * -0.25)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -1.7e+117)
		tmp = t_1;
	elseif (KbT <= -2.15e-194)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	elseif (KbT <= 9.2e-117)
		tmp = Float64(NaChar / Float64(exp(Float64(Ev / KbT)) + 1.0));
	elseif (KbT <= 3.6e+208)
		tmp = Float64(NdChar / Float64(2.0 - Float64(Float64(Float64(Float64(Float64(Ec - mu) - Vef) - EDonor) + Float64(Float64(-0.5 * Float64(t_0 * t_0)) / KbT)) / KbT)));
	else
		tmp = Float64(t_1 - Float64(Float64(Ec * Float64(NdChar / KbT)) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = EDonor + (Vef + (mu - Ec));
	t_1 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -1.7e+117)
		tmp = t_1;
	elseif (KbT <= -2.15e-194)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	elseif (KbT <= 9.2e-117)
		tmp = NaChar / (exp((Ev / KbT)) + 1.0);
	elseif (KbT <= 3.6e+208)
		tmp = NdChar / (2.0 - (((((Ec - mu) - Vef) - EDonor) + ((-0.5 * (t_0 * t_0)) / KbT)) / KbT));
	else
		tmp = t_1 - ((Ec * (NdChar / KbT)) * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -1.7e+117], t$95$1, If[LessEqual[KbT, -2.15e-194], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 9.2e-117], N[(NaChar / N[(N[Exp[N[(Ev / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 3.6e+208], N[(NdChar / N[(2.0 - N[(N[(N[(N[(N[(Ec - mu), $MachinePrecision] - Vef), $MachinePrecision] - EDonor), $MachinePrecision] + N[(N[(-0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(N[(Ec * N[(NdChar / KbT), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if KbT < -1.7e117

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 64.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 64.7%

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

   if -1.7e117 < KbT < -2.15000000000000003e-194

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 69.2%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 69.2%

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

   8. Taylor expanded in EAccept around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 26.8%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right)\right)\right)\right) \]

   10. Simplified 26.8%

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

   if -2.15000000000000003e-194 < KbT < 9.19999999999999978e-117

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 72.9%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 72.9%

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

   8. Taylor expanded in Ev around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 40.8%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Ev, KbT\right)\right)\right)\right) \]

   10. Simplified 40.8%

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

   if 9.19999999999999978e-117 < KbT < 3.60000000000000003e208

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 70.2%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 70.2%

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

   8. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right) \]

   10. Simplified 32.7%

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

   if 3.60000000000000003e208 < 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%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{4} \cdot \left(NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right) + \frac{1}{4} \cdot \left(NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

   7. Simplified 41.8%

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

   8. Taylor expanded in Ec around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{Ec \cdot NdChar}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{Ec \cdot NdChar}{KbT}\right)}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \left(Ec \cdot \color{blue}{\frac{NdChar}{KbT}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(Ec, \color{blue}{\left(\frac{NdChar}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 74.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(Ec, \mathsf{/.f64}\left(NdChar, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 74.0%

       \[\leadsto 0.5 \cdot \left(NaChar + NdChar\right) - \color{blue}{-0.25 \cdot \left(Ec \cdot \frac{NdChar}{KbT}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -1.7 \cdot 10^{+117}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq -2.15 \cdot 10^{-194}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 9.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Ev}{KbT}} + 1}\\ \mathbf{elif}\;KbT \leq 3.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\left(\left(\left(Ec - mu\right) - Vef\right) - EDonor\right) + \frac{-0.5 \cdot \left(\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - \left(Ec \cdot \frac{NdChar}{KbT}\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{Vef}{KbT}} + 1\\ t_1 := \frac{NaChar}{t\_0}\\ \mathbf{if}\;NaChar \leq -5200000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;NaChar \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\frac{NdChar}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ (exp (/ Vef KbT)) 1.0)) (t_1 (/ NaChar t_0)))
   (if (<= NaChar -5200000000.0)
     t_1
     (if (<= NaChar 7e-20) (/ NdChar t_0) t_1))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = exp((Vef / KbT)) + 1.0;
	double t_1 = NaChar / t_0;
	double tmp;
	if (NaChar <= -5200000000.0) {
		tmp = t_1;
	} else if (NaChar <= 7e-20) {
		tmp = NdChar / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp((vef / kbt)) + 1.0d0
    t_1 = nachar / t_0
    if (nachar <= (-5200000000.0d0)) then
        tmp = t_1
    else if (nachar <= 7d-20) then
        tmp = ndchar / t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = Math.exp((Vef / KbT)) + 1.0;
	double t_1 = NaChar / t_0;
	double tmp;
	if (NaChar <= -5200000000.0) {
		tmp = t_1;
	} else if (NaChar <= 7e-20) {
		tmp = NdChar / 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 / KbT)) + 1.0
	t_1 = NaChar / t_0
	tmp = 0
	if NaChar <= -5200000000.0:
		tmp = t_1
	elif NaChar <= 7e-20:
		tmp = NdChar / t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(exp(Float64(Vef / KbT)) + 1.0)
	t_1 = Float64(NaChar / t_0)
	tmp = 0.0
	if (NaChar <= -5200000000.0)
		tmp = t_1;
	elseif (NaChar <= 7e-20)
		tmp = Float64(NdChar / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = exp((Vef / KbT)) + 1.0;
	t_1 = NaChar / t_0;
	tmp = 0.0;
	if (NaChar <= -5200000000.0)
		tmp = t_1;
	elseif (NaChar <= 7e-20)
		tmp = NdChar / t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(NaChar / t$95$0), $MachinePrecision]}, If[LessEqual[NaChar, -5200000000.0], t$95$1, If[LessEqual[NaChar, 7e-20], N[(NdChar / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if NaChar < -5.2e9 or 7.00000000000000007e-20 < 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%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 71.7%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 71.7%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 48.2%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 48.2%

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

   if -5.2e9 < NaChar < 7.00000000000000007e-20

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 73.9%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 73.9%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 54.8%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 54.8%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;NaChar \leq -5200000000:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;NaChar \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{if}\;Vef \leq -2.3 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;Vef \leq 2.65 \cdot 10^{+84}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (/ NaChar (+ (exp (/ Vef KbT)) 1.0))))
   (if (<= Vef -2.3e+23)
     t_0
     (if (<= Vef 2.65e+84) (/ NdChar (+ (exp (/ EDonor KbT)) 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 = NaChar / (exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -2.3e+23) {
		tmp = t_0;
	} else if (Vef <= 2.65e+84) {
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = nachar / (exp((vef / kbt)) + 1.0d0)
    if (vef <= (-2.3d+23)) then
        tmp = t_0
    else if (vef <= 2.65d+84) then
        tmp = ndchar / (exp((edonor / kbt)) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	double tmp;
	if (Vef <= -2.3e+23) {
		tmp = t_0;
	} else if (Vef <= 2.65e+84) {
		tmp = NdChar / (Math.exp((EDonor / KbT)) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = NaChar / (math.exp((Vef / KbT)) + 1.0)
	tmp = 0
	if Vef <= -2.3e+23:
		tmp = t_0
	elif Vef <= 2.65e+84:
		tmp = NdChar / (math.exp((EDonor / KbT)) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0))
	tmp = 0.0
	if (Vef <= -2.3e+23)
		tmp = t_0;
	elseif (Vef <= 2.65e+84)
		tmp = Float64(NdChar / Float64(exp(Float64(EDonor / KbT)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = NaChar / (exp((Vef / KbT)) + 1.0);
	tmp = 0.0;
	if (Vef <= -2.3e+23)
		tmp = t_0;
	elseif (Vef <= 2.65e+84)
		tmp = NdChar / (exp((EDonor / KbT)) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Vef, -2.3e+23], t$95$0, If[LessEqual[Vef, 2.65e+84], N[(NdChar / N[(N[Exp[N[(EDonor / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if Vef < -2.3e23 or 2.6500000000000001e84 < 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%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 71.6%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 71.6%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 57.4%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 57.4%

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

   if -2.3e23 < Vef < 2.6500000000000001e84

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 61.2%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 61.2%

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

   8. Taylor expanded in EDonor around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EDonor}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 45.3%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(EDonor, KbT\right)\right)\right)\right) \]

   10. Simplified 45.3%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Vef \leq -2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{elif}\;Vef \leq 2.65 \cdot 10^{+84}:\\ \;\;\;\;\frac{NdChar}{e^{\frac{EDonor}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -6.5 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.3 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(Ec \cdot \frac{NdChar}{KbT}\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -6.5e+117)
     t_0
     (if (<= KbT 1.3e+187)
       (/ NaChar (+ (exp (/ Vef KbT)) 1.0))
       (- t_0 (* (* Ec (/ NdChar KbT)) -0.25))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -6.5e+117) {
		tmp = t_0;
	} else if (KbT <= 1.3e+187) {
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	} else {
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25);
	}
	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 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-6.5d+117)) then
        tmp = t_0
    else if (kbt <= 1.3d+187) then
        tmp = nachar / (exp((vef / kbt)) + 1.0d0)
    else
        tmp = t_0 - ((ec * (ndchar / kbt)) * (-0.25d0))
    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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -6.5e+117) {
		tmp = t_0;
	} else if (KbT <= 1.3e+187) {
		tmp = NaChar / (Math.exp((Vef / KbT)) + 1.0);
	} else {
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -6.5e+117:
		tmp = t_0
	elif KbT <= 1.3e+187:
		tmp = NaChar / (math.exp((Vef / KbT)) + 1.0)
	else:
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -6.5e+117)
		tmp = t_0;
	elseif (KbT <= 1.3e+187)
		tmp = Float64(NaChar / Float64(exp(Float64(Vef / KbT)) + 1.0));
	else
		tmp = Float64(t_0 - Float64(Float64(Ec * Float64(NdChar / KbT)) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -6.5e+117)
		tmp = t_0;
	elseif (KbT <= 1.3e+187)
		tmp = NaChar / (exp((Vef / KbT)) + 1.0);
	else
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -6.5e+117], t$95$0, If[LessEqual[KbT, 1.3e+187], N[(NaChar / N[(N[Exp[N[(Vef / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(Ec * N[(NdChar / KbT), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -6.5000000000000004e117

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 66.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 66.1%

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

   if -6.5000000000000004e117 < KbT < 1.2999999999999999e187

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 66.6%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 66.6%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 38.5%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 38.5%

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

   if 1.2999999999999999e187 < 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%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{4} \cdot \left(NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right) + \frac{1}{4} \cdot \left(NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

   7. Simplified 35.7%

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

   8. Taylor expanded in Ec around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{Ec \cdot NdChar}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{Ec \cdot NdChar}{KbT}\right)}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \left(Ec \cdot \color{blue}{\frac{NdChar}{KbT}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(Ec, \color{blue}{\left(\frac{NdChar}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 67.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(Ec, \mathsf{/.f64}\left(NdChar, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 67.2%

       \[\leadsto 0.5 \cdot \left(NaChar + NdChar\right) - \color{blue}{-0.25 \cdot \left(Ec \cdot \frac{NdChar}{KbT}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6.5 \cdot 10^{+117}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.3 \cdot 10^{+187}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{Vef}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - \left(Ec \cdot \frac{NdChar}{KbT}\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -2.35 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.02 \cdot 10^{+163}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(Ec \cdot \frac{NdChar}{KbT}\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -2.35e+117)
     t_0
     (if (<= KbT 1.02e+163)
       (/ NaChar (+ (exp (/ EAccept KbT)) 1.0))
       (- t_0 (* (* Ec (/ NdChar KbT)) -0.25))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -2.35e+117) {
		tmp = t_0;
	} else if (KbT <= 1.02e+163) {
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25);
	}
	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 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-2.35d+117)) then
        tmp = t_0
    else if (kbt <= 1.02d+163) then
        tmp = nachar / (exp((eaccept / kbt)) + 1.0d0)
    else
        tmp = t_0 - ((ec * (ndchar / kbt)) * (-0.25d0))
    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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -2.35e+117) {
		tmp = t_0;
	} else if (KbT <= 1.02e+163) {
		tmp = NaChar / (Math.exp((EAccept / KbT)) + 1.0);
	} else {
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -2.35e+117:
		tmp = t_0
	elif KbT <= 1.02e+163:
		tmp = NaChar / (math.exp((EAccept / KbT)) + 1.0)
	else:
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -2.35e+117)
		tmp = t_0;
	elseif (KbT <= 1.02e+163)
		tmp = Float64(NaChar / Float64(exp(Float64(EAccept / KbT)) + 1.0));
	else
		tmp = Float64(t_0 - Float64(Float64(Ec * Float64(NdChar / KbT)) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -2.35e+117)
		tmp = t_0;
	elseif (KbT <= 1.02e+163)
		tmp = NaChar / (exp((EAccept / KbT)) + 1.0);
	else
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -2.35e+117], t$95$0, If[LessEqual[KbT, 1.02e+163], N[(NaChar / N[(N[Exp[N[(EAccept / KbT), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(Ec * N[(NdChar / KbT), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -2.35000000000000003e117

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 64.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 64.7%

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

   if -2.35000000000000003e117 < KbT < 1.02e163

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 66.9%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 66.9%

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

   8. Taylor expanded in EAccept around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{EAccept}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 35.6%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(EAccept, KbT\right)\right)\right)\right) \]

   10. Simplified 35.6%

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

   if 1.02e163 < 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%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{4} \cdot \left(NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right) + \frac{1}{4} \cdot \left(NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

   7. Simplified 32.6%

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

   8. Taylor expanded in Ec around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{Ec \cdot NdChar}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{Ec \cdot NdChar}{KbT}\right)}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \left(Ec \cdot \color{blue}{\frac{NdChar}{KbT}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(Ec, \color{blue}{\left(\frac{NdChar}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 64.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(Ec, \mathsf{/.f64}\left(NdChar, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 64.5%

       \[\leadsto 0.5 \cdot \left(NaChar + NdChar\right) - \color{blue}{-0.25 \cdot \left(Ec \cdot \frac{NdChar}{KbT}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.35 \cdot 10^{+117}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.02 \cdot 10^{+163}:\\ \;\;\;\;\frac{NaChar}{e^{\frac{EAccept}{KbT}} + 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - \left(Ec \cdot \frac{NdChar}{KbT}\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := EDonor + \left(Vef + \left(mu - Ec\right)\right)\\ t_1 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -6.5 \cdot 10^{+165}:\\ \;\;\;\;t\_1 - \frac{Ev}{KbT} \cdot \left(NaChar \cdot 0.25\right)\\ \mathbf{elif}\;KbT \leq 3.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\left(\left(\left(Ec - mu\right) - Vef\right) - EDonor\right) + \frac{-0.5 \cdot \left(t\_0 \cdot t\_0\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \left(Ec \cdot \frac{NdChar}{KbT}\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (+ EDonor (+ Vef (- mu Ec)))) (t_1 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -6.5e+165)
     (- t_1 (* (/ Ev KbT) (* NaChar 0.25)))
     (if (<= KbT 3.6e+208)
       (/
        NdChar
        (-
         2.0
         (/
          (+ (- (- (- Ec mu) Vef) EDonor) (/ (* -0.5 (* t_0 t_0)) KbT))
          KbT)))
       (- t_1 (* (* Ec (/ NdChar KbT)) -0.25))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = EDonor + (Vef + (mu - Ec));
	double t_1 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -6.5e+165) {
		tmp = t_1 - ((Ev / KbT) * (NaChar * 0.25));
	} else if (KbT <= 3.6e+208) {
		tmp = NdChar / (2.0 - (((((Ec - mu) - Vef) - EDonor) + ((-0.5 * (t_0 * t_0)) / KbT)) / KbT));
	} else {
		tmp = t_1 - ((Ec * (NdChar / KbT)) * -0.25);
	}
	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 = edonor + (vef + (mu - ec))
    t_1 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-6.5d+165)) then
        tmp = t_1 - ((ev / kbt) * (nachar * 0.25d0))
    else if (kbt <= 3.6d+208) then
        tmp = ndchar / (2.0d0 - (((((ec - mu) - vef) - edonor) + (((-0.5d0) * (t_0 * t_0)) / kbt)) / kbt))
    else
        tmp = t_1 - ((ec * (ndchar / kbt)) * (-0.25d0))
    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 = EDonor + (Vef + (mu - Ec));
	double t_1 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -6.5e+165) {
		tmp = t_1 - ((Ev / KbT) * (NaChar * 0.25));
	} else if (KbT <= 3.6e+208) {
		tmp = NdChar / (2.0 - (((((Ec - mu) - Vef) - EDonor) + ((-0.5 * (t_0 * t_0)) / KbT)) / KbT));
	} else {
		tmp = t_1 - ((Ec * (NdChar / KbT)) * -0.25);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = EDonor + (Vef + (mu - Ec))
	t_1 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -6.5e+165:
		tmp = t_1 - ((Ev / KbT) * (NaChar * 0.25))
	elif KbT <= 3.6e+208:
		tmp = NdChar / (2.0 - (((((Ec - mu) - Vef) - EDonor) + ((-0.5 * (t_0 * t_0)) / KbT)) / KbT))
	else:
		tmp = t_1 - ((Ec * (NdChar / KbT)) * -0.25)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(EDonor + Float64(Vef + Float64(mu - Ec)))
	t_1 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -6.5e+165)
		tmp = Float64(t_1 - Float64(Float64(Ev / KbT) * Float64(NaChar * 0.25)));
	elseif (KbT <= 3.6e+208)
		tmp = Float64(NdChar / Float64(2.0 - Float64(Float64(Float64(Float64(Float64(Ec - mu) - Vef) - EDonor) + Float64(Float64(-0.5 * Float64(t_0 * t_0)) / KbT)) / KbT)));
	else
		tmp = Float64(t_1 - Float64(Float64(Ec * Float64(NdChar / KbT)) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = EDonor + (Vef + (mu - Ec));
	t_1 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -6.5e+165)
		tmp = t_1 - ((Ev / KbT) * (NaChar * 0.25));
	elseif (KbT <= 3.6e+208)
		tmp = NdChar / (2.0 - (((((Ec - mu) - Vef) - EDonor) + ((-0.5 * (t_0 * t_0)) / KbT)) / KbT));
	else
		tmp = t_1 - ((Ec * (NdChar / KbT)) * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(EDonor + N[(Vef + N[(mu - Ec), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -6.5e+165], N[(t$95$1 - N[(N[(Ev / KbT), $MachinePrecision] * N[(NaChar * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 3.6e+208], N[(NdChar / N[(2.0 - N[(N[(N[(N[(N[(Ec - mu), $MachinePrecision] - Vef), $MachinePrecision] - EDonor), $MachinePrecision] + N[(N[(-0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(N[(Ec * N[(NdChar / KbT), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -6.4999999999999999e165

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{4} \cdot \left(NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right) + \frac{1}{4} \cdot \left(NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

   7. Simplified 35.2%

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

   8. Taylor expanded in Ev around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{Ev \cdot NaChar}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \left(\frac{\frac{1}{4} \cdot \left(Ev \cdot NaChar\right)}{\color{blue}{KbT}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(Ev \cdot NaChar\right)\right), \color{blue}{KbT}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(Ev \cdot NaChar\right)\right), KbT\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(NaChar \cdot Ev\right)\right), KbT\right)\right) \]

      5. *-lowering-*.f64 66.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(NaChar, Ev\right)\right), KbT\right)\right) \]

   10. Simplified 66.3%

       \[\leadsto 0.5 \cdot \left(NaChar + NdChar\right) - \color{blue}{\frac{0.25 \cdot \left(NaChar \cdot Ev\right)}{KbT}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \left(\frac{\left(\frac{1}{4} \cdot NaChar\right) \cdot Ev}{KbT}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \left(\left(\frac{1}{4} \cdot NaChar\right) \cdot \color{blue}{\frac{Ev}{KbT}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot NaChar\right), \color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\left(NaChar \cdot \frac{1}{4}\right), \left(\frac{\color{blue}{Ev}}{KbT}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \frac{1}{4}\right), \left(\frac{\color{blue}{Ev}}{KbT}\right)\right)\right) \]

       6. /-lowering-/.f64 68.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \frac{1}{4}\right), \mathsf{/.f64}\left(Ev, \color{blue}{KbT}\right)\right)\right) \]

   12. Applied egg-rr 68.8%

       \[\leadsto 0.5 \cdot \left(NaChar + NdChar\right) - \color{blue}{\left(NaChar \cdot 0.25\right) \cdot \frac{Ev}{KbT}} \]

   if -6.4999999999999999e165 < KbT < 3.60000000000000003e208

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 62.8%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 62.8%

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

   8. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right) \]

   10. Simplified 30.9%

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

   if 3.60000000000000003e208 < 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%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{4} \cdot \left(NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right) + \frac{1}{4} \cdot \left(NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

   7. Simplified 41.8%

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

   8. Taylor expanded in Ec around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{Ec \cdot NdChar}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{Ec \cdot NdChar}{KbT}\right)}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \left(Ec \cdot \color{blue}{\frac{NdChar}{KbT}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(Ec, \color{blue}{\left(\frac{NdChar}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 74.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(Ec, \mathsf{/.f64}\left(NdChar, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 74.0%

       \[\leadsto 0.5 \cdot \left(NaChar + NdChar\right) - \color{blue}{-0.25 \cdot \left(Ec \cdot \frac{NdChar}{KbT}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6.5 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - \frac{Ev}{KbT} \cdot \left(NaChar \cdot 0.25\right)\\ \mathbf{elif}\;KbT \leq 3.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{NdChar}{2 - \frac{\left(\left(\left(Ec - mu\right) - Vef\right) - EDonor\right) + \frac{-0.5 \cdot \left(\left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right) \cdot \left(EDonor + \left(Vef + \left(mu - Ec\right)\right)\right)\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - \left(Ec \cdot \frac{NdChar}{KbT}\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -4.8 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 1.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{NdChar}{2 + Vef \cdot \left(\frac{Vef \cdot 0.5}{KbT \cdot KbT} + \frac{1}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -4.8e+26)
     t_0
     (if (<= KbT 1.1e-52)
       (/ NdChar (+ 2.0 (* Vef (+ (/ (* Vef 0.5) (* KbT KbT)) (/ 1.0 KbT)))))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -4.8e+26) {
		tmp = t_0;
	} else if (KbT <= 1.1e-52) {
		tmp = NdChar / (2.0 + (Vef * (((Vef * 0.5) / (KbT * KbT)) + (1.0 / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-4.8d+26)) then
        tmp = t_0
    else if (kbt <= 1.1d-52) then
        tmp = ndchar / (2.0d0 + (vef * (((vef * 0.5d0) / (kbt * kbt)) + (1.0d0 / kbt))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -4.8e+26) {
		tmp = t_0;
	} else if (KbT <= 1.1e-52) {
		tmp = NdChar / (2.0 + (Vef * (((Vef * 0.5) / (KbT * KbT)) + (1.0 / KbT))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -4.8e+26:
		tmp = t_0
	elif KbT <= 1.1e-52:
		tmp = NdChar / (2.0 + (Vef * (((Vef * 0.5) / (KbT * KbT)) + (1.0 / KbT))))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -4.8e+26)
		tmp = t_0;
	elseif (KbT <= 1.1e-52)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Vef * Float64(Float64(Float64(Vef * 0.5) / Float64(KbT * KbT)) + Float64(1.0 / KbT)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -4.8e+26)
		tmp = t_0;
	elseif (KbT <= 1.1e-52)
		tmp = NdChar / (2.0 + (Vef * (((Vef * 0.5) / (KbT * KbT)) + (1.0 / KbT))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -4.8e+26], t$95$0, If[LessEqual[KbT, 1.1e-52], N[(NdChar / N[(2.0 + N[(Vef * N[(N[(N[(Vef * 0.5), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision] + N[(1.0 / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -4.80000000000000009e26 or 1.10000000000000005e-52 < 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%

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

   5. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 45.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 45.2%

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

   if -4.80000000000000009e26 < KbT < 1.10000000000000005e-52

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 60.9%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 60.9%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 39.1%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 39.1%

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

   11. Taylor expanded in Vef around 0

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(2 + Vef \cdot \left(\frac{1}{2} \cdot \frac{Vef}{{KbT}^{2}} + \frac{1}{KbT}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \color{blue}{\left(Vef \cdot \left(\frac{1}{2} \cdot \frac{Vef}{{KbT}^{2}} + \frac{1}{KbT}\right)\right)}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(Vef, \color{blue}{\left(\frac{1}{2} \cdot \frac{Vef}{{KbT}^{2}} + \frac{1}{KbT}\right)}\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(Vef, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{Vef}{{KbT}^{2}}\right), \color{blue}{\left(\frac{1}{KbT}\right)}\right)\right)\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(Vef, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot Vef}{{KbT}^{2}}\right), \left(\frac{\color{blue}{1}}{KbT}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(Vef, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot Vef\right), \left({KbT}^{2}\right)\right), \left(\frac{\color{blue}{1}}{KbT}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(Vef, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Vef\right), \left({KbT}^{2}\right)\right), \left(\frac{1}{KbT}\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(Vef, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Vef\right), \left(KbT \cdot KbT\right)\right), \left(\frac{1}{KbT}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(Vef, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Vef\right), \mathsf{*.f64}\left(KbT, KbT\right)\right), \left(\frac{1}{KbT}\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 31.4%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(Vef, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Vef\right), \mathsf{*.f64}\left(KbT, KbT\right)\right), \mathsf{/.f64}\left(1, \color{blue}{KbT}\right)\right)\right)\right)\right) \]

   13. Simplified 31.4%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -4.8 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 1.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{NdChar}{2 + Vef \cdot \left(\frac{Vef \cdot 0.5}{KbT \cdot KbT} + \frac{1}{KbT}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.9% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -7.6 \cdot 10^{+165}:\\ \;\;\;\;t\_0 - \frac{Ev}{KbT} \cdot \left(NaChar \cdot 0.25\right)\\ \mathbf{elif}\;KbT \leq 3.95 \cdot 10^{+208}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{Vef - \frac{-0.5 \cdot \left(Vef \cdot Vef\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(Ec \cdot \frac{NdChar}{KbT}\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -7.6e+165)
     (- t_0 (* (/ Ev KbT) (* NaChar 0.25)))
     (if (<= KbT 3.95e+208)
       (/ NdChar (+ 2.0 (/ (- Vef (/ (* -0.5 (* Vef Vef)) KbT)) KbT)))
       (- t_0 (* (* Ec (/ NdChar KbT)) -0.25))))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -7.6e+165) {
		tmp = t_0 - ((Ev / KbT) * (NaChar * 0.25));
	} else if (KbT <= 3.95e+208) {
		tmp = NdChar / (2.0 + ((Vef - ((-0.5 * (Vef * Vef)) / KbT)) / KbT));
	} else {
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25);
	}
	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 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-7.6d+165)) then
        tmp = t_0 - ((ev / kbt) * (nachar * 0.25d0))
    else if (kbt <= 3.95d+208) then
        tmp = ndchar / (2.0d0 + ((vef - (((-0.5d0) * (vef * vef)) / kbt)) / kbt))
    else
        tmp = t_0 - ((ec * (ndchar / kbt)) * (-0.25d0))
    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 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -7.6e+165) {
		tmp = t_0 - ((Ev / KbT) * (NaChar * 0.25));
	} else if (KbT <= 3.95e+208) {
		tmp = NdChar / (2.0 + ((Vef - ((-0.5 * (Vef * Vef)) / KbT)) / KbT));
	} else {
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25);
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -7.6e+165:
		tmp = t_0 - ((Ev / KbT) * (NaChar * 0.25))
	elif KbT <= 3.95e+208:
		tmp = NdChar / (2.0 + ((Vef - ((-0.5 * (Vef * Vef)) / KbT)) / KbT))
	else:
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25)
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -7.6e+165)
		tmp = Float64(t_0 - Float64(Float64(Ev / KbT) * Float64(NaChar * 0.25)));
	elseif (KbT <= 3.95e+208)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(Vef - Float64(Float64(-0.5 * Float64(Vef * Vef)) / KbT)) / KbT)));
	else
		tmp = Float64(t_0 - Float64(Float64(Ec * Float64(NdChar / KbT)) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -7.6e+165)
		tmp = t_0 - ((Ev / KbT) * (NaChar * 0.25));
	elseif (KbT <= 3.95e+208)
		tmp = NdChar / (2.0 + ((Vef - ((-0.5 * (Vef * Vef)) / KbT)) / KbT));
	else
		tmp = t_0 - ((Ec * (NdChar / KbT)) * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -7.6e+165], N[(t$95$0 - N[(N[(Ev / KbT), $MachinePrecision] * N[(NaChar * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[KbT, 3.95e+208], N[(NdChar / N[(2.0 + N[(N[(Vef - N[(N[(-0.5 * N[(Vef * Vef), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(Ec * N[(NdChar / KbT), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if KbT < -7.59999999999999981e165

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{4} \cdot \left(NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right) + \frac{1}{4} \cdot \left(NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

   7. Simplified 35.2%

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

   8. Taylor expanded in Ev around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{Ev \cdot NaChar}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \left(\frac{\frac{1}{4} \cdot \left(Ev \cdot NaChar\right)}{\color{blue}{KbT}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(Ev \cdot NaChar\right)\right), \color{blue}{KbT}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(Ev \cdot NaChar\right)\right), KbT\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(NaChar \cdot Ev\right)\right), KbT\right)\right) \]

      5. *-lowering-*.f64 66.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(NaChar, Ev\right)\right), KbT\right)\right) \]

   10. Simplified 66.3%

       \[\leadsto 0.5 \cdot \left(NaChar + NdChar\right) - \color{blue}{\frac{0.25 \cdot \left(NaChar \cdot Ev\right)}{KbT}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \left(\frac{\left(\frac{1}{4} \cdot NaChar\right) \cdot Ev}{KbT}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \left(\left(\frac{1}{4} \cdot NaChar\right) \cdot \color{blue}{\frac{Ev}{KbT}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot NaChar\right), \color{blue}{\left(\frac{Ev}{KbT}\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\left(NaChar \cdot \frac{1}{4}\right), \left(\frac{\color{blue}{Ev}}{KbT}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \frac{1}{4}\right), \left(\frac{\color{blue}{Ev}}{KbT}\right)\right)\right) \]

       6. /-lowering-/.f64 68.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(NaChar, \frac{1}{4}\right), \mathsf{/.f64}\left(Ev, \color{blue}{KbT}\right)\right)\right) \]

   12. Applied egg-rr 68.8%

       \[\leadsto 0.5 \cdot \left(NaChar + NdChar\right) - \color{blue}{\left(NaChar \cdot 0.25\right) \cdot \frac{Ev}{KbT}} \]

   if -7.59999999999999981e165 < KbT < 3.95e208

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 62.8%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 62.8%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 39.4%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 39.4%

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

   11. Taylor expanded in KbT around -inf

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot Vef + \frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}}{KbT}\right)}\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{-1 \cdot Vef + \frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{-1 \cdot Vef + \frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot Vef + \frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot Vef + \frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}\right), KbT\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-1 \cdot Vef\right), \left(\frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}\right)\right), KbT\right)\right)\right)\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(Vef\right)\right), \left(\frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}\right)\right), KbT\right)\right)\right)\right) \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(Vef\right), \left(\frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}\right)\right), KbT\right)\right)\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(Vef\right), \left(\frac{\frac{-1}{2} \cdot {Vef}^{2}}{KbT}\right)\right), KbT\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(Vef\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot {Vef}^{2}\right), KbT\right)\right), KbT\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(Vef\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left({Vef}^{2}\right)\right), KbT\right)\right), KbT\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(Vef\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(Vef \cdot Vef\right)\right), KbT\right)\right), KbT\right)\right)\right)\right) \]

       12. *-lowering-*.f64 27.9%

           \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(Vef\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(Vef, Vef\right)\right), KbT\right)\right), KbT\right)\right)\right)\right) \]

   13. Simplified 27.9%

       \[\leadsto \frac{NdChar}{\color{blue}{2 + \left(-\frac{\left(-Vef\right) + \frac{-0.5 \cdot \left(Vef \cdot Vef\right)}{KbT}}{KbT}\right)}} \]

   if 3.95e208 < 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%

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

   5. Taylor expanded in KbT around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{4} \cdot \left(NaChar \cdot \left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right)\right) + \frac{1}{4} \cdot \left(NdChar \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)\right)}{KbT} + \left(\frac{1}{2} \cdot NaChar + \frac{1}{2} \cdot NdChar\right)} \]

   7. Simplified 41.8%

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

   8. Taylor expanded in Ec around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \color{blue}{\left(\frac{-1}{4} \cdot \frac{Ec \cdot NdChar}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{Ec \cdot NdChar}{KbT}\right)}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \left(Ec \cdot \color{blue}{\frac{NdChar}{KbT}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(Ec, \color{blue}{\left(\frac{NdChar}{KbT}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 74.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, NdChar\right)\right), \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(Ec, \mathsf{/.f64}\left(NdChar, \color{blue}{KbT}\right)\right)\right)\right) \]

   10. Simplified 74.0%

       \[\leadsto 0.5 \cdot \left(NaChar + NdChar\right) - \color{blue}{-0.25 \cdot \left(Ec \cdot \frac{NdChar}{KbT}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -7.6 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - \frac{Ev}{KbT} \cdot \left(NaChar \cdot 0.25\right)\\ \mathbf{elif}\;KbT \leq 3.95 \cdot 10^{+208}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{Vef - \frac{-0.5 \cdot \left(Vef \cdot Vef\right)}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right) - \left(Ec \cdot \frac{NdChar}{KbT}\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -5.4 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.15 \cdot 10^{-54}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Vef - -0.5 \cdot \frac{Vef \cdot Vef}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -5.4e+117)
     t_0
     (if (<= KbT 2.15e-54)
       (/ NaChar (+ 2.0 (/ (- Vef (* -0.5 (/ (* Vef Vef) KbT))) KbT)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -5.4e+117) {
		tmp = t_0;
	} else if (KbT <= 2.15e-54) {
		tmp = NaChar / (2.0 + ((Vef - (-0.5 * ((Vef * Vef) / KbT))) / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-5.4d+117)) then
        tmp = t_0
    else if (kbt <= 2.15d-54) then
        tmp = nachar / (2.0d0 + ((vef - ((-0.5d0) * ((vef * vef) / kbt))) / kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -5.4e+117) {
		tmp = t_0;
	} else if (KbT <= 2.15e-54) {
		tmp = NaChar / (2.0 + ((Vef - (-0.5 * ((Vef * Vef) / KbT))) / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -5.4e+117:
		tmp = t_0
	elif KbT <= 2.15e-54:
		tmp = NaChar / (2.0 + ((Vef - (-0.5 * ((Vef * Vef) / KbT))) / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -5.4e+117)
		tmp = t_0;
	elseif (KbT <= 2.15e-54)
		tmp = Float64(NaChar / Float64(2.0 + Float64(Float64(Vef - Float64(-0.5 * Float64(Float64(Vef * Vef) / KbT))) / KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -5.4e+117)
		tmp = t_0;
	elseif (KbT <= 2.15e-54)
		tmp = NaChar / (2.0 + ((Vef - (-0.5 * ((Vef * Vef) / KbT))) / KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -5.4e+117], t$95$0, If[LessEqual[KbT, 2.15e-54], N[(NaChar / N[(2.0 + N[(N[(Vef - N[(-0.5 * N[(N[(Vef * Vef), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -5.4000000000000005e117 or 2.15e-54 < 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%

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

   5. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 49.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 49.9%

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

   if -5.4000000000000005e117 < KbT < 2.15e-54

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 71.9%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 71.9%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 42.8%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 42.8%

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

   11. Taylor expanded in KbT around -inf

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot Vef + \frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}}{KbT}\right)}\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \left(2 + \left(\mathsf{neg}\left(\frac{-1 \cdot Vef + \frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \left(2 - \color{blue}{\frac{-1 \cdot Vef + \frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}}{KbT}}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \color{blue}{\left(\frac{-1 \cdot Vef + \frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(-1 \cdot Vef + \frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}\right), \color{blue}{KbT}\right)\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT} + -1 \cdot Vef\right), KbT\right)\right)\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT} + \left(\mathsf{neg}\left(Vef\right)\right)\right), KbT\right)\right)\right) \]

       7. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT} - Vef\right), KbT\right)\right)\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{-1}{2} \cdot \frac{{Vef}^{2}}{KbT}\right), Vef\right), KbT\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{{Vef}^{2}}{KbT}\right)\right), Vef\right), KbT\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left({Vef}^{2}\right), KbT\right)\right), Vef\right), KbT\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(Vef \cdot Vef\right), KbT\right)\right), Vef\right), KbT\right)\right)\right) \]

       12. *-lowering-*.f64 27.5%

           \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Vef, Vef\right), KbT\right)\right), Vef\right), KbT\right)\right)\right) \]

   13. Simplified 27.5%

       \[\leadsto \frac{NaChar}{\color{blue}{2 - \frac{-0.5 \cdot \frac{Vef \cdot Vef}{KbT} - Vef}{KbT}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -5.4 \cdot 10^{+117}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.15 \cdot 10^{-54}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Vef - -0.5 \cdot \frac{Vef \cdot Vef}{KbT}}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -6.8 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.15 \cdot 10^{-61}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{EDonor + \left(\left(Vef + mu\right) - Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -6.8e+25)
     t_0
     (if (<= KbT 2.15e-61)
       (/ NdChar (+ 2.0 (/ (+ EDonor (- (+ Vef mu) Ec)) KbT)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -6.8e+25) {
		tmp = t_0;
	} else if (KbT <= 2.15e-61) {
		tmp = NdChar / (2.0 + ((EDonor + ((Vef + mu) - Ec)) / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-6.8d+25)) then
        tmp = t_0
    else if (kbt <= 2.15d-61) then
        tmp = ndchar / (2.0d0 + ((edonor + ((vef + mu) - ec)) / kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -6.8e+25) {
		tmp = t_0;
	} else if (KbT <= 2.15e-61) {
		tmp = NdChar / (2.0 + ((EDonor + ((Vef + mu) - Ec)) / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -6.8e+25:
		tmp = t_0
	elif KbT <= 2.15e-61:
		tmp = NdChar / (2.0 + ((EDonor + ((Vef + mu) - Ec)) / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -6.8e+25)
		tmp = t_0;
	elseif (KbT <= 2.15e-61)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Float64(EDonor + Float64(Float64(Vef + mu) - Ec)) / KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -6.8e+25)
		tmp = t_0;
	elseif (KbT <= 2.15e-61)
		tmp = NdChar / (2.0 + ((EDonor + ((Vef + mu) - Ec)) / KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -6.8e+25], t$95$0, If[LessEqual[KbT, 2.15e-61], N[(NdChar / N[(2.0 + N[(N[(EDonor + N[(N[(Vef + mu), $MachinePrecision] - Ec), $MachinePrecision]), $MachinePrecision] / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -6.79999999999999967e25 or 2.1500000000000002e-61 < 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%

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

   5. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 44.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 44.7%

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

   if -6.79999999999999967e25 < KbT < 2.1500000000000002e-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%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 61.1%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 61.1%

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

   8. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right) \]

   10. Simplified 30.1%

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

   11. Taylor expanded in KbT around inf

       \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), \color{blue}{KbT}\right)\right)\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\left(Vef + mu\right), Ec\right)\right), KbT\right)\right)\right) \]

       5. +-lowering-+.f64 28.8%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(Vef, mu\right), Ec\right)\right), KbT\right)\right)\right) \]

   13. Simplified 28.8%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -6.8 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.15 \cdot 10^{-61}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{EDonor + \left(\left(Vef + mu\right) - Ec\right)}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.6% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -2.8 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{NdChar}{\frac{0.5 \cdot \left(Vef \cdot Vef\right)}{KbT \cdot KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -2.8e+117)
     t_0
     (if (<= KbT 2.7e-53)
       (/ NdChar (/ (* 0.5 (* Vef Vef)) (* KbT KbT)))
       t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -2.8e+117) {
		tmp = t_0;
	} else if (KbT <= 2.7e-53) {
		tmp = NdChar / ((0.5 * (Vef * Vef)) / (KbT * KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-2.8d+117)) then
        tmp = t_0
    else if (kbt <= 2.7d-53) then
        tmp = ndchar / ((0.5d0 * (vef * vef)) / (kbt * kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -2.8e+117) {
		tmp = t_0;
	} else if (KbT <= 2.7e-53) {
		tmp = NdChar / ((0.5 * (Vef * Vef)) / (KbT * KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -2.8e+117:
		tmp = t_0
	elif KbT <= 2.7e-53:
		tmp = NdChar / ((0.5 * (Vef * Vef)) / (KbT * KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -2.8e+117)
		tmp = t_0;
	elseif (KbT <= 2.7e-53)
		tmp = Float64(NdChar / Float64(Float64(0.5 * Float64(Vef * Vef)) / Float64(KbT * KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -2.8e+117)
		tmp = t_0;
	elseif (KbT <= 2.7e-53)
		tmp = NdChar / ((0.5 * (Vef * Vef)) / (KbT * KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -2.8e+117], t$95$0, If[LessEqual[KbT, 2.7e-53], N[(NdChar / N[(N[(0.5 * N[(Vef * Vef), $MachinePrecision]), $MachinePrecision] / N[(KbT * KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -2.79999999999999997e117 or 2.6999999999999999e-53 < 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%

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

   5. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 49.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 49.9%

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

   if -2.79999999999999997e117 < KbT < 2.6999999999999999e-53

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 58.8%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 58.8%

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

   8. Taylor expanded in KbT around -inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(2 + -1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(2, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right) + \frac{-1}{2} \cdot \frac{{\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right)}^{2}}{KbT}\right), KbT\right)\right)\right)\right) \]

   10. Simplified 29.2%

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

   11. Taylor expanded in Vef around inf

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(\frac{1}{2} \cdot \frac{{Vef}^{2}}{{KbT}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \left(\frac{\frac{1}{2} \cdot {Vef}^{2}}{\color{blue}{{KbT}^{2}}}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {Vef}^{2}\right), \color{blue}{\left({KbT}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({Vef}^{2}\right)\right), \left({\color{blue}{KbT}}^{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(Vef \cdot Vef\right)\right), \left({KbT}^{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(Vef, Vef\right)\right), \left({KbT}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(Vef, Vef\right)\right), \left(KbT \cdot \color{blue}{KbT}\right)\right)\right) \]

       7. *-lowering-*.f64 25.2%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(Vef, Vef\right)\right), \mathsf{*.f64}\left(KbT, \color{blue}{KbT}\right)\right)\right) \]

   13. Simplified 25.2%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -2.8 \cdot 10^{+117}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{NdChar}{\frac{0.5 \cdot \left(Vef \cdot Vef\right)}{KbT \cdot KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 31.1% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -9 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -9e+23)
     t_0
     (if (<= KbT 2e-83) (/ NdChar (+ 2.0 (/ Vef KbT))) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -9e+23) {
		tmp = t_0;
	} else if (KbT <= 2e-83) {
		tmp = NdChar / (2.0 + (Vef / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-9d+23)) then
        tmp = t_0
    else if (kbt <= 2d-83) then
        tmp = ndchar / (2.0d0 + (vef / kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -9e+23) {
		tmp = t_0;
	} else if (KbT <= 2e-83) {
		tmp = NdChar / (2.0 + (Vef / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -9e+23:
		tmp = t_0
	elif KbT <= 2e-83:
		tmp = NdChar / (2.0 + (Vef / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -9e+23)
		tmp = t_0;
	elseif (KbT <= 2e-83)
		tmp = Float64(NdChar / Float64(2.0 + Float64(Vef / KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -9e+23)
		tmp = t_0;
	elseif (KbT <= 2e-83)
		tmp = NdChar / (2.0 + (Vef / KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -9e+23], t$95$0, If[LessEqual[KbT, 2e-83], N[(NdChar / N[(2.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -8.99999999999999958e23 or 2.0000000000000001e-83 < 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%

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

   5. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 43.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 43.6%

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

   if -8.99999999999999958e23 < KbT < 2.0000000000000001e-83

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 61.1%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 61.1%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 38.6%

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 38.6%

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

   11. Taylor expanded in Vef around 0

       \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(2 + \frac{Vef}{KbT}\right)}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \left(\frac{Vef}{KbT} + \color{blue}{2}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \color{blue}{2}\right)\right) \]

       3. /-lowering-/.f64 22.2%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), 2\right)\right) \]

   13. Simplified 22.2%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -9 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\frac{NdChar}{2 + \frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 31.1% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{if}\;KbT \leq -3.35 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;KbT \leq 7 \cdot 10^{-58}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ NdChar NaChar))))
   (if (<= KbT -3.35e-15)
     t_0
     (if (<= KbT 7e-58) (/ NaChar (+ 2.0 (/ Vef KbT))) t_0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -3.35e-15) {
		tmp = t_0;
	} else if (KbT <= 7e-58) {
		tmp = NaChar / (2.0 + (Vef / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (ndchar + nachar)
    if (kbt <= (-3.35d-15)) then
        tmp = t_0
    else if (kbt <= 7d-58) then
        tmp = nachar / (2.0d0 + (vef / kbt))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double t_0 = 0.5 * (NdChar + NaChar);
	double tmp;
	if (KbT <= -3.35e-15) {
		tmp = t_0;
	} else if (KbT <= 7e-58) {
		tmp = NaChar / (2.0 + (Vef / KbT));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	t_0 = 0.5 * (NdChar + NaChar)
	tmp = 0
	if KbT <= -3.35e-15:
		tmp = t_0
	elif KbT <= 7e-58:
		tmp = NaChar / (2.0 + (Vef / KbT))
	else:
		tmp = t_0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = Float64(0.5 * Float64(NdChar + NaChar))
	tmp = 0.0
	if (KbT <= -3.35e-15)
		tmp = t_0;
	elseif (KbT <= 7e-58)
		tmp = Float64(NaChar / Float64(2.0 + Float64(Vef / KbT)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	t_0 = 0.5 * (NdChar + NaChar);
	tmp = 0.0;
	if (KbT <= -3.35e-15)
		tmp = t_0;
	elseif (KbT <= 7e-58)
		tmp = NaChar / (2.0 + (Vef / KbT));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := Block[{t$95$0 = N[(0.5 * N[(NdChar + NaChar), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[KbT, -3.35e-15], t$95$0, If[LessEqual[KbT, 7e-58], N[(NaChar / N[(2.0 + N[(Vef / KbT), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if KbT < -3.35e-15 or 6.9999999999999998e-58 < 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%

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

   5. Taylor expanded in KbT around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

      3. +-lowering-+.f64 44.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

   7. Simplified 44.0%

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

   if -3.35e-15 < KbT < 6.9999999999999998e-58

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 72.1%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 72.1%

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

   8. Taylor expanded in Vef around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\color{blue}{\left(\frac{Vef}{KbT}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 41.5%

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right)\right)\right)\right) \]

   10. Simplified 41.5%

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

   11. Taylor expanded in Vef around 0

       \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(2 + \frac{Vef}{KbT}\right)}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \left(\frac{Vef}{KbT} + \color{blue}{2}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(\left(\frac{Vef}{KbT}\right), \color{blue}{2}\right)\right) \]

       3. /-lowering-/.f64 20.9%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(\mathsf{/.f64}\left(Vef, KbT\right), 2\right)\right) \]

   13. Simplified 20.9%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;KbT \leq -3.35 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \mathbf{elif}\;KbT \leq 7 \cdot 10^{-58}:\\ \;\;\;\;\frac{NaChar}{2 + \frac{Vef}{KbT}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(NdChar + NaChar\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 22.9% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;NdChar \leq -3 \cdot 10^{+121}:\\ \;\;\;\;\frac{NdChar}{2}\\ \mathbf{elif}\;NdChar \leq 9.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{NaChar}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{NdChar}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (NdChar Ec Vef EDonor mu KbT NaChar Ev EAccept)
 :precision binary64
 (if (<= NdChar -3e+121)
   (/ NdChar 2.0)
   (if (<= NdChar 9.5e-85) (/ NaChar 2.0) (/ NdChar 2.0))))

C

double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -3e+121) {
		tmp = NdChar / 2.0;
	} else if (NdChar <= 9.5e-85) {
		tmp = NaChar / 2.0;
	} else {
		tmp = NdChar / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(ndchar, ec, vef, edonor, mu, kbt, nachar, ev, eaccept)
    real(8), intent (in) :: ndchar
    real(8), intent (in) :: ec
    real(8), intent (in) :: vef
    real(8), intent (in) :: edonor
    real(8), intent (in) :: mu
    real(8), intent (in) :: kbt
    real(8), intent (in) :: nachar
    real(8), intent (in) :: ev
    real(8), intent (in) :: eaccept
    real(8) :: tmp
    if (ndchar <= (-3d+121)) then
        tmp = ndchar / 2.0d0
    else if (ndchar <= 9.5d-85) then
        tmp = nachar / 2.0d0
    else
        tmp = ndchar / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double NdChar, double Ec, double Vef, double EDonor, double mu, double KbT, double NaChar, double Ev, double EAccept) {
	double tmp;
	if (NdChar <= -3e+121) {
		tmp = NdChar / 2.0;
	} else if (NdChar <= 9.5e-85) {
		tmp = NaChar / 2.0;
	} else {
		tmp = NdChar / 2.0;
	}
	return tmp;
}

Python

def code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept):
	tmp = 0
	if NdChar <= -3e+121:
		tmp = NdChar / 2.0
	elif NdChar <= 9.5e-85:
		tmp = NaChar / 2.0
	else:
		tmp = NdChar / 2.0
	return tmp

Julia

function code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0
	if (NdChar <= -3e+121)
		tmp = Float64(NdChar / 2.0);
	elseif (NdChar <= 9.5e-85)
		tmp = Float64(NaChar / 2.0);
	else
		tmp = Float64(NdChar / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(NdChar, Ec, Vef, EDonor, mu, KbT, NaChar, Ev, EAccept)
	tmp = 0.0;
	if (NdChar <= -3e+121)
		tmp = NdChar / 2.0;
	elseif (NdChar <= 9.5e-85)
		tmp = NaChar / 2.0;
	else
		tmp = NdChar / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := If[LessEqual[NdChar, -3e+121], N[(NdChar / 2.0), $MachinePrecision], If[LessEqual[NdChar, 9.5e-85], N[(NaChar / 2.0), $MachinePrecision], N[(NdChar / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if NdChar < -3.0000000000000002e121 or 9.49999999999999964e-85 < NdChar

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{\left(1 + e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EDonor + \left(Vef + mu\right)\right) - Ec}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EDonor + \left(Vef + mu\right)\right) - Ec\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EDonor + \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) - Ec\right)\right), KbT\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(\left(Vef + mu\right) + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \left(Vef + \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + -1 \cdot Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu + \left(\mathsf{neg}\left(Ec\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \left(mu - Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. --lowering--.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(NdChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(EDonor, \mathsf{+.f64}\left(Vef, \mathsf{\_.f64}\left(mu, Ec\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 72.6%

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

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NdChar, \color{blue}{2}\right) \]
   9. Step-by-step derivation

   10. Simplified 28.2%

       \[\leadsto \frac{NdChar}{\color{blue}{2}} \]

   if -3.0000000000000002e121 < NdChar < 9.49999999999999964e-85

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in NdChar around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

      14. --lowering--.f64 73.7%

          \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

   7. Simplified 73.7%

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

   8. Taylor expanded in KbT around inf

      \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{2}\right) \]
   9. Step-by-step derivation

   10. Simplified 25.2%

       \[\leadsto \frac{NaChar}{\color{blue}{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 28.4% 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%

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

5. Taylor expanded in KbT around inf

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

   1. distribute-lft-out N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(NaChar + NdChar\right)} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(NaChar + NdChar\right)}\right) \]

   3. +-lowering-+.f64 29.5%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(NaChar, \color{blue}{NdChar}\right)\right) \]

7. Simplified 29.5%

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

8. Final simplification 29.5%

   \[\leadsto 0.5 \cdot \left(NdChar + NaChar\right) \]
9. Add Preprocessing

Alternative 20: 18.8% accurate, 76.3× speedup

Math

\[\begin{array}{l} \\ \frac{NaChar}{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[NdChar_, Ec_, Vef_, EDonor_, mu_, KbT_, NaChar_, Ev_, EAccept_] := N[(NaChar / 2.0), $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%

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

5. Taylor expanded in NdChar around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{\left(1 + e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}}\right)}\right)\right) \]

   3. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\frac{\left(EAccept + \left(Ev + Vef\right)\right) - mu}{KbT}\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + \left(Ev + Vef\right)\right) - mu\right), KbT\right)\right)\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) - mu\right)\right), KbT\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(\left(Ev + Vef\right) + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

   7. associate-+r+ N/A

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(EAccept + \left(Ev + \left(Vef + -1 \cdot mu\right)\right)\right), KbT\right)\right)\right)\right) \]

   9. associate-+r+ N/A

      \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\left(EAccept + Ev\right) + \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(EAccept + Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + -1 \cdot mu\right)\right), KbT\right)\right)\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef + \left(\mathsf{neg}\left(mu\right)\right)\right)\right), KbT\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \left(Vef - mu\right)\right), KbT\right)\right)\right)\right) \]

   14. --lowering--.f64 61.3%

       \[\leadsto \mathsf{/.f64}\left(NaChar, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(EAccept, Ev\right), \mathsf{\_.f64}\left(Vef, mu\right)\right), KbT\right)\right)\right)\right) \]

7. Simplified 61.3%

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

8. Taylor expanded in KbT around inf

   \[\leadsto \mathsf{/.f64}\left(NaChar, \color{blue}{2}\right) \]
9. Step-by-step derivation

10. Simplified 18.1%

    \[\leadsto \frac{NaChar}{\color{blue}{2}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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))))))